Physics Reports vol.347

D. Atwood et al. / Physics Reports 347 (2001) 1}222 1 CP VIOLATION IN TOP PHYSICS David ATWOODa, Shaouly BAR-SHALOMb,...

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D. Atwood et al. / Physics Reports 347 (2001) 1}222

1

CP VIOLATION IN TOP PHYSICS

David ATWOODa, Shaouly BAR-SHALOMb, Gad EILAMc, Amarjit SONId a

b

Department of Physics and Astronomy, Iowa State University, Ames, IOWA 50011, USA INFN, Sezione di Roma and Department of Physics, University of Roma I, La Sapienza, Roma, Italy c Physics Department, Technion-Institute of Technology, Haifa 32000, Israel d Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 347 (2001) 1}222

CP violation in top physics David Atwood , Shaouly Bar-Shalom, Gad Eilam, Amarjit Soni Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA INFN, Sezione di Roma and Department of Physics, University of Roma I, La Sapienza, Roma, Italy Physics Department, Technion-Institute of Technology, Haifa 32000, Israel Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA Received September 2000; editor: J.V. Allaby Contents 1. Introduction 2. General discussion 2.1. De"nitions of discrete symmetries C, P and T 2.2. CP-violating observables: categorizing according to ¹ , 2.3. Partial rate asymmetries and the CP-CPT connection 2.4. Resonant = e!ects and CP violation in top decays 2.5. E!ective Lagrangians and observables 2.6. Optimized observables 2.7. The naked top 2.8. Elements of top polarimetry 3. Models of CP violation 3.1. CP violation and the standard model 3.2. Multi-Higgs doublet models 3.3. Supersymmetric models 4. Top dipole moments 4.1. Theoretical expectations 4.2. Arbitrary number of Higgs doublets and a CP-violating neutral Higgs sector 4.3. Expectations from 2HDMs with CP violation in the neutral Higgs sector 4.4. Expectations from a CP-violating charged Higgs sector

4 7 5. 7 9 13 6. 19 20 23 28 29 31 31 39 52 64 64 65 67 72

7.

8.

4.5. Expectations from the MSSM 4.6. Top dipole moments } summary CP violation in top decays 5.1. Partial rate asymmetries 5.2. Partially integrated rate asymmetries 5.3. Energy asymmetry 5.4. -polarization asymmetry 5.5. CP violation in top decays } summary CP violation in e>e\ collider experiments 6.1. e>e\PttN 6.2. e>e\PttN h, ttN Z, examples of tree-level CP violation 6.3. e>e\PttN g 6.4. CP violation via == fusion in e>e\PttN N C C CP violation in pp collider experiments 7.1. ppPttN #X: general comments 7.2. ppPttN #X: general form factor approach and the CEDM of the top 7.3. 2HDM and CP violation in ppPttN #X 7.4. SUSY and CP violation in ppPttN #X CP violation in ppN collider experiments 8.1. ppN PttN #X 8.2. ppN PtbN #X 8.3. ppN PtbN h#X, a case of tree-level CP violation

75 80 85 86 102 103 104 105 106 106 119 135 146 149 149 150 154 162 164 164 171 180

E-mail addresses: [email protected] (D. Atwood), [email protected] (S. Bar-Shalom), [email protected] (G. Eilam), [email protected] (A. Soni). Much of the work in this review was done while S. Bar-Shalom was at Physics Department, University of California, Riverside, CA, USA. 0370-1573/01/$ - see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 1 2 - 5

D. Atwood et al. / Physics Reports 347 (2001) 1}222 9. CP violation in collider experiments 9.1. PX: general comments 9.2. PttN and the top EDM 9.3. PttN and s-channel Higgs exchange in a 2HDM 10. CP violation in >\ collider experiments 10.1. >\PttN 10.2. CP violation in the #avor changing reaction >\PtcN

183 184 186 189 194 194

11. Summary and outlook 12. Note on literature survey Acknowledgements Appendix A. One-loop C functions Appendix B. Abbreviations References

3 205 212 212 213 213 214

202

Abstract CP violation in top physics is reviewed. The standard model has negligible e!ects, consequently CP violation searches involving the top quark may constitute the best way to look for physics beyond the standard model. Non-standard sources of CP violation due to an extended Higgs sector with and without natural #avor conservation and supersymmetric theories are discussed. Experimental feasibility of detecting CP violation e!ects in top quark production and decays in high energy e>e\, , >\, pp and pp colliders are surveyed. Searches for the electric, electro-weak and the chromo-electric dipole moments of the top quark in e>e\PttM and in ppPttM X are described. In addition, other mechanisms that appear promising for experiments, e.g., tree-level CP violation in e>e\PttM h, ttM Z, ttM and in the top decay tPb and CP C C O violation driven by s-channel Higgs exchanges in pp, , >\PttM , etc. are also discussed. 2001 Published by Elsevier Science B.V. PACS: 11.30.!j

4


1. Introduction Violations of the CP (charge conjugation combined with parity) symmetry are of great interest in particle physics especially since its origin is still unclear. Better understanding of this (so far) rare phenomenon can lead to new physics which may explain both the origin of mass and the preponderance of matter over anti-matter in the present universe. Indeed, reactions that violate CP are such a scarce resource that in over 30 years the only con"rmed examples of CP violation are those found in the decay of the K -meson. * The "rst experimental observation of CP violation was in 1964 by Christenson et al. [3], who observed a non-vanishing rate for the decay K P2 [4], * Br(K P2)"3.00$0.04;10\ . *

(1.1)

Since the dominant decay of K is to a 3 state of CP"!1, the above decay to a manifestly * CP"#1 state clearly violates this symmetry. Another example of CP violation which is well established in K is the di!erence between * (K Pl>l \) and (K Pl\l >), l"e, [4]: * * (K Pl>\)!(K Pl\>) * * "(3.27$0.012);10\ . (K Pl>\)#(K Pl\>) * *

(1.2)

All of these observations of CP violation in the K system can be explained by the CP violation * in the mixing of the neutral K mesons. Thus, K "((1# )K!(1! )KM )/(2(1# ) , *

(1.3)

K "((1# )K#(1! )KM )/(2(1# ) , 1

(1.4)

where the experimental value of is [4] "(2.263$0.023);10\, arg( )"(43.49$0.08)3 .

(1.5)

As is well known, this mixing can be accommodated in the standard model (SM) with three generations where the CP violation originates through a phase in the Cabibbo Kobayashi Maskawa (CKM) [5,6] matrix as will be discussed in some detail in Section 3.1. The SM further predicts that there is an additional CP violation in K P parameterized by * the quantity . The prediction is that / "O(10\); the theoretical di$culties in determining the hadronic matrix element prevent us from making a more precise estimate, see e.g., [7}14]. Experimentally, Re( / ) may be measured via [15]

1 , Re( / )K 1! 6 >\ For excellent recent books on CP violation see Refs. [1,2].

(1.6)


5

where GHH K 5 * , " (1.7) GH GHH K 5 1 and H is the relevant weak interaction Hamiltonian. 5 After some two decades of intensive e!orts, new and quite dramatic experimental developments have recently taken place that we would now brie#y like to mention. First of all, let us recall that a few years ago the CERN experiment NA31 gave the result [16]: Re( / )"(23$6.5);10\ ,

(1.8)

appreciably di!erent from zero. On the other hand, the Fermilab experiment E731 found it completely consistent with zero [17,18]: Re( / )"(7.4$6);10\ .

(1.9)

For the past many years improved experiments have been underway, at CERN (experiment NA48), and at FNAL (KTEV) with an expected accuracy of about O(10\), KTEV has recently announced their new results on / , based upon analysis of 20% data collected so far [19]: Re( / )"(28$4.1);10\ .

(1.10)

For the world average one would now "nd [18] Re( / )"(21$5);10\ ,

(1.11)

thus, conclusively establishing that / O0. Such a non-vanishing value formally lays to rest the phenomenological superweak model [20] of CP violation as it unambiguously predicts / "0. However, unless the computational challenges presented by strong interactions can be overcome, it is unlikely that the measured value of / would con"rm or refute the SM in any reliable fashion. Experiments involving B-mesons are more likely to have a quantitative bearing on the SM. Just as the SM indicates that the natural size of CP asymmetries in K physics is O(10\}10\), it also strongly suggests that the e!ects in the B system are much bigger; in many cases CP asymmetries are expected to be tens of percents. This expectation renders the B system ideal for a precise extraction of the CKM phase and, indeed for a thorough quantitative test of the SM through a detailed study of the unitarity triangle [21,22]. The asymmetric and symmetric B factories currently in the early stages of running at KEK, SLAC and Cornell and hadron machines, should have a very important role to play in confronting the experimental results with the detailed predictions of the SM. A recent CDF result [23] for CP violation in BPJ/K , though crude at 1 the moment, indicates that CP violation may indeed be large in the b system. Experiments at FNAL have decisively [24,25] demonstrated that the mass of the top quark is extremely large, i.e., the D0 and CDF average is now m &174 GeV [26,27]. This has some very R important consequences. First, the top rapidly undergoes two-body weak decay: tPb#=, with a time scale of about 10\ s, which is shorter by an order of magnitude than the typical QCD time scale necessary for hadronic bound states to be formed [28]. Thus, unlike the other "ve quarks, the top does not form hadrons. It means that the dynamics of top production and decay does not get

6


masked by the complications of non-perturbative, bound state physics, i.e., the `brown mucka. All of the CP violation phenomena relevant to the top are therefore of the `directa type. We should think of the top quark as an elementary fermion. For example, it therefore is sensible to ask for its dipole moment [29}32]. Unlike the other quarks the spin of the top quark becomes an extremely important observable. Indeed the decays of the top quark become very e!ective analyzers of its spin, see e.g., [33}35]. The SM predicts, however, that CP-violating e!ects in t-physics are very small. This is primarily due to the fact that its large mass in comparison to the other quarks renders the Glashow}Iliopoulos}Maiani (GIM) [36] cancellation particularly e!ective [37,38]. This being the case, what then is the motivation for the study of CP violation in the top quark system? There are two related reasons why one might expect to "nd such e!ects. First of all there is another important example of CP violation which the SM fails to explain, namely the excess of matter over anti-matter in the universe. It was shown by Sakharov [39] in 1967 that CP violation is one of the necessary conditions for baryon number asymmetry to appear in the early universe; baryon}anti-baryon asymmetry can be dynamically generated at early stages after the big bang even if the universe was `borna symmetric, provided that: (i) C and CP are violated, (ii) there are baryon number violating interactions and (iii) there is a deviation from thermal equilibrium. The basic idea is that, if CP is violated, then baryons and anti-baryons interact with di!erent rates at some point in the early universe. However, the CP violation due to the SM appears too weak to drive such an asymmetry [40,41]. In many cases, extensions of the SM such as the Two Higgs doublet model (2HDM) or the SUperSYmmetric (SUSY) extensions of the SM are able to supply the CP violation required to produce such a baryon asymmetry in the early universe. In fact, in some models [42,43], it is precisely the couplings of the top quark to CP-violating phases in beyond the SM physics which drive baryogenesis. Thus, the study of CP violation in top quark interactions in the laboratory could shed light on these primordial processes. The second motivation for investigating CP violation in top quark physics is that in many extensions of the SM, CP violation in the top quark can be particularly large. Indeed, because the SM contribution to CP violation in the top quark is so small, any observation of such e!ects would be a clear evidence of physics beyond the SM. The argument here parallels the search for the weak neutral current, in the 1970s, by looking for parity violation in deep-inelastic-scattering. The point is that the existing theory of the time, namely QED, could not cause parity violation in deepinelastic-scattering. Such an e!ect became an unambiguous signature for the existence of the weak neutral current. Since various extensions of the SM entail new CP-violating phase(s), we should seek the optimal strategies for searching each type of new phase. In this context, we "rst recall, what has been emphasized on the preceding page, that the b-quark is very sensitive to the CKM phase of the SM. Existing literature has revealed that top physics is very sensitive to several di!erent types of new phases. Upcoming high-energy colliders of the next decade can therefore serve as excellent laboratories for searching for new physics in top quark systems, and in particular, for studying CP-violating e!ects associated with those new CP-odd phases. The upgraded Tevatron pp collider (runs 2 and 3) at Fermilab which will be able to produce about 10}10 ttM /year, the CERN pp large hadron collider (LHC) will produce about 10}10 ttM /year, and about 10}10 ttM /year are expected at a future e>e\ next linear collider (NLC).


7

A CP-odd phase due to an extended neutral Higgs sector or vertex corrections arising in other extensions of the SM can endow the top quark with a large dipole moment form factor. Such an e!ect could be detected both at an e>e\ collider, such as the NLC or hadron colliders such as the LHC. A CP-violating phase in the neutral Higgs sector also causes large CP asymmetries in the reactions e>e\PttM H and e>e\PttM , both of which should be a prime target for the NLC. C C Moreover, CP violation in the neutral Higgs sector and in supersymmetry can have interesting e!ects in single top production at the upgraded Tevatron pp collider at Fermilab and in ttM pair production at the LHC. The transverse polarization of the in the three-body top decay tPb is extremely sensitive to a new phase from a charged Higgs sector in multi-Higgs doublets models (MHDMs). Finally, CP-odd phases in SUSY models have also interesting e!ects in partial rate asymmetry (PRA) in tP=>b versus tM P=\bM . These processes and others will be discussed in the subsequent sections. We will not consider CP violation phenomena in which the top quark is virtual rather than an external particle. It su$ces to recapitulate that, in the SM, CP violation is often dominated by the virtual top quarks in the loops. Let us also comment that the discovery of the top with the measurement of m , and the progress in determination of the CKM matrix elements as well as R considerable progress in theory, has in#uenced our understanding of CP violation in K and B physics within the SM [13] and beyond the SM [44]. Furthermore, for the electric dipole moments (EDMs) of the electron and the neutron [45], the virtual top quark also plays a crucial role in extended Higgs sector scenarios.

2. General discussion 2.1. Dexnitions of discrete symmetries C, P and ¹ Let us now review the de"nitions of the discrete symmetries C, P and ¹ and recall a few basic facts concerning their manifestation in relativistic quantum "eld theory. Under the parity transformation, P, the spatial coordinate axes are reversed, i.e., Px"!x. Thus for an ingoing particle, X, in a speci"c momentum and spin state X; P, S , the action of parity is to reverse the momentum, leaving the spin "xed as angular momentum is an axial vector de"ned by a cross product. Hence, PX; P, S PX;!P, S .

(2.1)

Under the Wigner de"nition of time reversal, ¹, the sign of both momenta and spins are reversed, and also, due to the anti-unitary nature of ¹, and states are interchanged. Thus, ¹X; P, S PX;!P,!S .

(2.2)

Under charge conjugation, C, each particle is replaced by its anti-particle, and so CX; P, S PXM ; P, S , where XM means that all charges and other additive quantum numbers are reversed.

(2.3)

8


It can be shown that local relativistic quantum "eld theories with the usual spin-statistics relations are invariant under the combined action of all three of these symmetries where [46}49]: CPTX; P, S "XM ; P,!S . Thus, such a theory violates ¹ if and only if it violates CP, where

(2.4)

CPX; P, S PXM ;!P, S , (2.5) and in this sense CP and ¹ violation are equivalent. Other well-known consequences of the CPT theorem are that masses of particles and antiparticles are the same, m "m M , and the total widths of particles and anti-particles are also equal, 6 6 " M . Note that it does not follow from CPT that decay rates to speci"c "nal states are the same. 6 6 In fact, partial width di!erences, i.e., a non-zero value of (XPA),(XPA)!(XM PAM ) ,

(2.6)

is a form of CP violation that we will discuss in more detail in Section 2.3. Clearly, it follows from " M that 6 6 (XPA)"0 , (2.7) where the sum is over all possible "nal states. The relationship between (XPA) and the other "nal states which compensate for it will also be discussed in detail in Section 2.3. In this report we are largely concerned with the violation of discrete symmetries in decay and scattering experiments. We, therefore, need to consider the implementation of CP and ¹ on the S-matrix. For C and P this is straightforward. Consider the initial state of n particles G i"P , P ,2, S , S ,2 , (2.8) ? @ ? @ and the "nal state of n particles D f "P , P ,2, S , S ,2 , (2.9) so that the S-matrix element is f Si"S

. DG The transformations P and C follow from the single-particle transformation

(2.10)

f "!P ,!P ,2, S , S ,2 , (2.11) . f "P , P ,2, S , S ,2 , (2.12) ! and likewise for i , i . The transformation of the S-matrix element under these symmetries is . ! thus . S S P , DG D. G. ! S . S P DG D! G!

(2.13) (2.14)


9

and thus !. S S P . (2.15) DG D!. G!. The nature of time reversal, however, requires the interchange of and states so that the e!ect on the S-matrix will be anti-unitary. Thus, f "!P ,!P ,2,!S ,!S ,2 , 2 i "!P ,!P ,2,!S ,!S ,2 , 2 ? @ ? @ such that

(2.16) (2.17)

2 S . (2.18) S P DG G2D2 Needless to say, because of the interchange of initial and "nal states, accelerator-based experiments seldom test ¹ directly. These symmetries, as they are de"ned in S-matrix theory are fundamental in that if the Lagrangian and the vacuum states respect C, P or ¹, then the corresponding symmetry of the S-matrix will apply. We will also "nd it useful to consider the symmetry ¹ } `naivea time reversal , } for which this is not true. The de"nition of ¹ is to apply ¹ to the initial and "nal states without , interchanging them 2, S . (2.19) S P D2 G2 DG Thus, ¹ is a `symmetrya which can be tested in accelerator-based scattering experiments, but, as , we shall see in the following section, it only corresponds to `truea time reversal (¹) operation at tree-level in perturbation theory. It is nonetheless useful in categorizing the various modes of CP violation. 2.2. CP-violating observables: categorizing according to ¹ , It is useful to divide CP-violating observables into two categories (see e.g., [29,32,35,50,51]), those that are even under `naivea time reversal (¹ ) and those that are odd. Recall that ¹ is , , de"ned as a transformation which reverses the momenta and spins of all particles without the interchange of the initial and "nal states. This contrasts with true time reversal, ¹, in that under ¹ initial and "nal states are also interchanged. The symmetry ¹ is not a fundamental symmetry like C, P and ¹ since the S-matrix under , ¹ need not follow from the transformation properties of the Lagrangian. Nevertheless, it is , a useful tool for categorization and as we shall presently show, observables which are CP-odd and ¹ -odd, i.e., are CP¹ -even, may assume non-zero expectation values in the absence of "nal state , , interaction (FSI) e!ects. In particular, tree-level processes in perturbation theory may lead to non-vanishing expectation values for these operators. On the other hand, CP-odd ¹ -even (i.e., , CP¹ -odd) operators may only assume non-zero expectation values if such FSI e!ects are present , giving a non-trivial phase to the Feynman amplitude. Such a phase, often called a strong phase or absorptive phase, may arise in several ways. It may be present in a loop diagram if the internal particle(s) can be on-shell. An interesting variation of this, which we will consider in Section 2.4, is

10


in the propagator of an unstable particle where the strong phase is the phase of the Breit}Wigner amplitude. Non-perturbative rescattering of "nal state particles can also give a strong phase though this is more of interest in CP studies in B and K physics. Indeed, ¹ is useful in understanding when CP-conserving observables depend on an absorptive , phase. In particular, a CP-even ¹ -odd observable (typically a CP-even triple product correlation , of momenta and/or spins) will only assume an expectation value if FSI e!ects are present. Thus, for instance, if one is looking for a CP-odd, ¹ -odd e!ect, one must have data both on the process of , interest and its CP conjugate (if they are di!erent) in order to distinguish from the possible background of CP-even ¹ -odd e!ects (see e.g., [52]). , In order to understand the role of ¹ , let us consider the unitarity relations of the S-matrix , (implied by conservation of probabilities). Following a derivation analogous to the optical theorem [53] we write the S-matrix in terms of the scattering amplitude T S"1#iT ,

(2.20)

where, for a given transition iPf, T is related to the `reduced scattering amplitudea, , by f Ti"(2)( p !p ) f i . D G Substituting Eq. (2.20) into the unitarity relation SRS"1 we obtain

(2.21)

T !TH "i TH T , (2.22) DG GD LD LG L where we denote aTb,T . In terms of this becomes ?@ !H "i(2) (p !p )H . (2.23) DG GD L G LD LG L Let us now assume that there are no rescattering e!ects and that (to the order of approximation considered) i and f are stable states so that " "0. Thus, for each possible intermediate state GG DD n, the rhs of Eq. (2.23) vanishes. Therefore, in the absence of rescattering, is hermitian "H . GD DG Now, if is CP invariant, then by the CPT theorem it is also ¹ invariant. Thus, f i" i f " f i H 2 2 2 2 and, therefore,

(2.24)

(2.25)

f i" f i . (2.26) 2 2 In fact, this equation means precisely that the modulus of f i is invariant under ¹ . Since, in the , absence of rescattering, the expectation value of any operator depends only on f i, Eq. (2.26) implies that if CP is conserved, then only ¹ -even operators can have a non-vanishing expectation , value. What we have shown therefore is that in the absence of rescattering e!ects (i.e., Im()"0, in which case the requirement of CPT invariance leads e!ectively to conservation of the scattering amplitude under CP¹ ) and in the absence of CP violation, ¹ -odd observables have zero , , expectation value. Thus, if such a ¹ -odd observable O has a non-zero expectation value, either CP ,


11

Table 1 Transformation properties under ¹ and CP and presence or absence of "nal state interactions (FSI). Here Y,FSI , present and N,FSI absent ¹ ,

CP-violating

CP-conserving

even odd

Y N

N Y

is violated and O is CP-odd (i.e., CP¹ -even) or there are rescattering e!ects present (implying , CPTOCP¹ ) and O is CP-even (i.e., CP¹ -odd). Conversely, let us suppose O is CP-odd and , , ¹ -even (i.e., CP¹ -odd). Again Eq. (2.26) implies that this operator can only assume a non-zero , , expectation value if rescattering e!ects are present. These properties of the operators are summarized in Table 1. From Table 1 we see that another consequence of Eq. (2.26) is that a ¹ -odd signal is only , a de"nite signal for ¹ violation and hence of CP violation in the absence of rescattering e!ects. To con"rm the CP-even or CP-odd nature of such a reaction one must therefore compare data from iPf with the charge conjugate channel iM PfM to explicitly verify CP violation or else rule out rescattering e!ects in some other way. Recall from the de"nition of time reversal that the spatial components of vectors representing momenta and spins are reversed. Thus, an observable is ¹ -odd if it is proportional to a term of the , form (v , v , v , v ), where v are 4-vectors representing spins or momenta of initial and "nal state G particles and is the Levi-Civita tensor. Consequently, ¹ -odd signals can only be observed in , reactions where there are at least four independent momenta or spins that can be measured. There are two important venues for the investigation of CP violation that we will deal with extensively. The "rst one is when CP nonconservation appears in decays of a particle and the second, is to search for scattering processes that can give rise to CP violation. The latter consist of two di!erent possibilities: either the CP-violating e!ect is due to the subsequent decay of the particle which is produced in the scattering process or the CP nonconservation is driven by an intrinsic property of the scattering mechanism itself. An observable which is CP-odd and ¹ -even, thus requiring an absorptive phase (as was shown , above), and which is widely used in the case where the CP e!ect appears in decays of a particle is called PRA (partial rate asymmetry). This observable is non-vanishing when a particle A decays to a state B with a partial width (APB) whereas the partial width of the conjugate process, i.e., (AM PBM ) is di!erent from (APB). Thus, de"ning

(APB)!(AM PBM ) , , .0 (APB)#(AM PBM )

(2.27)

it is easy to see that is odd under CP and CP¹ . For to receive non-vanishing .0 , .0 contributions, at least two amplitudes with di!erent (CP-even) absorptive phases as well as with di!erent CP-odd phases must contribute to APB. To see this explicitly, let us de"ne M and M to be the two possible amplitudes contributing to APB M,M(APB)"M e P e B #M e P e B , M M ,M(AM PBM )"M e\ P e B #M e\ P e B , (2.28)

12


where are CP-odd phases that change sign as one goes from APB to AM PBM , and are CP-even G G phases that can arise due to FSI. It is then easy to see that M!M M "!4M M sin( ! )sin( ! ) . (2.29) Clearly , being proportional to (M!M M ), will vanish if the two amplitudes do not have .0 a relative absorptive phase, i.e., ! O0 as well as a relative CP-odd phase, i.e., ! O0. Of course does not violate CPT [54] as the requirement of CPT applies only to total widths .0 (A)"(AM ). This equivalence (i.e., (A)"(AM )) is due to the fact that the absorptive phase of (APB) emanates from rescattering through an on-shell intermediate state C and vice versa, i.e., the absorptive phase for (APC) will emanate from the on-shell intermediate state B. This fact is an example of the well-known `CP-CPT connectiona [55}58]. The states B and C are referred to as compensating processes. Of course, one can in general have this compensation act between several "nal states. It is sometimes useful to de"ne a slightly di!erent asymmetry which also requires dealing with partial rates. This asymmetry is called the partially integrated rate asymmetry (PIRA) and is de"ned as (APB)! (AM PBM ) .' , (2.30) , .' .'0 (APB)# (AM PBM ) .' .' where is the partially integrated width for APB obtained by integrating only part of the full .' kinematic range of phase-space. Often such asymmetries can be larger than since the portion .0 of the "nal states not included in the integral may themselves be the compensating process. For example, in [59] it was shown that detecting CP violation e!ects in the process tPb through O is more e$cient than through as the former is driven by tree-level diagrams and the .'0 .0 latter by 1-loop diagrams. A related observable which is also CP-odd and ¹ -even is the energy asymmetry , E ! E M G , (2.31) , G # E # E M G G where E is the average energy of a particle i in a decay of the `parenta particle and E M is the G G average energy of the corresponding anti-particle iM in the decay of the conjugate state of the `parenta particle. Such an asymmetry becomes relevant when the decay involves three or more particles in the "nal state and may be regarded as a weighted PRA. A further generalization of the above constructions of CP-odd ¹ -even observables is by , considering combinations of dot products (thus being even under ¹ ) of measurable momenta or , spin vectors. Examples of such CP-odd ¹ -even observables will be given in the following sections. , As an example of the various types of operators discussed above let us consider the reaction

e\(p )#e>(p )Pt(p , s )#tM (p , s ) , (2.32) C C R R R R where p are 4-momenta and s are spins. Clearly, no ¹ -odd observable can be constructed G G , without the Levi-Civita tensor, , and since the momenta satisfy p #p "p #p , one needs to use C C R R the spins of the t and tM to construct such observables. Indeed no non-trivial CP-odd observable can


13

be constructed without knowing the spins either, so top polarimetry is essential for the study of CP violation in this reaction (see Section 2.8). As an example of a ¹ -even CP-odd operator consider in the c.m. frame of the e>e\: , O "(p !p ) ) (s !s )"!(P !PM ) ) (S !SM ) . C C R R C C R R

(2.33)

Clearly this is ¹ -even and also C-even. It is P-odd since S , SM are axial vectors while P , PM are , R R C C polar vectors. Thus, since O is CP-odd and CP¹ -odd, it will have an expectation value if both CP , violation and rescattering e!ects are present. Consider now the operator O " ?@ABp p s s J(P !PM ) ) (S ;SM ) . C? C@ RA RB C C R R

(2.34)

Clearly O is ¹ -odd and C-even but also P-odd. This CP-odd observable can therefore give , a signal of CP violation without the necessity of rescattering e!ects. Finally, the operator O " ?@ABp p (s !s ) (p #p ) J(P !PM ) ) [(S !SM );(P #PM )] , C? C@ R RA R RB C C R R R R

(2.35)

is ¹ -odd, P-even and C-even and so provides a signal of CP-conserving rescattering e!ects. , Similarly many more operators can be constructed with the symmetries of the above. Some consideration of the physics to be tested for is helpful in selecting the operator most useful in measuring possible CP-violating e!ects. We will consider that in more detail in Section 2.6. 2.3. Partial rate asymmetries and the CP-CPT connection As mentioned in the previous section, one of the most interesting and widely studied observable for testing CP violation is the di!erence between the partial width of a reaction, (APB), from that of the conjugate reaction, (AM PBM ). Thus, if (APB)O(AM PBM ), then CP is violated in the decay. In practice, it is better to work with the corresponding dimensionless ratio in Eq. (2.27), .0 called the PRA. Its use, speci"cally in the context of heavy quarks and the SM, "rst appeared in [60]. Since the CPT theorem demands that particle and anti-particle have identical life-times (or total widths), that theorem imposes important restrictions on the form of CP-violating PRAs; these were "rst recognized by GeH rard and Hou [56,57]. In perturbative calculations, which lead to PRAs, if all the diagrams are systematically included, the requirement of CPT } that the total rate and its conjugate be identical } should be manifest order by order. The compensating processes should also be evident as the internal states of loop diagrams. When simpli"cations are used in such calculations the constraint of the CPT theorem provides an important consistency check. Furthermore, these restrictions can be used to greatly facilitate the calculations of the PRA for a compensating process. A general formalism for maintaining the CPT constrains in calculating PRAs was given in [58]. In particular, it was shown that if one de"nes a partial width di!erence, into a particular "nal state, I, as ,(PPI)!(PM PIM ) , '

(2.36)

14


where P and PM are the decaying particle and its anti-particle, respectively, then equality of total widths (P)"(PM ) implies "! . ' ( ($' More speci"cally, in perturbation theory one can write at a given order

(2.37)

" (J) ' ' ($'

(2.38)

(J)"! (I) . ' (

(2.39)

and

Here (J) denotes a contribution to the partial width di!erence into the "nal state I which is ' driven by the "nal state J, and conversely, (I) is the contribution to the particle width di!erence ( into the "nal state J being driven by the "nal state I. Due to Eq. (2.39), summing up the partial width di!erences over all the "nal states, one gets "0 , (2.40) ) ) where K runs over all "nal states, I, J,2. Thus the requirements of CPT are automatically satis"ed. Two important conclusions that can also be drawn are: (a) Rescattering of a state on to itself cannot give rise to PRA. This follows immediately from Eq. (2.39) by setting J"I. (b) The knowledge of the PRA into some "nal state I that is driven by the absorptive cut across a "nal state J, can be used to deduce the PRA into the "nal state J arising over a corresponding cut across the state I. As mentioned before, such two processes are often called `compensating processesa. Let us "rst illustrate how these considerations apply to PRAs in b decays. Consider, for example, the process bPscc . The lowest order non-vanishing contribution to the PRA arises here, at order , from the interference of the tree graph in Fig. 1(a) with the penguin graph in Fig. 1(b) Q which has an absorptive cut across the u quark line. The compensating process is then bPsuu where, for this process, the absorptive part, Im(loop), is driven by a cut on the c quark line in the loop. Thus, (suu )# (scc )"0 , QSS QAA

(2.41)

where the compensating nature of these two processes is illustrated in Fig. 1(c). At this point, it is instructive to discuss in some detail how the cancellation follows from the Feynman diagrams combined in Fig. 1(c). Let us denote by ¹ , ¹ the tree-level contributions to QAA QSS bPscc , bPsuu , respectively. Likewise, let us denote by PO , PO the penguin contributions to QAA QSS bPscc , bPsuu , respectively, with q"u, c, t being the intermediate quark in the penguin. We also denote the conjugate amplitudes as ¹M , ¹M , PM O and PM O . QSS QAA QSS QAA


15

Fig. 1. (a) Tree-level diagram for bPscc , (b) 1-loop (order ) penguin diagram for bPscc , and (c) a diagrammatic Q description for the compensating nature of the contribution from bPscc and bPsuu to the PRAs. For bPscc : the dashed line indicates an absorptive cut along inner particles (uu ) in the penguin diagram and the solid line refers to an external phase-space cut (cc ). For bPsuu the role of the cuts are reversed: the dashed line is the external uu phase-space cut and the solid line indicates the cc absorptive cut in the penguin contribution.

These amplitudes may be represented in terms of their magnitude and phase as follows ¹ "e (A ¹ , ¹M "e\ (A ¹ , QAA QAA QAA QAA ¹ "e (S ¹ , ¹M "e\ (S ¹ , QSS QSS QSS QSS PO "e (O e HAO PO , PM O "e\ (O e HAO PO , QAA QAA QAA QAA PO "e (O e HSO PO , PM O "e\ (O e HSO PO . (2.42) QSS QSS QSS QSS Here is the CP-odd weak phase which has its origin in the Lagrangian. In particular, the SM O gives "arg(< }H> interference has non-vanishing contributions only for the scalar part of the =. This argument is most readily seen if one uses the Landau gauge for calculating this interference (for more details see Section 5). In that case, the scalar and vector components of the = propagator are cleanly separated according to their total angular momentum. Thus, graphs which pass through a vector = intermediate state will not interfere with graphs that pass through a Higgs state. The Higgs must therefore interfere only with the Goldstone propagator which corresponds to the decay of longitudinal = into fermion pairs cs , udM , e 2 which are all suppressed by powers of the fermion masses. Furthermore, the C Goldstone propagator shows none of the resonance enhancement associated with the vector component. Thus, the cs is the most important contributor to the scalar component of the =-boson `bubblea. The and cs can be thought of as the compensating processes. So, in fact, the PRA goes as m m & A O 5 , O m m m R R 5 and is extremely small [63].

(2.50)


19

However, as already mentioned before, just because PRA is vanishingly small does not, though, mean that there are no CP violation e!ects. Indeed, very important and large CPviolating asymmetries may arise in the decay tPb . First of all, one can construct an energy O asymmetry: E ! E O . (2.51) " O # E # E O O That is, compare, e.g., the average energy of the in tPb with that of the in tM PbM . Then will not su!er from the helicity suppression or constraints of CPT on and one expects # O m m #& R R , (2.52) m m O A O as explicit calculations con"rm. Indeed, a CP-violating asymmetry even bigger than the energy asymmetry, namely the transverse polarization asymmetry of the , resides in this =>}H> interference. In fact, the transverse polarization asymmetry is enhanced by another factor of m /m compared to the energy asymmetry R O as is shown in Section 5.1.3. 2.4. Resonant = ewects and CP violation in top decays The large mass of the top (m K174 GeV) means that it decays to a three-body "nal state R primarily through an on-shell =. This fact is of particular interest in the study of CP violation in such decays since there will be a large strong (i.e., CP-even) phase inherent in this =-propagator. In particular, since the =-width is substantial ( &2 GeV), the transverse modes of the = are 5 controlled by the Breit}Wigner propagator 1 , (2.53) G " 2 q !m #im 5 5 5 5 which will have a substantial strong phase. The enhancement of the imaginary part of G is evident by considering that, at q "m , 2 5 5 Im(G )"(m )\ . (2.54) 2 5 5 The real part swings through 0 at this point but in the vicinity of the resonance it will also be large. For instance, if q !m "m , then 5 5 5 5 Re(G )"(2m )\ . (2.55) 2 5 5 Since &O() this means that near qPm both the real and imaginary parts of the 5 5 amplitudes for decays such as tPbudM , bcbM ,2 behave as if they are O(1) in the gauge coupling constant. This phenomena is what we mean by `resonance enhancementa. The imaginary part here then provides the needed absorptive part (i.e., FSI phase) to lead to enhancement of CP-odd ¹ -even observables. Likewise, near q&m , the real part can magnify the e!ect of , 5 CP-odd ¹ -odd observables. ,

20


The basic idea of FSI phase driven by particle widths in decays was discussed in [64,65]. Although originally the discussion [64,65] took place in the context of the SM in conjunction with the CKM phase, it should be completely clear that they can be equally well used with non-standard sources of CP violation. Indeed, as dealt in Section 5.1.3, the =-resonant e!ects provide a signi"cant enhancement of ¹ -even and ¹ -odd CP-violating e!ects in top decays in the context of an , , extended Higgs sector. 2.5. Ewective Lagrangians and observables One tool that is often used to catalog the e!ects of new physics at an energy scale, , much higher than the electroweak scale, is the e!ective Lagrangian (L ) method. If the underlying extended theory under consideration only becomes important at a scale , then it makes sense to expand the Lagrangian in powers of \ where the term is the SM Lagrangian and the other terms are the e!ective Lagrangian terms. Simple dimensional arguments tell us that the operator which multiplies \L must be of dimension n#4. This restriction together with symmetry considerations implies that at each order in \ there are only a "nite number of possible terms. Conversely, this implies that experimental tests for the existence of speci"c terms in L is a relatively model independent [71] way to search for new physics. Here we are interested in top quark physics which violates CP and so, in the e!ective Lagrangian approach, the operators of interest are further restricted. For example, in Section 4 we discuss the top electric dipole moment (and related e!ects) which can arise from a dimension 5 term in the e!ective Lagrangian: L JtM tFIJ . (2.56) IJ At dimension 6, photons can interact with the top quark via a CP-violating operator such as L J(tM t)(FIJF ) , (2.57) IJ which could arise, for instance, via a SUSY box diagram. Let us now consider as a speci"c example e!ective Lagrangian terms which would contribute to the process ggPttM . It is useful to recall that in such an expansion, operators that are proportional to the QCD equations of motion for the top or the gluon "elds may be eliminated by a "eld rede"nition and are therefore redundant [71]. There are a number of requirements that an operator has to satisfy for it to be relevant to CP violation in top production in hadronic collisions. These are: 1. It must violate CP. 2. Its Feynman rules must include couplings to two or fewer gluons. 3. It must not be proportional to q of one of the on-shell gluons in the initial state. For early references to resonance enhancement of CP violation in scattering processes, see [66,67]. For later references, see [68]. For subtelties resulting from non-gauge-invariance of Breit}Wigner form of the = propagator, see [69,70] and references therein.


21

The need for the "rst requirement is clear. The second requirement is present since the events in question have two gluons in the initial state and no gluons in the "nal states. If one wanted to consider experiments where additional gluon jets were detected in the "nal state, clearly one would have to generalize this requirement. In constructing a basis of operators which satisfy the above conditions it can be shown that [72] one can eliminate, without loss of generality, any operator which is equal to 0 modulo the equations of motion. Equivalently, if the di!erence of two operators is 0 modulo equations of motion, then only one need to be included. Here is a set of operators that we choose which satisfy the requirements mentioned above and are of dimension six or less O "tM i[ f (!䊐)FIJ] ¹G t , G IJ ? ? O "tM i[ f (!䊐)FIJFG ] t , @ @ G IJ O "tM [ f (!䊐)F?@FAB]t , A ?@AB A G G O "tM i[ f (!䊐)FIJFI ] dGHI¹Gt , B B H IJ O "tM [ f (!䊐)F?@FABdGHI]¹G t , (2.58) C ?@AB A H I where F's are the gluon "eld strength tensor, 䊐"DID , the analytic functions f ,2, f are form I ? C factors and ¹G"G/2, G being the Gell}Mann color matrices. Note, that in general, higher order terms in 䊐 will imply the existence of couplings to additional numbers of gluons. As an illustration, let us consider further the operator O . This operator essentially corresponds ? to the chromo-electric dipole moment (CEDM) form factor. The experimental implications of the static analog of this quantity were considered in [35]. We can expand the operator to obtain the Feynman rule. The vertex for the one gluon interaction is i f (q)tM IJ ¹Htq H , (2.59) ? I J here is the polarization vector of the gluon. This is completely analogous to the EDM form I factor. However, O now also gives rise to a two gluon coupling given for on-shell gluons by ? h (q)!h (0) ? (q ) H q I !q ) I q H ) , (2.60) g tM IJ ¹GtFGHI h (q) H I # ? ? I J I J I J Q q

where q"q #q . Note that the second term that appears is needed to maintain gauge invariance. An important feature of f (as well as of the other form factors) is that the constant piece f (0) ? ? must be real while, at qO0, f may have an imaginary part due to the possibility of thresholds ? giving rise to absorptive pieces. Indeed these type of e!ects have also been considered in some particular extensions of the SM (see Section 4). The phenomenology of the static CEDM (i.e., for qK0) was considered extensively in [35]. It was shown that in a hadron collider with &10 ttM pairs, both of which decay leptonically, a precision of about 5;10\g cm for f (0) could be achieved. We can extend this consideration Q ? with a simplifying assumption that f is approximately constant above the ttM threshold. We can ? then introduce the quantity f "f (4m)!f (0). With this assumption and approximation we "nd ? R ?

22


that, under ideal conditions, Re( f ) and Im( f ) can also be measured to a precision of about (2}3);10\g cm. Q The Weinberg model with an extended Higgs sector provides a speci"c example of non-standard physics where one can study this general feature of the operator analysis above. We recall that the source of CP violation now are the charged Higgs exchanges (see Section 3.2.4). Since O is the only ? operator which gives a one gluon Feynman rule, the electromagnetic form factor calculated in [31] is the quantity f except for the replacement of g with e, the electric charge. The result of that ? Q reference is that thus far QCD yields a limit around 5;10\g cm. In that work, it is also Q explicitly shown (see Section 4) that the q dependence is rather mild. Furthermore, above threshold the Im( f ) was also shown to have roughly the same ball park value. Thus, studies at ? a hadron collider could exhibit CP-violating signals although admittedly the experimental challenges are formidable. Another useful way to characterize the amplitude for ggPttM is to express it in terms of form factors. There are three possible color structures which such an amplitude can have. If A, B are the color indices of the gluon and i, j the indices of the t and tM , these color structures are " , D"d !¹! , F"f !¹! . GH GH GH

(2.61)

Let us de"ne P , P to be the momenta of the gluons and P , P to be the momenta of the t,tM E E R R quarks, respectively. Let us further de"ne the variables s"(P #P ), t"(P !P ), u"(P !P ) , E E E R E R z"(t!u)/(s!2m) . R

(2.62)

Let E and E be the polarizations of the gluons in a gauge where E ) P "E ) P "0. Here, we are interested in amplitudes which violate CP. These amplitudes must also be symmetric under the interchange of the two gluons. The helicity amplitudes which satisfy these conditions are aL "f L (s, z)(E ) E )(tM t)[D, , Fz] , aL "f L (s, z)( EI EJ PN PM )(tM t)[D, , Fz] , IJNM E E aL "zf L (s, z)( )EI EJ PN PM (tM t)[D, , Fz] , IJNM R R aL "zf L (s, z)EI EJ (tM t)[D, , Fz] , IJ aL "f L (s, z)( EI EJ (P #P )N)(tM Mt)[D, , Fz] , IJNM E E

(2.63)

where f is a function of s and z and the notation [D, , Fz] means that the term may be multiplied G by any of these color structures, the index n"1, 2, 3, respectively, depending on which of these color structures apply. As an explicit example, at tree-level, the CEDM operator discussed above will contribute to the amplitude a .


23

2.6. Optimized observables Due to the exceedingly short lifetime of the top quark, measurement of its couplings requires considering top production and decay simultaneously. Consider, for example, the production and decay of ttM in e>e\ annihilation (see Fig. 3), e>(p )#e\(p )Pt(p )#tM (p M ) > \ R R

(2.64)

t(p )Pb(p )#=>(p > ) R @ 5

(2.65)

with

and tM (p M )PbM (p M )#=\(p \ ) . @ 5 R Indeed each of the =! also decays leptonically or hadronically (i.e., into jets). Thus,

(2.66)

=>(p > )Pl>(pl> )#l (p ) , 5 J =\(p \ )Pl\(pl\ )#l (p ) , J 5 =!Pj (p )#j (p ) . (2.67) H H This allows one to construct a multitude of observables, involving momenta of the initial beam and various decay products, to probe the presence of anomalous vertices in the top interactions. The case of the CP violating dipole moment interactions is especially interesting to this work. Such anomalous terms could occur at the tM t, tM tZ or tM tg vertices corresponding to electric, weak or chromo-electric dipole moment of the top quark. Of experimental interest are the values of the corresponding form factors at q"s, the square of the c.m. energy rather than the moments (q"0) themselves. It is possible, therefore, that the form factor is complex. The real and imaginary parts can thus lead to distinct experimental e!ects. Two examples of simple (or `naivea) observables that can be used to probe the presence of imaginary part of the dipole moment form factor are El> ! El\ , El> # El\

(2.68)

E ! E M @ @ . E # E M @ @

(2.69)

Fig. 3. Feynman diagram describing the process e>e\PttM followed by the ttM decays tPb=>Pbl>l and tM PbM =\PbM l\l .

24


Examples of simple (or `naivea) observables that probe the presence of the real part of the dipole moment form factor are pI pJ pM pN IJMN l> l\ @ @M , (pl> ) pl\ p ) p M ) @ @ (pVl> pWl\ !pWl> pVl\ ) sgn (pXl> !pXl\ ) . (pl> ) pl\ )

(2.70) (2.71)

The ability to polarize the electron beams at a future e>e\ collider (e.g., the NLC) allows us to construct additional observables involving beam polarization. Clearly, while many observables can be constructed to probe the dipole moment, we may ask whether it is possible to construct an òptimal observablea i.e., one which will be the most sensitive or will have the largest `resolving powera. A general procedure for constructing an optimal observable was given in [29]. Here we will brie#y review the method. Let us write the di!erential cross-section as a sum of two terms ()" ()# () , (2.72) where is a parameter (e.g., dipole moment or magnetic moment form factor) and is some phase-space variable (including angular and polarization variables). For an ideal detector that accurately records the value of for each event that occurs, any method for determining the value of amounts to weighting the events with a phase-spacedependent function f () which we assume is CP-odd. Let us de"ne

f ()" f ()() d .

(2.73)

Thus, the change due to the contribution from the presence of is

f ()" f () () d, f ("1) .

(2.74)

f () then has to be compared to the error in its measurement. If n events are recorded, the error is

f"

f !( f ()) f + , n n

(2.75)

where

" d" d ,

(2.76)

is the total cross-section and

f " f d" f d .

(2.77)

Note that f ()J but f and do not depend on to "rst order; we will assume that is su$ciently small so that the above approximation is valid as well as f S (q, q ) HI HI I0 H* !\ I0 H* HI

(3.2)

32


where we introduced the multiplets of the quark weak eigenstates q H* , q , q , H0 H0 q

H*

j"1, 2,2, N ,

(3.3)

1# q, q " 0 2

1! q . q " * 2

(3.4)

and

Also, j, k are family indices, N denotes the number of families and >S , >B are the Yukawa HI HI couplings, which are arbitrary complex numbers. In our discussion we will consider N"3 corresponding to the SM with the three known families of fermions. This Lagrangian has no fermion mass term; fermion masses must therefore be induced by spontaneous symmetry breaking (SSB) of the SU(2);;(1) symmetry of the scalar potential term S , MB " >B , MS " GH (2 GH GH (2 GH

(3.5)

where v is the VEV of the Higgs doublet. In general, the mass matrices MBS are not hermitian, and each one depends on 9 complex unknown parameters. Since an arbitrary matrix M can be written as M"Hu, with H hermitian and u unitary, there exists a "eld rede"nition such that MS and MB are hermitian, i.e., that MS"MSR and MB"MBR [88}90]. A unitary transformation on the u and d quark "elds gives the physical basis where the mass matrices are diagonal ;R MS; "diag(m , m , m ) , (3.6) 0 * S A R DR MBD "diag(m , m , m ) , (3.7) 0 * B Q @ where ; , ; , D and D are unitary matrices that relate the weak eigenstates to the physical 0 * 0 * eigenstates. It is worth mentioning already at this point that all CP violation in the SM emanates from the apparent mismatch between the gauge and mass (physical) eigenstates of the quark "elds. For the physical states thus de"ned, there is no longer an exact SU(2) identity between the left handed d and u quarks (since they are no longer the gauge eigenstates). To see this, write the Lagrangian in terms of the physical "elds and drop the numerical factors and the coupling constants. Thus one is left with the CP-violating charge current terms X "=>JI #h.c. , ! I ! where => is the charged, spin 1, SU(2) vector-boson and

(3.8)

d

JI "(u , c , tM ) I< s ! * b

*

.

(3.9)


33

Here u, c, t, d, s and b are the quark mass eigenstates. The 3;3 unitary matrix < will therefore be the product of the unitary diagonalizing matrices since

< < < SB SQ S@ (3.10) , "0.33> . (3.40) \ \ The corresponding 68% and 95% CL contours are shown in Fig. 5. In the future, the measurement of B !BM oscillations will provide an extremely important test Q Q of the SM. The point is that the ratio of the mass di!erences m f B < m B " B B B B RB , (3.41) m Q f Q B Q Q < m RQ Q will involve signi"cantly less uncertainty due to hadronic matrix elements (i.e., f B). Aside from the CKM elements, the most uncertain factor on the rhs of Eq. (3.41) is f B r , Q QB f B B

Q B

.

(3.42)


39

However, theoretical uncertainties in extracting < from r are expected to become smaller RB QB [110,111] in comparison to the errors in extracting < from m alone. RB B The experiments at LEP have already made signi"cant progress in studying B !BM oscillations. Q Q A combined analysis of ALEPH, DELPHI and OPAL leads to [95}97,112]: m '8.0ps\ at 95% CL . (3.43) Q Incorporating this along with , m and bPul, into the , constraints one "nds [95}97]: ) B "0.11> , "0.33$0.06 . (3.44) \ Comparing this with Eq. (3.40) we see that the LEP bound on m is already reducing the negative Q error on appreciably [95}97]. Further slight changes (see the update of Buras in [14]) result from improvement, obtained by combining LEP/SLD/D0 data, of the limit on m to read: Q m '12.4ps\ at 95% CL , (3.45) Q and from the small increase of < /< to [113] S@ A@ < /< "0.091$0.016 . (3.46) S@ A@ Translated to and (see Fig. 4) yields sin 2"0.71$0.13,

sin "0.83$0.17 .

(3.47)

The above value of sin2 is consistent with the recent CDF result [23,114] sin 2"0.79> . \ 3.2. Multi-Higgs doublet models In the SM the interaction of the only neutral Higgs-boson with fermions is automatically P and C conserving as well as #avor conserving. This property is not valid in general in models beyond the SM. In this section we consider extensions to the SM involving the addition of extra Higgs doublets. CP-violating e!ects in such models can originate in the scalar sector and be manifested in the physics of fermions, particularly the top quark. For such an e!ect to occur, two or more complex SU(2) doublets of Higgs "elds are required; this was "rst pointed out by Lee [115,116]. However, the mere presence of more than one doublet does not guarantee CP violation in the Higgs sector. For instance, a CP-violating phase in the case of models with two Higgs doublets (2HDM) can be rotated out of the Higgs sector entirely if one imposes various discrete symmetries as will be discussed below. But if such phase(s) cannot be rotated away, this approach leads to CP violation from neutral Higgs-boson interactions, from charged Higgs-boson interactions, and perhaps in addition from the presence of a non-vanishing phase in the CKM matrix. In all cases, the various types of CP violation are presumably related at a fundamental level to CP violation in the Higgs potential, but because of our ignorance of the Higgs sector, in practice the parameters of each type of interaction are independent and should be separately measured. Another general feature of CP violation in an extended Higgs sector is that larger e!ects are expected in heavier quarks systems (compared to the usual SM approach), because Higgs-boson

40


couplings to fermions are proportional to the fermions masses. This makes the top quark system an especially good testing ground for such phenomena. CP violation in the Higgs sector can arise in models where the Higgs potential may contain complex couplings. This might lead directly to a CP-violating interaction or to complex VEVs of the Higgs "elds which can induce CP-violating e!ects. In addition, as we shall see in some examples below, it is also possible that a real potential can lead to a ground state with a complex VEV, in which case CP is broken spontaneously. In any case, there are generally a large number of parameters in these models so that considerable experimental e!ort will eventually be required to determine them all. In particular, it is important to consider which predictions of such models di!er from the SM, so that might lead to early signs that extra scalar "elds are present. CP violation in top physics is especially useful since the SM contributions to CP violation in top quark reactions are negligible and the mass-dependent coupling of the Higgs means that top quark physics is very sensitive to such e!ects. It is convenient to classify CP symmetry-breaking in the scalar sector into three di!erent categories: hard (intrinsic), soft and spontaneous. Hard or intrinsic CP violation refers to symmetry-breaking terms with dimension four, for example, terms in the Lagrangian with complex Yukawa coupling constants, or with self-coupling of scalar "elds. Soft breaking is associated with terms in the Lagrangian with canonical dimension less than four. If the Lagrangian starting from the outset is CP invariant, CP violation can still be achieved by introducing complex phases from the VEVs of the scalar "elds (i.e., spontaneous CP violation). In the following, we will consider simple versions of 2HDM and three Higgs doublets model (3HDM) in which CP violation is manifested in the interactions of neutral and charged Higgs particles with fermions. 3.2.1. Two Higgs doublet models We start with a description of the most general 2HDM. The Higgs potential for such a 2HDM is given by [117,118] /v is G G X #X !X !i> !X #i> m" !X #i> X #X !X !i> , (3.80) !X #i> !X #i> X #X where

X "[b #c cos( ! )]v v , GH GH GH G H G H

(3.81)

and >"!c v v sin 2( ! )"!c v v sin 2( ! ) "c v v sin 2 . (3.82) Since the parameter > is, in general, non-zero, it is evident that CP violation in the charged Higgs-boson sector comes from the imaginary part of the o!-diagonal Higgs-boson mass matrix elements. The unitary matrix which relates the weak eigenstates > to the physical charged states H> is G G de"ned by

> G> (3.83) > ";> H> , > H> where G> is the charged Goldstone-boson which is absorbed into the =>. ;>, which has three arbitrary phases, of which two can be removed by a rede"nition of H> and H>, can be parameterized exactly in the same way as the CKM matrix [161,162]:

c

s c s s ;>, !s c (3.84) c c c #s s e B& c c s !s c e B& , !s s c s c !c s e B& c s s #c c e B& with s , sin I , c , cos I and are the charged Higgs mixing angles and phase, respectively. G G G G & From the gauge sector, it is straightforward to show that

(2(2G )\"v #v #v "v , $

(3.85)

50


or that 1 G>" (v >#v >#v >) . v

(3.86)

It follows from Eqs. (3.83) and (3.84) that v "c v, v "!s c v, v "!s s v . (3.87) Therefore, the mixing angles I , I are determined by the VEVs v , v and v whereas I and depend on the parameters of the Higgs potential. & In terms of the Higgs mass eigenstates, the Yukawa interactions in Eq. (3.78) become L>"(2(2G ) ( ;M KM D # ;M M KD # NM M E )H>#h.c. , $ G * " 0 G 0 3 * G * # 0 G 7 G

(3.88)

with c c c #s s e B& c s c !c s e B& s c , " , " , " s c s s c s s c c s !s c e B& c s s #c c e B& " , " , " , c s c s s and we see that

(3.89)

Im( H)"!Im( H), Im( H)"!Im( H) , Im( H)"!Im( H) . (3.90) As in the charged Higgs case, we can write down an analogous 6;6 real mass matrix for the neutral scalar states. Then the neutral Higgs-boson Yukawa interactions in Eq. (3.79) become [160]: L "(2(2G ) (g DM M D#g DM M i D#g ;M M ; 7 $ G " G " G 3 G # g ;M M i ;#g EM M E#g EM M i E)H , (3.91) G 3 G # G # G where the couplings g are real. Since M and M i have opposite P, ¹ and CP transformation G properties, P and CP can be violated through the exchange of neutral Higgs-bosons. We will now brie#y mention some of the more notable constraints on this class of models [59,165}167]. One class of restrictions that are of some importance follow (as in the case of the 2HDM) if we further assume that the Higgs sector of the theory is perturbative [131]. These lead to :120, :6 . (3.92) G G Since these complex coupling constants arise from the diagonalization of the charged scalar mixing matrix, they obey the relation H"1 . G G G

(3.93)


51

Thus [165], Im( H)"!Im( H) ,

(3.94)

and Im( H)4 :720 . (3.95) G G G G Assuming for simplicity that one of the H>, for instance H>, is very heavy, then B}BM mixing imposes an important constraint on [131,168]: :2 for m &m & 8 (3.96) :3 for m &2m . 8 & Using BPX, a constraint on is deduced [152,169]: (3.97) :2m /GeV . & A direct bound on Im(H) comes from the electric dipole moment of the neutron (d ) [170,171]: L Im(H):20 for m &m & 8 :100 for m &2m . (3.98) & 8 Interestingly enough, the strongest constraint so far actually comes from a CP-conserving process bPs [165,167]. The amplitude for bPs receives contributions from terms proportional to Im( H) that do not interfere with the other terms. G G Thus, these terms only enter quadratically in the expression for the rate for bPs. A conservative bound on Im(H) (where Im(H)"Im( H)"!Im( H)) is obtained by assuming that such a contribution saturates the measured rate for bPs. These constraints are displayed in Fig. 8 [132,151] as a function of the light charged Higgs mass (m ) for various values of the heavier & charged Higgs mass (m ), subject to the restriction m 4m . In the "gure, the bottom solid & & & curve corresponds to the case when m <m so that the contribution of the second charged & & Higgs is neglected. We note that the constraints depend strongly on m and they essentially & disappear when m Km due to a cancellation between the two contributions [132,151] & & resulting from a GIM-like mechanism. It is useful to note how stringent these constraints are. For example, Im(H):1.5 for m &m & 8 (3.99) :2.5 for m &2m , 8 & for m &500 GeV. A very important consequence of these tight bounds on Im(H) is that the & charged scalar exchange can only make a negligible contribution to the CP violation parameters in KP2, i.e., or . Therefore, CP violation in the 3HDM cannot be the sole source of the observed CP violation. We should note, though, that in the original Weinberg model for three Higgs doublets, CP is not assumed to be an a priori symmetry of the model. Thus CP violation arises from complex quartic terms in the Higgs potential as well as from the phase di!erences of the VEVs. In addition, one has the complex CKM phase as an independent source of CP violation which may be able to accommodate the CP violation in the Kaon system.

52


Fig. 8. Constraints on Im(X>H) (,Im(H) in our notation) as a function of the lightest charged Higgs-boson mass M > ,m , with m "100, 250, 500 and 750 GeV corresponding to (from left to right) the dashed, dashed-dotted, solid & & & and dotted curves, respectively. The bottom dashed curve represents the case where the H! contributions have been neglected. The allowed region lies to the right and below the curves. m "175 GeV is used. Updated "gure from [132] R (see [151]).

3.3. Supersymmetric models Needless to say, the minimal SUSY extension of the SM is a very appealing theory (for reviews on supersymmetry see e.g., [172}177]). Among its compelling features are: it allows for radiative electroweak symmetry breaking (REWSB), it uni"es the gauge coupling constants, with masses of superpartners not much heavier then a TeV, it gives a well-grounded explanation to the hierarchy problem and it provides a good dark matter candidate } the lightest SUSY particle. New non-SM mechanisms of CP violation are introduced in each version of such SUSY models [44,172}179]. It is again the top quark sector in these models that may exhibit large CP-violating e!ects due to its very large mass. In particular, the supersymmetric partners of the top quark (these two scalar particles are often referred to as the stop and denoted by tI ), can be responsible for relatively large CP-violating phenomena. Such e!ects are enhanced by the possibility of having large mass splittings between the two stops which is in turn due to the relatively large top mass. This type of SUSY CP violation in top quark reactions has received considerable attention in the past few years. They are all strongly dependent on the magnitude of the low-energy phase of the soft trilinear breaking term A in the SUSY Lagrangian and we will describe some of these works in R the following sections. Although the top quark provides a good laboratory to investigate CP violation in SUSY models, 2-loop e!ects can also induce CP violation in the neutron electric dipole moment (NEDM) as well as in the electron electric dipole moment, and thus provide important constraints on such models as has been considered in [180,181].


53

Another possible manifestation of the CP-violating phase arg(A ) is Baryogenesis in the early R universe. It was shown that with arg()P0, t squarks can mediate the charge transport mechanism needed to generate the observed baryon asymmetry, even with squark masses & hundred GeV, provided that arg(A ) is not much suppressed [43]. We will therefore emphasize here the phenomR enological importance of possible CP-violating e!ects which may reside in tI !tI mixing and are * 0 therefore proportional to arg(A ). Indeed, due to experimental constraints on the NEDM, the R possible phase in the Higgs mass term, i.e., arg(), is expected to be small (see below). Thus, arg(A ) R should be practically the only important SUSY CP-odd phase observable in high-energy reactions. Of course, the most natural place to look for such e!ects, driven by arg(A ), is high-energy processes R involving the top quark. Thus, CP-violating e!ects of the top quark observable in the laboratory may have direct bearing on Baryogenesis in the early Universe. It was stated in [182] that using the relations obtained from the renormalization group equations (RGE) of the imaginary parts in the SUSY Lagrangian, combined with the severe constraint on the low-energy phase of the Higgs mass parameter, , from the present experimental limit [183] on the NEDM, the phase in A at low-energy scales is likely to be very small provided R one imposes some de"nite boundary conditions for the SUSY soft breaking terms. As a consequence, at high energies, any CP-non-conserving e!ect that is driven by arg(A ) will then be R suppressed leaving top quark reactions almost insensitive to CP-violating e!ects of a SUSY origin in models with these assumptions. On the other hand, we will describe below the key phenomenological features of a general MSSM and a GUT-scale N"1 minimal SUperGRAvity (SUGRA) model. We will demonstrate that the prediction made in [182] depends on the GUT-scale boundary conditions, and therefore may be signi"cantly relaxed to yield a large CP-violating phase in the A term compatible with the R existing experimental limit on the NEDM. This should encourage SUSY CP violation studies in top quark systems as they may well be the only venue for constraining arg(A ) in high-energy R experiments at colliders in the foreseeable future. 3.3.1. General description and the SUSY Lagrangian The most general low-energy softly broken minimal SUSY Lagrangian which is invariant under SU(3);SU(2);;(1) consists of three generations of quarks and leptons, two Higgs doublets and the SU(3);SU(2);;(1) gauge "elds, along with their SUSY partners, can be written as [172}177, 184}186]:

L"kinetic terms# d=#L

.

(3.100)

Here = is the superpotential and is given by =" (g'(QK G HK H ;K #g'(QK G HK H DK A #g'(¸K G HK H RK A #HK G HK H ) . (3.101) GH 3 ' ( " ' ( # ' ( is the antisymmetric tensor with "1 and the usual convention was used for the super"elds GH QK , ;K , ¸K , RK and HK [184]. I, J"1, 2 or 3 are generation indices and i, j are SU(2) indices. We do not include R-parity violating terms in the SUSY Lagrangian below, since we do not discuss in this review any CP-violating e!ect which may be driven by such terms. We only brie#y mention in Section 11 the possible impact of R-parity violating SUSY interactions on CP violation studies in the top quark system.

54


L consists of the soft breaking terms and can be divided into three pieces L ,L #L #L , (3.102) which are the soft supersymmetry breaking gaugino, scalar mass terms and the trilinear coupling terms. These are given by [185] L "(m #m ? ? #m @ @ ) , (3.103) 5 5 % % L "!m H!m H!m ¸G!m R!m QG!m D!m ; , & G & G * 0 / " 3 (3.104) L " (g A QGHH ;#g A QGHH D#g A ¸GHH R#BHG HH ) , (3.105) GH 3 3 " " # # where we have omitted the generation indices I and J in the soft breaking terms. The above scalar "elds correspond to the super"elds which were indicated in our notation by a `hata. , ? (with 5 a"1, 2 or 3) and @ (with b"1,2, 8) are the gauge superpartners of the ;(1), SU(2) and SU(3) % gauge-bosons, respectively. Also we remark that proportionality of the trilinear couplings to the Yukawa couplings (i.e., g , g and g ) is imposed in Eq. (3.105). 3 " # 3.3.2. CP violation in a general MSSM We now turn to a discussion of the CP-odd phases in the theory. In general, when no further assumptions are imposed on the pieces of the Lagrangian in Eqs. (3.101) and (3.103)}(3.105), there are several possible new sources (apart from the usual SM CKM and strong phases) of CP non-conservation at the scale } where the soft breaking terms are generated. These are 1 [44,172}178]: the trilinear couplings A (i.e., F";, D or E), the soft breaking Higgs coupling B, $ the gauginos mass parameters m (a"1, 2 or 3) and the Higgs mass parameter in the superpoten? tial. However, not all of them are physical and by a global phase change of one of the Higgs multiplets one can set arg(B)"0 ensuring real VEVs of the Higgs doublets and "xing the phase of to be arg()"!arg(B). Moreover, in the absence of the soft breaking Lagrangian, the MSSM possesses an additional ;(1) R-symmetry [187]. Thus, with an R-transformation one can remove an additional phase from the theory, say from one of the soft gaugino masses m . The remaining ? physical phases are: one phase for each arg(A ) (corresponding to a fermion f ), arg(B) and arg(m ), D ? say for a"1, 2. In the most general MSSM scenario, these remaining complex parameters at the -scale cannot simultaneously be made real by rede"ning the phases of "elds without introducing 1 phases in the other couplings. Of course, once the above phases are set to their -scale values, they feed into the SUSY 1 parameters of the theory at the EW-scale through the RGE. Instead of studying the RGE for the full theory, one needs to consider only a complete subset of the RGE of the complex parameters in the e!ective theory. Taking only the top and bottom Yukawa couplings and neglecting small e!ects from the other Yukawa couplings, such a complete subset was given in [182]: dm ? "2b m , ? ? ? dt

(3.106)

dA R "2c m #12 A #2 A , ? ? ? R R @ @ dt

(3.107)


55

dA @ "2c m #12 A #2 A , ? ? ? @ @ R R dt

(3.108)

dA SA "2c m #6 A , ? ? ? R R dt

(3.109)

dA BQ "2c m #6 A , ? ? ? @ @ dt

(3.110)

dB "2c m #6 A #6 A , ? ? ? @ @ R R dt

(3.111)

d R "2 (!c #6 # ) , R ? ? R @ dt

(3.112)

d @ "2 (!c #6 # ) , @ ? ? @ R dt

(3.113)

d ? "2b , ? ? dt

(3.114)

where t,ln(Q/ )/4, a is summed from 1 to 3 and b "(, 1,!3), c "(, 3, ), c "( , 3, ), 1 ? ? ? c"(, 3, 0). Also, and are related to the corresponding quark masses via R @ ? 1 g m . (3.115) " R@ R@ 8 m sin (cos ) 5 We remark that in the general MSSM framework with arbitrary CP-violating phases at , the above RGE are of less importance and with such boundary conditions almost any 1 low-energy CP-violating scenario can be generated. In particular, large CP-violating phases at the EW-scale are not excluded in this unconstrained scenario. However, in a more constrained SUSY version, one can reduce the number of the physical CP-odd phases in the theory. In this case the RGE given above are crucial for determining the SUSY CP-violating phases at the EW-scale [182]. We will return to this point later when we discuss the N"1 minimal low-energy SUGRAGUT model. Let us consider now the phenomenological consequences of CP non-conservation in such a general SUSY model. First, we need to describe brie#y how these new CP-violating phases enter in reactions which are driven by supersymmetric particles. As it turns out, all CP violation in the low-energy SUSY vertices is driven by diagonalization of the complex mass matrices of the sfermions, charginos and neutralinos. For more detailed investigations of the diagonalization procedure and extraction of the mass spectrum and CP-violating phases from these mixing matrices we refer the reader to the existing literature, see e.g., [172}177] and [185,188}196]. Here we only wish to brie#y describe the key features of the formulation and the de"nitions. We denote by MI the mass squared matrix of the scalar partners of a fermion, and M and Q D M are the mass matrices of the charginos and neutralinos, respectively. Then, with the rotation Q

56


matrices Z , Z , Z> and Z\, we can de"ne D , ZR MI Z "diag(mI , mI ) , D D D D D (Z\)RM Z>"diag(m , m ) , Q Q Q Z2 M Z "diag(m , m , m , m ) . Q Q Q Q , Q , MI is then given by D m !cos 2(¹ !Q sin )m #mI * D D 5 8 D MI " D D !m (R H#A ) D D D

(3.116) (3.117) (3.118) !m (R #AH) D D D m !cos 2Q sin m #mI 0 D D 5 8 D

, (3.119)

where m is the mass of the fermion f, Q its charge and ¹ the third component of the weak D D D isospin of a left-handed fermion f. mI * (mI 0 ) is the low-energy mass squared parameter for the left D D (right) sfermion fI ( fI ). R "cot (tan ) for ¹ "(!) where tan "v /v is the ratio between * 0 D D the two VEVs of the two Higgs doublets in the model. M and M are given by Q Q m (2m sin 5 M " , (3.120) Q (2m cos 5

m

0

M " Q !m cos sin 8 5 m sin sin 8 5

0 m

m cos cos 8 5 !m sin cos 8 5

!m cos sin 8 5 m cos cos 8 5 0 !

m sin sin 8 5 !m sin cos 8 5 , ! 0

(3.121)

where m (m ) is the mass parameter for the ;(1)(SU(2)) gaugino. Because of their relatively simple form, we will discuss below the way of parameterizing the CP-violating phases only of the sfermions and charginos diagonalizing matrices Z and Z>, Z\, D respectively. The diagonalization of the 4;4 neutralino mixing matrix with complex entries is more involved and may be estimated numerically, although in some limiting cases it may be approximated analytically (see e.g., [182]). If all elements of Z are real then the diagonalization , procedure can also be done analytically (see e.g., [191,193]). The mixing matrix of the sfermions is parameterized as

cos !e\ @D sin D D , (3.122) Z " D cos e @D sin D D where is the mixing angle and is the phase responsible for CP-violating phenomena in D D sfermions interactions with other particles in the theory, and is given by R sin !A sin D I D D , tan "! D R cos #A cos D I D D

(3.123)


57

where we have used A "A e ?D and "e ?I . Recall that R "cot (tan ) for ¹ "(!) D D D D and tan "v /v . Also, the mixing angle is given by D !2m R #AH D D D . (3.124) tan " D cos 2(2Q sin !¹ )m #mI !mI D 5 D 8 D* D0 It is obvious from Eq. (3.123) and (3.124) that in the limit where all the quark masses are small except for m , only the phase of A leads to CP-violating e!ects (the limit m P0, fOt, is useful R R D when considering high-energy reactions). In particular, the other A-terms are multiplied by the light fermion masses (see also Eq. (3.119)) and, therefore, they have negligible e!ect on any physical quantity evaluated at high enough energies. That is, the o!-diagonal elements of Z are zero in this D limit (i.e., from Eq. (3.124) we see that sin P0 when m P0) and there is no mixing between the D D left and right components of the superpartners of light quarks. Of course, this is not the case for the NEDM which is particularly sensitive to the slight deviation from degeneracy of the supersymmetric partners of the u and the d quarks. We will return to a more detailed discussion on the `SUSY-CP problema of the NEDM in the following section. For a sfermion fI , it is useful to adopt a parameterization for its fI !fI mixing such that the * 0 sfermions of di!erent handedness are related to their mass eigenstates through the transformation fI "ZfI #ZfI , * D D fI "ZfI #ZfI , (3.125) 0 D D where fI are the two mass eigenstates. We note that in the case where all CP violation arises from fI !fI mixing, i.e., from the complex entries in the sfermion mixing matrix Z , it has to be * 0 D proportional to (!1)G\ sin 2 sin . Im(G ),Im(ZGHZG)" D D D D D 2

(3.126)

Clearly, G P0 if there is no mixing between the left and right sfermions such that they are nearly D degenerate. We will describe in the next sections CP-non-conserving e!ects in top quark systems which are driven by the possibly large mass splitting between the two stop mass eigenstates and are therefore proportional to Im(G ). R The charginos mixing matrices are given by Z!"P!O! ,

(3.127)

where

1

P>"

0

0 !e ?

, P\"

e ?

0

0

!1

,

(3.128)

and

cos !e\ @> sin > > , cos e @> sin > > e A cos !e\ @\ \A sin \ \ . O\" e A cos e @\ >A sin \ \ O>"

(3.129) (3.130)

58


Here "arg(m ) and the CP-violating angles and above are given by ! sin tan ,! , (3.131) > cos #m cot / sin tan , , (3.132) \ cos #m tan / sin , (3.133) tan ,! cos #2m (m !)/gv v Q sin tan , , (3.134) cos #gm v v /2(m !m ) Q where , # and m G (i"1, 2) are the masses of the two charginos. and are also I Q > \ functions of m , , v and v (see [189,190]). 3.3.3. CP violation in a GUT-scale N"1 minimal SUGRA model Let us proceed by describing a more constrained supersymmetric model. In particular, we want to consider a spontaneously broken N"1 SUGRA, which apart from gravitational interactions, is essentially identical at low energies to a theory with softly broken supersymmetry with GUT motivated relations at the GUT mass scale. One of the most appealing consequences of such a constrained SUSY scenario is that it allows REWSB of SU(2);;(1) with the fewest number of free parameters. According to conventional wisdom, complete universality of the soft supersymmetric parameters at the GUT-scale (or at the scale where the SUSY soft breaking terms are generated) is assumed. More explicitly, a common scalar mass m and a common gaugino mass M at the GUT-scale m "m "m "m "m "m "m "m , & 0 * / 3 " & m "M , a"1, 2 or 3 , (3.135) ? and also universal boundary conditions for the soft breaking trilinear terms are assumed A "A "A ,A% . (3.136) # " 3 Of course, the above relations do not survive after renormalization e!ects from the GUT-scale (which is usually taken to be M &2;10 GeV) to the EW-scale are included. % It then follows that the universal parameters of the minimal SUGRA model at the GUT-scale are: m , M , A%, B%, % and tan . This is the most general set of independent parameters before REWSB. However, a bonus of this economical framework is that REWSB occurs and the parameters % and B% are no longer taken as independent but are set by m , M and tan (the magnitude of is adjusted to give the appropriate Z-boson mass but the sign of remains as an independent parameter). Thus, the number of independent parameters is reduced to "ve, namely m , M , A%, sign() and tan . In this GUT motivated SUGRA theory there are four possible new sources of CP nonconservation at the GUT-scale. These are [178]: the universal trilinear coupling A%, the Higgs mass


59

parameter %, the gauge mass parameter M and the parameter B%. However, as was mentioned before, M can be made real by an R-transformation and by using one remaining phase freedom, a rede"nition of the Higgs "elds can set the product B%% to be real so that the VEVs of the two Higgs "elds in the theory are also made real. We therefore have arg(%)"!arg(B%). With this choice we are left with only two new SUSY CP-odd phases at the GUT-scale which are carried by A% and %, apart from the usual CKM phase that originates from the Yukawa couplings in the theory for three generations. One can proceed by choosing a more constrained CP-violating sector by setting one of these two phases to zero (we will address to this possibility in the next section), or even a `supera constrained CP-violating sector by taking arg(%)"arg(A%)"0, thus being left (at the GUT-scale) only with the usual CKM phase present as in the SM. Note that having a universal phase for all trilinear couplings of sfermions at the GUT-scale, does not necessarily mean that all sfermions will have the same phase (driven by their trilinear coupling A ) at the EW-scale. That is, the GUT-scale phases arg(A%) and arg(%) feed into the other D parameters of the theory through renormalization e!ects from the GUT-scale to the EW-scale. In particular, these GUT-scale phases can produce di!erent phases for di!erent squarks and sleptons of di!erent generations. However, irrespective of what those di!erent phases are at the EW-scale, they can all be expressed, in principle, with the two new CP-odd phases of A% and % through the RGE in Eqs. (3.106)}(3.114). Thus, it is evident from the very simple structure of the evolution equations of the gaugino masses (see Eq. (3.106)) that m , a"1}3, are left with no phase at any ? scale. Moreover, the di!erence between the three real low-energy gaugino mass parameters comes from the fact that they undergo a di!erent renormalization as they evolve from the GUT-scale to the EW-scale, due to the di!erent gauge structure of their interactions. In particular, they are related, at the EW-scale, by (see e.g., [191]) m m 3 cos m 5 "sin " , (3.137) 5 5 Q where is the weak mixing angle and m is the low-energy gluino mass. 5 Notice now that with arg(m )"0, the RGE (Eqs. (3.106)}(3.114)) simplify to a large extent, thus ? we have only to consider the evolution of the A%'s and B% (or equivalently %) to the EW-scale. This D constrained version of the MSSM has strong implications on the low-energy phase in A as was R suggested in [182]. In particular, it was shown that with and without universal trilinear couplings at the GUT-scale and with some de"nite boundary conditions for them (for example, arg(%)"arg(A%)"0 for fermion species f ), the low-energy CP-violating phase of A induces D R potentially large phases in A , A and B at the EW-scale (through relations obtained from the S B above set of the RGE). This in turn gives rise to a large NEDM which is ruled out by the present experimental limit. Therefore, severe constraints on the low-energy phases of A and where D obtained. We will return to this point in the next section. 3.3.4. A plausible low-energy MSSM framework and the `SUSY-CP problema of the NEDM The EDM of the neutron, d (for reviews on fermion electric dipole moments see e.g., [197}201]) L imposes important phenomenological constraints on SUSY models, see e.g., [178,182,189,190, 192,194}196,202], [203}214] and references therein. In particular, with a low-energy MSSM that originates from a GUT-scale SUGRA model (with complete universality of the soft breaking terms),

60


keeping d within its allowed experimental value (i.e., d :10\ e cm [183]) requires the `"ne L L tuninga of the SUSY phases to be less than or of the order of 10\}10\ for SUSY particle masses of the order of the EW symmetry breaking scale. We remark, though, that it has recently been claimed in [194}196], that in some regions of the SUSY parameter space, cancellations among the di!erent components of the neutron EDM may occur and such a severe "ne tuning for either the SUSY masses or the SUSY CP-violating phases (i.e., at the order of 10\) may not be necessary. However, for such cancellations to occur SUSY parameters have to be suitably arranged and, also, several large SUSY phases have to be present, which renders this scenario less predictive and less attractive as well. The NEDM can be written schematically in any low-energy MSSM scenario in which the CP-phases originate from the and the A terms as [182] D A A d L "XI sin #XS S sin #XB B sin , (3.138) I S B M M M 10\ e cm 1 1 1 where M is the typical SUSY mass scale which may be used to describe the typical squark masses. 1 It was found in [182] that typical values of the XG's in almost every such low-energy SUSY realization are: XS B 'O(1) and XI'O(10) for M :500 GeV. Therefore, with A + 1 D +M , only moderate bounds can be put on sin and sin . In contrast, an unambiguous 1 S B severe constraint is obtained on the low-energy phase of the Higgs mass term , namely sin (O(10\) for M :500 GeV (see also [44,189,190,202] and references therein). I 1 The important "nding in [182] is that if a complete universality of the soft breaking terms is imposed at the GUT-scale and the two GUT phases are zero or very small, i.e., arg(A%)+ arg(B%):0.1, then this severe constraint on sin combined with the relations between Im(A ) and I R Im(), obtained from the RGE, leads to the comparable constraint sin (O(10\). With a univerR sal A term at the GUT-scale this will also imply (through relations obtained from the RGE) sin (O(10\). Moreover, the above strong constraints on the SUSY CP-violating phases hold SB even if the universality of the A terms is relaxed at the GUT-scale as long as the GUT-scale CP-violating phases are kept very small. This strong constraint on the low-energy phase of A would also eliminate any possible SUSY CP-non-conserving e!ect in top quark systems. R While this scenario with very small CP-violating phases at the GUT-scale provides an explanation of the smallness of the NEDM (in fact, the above constraints will drive the NEDM to a value of the order of 10\!10\ e cm), an unavoidable question then arises: why do the CP-violating phases happen to be so small wherever they appear? If so, then an underlying theory that screens the CP-violating phases is required. We therefore feel that a somewhat di!erent phenomenological approach is needed, namely, the implication that sin (O(10\) should be specially scrutinized in the top quark sector. The latter, R being very sensitive to sin at high energies, may serve as a unique probe for searching for R signi"cant deviations from the above upper bound, sin (O(10\) (see e.g., [215]). Indeed, if at R the GUT-scale the universality of the A terms is relaxed and no assumption is made on the magnitude of the CP-violating phases, then one can construct a plausible low-energy MSSM framework which incorporates an O(1) low-energy phase for A , while leaving the NEDM within its R experimental limit. The crucial di!erence in assuming non-universal boundary conditions for the Since M is the only SUSY mass scale associated with the squarks sector, it is natural to choose the soft breaking 1 terms to be of O(M ). 1


61

soft breaking trilinear A terms is that, in this case, there is no a priori reason to believe that the low-energy phases associated with the di!erent A terms are related at the EW-scale. In particular, D the bounds on sin obtained from the experimental limit of the NEDM may not be used to SB constrain sin . In addition, when no further assumption is made on the magnitude of the R CP-violating phases at the GUT-scale, a value of sin P0 may be realized without contradicting I any existing relation from the RGE. As in [215] we therefore take the following phenomenological point of view in constructing a plausible low-energy set of the MSSM CP-violating phases and mass spectrum: 1. sin P0 as strongly implied from the analysis of the NEDM. I 2. sin , sin and sin are not correlated at the EW-scale, which implicitly assumes nonS B R universal boundary conditions at the GUT-scale. In particular, sin , sin may then be S B constrained only from the NEDM experimental limit, with no implications on the size of sin . R 3. Motivated by the theoretical prediction of the uni"cation of the SU(3), SU(2) and ;(1) gauge couplings when SUSY particles with a mass scale around 1 TeV are folded into the RGE, one may follow only the following traditional simplifying GUT assumption: there is an underlying grand uni"cation. As mentioned before, this leads us to have a common gaugino mass parameter de"ned at the GUT-scale which can be made real by an R-rotation. Thus, m , a"1}3 ? are left with no phase at any scale. Moreover, using the relation of Eq. (3.137), once the gluino mass is set at the EW-scale, the SU(2) and ;(1) gaugino masses m and m , respectively, are determined. 4. The typical SUSY-scale is M and all the squarks except for the light stop, are assumed to be 1 degenerate with mass M . 1 Note that in this low-energy framework one is only left with the phases of the various A terms at the EW-scale, out of which only A plays a signi"cant role in any high-energy D R reaction. For m P0 the superpartners of the light quarks are practically degenerate and D therefore the CP-violating e!ects from the phases of the other A terms, corresponding to the D light quarks, can be safely neglected. Note again that our approach, the EWPGUT approach, assumes a set of phases at the EW-scale, subject to existing experimental data, which implicitly assumes arbitrary phases at the scale in which the soft breaking terms are generated. As was mentioned above, this low-energy CP violation scenario can naturally arise from a GUT-scale SUGRA model if only the universality of the A terms is relaxed and no assumption is made on the magnitude of the GUT-scale phases. Also, the mass matrices of the neutralinos, M , and charginos, M , depend on the low-energy Q Q Higgs mass parameter , the two gaugino masses m and m (which are resolved by the gluino mass) and tan (see previous sections) and are therefore real in this scenario. Thus, once , m and tan are set to their low-energy values, the four physical neutralino species m L (n"1}4) Q and the two physical chargino species m K (m"1, 2) are extracted by diagonalizing the real Q matrices M and M . Q Q 5. Finally, the resulting SUSY mass spectrum may now be subject to the existing experimental limits, i.e., limits on the masses of squarks, gluino, neutralinos, charginos, etc. (for recent phenomenological reviews of supersymmetry and reported limits obtained and expected in existing and future experiments, see e.g., [216}223]).

62


With these assumptions, when arg()P0, the leading contribution to a light quark EDM comes from gluino exchange, which with the approximation of degenerate u and dI squark masses (which we will denote by m ), can be written as [189,190] O A sin 2 O (rK(r) , d (G)" Q Q em O O O O m 3 O

(3.139)

where m (m ) is the quark(squark) mass and Q is its charge. Also, r,m /m (for the rest of this O % O O O section we denote the gluino mass by m ) and K(r) is given by %

1 r(2#r) K(r)" #r# ln r . (r!1) 1!r

(3.140)

Then, within the naive quark model, the NEDM can be obtained by relating it to the u and d quarks EDM's (i.e., d and d , respectively) as S B d "(4d !d )/3 . L B S

(3.141)

We now consider arg(A ) and arg(A ) to be free parameters of the model irrespective of arg(A ). In S B R Figs. 9(a) and (b) we have plotted the allowed regions in the sin !sin plane for d not to S B L exceed 1;10\ e cm (for the present experimental limit see [183]) and 3;10\ e cm. In calculating d , we assumed that the above naive quark model relation holds. Although there is no doubt L that it can serve as a good approximation for an order of magnitude estimate, it may still deviate from the true theoretical value which involves uncertainties in the calculation of the corresponding hadronic matrix elements. Note also that it was argued in [224] that the naive quark model overestimates the NEDM, as the strange quark may carry an appreciable fraction of the neutron spin which can partly screen the contributions to the NEDM coming from the u and the d quarks. To be on the safe side, we therefore slightly relax the theoretical limit on d in Fig. 9(b) to be L 3;10\ e cm. We have used, for these plots, m I "m "M "400 GeV, m "500 GeV and for simplicity we S 1 % B also took A "A "M . As remarked before, it is only natural to choose the mass scale of the S B 1 soft breaking terms according to our typical SUSY mass scale M . Also, we took the values for 1 current quark masses as m "10 MeV, m "5 MeV and (m )"0.118. B S Q 8 From Fig. 9(a) and in particular Fig. 9(b), it is evident that M "400 GeV and m "500 GeV 1 % can be safely assumed, leaving ènough rooma in the sin !sin plane for d not to exceed S B L 1}3;10\ e cm. We observe that while sin is basically not constrained, !0.35:sin :0.35 is S B needed for d (1;10\ e cm and !0.55:sin :0.55 is needed for d (3;10\ e cm. L B L Moreover, varying m between 250 and 650 GeV has almost no e!ect on the allowed areas in the % sin !sin plane that are shown in Figs. 9(a) and (b). That is, keeping M "400 GeV and S B 1 lowering m down to 250 GeV, very slightly shrinks the dark areas in Figs. 9(a) and (b), whereas, % Note that the function K(r) in Eq. (3.140) is slightly di!erent from that obtained in [182]. However, we "nd that numerically the di!erence is insigni"cant and does not change our predictions below. We will take M "400 GeV and vary m in this range in some of the CP-violating e!ects in collider experiments to 1 % be discussed in the following sections.


63

Fig. 9. The allowed regions in the sin !sin plane for the NEDM not to exceed (a) 1;10\ e cm and S B (b) 3;10\ e cm. M "400 GeV and m "500 GeV is used. The shaded areas indicate the allowed regions. Figure 1 % taken from [215].

increasing m up to 650 GeV slightly widens them. Of course, d strongly depends on the scalar % L mass M } increasing M enlarges the allowed regions in Figs. 9(a) and (b) as expected from 1 1 Eq. (3.139). It is also very interesting to note that, in some instances, for a cancellation between the contributions of the u and d quarks to occur, sin , sin '0.1 is essential rather than being just S B possible. For example, with sin 90.75, sin 90.1 is required in order to keep d below its S B L experimental limit. We can therefore conclude that CP-odd phases in the A and A terms of the order of few;10\ S B can be accommodated without too much di$culty with the existing experimental constraint on the NEDM even for typical SUSY masses of :500 GeV. Therefore, we restate what is emphasized in [215]: somewhat in contrast to the commonly held viewpoint we do not "nd that a `"ne-tuninga at the level of 10\ is necessarily required for the SUSY CP-violating phases for squark masses of a few hundreds GeV or slightly heavier. 3.3.5. CP and the pure Higgs sector of the MSSM Since the superpotential is required to be a function of only left (or only right) chiral super"elds, it forbids the appearance of HK H and HK H in the superpotential = in Eq. (3.10). Therefore, since a QK G HK H ;K coupling in = is prohibited by gauge invariance, only H is responsible for giving mass ' ( to up quarks and H to down quarks [188]. As a consequence, the requirement that there will be no `harda breaking terms of the symmetry P! in the Higgs potential is automatically G G satis"ed in a minimal supersymmetric model. That is, " "0 in Eq. (3.48), for the Higgs

64


potential in the MSSM. Moreover, no term of the form ( R )#h.c. , (3.142) appears in the Higgs potential of the MSSM [188], thus implying "0 in Eq. (3.48). It is then straightforward to observe that with the above constraints on the pure Higgs sector of the MSSM, any phase which may appear in a complex soft breaking parameter (i.e., proportional to in Eq. (3.48)) can be removed by a rede"nition of one of the Higgs doublet "elds, thus also setting the relative phase between the two VEVs to zero. Therefore, the pure Higgs sector in the MSSM possesses no CP violation [188]. Of course, CP violation may emerge in interactions of the Higgs "elds with the other "elds in the theory due to the CP-violating phases carried by these latter "elds.

4. Top dipole moments 4.1. Theoretical expectations A non-vanishing value for the EDM of a fermion is of special interest as it signi"es the presence of CP-violating interactions. We recall that the search for the EDM of the neutron and that of the electron have intensi"ed in recent years. Since the top is such an unusual fermion, in fact so heavy that it is very unlikely to exist as a bound state with another quark, it is clearly important to ask: What is its EDM? How can we measure it, if at all? This topic has been of interest to many for the past several years. In the SM quarks cannot have an EDM at least to three loops (for reviews on fermion electric dipole moments see [197}201]). For the electron the three loop contribution has been estimated to be dA (0)&10\ e cm. Simple-dimensional scaling then suggest for the top the value C dA(0)&10\}10\ e cm, much too small to be observable. R In contrast, in extensions of the SM, e.g., MHDMs and SUSY models, this situation changes sharply and the top dipole moment (TDM) can arise at the 1-loop level and as a result, the typical TDM is of the order of 10\!10\ e cm which is larger than the SM prediction by more than 10 orders of magnitude. The enhancement due to the large top mass is particularly evident in some models with an extended Higgs sector for which the dipole moments often scale as m . Since D at 1-loop light quarks (or neutron) in these models can get dipole moment of order 10\ e cm, the TDM could easily reach 10\ e cm or even more. It is at that level that measurable consequences can arise. Because of the unique importance of the top quark, it should be clear that measurements of the TDM will be extremely important. However, it should also be clear that due to the extraordinary short lifetime of the top quark (:10\ s) it will be extremely di$cult to actually measure the static

Strictly speaking, the term dipole moment refers to the static form factor (i.e., at q"0). Here we will mostly concern ourselves with the dipole moment form factor at qO0; for simplicity, we will still use the term TDM throughout. We will use the abbreviation TDM in general for dA8E denoting the top quark EDM, weak-EDM (ZEDM) and R Chromo-EDM (CEDM), unless we need to explicitly separate the type.


65

(i.e., at q"0) TDM. Measurements of some of the e!ects driven by the presence of a dipole moment form factor may have a better chance. In fact, the TDM may be considered as a CP-odd form factor in the ttM , ZttM or gttM vertex that probes the interactions of a short-lived top quark with an o!-shell , Z or a gluon, respectively, and can be represented by A8E"idA8E(s)u (p ) pJv M (p M ) , I R R R IJ R R

(4.1)

where s"q, q"p #p M and color indices for E were suppressed. Therefore, depending on the I R R masses in the loops, the TDM form-factor can also develop an imaginary part (contrary to the static EDM of the electron or the quark) if the energy of the o!-shell , Z or gluon is su$cient for the particles in the loop to be on-shell, i.e., there is an absorptive cut. Since the dipole moment characterizes the e!ective coupling between the spin of the fermion and the external gauge "eld, to extract the dipole moment one needs information on the spin polarization of the top quark. Fortunately, the left-handed nature of the weak decays of the top allows us to determine its polarization quite readily. As discussed in Section 2.8, top quark decays can analyze the initial polarization of the top quark. For instance in the leptonic decay tPe> b, C in the rest frame of the top quark the top is 100% polarized in the direction of the e> momentum. This greatly simpli"es calculation of the top quark production followed by its subsequent decay. The problem is then essentially reduced to calculating the production of a polarized top quark. It is then straightforward to fold in the decay to the spin indices of the top quark. A serious limitation to be kept in mind about this procedure is that it is only valid when the decays are governed by the SM, since they assume the helicity structure of the SM. If non-standard interactions make large contributions to the decay, the decay distributions may be modi"ed to the point that the polarimetry we have discussed is only approximate. This point must be borne in mind when considering the e!ects of new physics. A detailed discussion of the feasibility of extracting the TDM in future collider experiments such as e>e\, ppPttM , will be given in subsequent sections. In this section we consider the contribution to the TDM which arises in extended Higgs sectors and SUSY models. 4.2. Arbitrary number of Higgs doublets and a CP-violating neutral Higgs sector It is instructive to calculate the TDM in models with an arbitrary number of Higgs doublets and singlets satisfying NFC (natural #avor conservation) constraints. CP violation arises as a result of scalar exchanges between quarks, being driven by the imaginary parts of the complex quantities (e.g., ZI ) de"ned as [119]: L (2G ZI $ L , , q!m L (v ) O & L

(4.2)

(2G Z 1 $ L , >H> , (v vH) O q!m L &Y L

(4.3)

1

where v and v are the VEVs of the neutral Higgs "elds , and the summation runs over all the mass eigenstates of neutral or charged scalars in the theory (H or H , respectively). Also, L L O

66


Fig. 10. Feynman diagrams that contribute to the electric and weak top dipole moments in a two Higgs doublets model with CP-violating interaction of a neutral Higgs (h) with a top quark.

stands, for any pair of scalar "elds, and , for the momentum-dependent quantity

" dx 0¹[(x)(0)]0 e\ OV . O

(4.4)

CP violation in the neutral Higgs sector then generates the dominant 1-loop contribution to the EDM of the top, i.e., dA, through the 1-loop graph with the external photon line in Fig. 10(a) (for this R discussion, h,H in Fig. 10(a)). This contribution is given by L

m 2(2 G m e Im ZI f &L dA(0)" $ R L R 3(4) m R L m m R Im ZI f &L "(1.4;10\ e cm) L m GeV R L where

,

(4.5)

2!r r r r!2r 1! ln r# arctan #arctan 2 (r(4!r) (r(4!r) (r(4!r)

f (r)" 3!4 ln 2

if r(4 , if r"4 ,

r (r!(r!4 r!2r 1! ln r! ln 2 2 (r(r!4)

(4.6)

if r'4

and r"m L /m for any value of n. For r)!(tM PdM =\) I I , A , I (tPd =>)#(tM PdM =\) I I

(5.4)

in the SM, with k"3. The two interfering amplitudes, Fig. 20(a) and (b), have a relative CP-odd phase and Fig. 20(b) has the required absorptive phase. They are given by A? "

c c s !s c e B& , " s c c s s #c c e B& , " s s

(5.20) (5.21)

92


Fig. 22. Tree-level diagrams and a representative set of box diagrams considered for PRA in tPb, within the 3HDM.

where s ,sin(I ) and c ,cos(I ), and I , are parameters of the model. CP violation will be G G G G G & proportional to combinations such as Im(;)"Im(H ) . (5.22) tPb> : Let us de"ne the PRA O (tPb> )!(tM PbM \ ) O O . (5.23) A" O (tPb> )#(tM PbM \ ) O O The lowest-order contribution to A arises due to the interference of the SM = mediated tree O diagram (see diagram (a) in Fig. 22) with the 3HDM H> mediated tree diagram (diagram (c) in Fig. 22). We assess the contribution of the second graph making two simplifying assumptions: (i) m > ,m ;m > , this allows us to neglect the e!ect of the heavier charged Higgs, H . Further& & & more, we also assume that (ii) m > 'm , thus the H> width becomes irrelevant. A will then be R O & proportional to Im(=!tree);Re(H!tree), where, in analogy to our previous discussion in Section 5.1.1, the CP-violating CKM-like angular function was factored out. It is easy to see that, because of the chirality miss match, only the longitudinal part of the =-propagator contributes to A . In other words, decomposing the =-propagator in the unitary gauge as O q q q q (5.24) D5 "i !g # I J G #i I J G , IJ 2 IJ q q *

only Im G will appear in A [242,243]. In fact A , obtained from the interference between * O O diagrams (a) and (c) in Fig. 22, is proportional to

KR \K@

f (q)Im GK , (5.25) * KO where GK indicates that is missing from G to respect CPT invariance and f (q) is a phase-space * * function. Now, while the transverse part of the =-propagator in Eq. (5.24) resonates, i.e., A JIm(;) O

1 , G K 2 q!m #i m 5 5 5

Note a typographical error in in [161,162].

(5.26)


93

for qKm , there is no such enhancement for G . This is one of the reasons for an extremely small 5 * asymmetry, 10\ [243] or smaller [63], from tree;tree interference. The other reasons are the proportionality of both the Higgs coupling and Im GK to small fermion masses. * The next logical step is to capitalize on the resonance behavior of Im G and the fact that, unlike 2 Im G , it is not proportional to small masses, by considering interferences that are higher order in * the weak interaction coupling constant [59]. Thus, PRA from interferences of the type Im(a);Re(b) and Re(c);Im(d), is calculated; (a)}(d) denote the diagrams in Fig. 22, where (b) and (d) represent all box diagrams. Bremsstrahlung is included, but diagrams yielding A P0 for O G P0, see below, were neglected. The asymmetry is then given by * [Im(;)/512m]dq du Im GK Re(tree;box) R 2 , (5.27) AK O (tPb=Pb) where Im(;) is de"ned in Eq. (5.22) and u"(p #p ), q"(p #p ) , O @ J O m K 5 5 Im GK "! 2 (q!m )#( m ) 5 5 5

(5.28) (5.29)

with K " ! . (5.30) 5 5 5OJ The maximal value of A turns out to be negative, and of order 10\. Subsequently, the following O contributions to A , explicitly neglected in [59], were calculated in [243]: O E Since the integration in Eq. (5.25) reaches up to (m !m ), it includes a region with q'm , for R @ 5 which a =! loop in G has to be taken into account. * E Imaginary parts can also appear in box and vertex diagrams corresponding to tPb=. Both new terms are non-resonant and do not su!er small mass suppression from fermion loops in G . They turn out to give a large correction, of about 50% and of the same sign, as compared to the * value of A calculated in [59] using only G . The minimal number of t-quarks required to observe O 2 CP violation in PRA within the 3HDM is } although many orders of magnitudes smaller than the best leptonic SM result (i.e., tPdl [64,65]) } of the order of 10}10 and thus not very promising. As we will see later, one can do much better in the 3HDM by considering PIRA rather than PRA. 5.1.4. PRA in the MSSM An extremely interesting possibility, investigated in [215,235,246,247], is that the CP-violating PRA in two-body modes (that was found to be extremely small in the SM, 2HDM and in the 3HDM) may receive appreciable contribution from new SUSY CP-odd phases. For example, While there is agreement in the literature as to the above facts, there is controversy } into which we do not enter here (due in part to the fact that it has no observational consequences) } regarding the form of Im G [63,242}245] to be * inserted in Eq. (5.25).

94


Fig. 23. The SUSY-induced 1-loop Feynman diagrams that contribute to CP violation in the main top decay tPb=. is the chargino, is the neutralino, g is the gluino and tI , bI are the stop and sbottom particles, respectively.

consider the PRA A in Eq. (5.4) for the main top decay tPb=; the Feynman diagrams that can potentially contribute to A in the MSSM are depicted in Fig. 23. Recall that since A is ¹ -even, it requires an absorptive phase in the Feynman amplitude. This , necessitates radiative corrections to the tPb= to at least 1-loop order and, in particular, the SUSY particles exchanged in the loops have to be light enough such that absorptive cuts will arise. Of course, in addition to the strong phase from Final state interactions (FSI), a CP-odd phase is needed. We recall that, in the MSSM, with the most general boundary conditions for the soft breaking parameters at the scale where they are generated and ignoring generation mixing, only three places remain in the SUSY Lagrangian that can give rise to CP phases that cannot be rotated away: The superpotential contains a complex coe$cient in the term bilinear in the Higgs super"elds and the soft-supersymmetry breaking operators introduce two further complex terms, the gaugino masses m and the left- and right-handed squark mixing terms. The latter, being proportional to the trilinear soft breaking terms (i.e., the A terms) and to , may be complex O in general (for more details see Section 3.3). It is clear then that, in general, there are many sources of CP-violating phases. Therefore, reliable predictions cannot be made unless we make some simplifying assumptions. Let us "rst describe a convenient way to derive the PRA A . Following [215], the tPb=> and tM PbM =\ decay vertices can be parameterized as follows

D. pI g I R #D. I Pu , (5.31) JIR,i 5 u I R I m (2 .*0 @ R D M . pI g I R #DM . I Pv , JIRM ,i 5 v (5.32) I I @ m (2 .*0 R R where D*0 and D*0 de"ned in Eqs. (5.31) and (5.32), contain the CP-violating phases as well as I I the absorptive phases of the decay diagram (k) (k"a, b, c or d corresponding to diagrams (a)}(c) or (d) in Fig. 23). The important contributions to the D's above are likely to come from those diagrams in which one of the two on-shell superparticle is the lightest supersymmetric particle (LSP), e.g., the neutralino in our case. Such is the case for diagrams (b) and (d) in Fig. 23. Also, with a very light stop (i.e., &50 GeV) an absorptive cut can arise from diagram (a) in Fig. 23 if the gluino mass is below :130 GeV. The current experimental bounds on the superparticles involved in the loop of diagram (c) in Fig. 23 are already stringent enough that they are unlikely to have absorptive parts for m &175 GeV, see e.g., [216}223]. R


95

Let us now write A in the most general case with no assumptions on the masses of the SUSY particles and taking into account all four diagrams in Fig. 23. In terms of the scalar (D ) and I vector (D ) form factors, the PRA A is given by I (x!1) A " Re(D0 #DM * )#Re(D* !DM * ) , (5.33) I I I I 2(x#2) I where x,m/m and the sum is carried out over all decay diagrams in Fig. 23 (i.e., k"a, b, c and R 5 d). It is easy to show that if one de"nes

I Q ;e BU , D0 &e BI (5.34) I I Q ;e BU , D* &e BI (5.35) I where I, I are the CP-even absorptive phases (i.e., FSI phases) and I, I are the CP-odd Q Q U U phases associated with diagrams (a)}(d) in Fig. 23, then I Q ;e\ BU , DM * &!e BI I I Q ;e\ BU . DM * &e BI I We then get for the scalar form factors in Eq. (5.33)

8 Re(D0 #DM * )"! Q m m OIm C? , ? ? 3 R E ? m [m OIm(C@ !C@ )#m L OIm C@ ] , Re(D0 #DM * )"! R R @ Q @ @ @ sin 5 m [m OIm(CA !CA )!m K OIm(CA !CA ) Re(D0 #DM * )" R R A Q A A A sin 5 #m L OIm(CA #CA )] , Q A Re(D0 #DM * )"Re(D0 #DM * )(m L P!m K , m K Pm L , Q Q Q B B A A Q O PO , Im CA PIm CB ) , A B GH GH while the vector form factors in Eq. (5.33) are given by

(5.36) (5.37)

(5.38) (5.39)

(5.40)

(5.41)

Re(D* !DM * )"0 , ? ?

(5.42)

OIm C@ , Re(D* !DM * )"! @ @ sin @ 5

(5.43)

In the discussion to follow we will evaluate A within a plausible set of the low-energy SUSY parameter space. We note that in [215] there is a misprint in one of the terms proportional to m in the form factor Re(D* !DM * ). 5 A A The correct form of this term is given in Eq. (5.44).

96


1 Re(D* !DM * )" [O(mIm(CA !CA ) A A A R 2 sin 5 #m Im(CA #CA !CA !CA )#2Im CA ) 5 #m m K OIm(CA !CA ) R Q A !m m L OIm(CA #CA !CA ) R Q A !m K m L OIm CA ] , (5.44) Q Q A Re(D* !DM * )"Re(D* !DM * )(m L P!m K , m K Pm L , OG POG , Im CA PIm CB ) . B B A A Q Q Q Q A B GH GH (5.45) Here Im CI , x30, 11, 12, 21, 22, 23, 24 and k"a!d, are the imaginary parts, i.e., absorptive V parts, of the three-point form factors associated with the 1-loop integrals in diagrams (a)}(d) in Fig. 23. The CI are given by [215]: V (5.46) C? "C (mI H , mI G , m , m , m, m) , V V @ R % 5 R @ (5.47) C@ "C (mI H , mI G , m L , m , m, m) , V V @ R Q 5 R @ (5.48) CA "C (m L , m K , mI H , m , m, m) , @ 5 R @ V V Q Q CB "C (m K , m L , mI G , m , m, m) , (5.49) V V Q Q R 5 R @ and C (m , m , m , p , p , p ) is de"ned in Appendix A. The indices i, j"1, 2 stand for the two stop, V sbottom mass eigenstates, respectively, and m"1,2 and n"1}4 correspond to the two charginos and four neutralinos mass eigenstates, respectively; also, m is the gluino mass. % The OG 's in Eqs. (5.38)}(5.45) contain the SUSY CP-odd phases for the decay diagrams and they I were given in [215]. There, also the required Feynman rules for calculating the above PRA were given. For example, OG , for i"1}4, containing the SUSY CP-odd phases which appear in diagram B (d) are O"!Im(K\M) , B O"Im(K\M) , B O"Im(K>M) , B O"!Im(K>M) . B Here we have de"ned 1 ZLHZ\ , K>,ZLHZ\ # , K (2 , K As will be shown below, in our case this diagram will give rise to the leading contribution to A .

(5.50) (5.51) (5.52) (5.53)

(5.54)


1 K\,ZLZ>H! ZLZ>H , , K (2 , K

97

(5.55)

m 1 R (Z> ¸>HGH!(2Z> ZLHG ) M, R K , R (2 M5 sin K 1 # (2

m R ZGZ> ZLH!(2ZGZ> ¸>H , R K , R K M sin 5

(5.56)

m 4 1 R tan ZGZ> ZL#(2ZGZ> ZL M, 5 R K , R K , (2 M5 sin 3

!

m 4 R Z> ZLGH# tan Z> ZLG , K , R 5 K , R M sin 3 (2 5 1

(5.57)

and G ,ZGHZG , R R R

(5.58)

1 ¸!, tan ZL$ZL . 5 , , 3

(5.59)

In Eqs. (5.54)}(5.59), Z , Z and Z\, Z> are the mixing matrices of the stops, neutralinos and R , charginos, respectively (i.e., with indices i, n and m), which are de"ned in Section 3.3.2. Obviously, to obtain an estimate of the numerical value of the asymmetry, one needs to know the de"nite form for the mixing matrices and various other parameters. Not knowing these makes it very di$cult to give a reliable quantitative prediction for the asymmetry. Therefore, one has to choose a reference set of the SUSY spectrum subject to theoretically motivated assumptions as well as experimental data. Such a reference set which constructs a plausible low-energy MSSM framework was described in [215] (and is also described in Section 3.3.4). The key assumptions made there are: E There is an underlying grand uni"cation which leads to the relation in Eq. (3.137) between ;(1) and SU(2) gaugino masses and the gluino mass. E All squarks except the lighter stop (with a mass denoted hereafter by m ) are degenerate with J a mass M ; in the analysis below we set M "400 GeV. 1 1 E The gluino mass is varied subject to m '250 GeV [216}223]. % E The parameters are chosen subject to the upper limit on the NEDM, d (1.1;10\ e cm [4]. L In particular, the Higgs parameter is chosen to be real as strongly implied from this upper bound on the NEDM when the squark masses are below &1 TeV. With the above criteria one is left with only one CP-odd phase arising from tI !tI mixing. That is, * 0 when is real all the elements in OG above except from G , de"ned in Eq. (5.58), are real. Recall that I R the stops of di!erent handedness are related to their mass eigenstates tI , tI through the following > \

98


transformations (see Section 3.3.2): tI "cos tI !e\ @R sin tI , * R \ R > tI "e @R sin tI #cos tI . 0 R \ R > The asymmetry is thus proportional to the quantity (see also Eqs. (3.126) and (4.43))

(5.60)

R ,2G "sin(2 ) sin( ) , (5.61) !. R R R where G is de"ned in Eq. (5.58). R Although the CP-odd phases in the squarks sector generate the NEDM, the resulting restrictions on the CP-phases in the tI }tI mixing are rather weak. As we have demonstrated in Section 3.3.4, * 0 the main contribution to the NEDM (when is real) comes from the mixing of the superpartners of the lighter squarks. Therefore, if the trilinear soft breaking terms A , A and A are not correlated S B R at the EW-scale, as is the case in our low-energy MSSM framework described in Section 3.3.4, then it is not unreasonable to study the e!ects of maximal CP violation in the stop sector, i.e., R "1 !. without contradicting the current limit on the NEDM. With no further assumptions, the reference parameter set consists of M , m , m , , tan and R . 1 J % !. The neutralinos and charginos masses are extracted by diagonalizing the corresponding mass matrices which are functions of , m and tan (see Section 3.3.2). Note that the consequences of % such a low-energy MSSM scenarios on the various diagrams in Fig. 23 that can potentially contribute to the PRA, A , are: E For m 9250 GeV diagram (a) does not have the needed absorptive cut and thus does not % contribute to the PRA. E Diagram (c) does not have a CP-violating phase when arg()"0. This simpli"es our discussion to a great extent and we are therefore left with only two diagrams that can contribute to A . These are diagrams (b) and (d), where in fact we "nd that, by far, the leading contribution comes from diagram (d). In particular, we have calculated the PRA e!ect, A , arising from diagrams (b) and (d), for arg()"0, m "M "400 GeV and subject to 1 O m '50 GeV, m '250 GeV, the LSP (in our case the neutralino) mass to be above 20 GeV and J % the mass of the lighter chargino to be above 65 GeV. In Figs. 24 and 25, we plot A for two values of tan which correspond to a low (tan "1.5) and high (tan "35) tan scenarios, where the SUSY mass parameters are varied subject to all the above constraints and maximal CP violation is taken in the sense that R "1, thus presenting !. A in units of sin 2 sin . In particular, in Figs. 24(a) and (b) we plot the asymmetry as a function R R of for several values of m and for tan "1.5 and tan "35, respectively. In Figs. 25(a) and (b) % the asymmetry is plotted as a function of the gluino mass m for several values of and for % tan "1.5 and tan "35, respectively. In both "gures, we set M "400 GeV and m "50 GeV. 1 J Evidently, from Figs. 24 and 25 we see that a PRA in tPb= is very small over the whole range of our SUSY parameter space. In particular, we always "nd A (0.3% . (5.62) Of course, the asymmetry further drops as the mass of the lighter stop, m , is increased and vanishes J when m 9130 GeV since in that case there is no absorptive cut in the relevant contributing J


99

Fig. 24. The SUSY-induced PRA A in the main top decay tPb=, as a function of , for several values of m and for (a) % tan "1.5 and (b) tan "35. M "400 GeV, m "50 GeV is used. Figure taken from [215]. 1 J

Fig. 25. The SUSY-induced PRA A in the main top decay tPb=, as a function of m , for several values of and for (a) % tan "1.5 and (b) tan "35. M "400 GeV, m "50 GeV is used. Figure taken from [215]. 1 J

100


diagrams. Also, we "nd that the PRA is almost insensitive to tan in the range tan 910 and that A &0.3% become possible only for tan &O(1). The asymmetry we "nd is therefore somewhat small compared to the estimates of Grzadkowski and Keung (GK) [235] and of Christova and Fabbrichesi (CF) [246]. In the GK limit only the gluino exchange of diagram (a) was considered. They utilized the CP-violating, quark-squarkgluino interaction, occurring with coupling strength g (the QCD coupling) and the =tI bI Q interaction ig R tI =\I#h.c. . L "i(2g [tI H¹?(M ?t )#tI H ¹?(M ?t )]#(t b)! 5 < bI >@ Q * * 0 0 OOH (2 R@ * I *

(5.63)

As in our case, the most important source of CP violation is then the phase in the tI !tI mixing * 0 and, therefore, their e!ect is also proportional to R de"ned above. However, the GK limit is !. applicable only if m 'm #m I , so that an absorptive cut can occur in diagram (a). In the best case, R % R GK found a &1% asymmetry for m "m I "100 GeV. % @ On the other hand, in the CF limit, numerical results were given only for the neutralino exchange diagram (i.e., diagram (b)) wherein the CP-odd phase was chosen to be proportional to arg() and maximal CP violation with regard to arg() was taken. This can be parameterized by introducing a single CP-violating phase [246]: 1 f @ NH K sin , !. L L 2

(5.64)

where f @ and N appear in the q q Lagrangian I I L

1 (2m D NH (1! ) q ¸ " g q fD(1# )! 5 D L L D* OOQ 2m B L\D 2 5 D LD

1 (2m D NH # g q gD(1! )! (1# ) q 5 D L L D0 2 2m B L\D 5 D LD #h.c. ,

(5.65)

and are de"ned, together with gD and B , in [246]. For maximal CP violation, i.e., sin "1, and L D !. with m "m > "100 GeV and m "18 GeV, CF "nd A K2%. So an asymmetry in the main Q Q O two-body mode, tPb=, of a few percent can occur in their limit. However, these relatively large PRAs, reported by GK and CF in [235,246] su!er from the following drawbacks: E For the GK limit, m #m I (m is now essentially disallowed by the current experimental R % R bounds. E For the CF limit, arg()910\ is an unnatural choice in view of the stringent constraints on this phase coming from the experimental limits on the NEDM as discussed in Section 3.3.4. E For both the GK and CF limits, the large asymmetry arises once the masses of the superpartners of the light quarks are set to 100 GeV. Again, this is a rather unnatural choice as it is theoretically


101

very hard, if at all possible, to meet the NEDM experimental limits when the masses of the squarks (except for the lighter stop) are of the order of 100 GeV. Besides, the current experimental limits disfavor down squarks lighter than about 200 GeV. We also remark that PRA in tPb= within the more constrained N"1 SUGRA model was investigated in [248], where similar numerical results (i.e. A (0.3%) for A were obtained. To conclude this section, although PRAs in the range of &10\}10\ in the main two-body decays of t,tM are appreciable, their measurements is likely to be very hard. Presumably an e>e\ collider (NLC) could be suitable due to its cleanliness. However, there may be about 10,000 to 50,000 ttM events a year. Therefore, bearing experimental e$ciency factors, under the best of circumstances only an asymmetry of the order a few percent could be measured in the NLC. The LHC, being able to produce 10}10 ttM pairs, might seem more appropriate for a measurement of such a small PRA. However, for a measurement of a&0.3% asymmetry, experimental systematics can pose serious limitations. 5.1.5. PRA within the form factor approach tPb=: The basic idea in the form factor approach is to write a model independent coupling, then investigate the dependence of various asymmetries on the form factors involved [34,232,233,249}251]. Thus one can write the amplitude for tPb=> as the sum M ,M #M , R@5 R@5 R@5 where M is the amplitude at the lowest order in the SM which is given by R@5 g M "! 5 < H(p > )u (p )I¸u (p ) . R R R@5 (2 R@ I 5 @ @

(5.66)

(5.67)

In this equation, (p > ) is the polarization vector of => with four momentum p > and p , p are 5 @ R I 5 the four momenta of b, t, respectively. M contains the new CP-violating interactions and can be R@5 written in general (for on-shell => and in the limit m "0) as follows: C f. g f . IP#i IpJ > P u (p ) , (5.68) M "! 5 < H(p > )u (p ) R@ I 5 @ @ J 5 R R R@5 m (2 5 .*0

where P"¸ or R, ¸(R)"(1!(#) )/2 and the form factors f . and f . are complex, in general, they can both have an absorptive phase and a CP-violating phase. Note also that, in the SM, f * "1; f 0"f * "f 0"0, at tree level. Similarly, the non-standard part of the amplitude for tM PbM =\ is de"ned as

fM . fM . IP#i IpJ \ P v (p M ) . @ @ m J 5 5 .*0 In general, the form factors f . and fM . can be further simpli"ed to the form G G f .,f . ;f . , G G!.! G!.4 fM .,fM . ;fM . , G G!.! G!.4 g M M "! 5 and those related to tM PbM =\) f* "fM * !.! !.!

and f 0 "fM 0 , !.! !.!

(5.72)

f* "( fM * )H and f 0 "( fM 0 )H , !.4 !.4 !.4 !.4

(5.73)

f* "fM 0 !.! !.!

(5.74)

and f 0 "fM * , !.! !.!

f* "( fM 0 )H and f 0 "( fM * )H . !.4 !.4 !.4 !.4

(5.75)

Using the relations above it is easy to show that any CP-violating observable must always be proportional to any one of the combinations: ( f * !fM * ), ( f 0 !fM 0), ( f * !fM 0) or ( f 0!fM * ). In particular, a CP-odd, ¹ -even quantity (like the PRA) will be proportional to the real parts of these , combinations, e.g., Re( f * !fM 0), but a CP-odd, ¹ -odd quantity will be proportional to their , imaginary parts, e.g., Im( f * !fM 0). Assuming these form factors to be purely CP-violating, i.e., Re( f . )"0 and Re( fM . )"0, G!.4 G!.4 the PRA de"ned in Eq. (5.4) for tPb= can be expressed as A " a.Re( f .) . G G G .*0

(5.76)

In this context it was found that [232] a* K0.7, a0K!0.04, a* K0.04 and a0K!0.7. We thus see that the PRA is more sensitive to f * and f 0 than to f 0 and f * . 5.2. Partially integrated rate asymmetries In the rest of the section we will discuss CP asymmetries for the tPb decay within the O 3HDM, starting with PIRA. This asymmetry is de"ned as follows: A

.'0

(tPb> )! (tM PbM \ ) O .' O , , .' (tPb> )# (tM PbM \ ) .' O .' O

(5.77)

where stands for the partially integrated width, i.e., the width obtained by integrating over only .' a part of phase-space, rather than over the full kinematic range available to u and q, de"ned in Eq. (5.28). It is easy to see that, unlike the PRA, the PIRA is non-vanishing even for m P0 ( fO). D The reason is that in the calculation of the PRA, when the integration over the full range of u is

Note the slight di!erence between our de"nition of the form factors f ., fM . and the de"nition presented in [232]. G G


103

performed, pI from the loop sandwiched between the = and the H>, necessarily gets replaced O O by qI. But then, the contribution of the transverse part of the =-propagator (i.e., G ) vanishes since 2 qIIJ"0 (IJ,!gIJ#qIqJ/q). On the other hand, when we calculate the PIRA, then the 2 2 relevant integration is over only a part of the full kinematic range of u which allows G to 2 contribute even to the =>-tree;H>-tree interference. Let us consider the integration over u for a "xed q in the rest frame of the =-boson, i.e., q"0. The integration over u is now equivalent to that over the angle between (!p ) and p . De"ne the O @ PIRA over positive values of cos to be A . Then, explicit calculation of the =>-tree;H>-tree > interference (i.e., diagrams (a) and (c) in Fig. 22) yields [59]: G mr Im(;) (2 $ O 5& , (5.78) A " > 4 (2#r )(1!r )B(=P ) 5R 5& O where Im(;) is de"ned in Eq. (5.22), r "m /m and r "m /m . For 200:m : 5R 5 R 5& 5 & & 300 GeV, A & a few ;10\ (see Table 4) so it is enhanced by two orders of magnitude over the > PRA, A , in Eq. (5.23). Even larger asymmetries are likely for m (m . O & R A related PIRA was investigated in [252]. There, a PRA for tPb was de"ned, speci"cally for O ggPtM t in a hadron collider. The imposition of experimental cuts (for details see [252]), turns the asymmetry into PIRA. Looking for the maximal CP-violating e!ect by choosing the most favorable values for the three CKM-like angles and "/2, subject to experimental constraints, & asymmetries of the order of a few;10\ were obtained from tree;tree interference with a resonant =. 5.3. Energy asymmetry Another explicit example of how an interesting CP-violating asymmetry can be sizable even when the PRA is vanishingly small is the energy asymmetry. Speci"cally, let us de"ne [59]: E > ! E \ O , (5.79) A " O # E > # E \ O O where E > is the average of the > energy in tPb>, etc. In the calculation of E > the O O integrand is of course equal to that for the PRA with just an additional factor of E > . Now G does 2 O Table 4 Results for the PIRA A , see Eq. (5.78), and (A B )\, in tPb within the Weinberg model for CP violation. > > > O s ,sin(I ), where I , "/2, are CKM-like angles chosen to maximize the charged Higgs coupling Im(;) (see G G G & Eq. (5.22)). Note (A B )\ is the number of ttM pairs required to observe the asymmetry to 1-. B K0.04 is the > > > appropriate branching fraction. Table taken from [59], updated to m "180 GeV R m > &

s

200 300

0.252 0.210

s (;10\)

s

8.29 9.99

0.707 0.707

A (;10\) >

(A B )\ > >

2.9 1.5

3.0;10 1.2;10

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contribute even when the integration is over the full range of u. Explicitly one "nds G mr (1!r )Im(;) (2 $ O 5& 5R A " . # 12 (1#3r #2r )(1!r )[B(=P )] 5R 5R 5& O The energy asymmetry is closely related to the PIRA and in fact numerically [59]:

(5.80)

A &A /3 . (5.81) # > Indeed, both are weighted CP-odd observables constructed from the outgoing momenta. Observables constructed in this way have the drawback that they are proportional to m . This factor of O m is in addition to a factor of m in the Yukawa coupling. The latter cannot be dispensed with, as O O long as we are dealing with tPb. However, the additional power of m entering these asymmetO ries can be overcome by examining the transverse polarization of the as we will discuss next. 5.4. -polarization asymmetry The advantage of using a polarization asymmetry over an energy or a rate asymmetry is that the latter asymmetries go as m/m , where one power of m comes from the Yukawa coupling at the O & O H vertex. The second power of m comes from the trace over the lepton loop in =>}H> O O interference, i.e., ¹r[I(p/ #m )(1! )p/ ]"4m pI . (5.82) O O J O J The only way to avoid this power of m is to avoid summing over the spin (s ) of in the preceding O O trace. Then the trace will take the form O 4i (, s , p , p )2 ¹r[I(p/ #m )(1# s. )(1! )p/ ]KP O O J O O O J

(5.83)

Thus the =>-tree;H>-tree interference will make a contribution to the transverse polarization of the , i.e., to s ) (p ;p ) without su!ering a suppression by an additional power of m (i.e. in O O J O addition to the Yukawa coupling) so this asymmetry will be enhanced over the PIRA and energy asymmetries by a factor of about m /m &100! R O We will consider the following CP asymmetries that involve the polarization [166]: >(!)!>( )#\(!)!\( ) , A , W >(!)#>( )#\(!)#\( )

(5.84)

>(!)!>( )!\(!)#\( ) A , , X >(!)#>( )#\(!)#\( )

(5.85)

where for A (A ) the arrows indicate the spin up or down in the direction y(z). The reference W X frame is de"ned to be the rest frame, such that the t momentum is in the !x direction (i.e. the x-axis is the boost axis from the top to the frame), the y-axis is de"ned to be in the decay plane with a positive y component for the b momentum. The z-axis is de"ned by the right-hand rule. Also, A (A ) is CP-odd, ¹ -even (CP-odd, ¹ -odd), and is therefore proportional to the absorptive W X , ,


105

(dispersive) part of the bubble in the =-propagator. When integrated over the entire phase-space, A and A give W X 9 g Im(;)(x x 5 O 5 , A "! W 64 (1#2x )(x !x ) 5 & 5

(5.86)

9 g Im(;)(x 5 J A " f (x , y , x ) . X 64 (1!x )(1#2x )x 5 5 & 5 5 &

(5.87)

Recall that Im(;) is given by Eqs. (5.20)}(5.22) x ,m/m and y , /m. Also, f is de"ned as H H R 5 5 R the integral

f (x , y , x ), 5 5 &

(!x )x (1!)( 5 & d , [(!x )#x y ](x !) 5 5 5 &

(5.88)

and ,(p #p )/m. C J R A fully integrated polarization asymmetry as in Eq. (5.87), but weighting events di!erently for di!erent ranges of the invariant mass, was also given in [166]: O 9 g Im(;)(x 5 J A " f (x , y , x ) , X 64 (1!x )(1#2x )x 5 5 & 5 5 &

(5.89)

where f is the integral of Eq. (5.88) except that !x is replaced by !x . 5 5 The results for the above asymmetries, where experimental constraints were imposed on the relevant 3DHM parameters (for details see [166]), are a few percents for A and A , and W X a few tens of percents for A . As expected, the -polarization asymmetries are much larger and, X therefore, perhaps better suited to look for CP violation within the 3HDM, than any other asymmetry. 5.5. CP violation in top decays } summary As discussed in Section 2, CP violation can manifest in decays of particles. Such CPviolating signals may be driven by new physics containing new heavy particles. Thus, the large mass of the top may cause enhancements of CP violation in top decays as compared to the situation in light quarks decays. CP-odd signals in top decays, therefore, are attractive venues for such studies. In this section, we have discussed several types of CP-violating asymmetries in two- and three-body top decays. In particular, PRA, PIRA, energy asymmetry in the top decay products and -polarization asymmetries in the three-body decay tPb. In the SM, the CP violation in the top decays is found to be vanishingly small. This fact makes CP violation in top decays an extremely useful place for searching for new physics. We have, of course, also considered CP violation in top decays in extensions of the SM such as MHDMs and SUSY. We found that a sizable CP-violating PRA can arise in the main top decay, tPb=, in SUSY models. In particular, a stop-neutralino-chargino loop in the tb= vertex can give rise to a PRA of

106


the order of 0.1% if the SUSY parameter space turns out to be favorable. Such a PRA is, in principle, within the reach of the LHC provided that detector systematics can be kept su$ciently under control. A much bigger CP-violating signal is expected in the three-body decay, tPb, in MHDMs with new CP-odd phases in the charged Higgs sector, e.g., a 3HDM. Indeed such CP violation may arise already at tree-level and is best observed through a CP-violating transverse -polarization asymmetry. In a favorable scenario this asymmetry may be as large as a few tens of percents, requiring &1000 top quarks for its detection. This is a particularly gratifying result since over 10 top quark pairs are expected to be produced in the future colliders. The decay tPb is therefore a very promising place to look for new signals of CP violation in top decays. 6. CP violation in e>e\ collider experiments The energy of circular e>e\ machines cannot be increased beyond the energy of LEP-II, due to heavy losses to the synchrotron radiation. Therefore, the next step in e>e\ physics will involve linear colliders only, the èxistence proof a of which has been demonstrated at the 100 GeV scale by SLC. For recent reviews on linear colliders see [253}255]. The luminosity of future e>e\ colliders is projected to be L+10 cm\ s\, corresponding to a yearly integrated luminosity of L+100 fb\. The working assumption usually is to take it as tens of fb\ at the lower end of the scale of c.m. energies, and as hundreds of fb\ at its upper scale, corresponding to higher c.m. energies, to compensate for the decreasing cross-sections. In the "rst stage, the c.m. energy will cover the range approximately between LEP-II and 500 GeV, eventually reaching perhaps 1.6 TeV and hopefully even 2 TeV. Furthermore, beam polarization } which can help in clarifying many of the physics issues } is an interesting option. In this context recall that the SLC achieved polarization as high as 70}80%. 6.1. e>e\PttM In an high energy e>e\ collider running with c.m. energies of 500}2000 GeV and an integrated luminosity of L&O(100) fb\, 10}10 pairs of ttM will be produced mainly through the simple reaction e>e\P, ZPttM . This facility, especially due to its relatively clean environment, may therefore be thought of as a very e$cient `top factoryaand it is expected that many of the rare phenomena associated with top quark systems will be intensely studied there. Here we will focus on CP violation in the overall reaction P b=> e>e\Pt#tM . bM =\ P

(6.1)

Decays of the = also need to be included; the leptonic channels (=Pll , l"e, ) are perhaps the cleanest although experimental simulations suggest that ='s could be detected through jet topologies as well [256]. In what follows, we will not entertain the theoretical possibility that there is additional CP violation in =>, =\ decays and will focus only on e!ects directly related to the top quark.


107

In general, in the limit m "0, the (or Z)ttM vertex can be modi"ed to include the top magnetic C and electric(or weak) dipole moments !i[I(A4#B4 )#IJq (ic4#d4 )] , (6.2) R R J R R where c4 and d4, for e\ couplings in the notation of Eq. (6.2) are AA "!e , C BA "0 , C

(6.3)

1 e ! #sin , (6.4) A8" 5 C sin cos 4 5 5 e B8" . C 4 sin cos 5 5 The magnetic form factor, which is CP-conserving, has a signi"cant SM contribution at 1-loop due to QCD corrections and therefore is of lesser interest. Since in e>e\PttM we have q"s'4m, R these form factors are in general complex. In particular, with regard to the EDM form factors: Re dA8(q) is ¹ -odd, and Im dA8(q) is ¹ -even and, of course, all of these four quantities are R , R , CP-odd. Similar to the production vertex, the tb=> and tM bM =\ decay amplitudes may have CP-violating pieces. In order to take into account this possibility, for on-shell => and in the limit m "0, the C decay amplitudes for tPb=> may be decomposed with the most general form factors as (see also Section 5.1.5)

f. f . IP#i I pJ > P u (p ) , (6.5) J 5 R R m 5 .*0 where H(p > ) is the polarization vector of => with four momentum p > and p , p are the four I 5 5 R @ momenta of the t, b, respectively. P"¸ or R, where ¸(R)"(1!(#) )/2 and the form factors f . and f . are complex in general. g M "! 5 < H(p > )u (p ) R@5 (2 R@ I 5 @ @

108


Fig. 26. Tree-level (a) and CP-violating amplitudes (b)}(d) to leading order in the SM couplings and in CP-violating form factors.

Similarly, the amplitude for tM PbM =\ is de"ned as

fM . fM . IP#i IpJ \ P v (p M ) . (6.6) @ @ m J 5 5 .*0 Furthermore, some useful relations exist (see Eqs. (5.70)}(5.75)) between pairs of ( f ., fM .) in G G terms of their CP-conserving and CP-violating parts. In particular, CP-violating observables associated with top decays must always be proportional to any one of the combinations: ( f * !fM * ), ( f 0!fM 0), ( f * !fM 0) or ( f 0!fM * ), such that a CP-odd, ¹ -even quantity will be propor , tional to the real parts of these combinations, but a CP-odd, ¹ -odd quantity will be proportional , to their imaginary parts (for details see Section 5.1.5). CP violation e!ects in e>e\PttM Pb=>bM =\ may thus enter in both the production and the decay vertices of the top and the anti-top. To leading order in the CP-violating form factors present in the production or the decay of the top, one has to include interferences of diagrams (b)}(d) with the SM diagram (a) in Fig. 26. In principle, in order to experimentally separate CP-non-conserving e!ects in the production vertex from the decay vertex, one has to construct appropriate observables with sensitivity to only one CP-violating vertex, i.e., either production or decay (see e.g., [258]), or alternatively some simplifying assumptions have to be made. It is important to note that the description of the e\PttM in terms of Eq. (6.2) is not necessarily su$cient. For example, in the MSSM, 1-loop box diagrams with exchanges of SUSY particles may contribute to CP violation in e>e\PttM . In these type of diagrams, a e\PttM , in the limit m "0. C g M M "! 5 e\PttM Pb=>bM =\, CP violation can arise from both the production (see Eq. (6.2)) and the decay (see Eqs. (6.5) and (6.6)) of the top. In this problem, For a comprehensive treatment of the helicity amplitudes for e>e\PttM and for the subsequent top decays tPbl J in the presence of the CP-violating couplings in Eqs. (6.2), (6.5) and (6.6), see e.g., [34,257].


109

many momenta are available, so several ¹ -odd triple correlations can be constructed all of which, , in principle, can have non-vanishing expectation values that are proportional to Re dA8(q). R Similarly several ¹ -even (CP-odd) observables can be constructed to measure Im dA8(q). The , R sensitivity to Re dA8(q) or Im dA8(q) can vary considerably amongst the observables. It is, R R therefore, useful to devise a general procedure that represents a rough measure of the sensitivity of the observables in such situations. Thereby, one is led to consider the possibility of constructing òptimized observablesa, i.e., observables that have the maximum statistical sensitivity. Recall that for a given number of ttM events, the optimized observables will yield the smallest attainable limit on the real and the imaginary parts of dA8(q). The basic idea of the optimized observables was "rst R outlined in [29]; the general recipe for construction of such observables is given in Section 2.6. Optimal observables have by now been used extensively in [29,260}264]. In [29,260,261] CP violation in the top decays was ignored and the CP-odd e!ect was attributed solely to the EDM (dA) R and ZEDM (d8) of the top in the ttM and ZttM production vertex. Indeed, in [227] it was shown that, R in model calculations such as 2HDM and MSSM, the dipole moment in the ttM production leads to larger CP-non-conserving e!ects than what might be expected in the top decays. In [262}264] the optimization technique was employed to the overall reaction e>e\PttM Pb=>bM =\, where CP violation from both the production and the decay vertices of the top were investigated. Using the general e\PttM may be expressed as () d" () d# (Re d4(s)R 4R ()#Im d4(s)I 4R ()) d . B R

B R 4A8

(6.7)

As was explained in [29], the simplest optimized observables for the real and imaginary parts of dA8(q) are R / , OA8"I A8 / . (6.8) OA8"R A8 BR

BR ' 0 These optimal observables are constructed simply from the available four momenta in e>e\PttM and the subsequent decays. It was found in [29] that, for example, with 10 ttM events in an NLC running at c.m. energies of (s"500 GeV, Re(dA8) and Im(dA8) of about &few;10\ e cm R R become accessible at the 1- signi"cance level. Recall that in model calculations, such as MHDMs and SUSY, the size of top EDM and ZEDM are typically at the level of :10\ e cm (see Section 4) if one pushes the CP-violating phases of these models to their largest allowed values. Thus, the 1- limit obtained in [29] is at least one order of magnitude above the theoretical expectation for these dipole form factors within extensions of the SM. Now, consider the reaction e>e\PttM Pb=>bM =\, where the =>, =\ further decay leptonically via =Pl or hadronically, i.e., to up and down quark jets. Of course, as is well known, the J top quark decay occurs in an extremely short time and one measures directly only the momenta of the decay products of the top. The optimization technique may be therefore improved to include all available 4-momenta in a given decay scenario of the ttM as, for example, was done in [260,261]. The basic idea there was to translate the CP-odd top spin correlations generated by the dipole moments in e>e\PttM to correlations among momenta of the decaying products of the t and tM . For this purpose, the most promising decay scenario is the single-leptonic decay channels, i.e., when one, say

110


the t, decays leptonically and the other, i.e., tM , decays hadronically or vice versa ttM Pl>(q )#l #b#XM (q ) , (6.9) > 6M (6.10) ttM PX (q )#l\(q )#l #bM . 6 \ The decay scenarios in Eqs. (6.9) and (6.10) (each of which has a branching ratio, Bl &0.15, if l"e, ) allow for the reconstruction of the tM and t momentum, respectively, which, in turn, gives the rest frames of these quarks. The fact that the rest frames of t and tM may be accessible in these decay modes allows one to use CP-odd observables in terms of lepton unit momenta q( H in the ! corresponding top rest frames, instead of q( } de"ned in the e>e\ c.m. frame. ! The optimal observables that they used are, again, simply the ratio between the CP-odd and CP-even di!erential cross-sections and are given in [260]. However, in their optimal observables, the di!erential cross-section corresponds to the overall production and decay of the ttM and the leptonic momenta are taken in the corresponding t, tM rest frames. With the optimal observables they [261] calculated the best 1- sensitivity to the CP-violating dipole moments form factors Re(dA), Re(d8), Im(dA), Im(d8) assuming 100% tagging e$ciency of the single-leptonic decay modes R R R R of ttM in Eqs. (6.9) and (6.10); these are given in Table 5. We thus see that beam polarization (of the incoming electrons), denoted here by P , may increase the sensitivity to Re(dA) and Im(d8) by C R R almost an order of magnitude. Evidently, the results in Table 5 imply that a NLC running with a c.m. energy of (s"500 GeV and an integrated luminosity of 20 fb\, or (s"800 GeV and an integrated luminosity of 50 fb\, will be able to probe, at 1- and in the best cases, real and imaginary parts of the TDM, typically of the order of a few;10\}10\. As in their analysis, this may be achieved by investigating the single-leptonic decay mode of ttM . This improves the limits obtained in [29] by about one order of magnitude. However, at the 3- signi"cance level, the corresponding sensitivities are typically few;10\}10\ and so are still about an order of magnitude above the expectations from the models such as MHDMs and SUSY. It should be noted again that a CP-violating dipole moment at the ttM and ttM Z vertices may not, in general, account for the entire CP violation e!ect in ttM production. As mentioned before, this will, for example, be the case in the MSSM where 1-loop CP-violating box diagrams can cause an additional CP-odd e!ect [259]. We note, however, that speci"cally in the MSSM, these box contributions to CP violation in ttM cannot signi"cantly enhance the CP-violating signal. In fact, in some ranges of the relevant SUSY parameter space, the contribution of the box graphs comes with

Table 5 Attainable 1- sensitivities to the CP-violating dipole moment form factors in units of 10\ e cm, with (P "$1) and C without (P "0) beam polarization. m "180 GeV. Table taken from [261] C R P "0 C (Re dA) R (Re d8) R (Im dA) R (Im d8) R

4.6 1.6 1.3 7.3

20 fb\, (s"500 GeV P "#1 P "!1 C C 0.86 1.6 1.0 2.0

0.55 1.0 0.65 1.3

P "0 C 1.7 0.91 0.57 4.0

50 fb\, (s"800 GeV P "#1 P "!1 C C 0.35 0.85 0.49 0.89

0.23 0.55 0.32 0.58


111

an opposite sign relative to the top dipole moments, such that cancellations may occur, thus decreasing the net CP violation e!ect in ttM production [259]. A more complete investigation was carried out in [262}264]. There the CP-violating form factors from both the production and the decay amplitudes of the t and tM were included. For this purpose, they used the single-leptonic energy spectrum 1 d! " c!f (x) , G G ! dx G where

(6.11)

c!"1, c!"G, c>"!Re( f 0), c\"!Re( fM * ) . (6.12) The functions f (x) are all given in [262] and f 0, fM * are de"ned in Eqs. (6.5) and (6.6). Also, G $ indicates the charge of the lepton and

El>(El\) (1!) , (6.13) m (1#) R El>(El\) being the energy of l>(l\) in the e>e\ c.m. frame and "(1!4/s. Speci"cally, for R m "180 GeV and (s"500 GeV R 1.76;10 K! ;(1.06Im(dA)#0.18Im(d8)) . (6.14) R R [e cm] x(x )"2

In [262], optimal observables that may be used to separately measure the CP-violating form factors in the production or the decay vertex were given. Their optimal observables utilize the single-leptonic energy spectrum in Eq. (6.11). With the optimization technique they showed that and Re( f 0}fM * ) may be extracted individually from the di!erence in the l> versus l\ energy spectra by convoluting the di!erential energy spectrum with approximately chosen kernel functions

1 d\ 1 d> 1 ! , " dx K \ dx > dx 2

(6.15)

1 d> 1 d\ ! , Re( f 0!fM * )" dx D > dx \ dx

(6.16)

where , are functions of f (x) de"ned in Eq. (6.11) and are given in [263,264]. The minimal D K G values of and Re( f 0!fM * ) that can be obtained with a statistical signi"cance NRR4 and NR@5, 1" 1" respectively, at the NLC with (s"500 GeV and m "180 GeV, can then be computed [262]: R NRR4 1" , (6.17) "11.3 ;pb * NR@5 1" Re( f 0!fM * )"13.1 , (6.18) ;pb * Notice the di!erence in our notation (Eq. (6.2)) and the one used in [262] for the top dipole moments. The translation is D "(4m sin /e);idA8. A8 R 5 R

112


Here represents the square root of the e!ective luminosity for the single-leptonic ttM pairs at the * NLC. Thus, ,( L, where L is the integrated luminosity at the NLC and is the tagging RR * RR e$ciency for the single-leptonic mode. Note that, in the best case, "Bl +15%, if one assumes RR 100% e$ciency in measuring the single leptons from ttM . We see from Eqs. (6.17) and (6.18) that, for example, with L"100 fb\ and "0.15 we have RR +122.5 pb\ (note that with these values one has &9000 single-leptonic ttM events). Therefore, * with this number, a 3- detection of will be possible for +0.28. Using the relation between and Im(dA8) in Eq. (6.14) we then get the following 3- equality: R 1.06Im(dA)#0.18Im(d8)K1.6;10\ e cm . (6.19) R R Eq. (6.19) implies that Im(dA8) of the order of &10\ e cm may be detected at the 3- level at the R NLC, running with c.m. energy of (s"500 GeV and with an integrated luminosity of L"100 fb\, using the optimal observables suggested in [262]. This result is again about one order of magnitude better compared to the results obtained in [29] and it is comparable to the results shown in Table 5 which were obtained in [261]. As for the CP-violating form factors in the decay amplitude, we can use Eq. (6.18) to get the 3- limit on Re( f 0}fM * ). For +122.5 pb\ * Re( f 0!fM * )+0.32 . (6.20) In Section 5, we have discussed the theoretical expectations for couplings such as f 0 and fM * in the SM and its extensions. The SM prediction for such form factors, induced by the CKM matrix, is much too small to be observed. Moreover, even within MHDMs and the MSSM the resulting 3- limit in Eq. (6.20) falls short by at least one order of magnitude. Finally, in [263,264] the single-leptonic channel was compared to the double-leptonic mode, i.e., when both t and tM decay leptonically, using the optimization technique. It was found there that the single-leptonic mode comes out favorable by about a factor of 2. 6.1.2. Naive observables constructed from momenta of the top decay products Various types of `naivea observables to deduce the real and imaginary parts of the non-standard form factors in top production and decay were considered in [29,30,227,249,260,261,265}267]. These observables are constructed simply from correlations between momenta of the decaying products of the top quark. The basic idea again utilizes the fact that weak decays of the top quark act as very e$cient analyzer of the top spin. So the momenta of the decay products (via tPbl) can be used to construct the observables with the right transformation properties. In general, as expected, the `naivea operators are less e!ective than the optimal ones, sometimes by as much as an order of magnitude. However, it is important to bear in mind that this advantage pertains only with respect to statistical errors; in actual experimental considerations, systematic errors will also need to be taken into account and that could o!set some of the advantage of the optimized observables. In [29], amongst the various possible correlations, the best simple operators that they found are PIQJ H>NH\M for Re dA , IJNM @ 8 R PIQJ H>NH\M for Re d8 , IJNM C 8 R H\ ) Q for Im dA and Im d8 , 8 R R

(6.21)


113

where the momenta above are given by P "p !p M , @ @ @ P "p>!p\ , C C C Q "p>#p\ , 8 C C H!"2(E> ) p )E> $2(E\ ) p )E\ , (6.22) 5 R 5 5 R 5 and E is the =-boson polarization for the reaction =(p )Pl(pl )l (p ). Because of the left5 5 J handed nature of the coupling of = to leptons its polarization can be constructed from the momenta of the decay products as ¹r[p/ . p/ l I(1! )] J . (6.23) EI " 5 4(p ) pl ) J Here is an arbitrary light-like vector that determines the phase convention for the polarization. The expression above requires the momenta. Recall that the "nal state consists of six particles bl>l bM l\l . Of these, only four are directly observable as the neutrinos escape detection. However, by imposing the following conditions, the momenta of the missing neutrinos may in fact be inferred. The conditions that need to be imposed are: (1) conservation of four-momentum together with the conditions that (2) the lepton and a neutrino reconstruct to the =! mass, (3) the b-quark together with a lepton and neutrino reconstruct the t, tM mass and (4) the neutrinos are massless. It was then found in [29] that the use of the = polarization in the operators of Eq. (6.21) can easily improve the sensitivity to the dipole moments by factors of 10}50 when compared to simple correlations which do not involve the = polarization vector. As compared to the optimal observables discussed in their work, the observables in Eq. (6.21) are less e!ective, typically, by about a factor of 2}10. In [30,249], the following CP-odd and ¹ -odd correlations were considered , (q( ;q( ) > H #(i j) , (6.24) ¹K "(q( !q( ) \ GH \ > G q( ;q( \ > (q( ;q( ) > , (6.25) AK "p( ) \ > q( ;q( \ > where p( is the unit momentum of the incoming positron and q( ! are the unit momenta of > a charged decay product from tPA, tM PBM in the overall c.m. system. Thus, q( are the directions of ! a charged lepton or a b jet. An interesting property of these correlations is that they are not sensitive to CP-violating e!ects in the t and tM decays and, therefore, they can be expressed in terms of only the real part of the EDM and ZEDM form factors, Re(dA8). The mean values of ¹K and R GH AK plus the conjugate ones are given by [249] (s (c Re dA#c Re d8)s , (6.26) ¹K M # ¹K M "2 GH R 8 R GH GH e A (s (r Re dA#r Re d8) , AK M # AK M "2 R 8 R e A

(6.27)

114


where, identifying the z-axis with the e> beam axis s "(p( p( !)"diag(!,!, ) . (6.28) GH >G >H The coe$cients c and r depend on the speci"c decay channel and were calculated as a function A8 A8 of s in [249] for correlations among AB"l>l\, bbM and l>b#l\bM . Possible CP-odd, ¹ -even correlations that use the momenta of the decay products of t and , tM were also examined in [249]: QK "(q( #q( ) (q( !q( ) #(i j) , (6.29) GH > \G \ >H AK "p( ) (q( #q( ) . (6.30) > > \ In contrast to the ¹ -odd observables in Eqs. (6.24) and (6.25), the ¹ -even observables QK and AK , , , GH which acquire absorptive phases, are sensitive also to the combinations Re( f . !fM .) (recall that G G P"¸ or R), e.g., Re( f0}fM * ), of the form factors in the decay amplitudes of Eqs. (6.5) and (6.6). In [249], CP-odd e!ects in the decay process were neglected when evaluating the two CP-odd, ¹ -even observables in Eqs. (6.29) and (6.30). In terms of the imaginary parts of the EDM and , ZEDM of the top they obtained (s QK M # QK M "2 (q Im dA#q Im d8)s , GH GH R 8 R GH e A

(6.31)

(s (p Im dA#p Im d8) . AK M # AK M "2 R 8 R e A

(6.32)

Here, again, the coe$cients q and p depend on the speci"c channel and were given as A8 A8 a function of s in [249] for correlations among AB"l>l\, bbM and l>b#l\bM . From the simultaneous measurement of the pairs ¹K , AK and QK , AK and for a given GH GH c.m. energy, Re(dA8) and Im(dA8) can be disentangled, respectively. Assuming 10 available ttM R R events at each c.m. energy, the 1- statistical sensitivity to Re dA and Re d8, using only the 33 R R component of ¹K , are GH ( (3r ¹K !c AK ) e 8 8 , (6.33) (Re dA)" R c ) r !c ) r (sN A 8 8 A ( (3r ¹K !c AK ) e A A , (6.34) (Re d8)" R c ) r !c ) r (sN 8 A A 8 where N "10;Br(tPA);Br(tM PBM ). Similar relations can be obtained for Im dA and Im d8 R R using QK and AK . Using this formalism, Ref. [249] presents the one standard deviation accuracies in measuring the real and imaginary parts of dA8, again assuming 10 ttM events at c.m. energy (s"500 GeV and R with m "175 GeV R Re dA+1;10\, Im dA+1.2;10\ , (6.35) R R Re d8+5;10\, Im d8+7.5;10\ , (6.36) R R


115

where the best sensitivity was obtained for correlations between the l>l\ momenta. We note that the sensitivity to Re dA8 is slightly better than to Im dA8. R R Cuypers and Rindani [268] have investigated the e!ect of polarized incoming electron beams. They found that the sensitivity of observables of the type A and A in Eqs. (6.25) and (6.30) to the real and imaginary parts of dA8, respectively, can be enhanced if the incoming electron beam is R longitudinally polarized. Note also [260], the type of observables in Eqs. (6.24), (6.25), (6.29) and (6.30) may be improved (with respect to their sensitivity to the dipole moments) if one replaces the momentum of the charged lepton in the e>e\ c.m. frame by its momentum in the top (or anti-top) rest frame, in the single-leptonic ttM channel. As was shown in the previous section, the CP-violating e!ects in the top decay tPb=> and its conjugate may be isolated using the optimization procedure. This may also be achieved in some limiting cases using naive observables. In [227,249] a naive observable that projects onto CP violation in the top decay vertex was suggested

(q( lM ;q( ) (q( l ;q( M ) @ ! @ . O "p( ) > q( lM ;q( q( l ;q( M @ @

(6.37)

Although, in general, the expectation value of O above receives contributions both from the CP violation in ttM production and in the decay amplitude, it was shown in [249] that close to threshold, i.e., (sK2m , the contribution from the ttM production vertex vanishes. For example, with the R correlation O , for an appropriate e>e\ collider with m "150 GeV and c.m. energy (sK2m , R R they "nd O +0.15;Im( f 0!fM * ) ,

(6.38)

where f 0, fM * are form factors in the decay amplitudes de"ned in Eqs. (6.5) and (6.6). Assuming 3;10 ttM events in which tPbl>l and tM PbM l\l , they found that, to 1-, (Im( f 0!fM * ))+0.1 ,

(6.39)

can be determined from a measurement of O . Again, the limit in Eq. (6.39) falls short from model predictions for these form factors (see Section 5). This is easily understood from Eq. (6.38) which implies an asymmetry of the order of &10\ for Im( f 0!fM * )&0.1. Recall that the typical asymmetries in extensions of the SM that we have described in Section 5 are : a few times 10\ and in the SM they are a lot smaller than that. A related CP-odd and ¹ -odd asymmetry that projects only onto CP-violating e!ects , in the decay processes of the t and tM was suggested by Grzadkowski and Keung [251]. This asymmetry was de"ned by partially integrating over the azimuthal angle of l>(l\) in the =>(=\) rest frame and subtracting the integration in the range !, 0 from the integration in the range 0, and it is essentially proportional to the triple product pl ) (p ;p ). When evaluated within the MSSM, the asymmetry was found to be &10\}10\ @ 5 which again falls short by at least one order of magnitude from the limits that are anticipated to be obtainable, through the study of e>e\PttM Pl>l\l l bbM in a future e>e\ high-energy collider.

116


6.1.3. Improved sensitivity using energy and angular distributions of top decay products and polarized electron beams Several di!erential leptonic asymmetries with respect to the charged lepton in tPbll and for longitudinal electron (positron) beam polarizations, P (P ), have been suggested in [29,262}264, C C 269}271]. Consider the CP-violating t!tM spin correlation p( ) (s !s M ). This spin correlation simply R R R translates to the asymmetry [N(t tM )!N(t tM )] * * 0 0 , N " *0 all ttM

(6.40)

suggested "rst by Schmidt and Peskin (SP) in [33] in the context of ttM production in hadron colliders (the SP e!ect will be discussed in more detail in Section 7). Now, N is related to the asymmetry in the energy spectrum de"ned as [32,269}271]: *0 d 1 d ! , (6.41) A (x)" # dx(l>) dx(l\)

through a simple multiplication by kinematic functions present in the lepton energy distribution functions (see [262]). The energy asymmetry in Eq. (6.41) is between distributions of l> and l\ at the same value of x"x(l>)"x(l\)"4 E(l!)/(s. Note that when CP violation is present in the top production and not in the top decay, then A (x)JN J, where is de"ned in Eq. (6.14). # *0 An up}down asymmetry, A , was also studied in [32,269,270]: SB > A () d cos , (6.42) A " SB SB \ where

1 d(l>, up) d(l>, down) d(l\, up) d(l\, down) A ()" ! # ! , SB 2 d cos d cos d cos d cos

(6.43)

and up/down refers to (p ! ) I0, (p ! ) being the y component of p ! with respect to a coordinate J W J W J system chosen in the e> e\ c.m. frame so that the z-axis is along p , and the y-axis is along p ;p . R C R The ttM production plane is thus the xz plane. refers to the angle between p and p in the c.m. R C frame. Note that the asymmetry A is related to spin components of the top transverse to the SB production plain and, therefore, it is a ¹ -odd quantity. , Three additional asymmetries were considered in [269,270]. The combined up}down and forward}backward asymmetry

A () d cos ! A () d cos , SB SB \ with A () given in Eq. (6.43). The left}right asymmetry SB > A " A () d cos , JP JP \ A " SB

(6.44)

(6.45)


117

where

1 d (l>, left) d (l>, right) d (l\, left) d (l\, right) A ()" ! # ! , JP 2 d cos d cos d cos d cos

(6.46)

and left/right refers to (p ! ) I0. The combined left}right and forward}backward asymmetry J V A" A () d cos ! A () d cos (6.47) JP JP JP \ with A () given in Eq. (6.46). JP To leading order in the dipole form factors and on ignoring CP violation in the t and tM decays, all the above "ve CP-odd asymmetries are linear functions of dA and d8. The asymmetries A , A , A R R # JP JP are ¹ -even and are therefore proportional to Im dA8, while A , A are ¹ -odd and are , R SB SB , proportional to Re dA8. The ¹ -even asymmetries can be symbolically written as R , (6.48) A ,aA(P , P )Im dA#a8(P , P )Im d8 . R G C C R G G C C Similarly, for the ¹ -odd observables, one obtains , B ,bA(P , P )Re dA#b8(P , P )Re d8 , (6.49) G G C C R G C C R where A "A , A or A and B "A or A . The functions aA8, bA8 depend, among other G # JP JP G SB SB parameters, on the polarizations of the incoming electron and positron beams, P and P , C C respectively, and are explicitly given in [269,270]. It is evident from Eqs. (6.48) and (6.49) that, without beam polarization, by measuring only one asymmetry of the type A and/or B , one can extract information only on one combination of G G Im dA, Im d8 and/or Re dA, Re d8, respectively. However, any two asymmetries with the same R R R R ¹ property can be used to determine two independent combinations of the corresponding real or , imaginary parts of dA and d8, thus, giving Im dA and Im d8 or Re dA and Re d8 independently. Such R R R R R R an analysis was described in the previous section where the observable pairs ¹K , AK and QK , AK were used to "nd the sensitivity of a high-energy e>e\ collider to Re dA, Re d8 and Im dA, Im d8, R R R R respectively. However, it was suggested in [269,270] that if, in addition, beam polarization is included, then one ¹ -even(¹ -odd) asymmetry is su$cient to determine Im dA and Im d8(Re dA , , R R R and Re d8) independently by measuring this asymmetry for di!erent polarizations. Both apR proaches were adopted in [269,270]. Their best results are summarized in Table 6 where 90% con"dence level limits are given for (s"500 GeV and m "174 GeV. We can see from Table 6 R that the second approach, of incorporating beam polarization, increases the sensitivity to both the real and imaginary parts of the EDM and ZEDM of the top, in some cases, by about one order of magnitude. With 50% beam polarization, the 90% con"dence level limits for Im dA, Im d8, Re dA R R R and Re d8 are again at best around &few;10\ e cm. R An interesting di!erential asymmetry that combines information from both the production and decay vertices of the top was suggested in [263,264]:

( dx dx d/dx dx ! dx dx d/dx dx ) VV . All , VV dx dx d/dx dx

(6.50)

118


Table 6 90% con"dence limits on the real and imaginary parts of the top dipole form factors dA and d8, in units of 10\ e cm, R R from di!erent asymmetries. In the unpolarized case, the asymmetries A , A are together used to get limits on Re dA8 SB SB R and A , A to obtain the limits on Im dA8. In the polarized case, the limits obtained from A and A are denoted by (a) JP JP R SB JP and the ones from A and A are denoted by (b). The numbers are for the single-leptonic ttM mode, for m "174 GeV, SB JP R (s"500 GeV and an integrated luminosity of L"10 fb\ Case

Re dA R

Re d8 R

Im dA R

Im d8 R

unpolarized (a) polarized(P "$0.5) C (b) polarized(P "$0.5) C

54.4 2.3 12.5

15.9 2.3 9.1

7.9 2.3 2.3

62.4 9.1 7.9

This asymmetry utilizes the double-leptonic energy distribution in e>e\PttM Pl>l\l l bbM 1 d " c f (x, x ) , G G dx dx G where x, x are de"ned in Eq. (6.13) and

(6.51)

(6.52) c "1, c ", c "!Re( f 0!fM * ) . f 0, fM * and are de"ned in Eqs. (6.5), (6.6) and (6.14), respectively. In terms of (or equivalently of Im dA8) and Re( f 0!fM * ) and with (s"500 GeV, m "180 GeV and the SM parameters they R R obtained the simple relation (6.53) All "!0.34!0.31Re( f 0!fM * ) . Given a number of available double-leptonic ttM events in the NLC and using the relation in Eq. (6.53), one can now plot the 1-, 2- and 3- detectable regions in the Im dA8!Re( f 0!fM * ) R plane. These regions are shown in Fig. 27 for &700 double-leptonic ttM events in a 500 GeV collider. We see from Fig. 27 that a 3- detection of CP violation through All is possible in a wide range of the parameters and f 0, fM * . However, if one parameter is very small, then it requires the other to be relatively large. Thus, for example, if Re( f 0!fM * )"0 then, at 3-, Im(dA8)91.7;10\ e cm R (or, with the notation used in [263,264], Re(D )90.3, see also Fig. 27). This is again comparable A8 to obtainable limits from other observables described in this section and, thus, falls short by about one order of magnitude when compared to model dependent predictions for the top dipole moment. Finally, interesting CP-violating asymmetries which involve correlations among b-quarks from ttM Pb=>bM =\ were suggested by Bartl et al. in [265}267]. They utilized the angular [265] and energy [266] distributions of b and bM with initial beam polarization, in e>e\PttM followed by tPb=> and tM PbM =\, to construct CP-violating asymmetries that can disentangle CP violation in the ttM production mechanism from CP violation in the top decay. Unfortunately, their asymmetries, when evaluated within the MSSM, range from 10\ to 10\ at best, and therefore also seem to be too small to be detectable at a future NLC. Also, in an interesting study of the CP-violating e!ects of the EDM, ZEDM and CEDM of the top in the ttM threshold region [272], it was suggested that at an e>e\ collider CP violating


119

Fig. 27. One can verify the asymmetry All in Eq. (6.50) to be non-zero at 1-, 2- and 3- level when the parameters Re(D ) (horizontal axis) and Re( f 0!fM *) (vertical axis) are outside the two solid lines, dashed lines and dotted lines, A8 respectively, given &700 double-leptonic ttM events in a 500 GeVe>e\ collider. Recall that D is related to dA8 via: A8 R D "(4m sin /e);idA8, i.e., Re(D ) corresponds to Im dA8. Figure taken from [263,264]. A8 R 5 R A8 R

correlations may be somewhat enhanced. Using CP-odd correlations of the top polarization projected onto the charged leptons from the top decays, they found that at the threshold region for ttM production, an e>e\ collider can be sensitive to an EDM (and ZEDM) at the level of &10\ e cm. Although this is comparable to the predicted sensitivity away from threshold, it has some advantage in that it may be achieved at the lowest possible energy needed for ttM production. 6.2. e>e\PttM h, ttM Z, examples of tree-level CP violation Let us now turn our attention to the reactions e>(p )#e\(p )Pt(p )#tM (p M )#h(p ) , F > \ R R

(6.54)

e>(p )#e\(p )Pt(p )#tM (p M )#Z(p ) , 8 > \ R R

(6.55)

120


which may exhibit large CP violation asymmetries in a 2HDM [78,273}275]. Note that for reasons discussed below, for the CP-violating e!ects in e>e\PttM h as well as in e>e\PttM Z, only two out of the three neutral Higgs of the 2HDM are relevant. We denote these two neutral Higgs particles by h and H corresponding to the lighter and heavier Higgs-boson, respectively. In some instances, we denote a neutral Higgs by H, then H"h or H is to be understood. A novel feature of these reactions is that the e!ect arises at the tree-graph level. As a consequence, one can construct new type of asymmetries which are J (tree;tree)/(tree;tree) and are therefore a priori of O(1). This stands in contrast to loop-induced CP-violating e!ects in ttM production for which the CP asymmetries, in general, are J(tree;loop)/(tree;tree) and are therefore suppressed by additional small couplings to begin with, i.e., at 1-loop, typically, by } the "ne structure constant. Indeed, we will show below that CP violation at the level of tens of a percent is possible in the reactions in Eqs. (6.54) and (6.55). Basically, for the ttM h(ttM Z) "nal states, Higgs(Z) emission o! the t and tM interferes with the Higgs(Z) emission o! the s-channel Z-boson (see Fig. 28) [78,273,274]. We "nd that the processes e>e\PttM h and e>e\PttM Z provide two independent, but analogous, promising probes to search for the signatures of the same CP-odd phase, residing in the ttM -neutral Higgs coupling, if the value of tan (the ratio between the two VEVs in a 2HDM) is in the vicinity of 1. In particular, they serve as good examples for large CP-violating e!ects that could emanate from t systems due to the large mass of the top quark and, thus, they might illuminate the role of a neutral Higgs particle in CP violation. Although these reactions are not meant (necessarily) to lead to the discovery of a neutral Higgs, they will, no doubt, be intensely studied at the NLC since they stand out as very important channels for a variety of reasons. In particular, they could perhaps provide a unique opportunity to

Fig. 28. Tree-level Feynman diagrams contributing to e>e\PttM h (left-hand side) and e>e\PttM Z (right-hand side) in the unitary gauge, in a 2HDM. For e>e\PttM Z, diagram (a) on the right-hand side represents 8 diagrams in which either Z or are exchanged in the s-channel and the outgoing Z is emitted from e>, e\, t or tM .


121

observe the top-Higgs Yukawa couplings directly [73,276}282]. In [73,279], using a generalization of the optimal observables technique outlined below (see also Section 2.6), Gunion et al. have extended the initial work [78] on CP violation in e>e\PttM h to include a detailed cross-section analysis such that all Higgs Yukawa couplings combinations are extracted (see below). A similar analysis which also uses the optimized observable technique for e>e\PttM Z is given in [275]. A detailed cross-section analysis of the reaction e>e\PttM Z in the SM was performed by Hagiwara et al. [280,281]. There, it was found that the Higgs exchange contribution of diagram (b) on the right-hand side of Fig. 28 will be almost invisible in a TeV e>e\ collider for neutral Higgs masses in the range m (2m . Interestingly, we will show here that, if the scalar sector is doubled, then the F R lightest neutral Higgs (h) may reveal itself through CP-violating interactions with the top quark even if m (2m . Obviously, a non-SM (e.g., larger) top-Higgs Yukawa coupling can cause an F R enhancement in the rates for both the ttM h and ttM Z "nal states. Thus, a `simplea cross-section study for these reactions may also come in handy for searching for new physics. However, one should keep in mind that, from the experimental point of view, asymmetries, i.e., ratios of cross-section, are easier to handle and, in particular, CP-violating signals are very distinctive evidence for new physics. This section will be divided to three parts. In the "rst part, we present a detailed analysis of the tree-level CP violation in the reactions e>e\PttM h and e>e\PttM Z which manifests itself as a ¹ -odd correlation of momenta. In the second part, we will consider the generalized optimization , technique developed by Gunion et al. and its application to the reaction e>e\PttM h. In the last part, we will discuss CP violation in the Higgs decay hPttM , where we take the Higgs to be produced through the Bjorken mechanism e>e\PZh. 6.2.1. Tree-level CP violation In the unitary gauge, the reactions in Eqs. (6.54) and (6.55) can proceed via the Feynman diagrams depicted in Fig. 28. We see that for e>e\PttM Z, diagram (b) on the right-hand side of Fig. 28, in which Z and H are produced (H"h or H is either a real or a virtual particle, i.e. mH '2m or mH (2m , respectively) followed by HPttM , is the only place where new R R CP-non-conserving dynamics from the Higgs sector can arise, being proportional to the CP-odd phase in the ttM H vertex. As mentioned above, in both the ttM h and the ttM Z "nal state cases, CP-violation arises due to interference of the diagrams where the neutral Higgs is coupled to a Z-boson with the diagrams where it is radiated o! the t or tM . We note that in the ttM Z case there is no CP-violating contribution coming from the interference between the diagrams with the ZZH coupling and the diagrams where the Z-boson is emitted from the incoming electron or positron lines (not shown in Fig. 28). The relevant pieces of the interaction Lagrangian involve the ttM HI Yukawa and the ZZHI couplings and are given in Eqs. (3.70) and (3.71). There, HI (k"1, 2 or 3) are the three neutral Higgs scalars in the theory. As usual the three couplings aI, bI and cI in Eqs. (3.70) and (3.71) are R R functions of tan ,v /v (the ratio of the two VEVs) and of the three mixing angles , , which characterize the Higgs mass matrix in Eq. (3.73) (for details see Section 3.2.3). As was also mentioned in Section 3.2.3, only two out of the three neutral Higgs can simultaneously have a coupling to vector-bosons and a pseudoscalar coupling to fermions. Therefore, only those two neutral Higgs particles are relevant for the present discussion and, without loss of generality, we denote them as H"h and H"H with couplings aF, bF, cF and a&, b&, c&, R R R R

122


corresponding to the light, h, and heavy, H, neutral Higgs, respectively. This implies the existence of a `GIM-likea cancellation, namely, when both h and H contribute to CP violation, then all CP-non-conserving e!ects, being proportional to bFcF#b&c&, must vanish when the two Higgs R R states h and H are degenerate. In the following, we set the mass of the heavy Higgs, H, to be m "750 GeV or 1 TeV. & In the process e>e\PttM h, a Higgs particle is produced in the "nal state, therefore, the heavy Higgs-boson, H, is not important and this `GIM-likea mechanism is irrelevant. Note that there is an additional diagram contributing to e>e\PttM h, which involves the ZhH coupling and is not shown in Fig. 28. This diagram is, however, negligible compared to the others for the large m values used here. In contrast, in the process e>e\PttM Z, the Higgs is exchanged as a virtual or & a real particle and the e!ect of H is, although small compared to h, important in order to restore the `GIM-likea cancellation discussed above. For both the ttM h and ttM Z "nal states processes, we denote the tree-level polarized di!erential cross-section (DCS) by , where f"ttM h or f"ttM Z corresponding to the ttM h or ttM Z "nal states, HD respectively, and j"1(!1) for the left(right) polarized incoming electron beam. can be HD subdivided into its CP-even ( ) and CP-odd ( ) parts >HD \HD " # . (6.56) HD >HD \HD The CP-even and CP-odd DCS's can be further subdivided into di!erent terms which correspond to the various Higgs coupling combinations and which transform as even or odd (denoted by the letter n) under ¹ . For both "nal states, f"ttM h and f"ttM Z, we have , " gGL FGL , CP-even , >HD >D >HD G " gGL FGL , CP-odd , (6.57) \HD \D \HD G where gGL , gGL , n"#or!, are di!erent combinations of the Higgs couplings aH, bH, cH and R R >D \D FGL , FGL , again with n"# or !, are kinematical functions of phase space which transform >HD \HD like n under ¹ . , Let us "rst write the Higgs coupling combinations for the CP-even part. In the case of e>e\PttM h, neglecting the imaginary part in the s-channel Z-propagator, we have four relevant coupling combinations [73,78,274]: "(aF), g> "(bF), g> "(cF), g> "aFcF . (6.58) g> R >RRM F R >RRM F >RRM F R >RRM F In the case of e>e\PttM Z, apart from the SM contribution, which corresponds to interference terms among the four SM diagrams represented by diagram (a) on the right-hand side of Fig. 28, and keeping terms proportional to both the real and imaginary parts of the Higgs propagator, H , we get [273,274] "(aHcH)Re(H ), g\ "(aHcH)Im(H ) , g> >RRM 8 R R >RRM 8 H H "(a c )Re(H ), g> "(aHcH)Im(H ) , g> R R >RRM 8 >RRM 8 H H "(b c )Re(H ), g> "(bHcH)Im(H ) , g> R R >RRM 8 >RRM 8

(6.59)


123

where H ,(s#m !mH !2p ) p #imH H )\ , 8 8 p,p #p and H is the width of H"h or H. \ > The CP-odd coupling combinations are

(6.60)

g\ "bFcF , \RRM F R for the ttM h "nal state and

(6.61)

(6.62) "bHcHRe(H ), g> "bHcHIm(H ) , g\ \RRM 8 R R \RRM 8 for ttM Z "nal state. The CP-even pieces, , yield the corresponding cross-sections (recall that f"ttM h or ttM Z) >HD

( ) d , " >HD HD

(6.63)

where stands for the phase-space variables. In Fig. 29(a) and (b), we plot the unpolarized cross-sections, M and M as a function of m and (s, for Model II (i.e., 2HDM of type II as RR8 F RRF de"ned in Section 3.2.3), with m "750 GeV and the set of values , , "/2, /4, 0 which & we denote as set II. Set II is also adopted later when discussing the CP-violating e!ect. Furthermore, for the ttM h "nal state we choose tan "0.5 while for ttM Z we choose tan "0.3. Afterwards,

Fig. 29. The cross-sections (in fb) for: (a) the reaction e>e\PttM h with tan "0.5 and (b) the reaction e>e\PttM Z with tan "0.3, assuming unpolarized electron and positron beams, for Model II with set II and as a function of m (solid and F dashed lines) and (s (dotted and dotted-dashed lines). Set II means , , ,/2, /4, 0. Figure taken from [274].

124


we will discuss the dependence of the CP-violating e!ect on tan in the ttM h and ttM Z cases. One can observe the dissimilarities in the two cross-sections M and M : while M is at most &1.5 fb, RR8 RRF RRF M can reach &7 fb at around (s+750 GeV and m 92m . M drops with m while M grows in RR8 F RR8 F R RRF the range m :2m . M peaks at around m 92m and drops as m grows further. Moreover, F R F F R RR8 M peaks at around (s+1(1.5) TeV for m "100(360) GeV, while M peaks at around F RR8 RRF (s+750 GeV for both m "100 and 360 GeV. As we will see later, these di!erent features of the F two cross-sections are, in part, the cause for the di!erent behavior of the CP asymmetries discussed below. Let us now concentrate on the CP-odd, ¹ -odd e!ects in e>e\PttM h, ttM Z, emanating from the , , , respectively. From Eqs. (6.61) and (6.62), it is clear that the ¹ -odd pieces in , \HRRM F \HRRM 8 , have to be proportional to bFcF (in the ttM Z case there is an CP-violating pieces R \HRRM F \HRRM 8 additional similar piece corresponding to the heavy Higgs H). The corresponding CP-odd kinematic functions, F\ M , F\ M , being ¹ -odd, are pure tree-level quantities and are proportional , \HRRF \HRR8 to the only non-vanishing Levi}Civita tensor present, (p , p , p , p M ), when the spins of the top are \ > R R disregarded. The explicit expressions for F\ are (recall that j"1(!1) for left(right) polarized \HD incoming electron beam)

1 g m 5 R ¹c8 (p , p , p , p M ) F\ M "! \HRRF (2 c5 m8 8F 8 R H \ > R R ;j(F#FM )[(s!s !m)(3w\!w>)#m (w\!w>)] R R R F H H 8 H H # ¹c8 (F!FM ) f , (6.64) R H 8 R R 2g m 5 R ¹c8 (p , p , p , p M ) F\ M "!(2 \HRR8 c m 8 R H \ > R R 5 8 (6.65) ;j(8#8M )[m w\#(s !s)w>]# ¹c8 (8!8M ) f , 8 H R H R H 8 R R R R where s,2p ) p is the c.m. energy of the colliding electrons, s ,(p #p M ) and f, \ > R R R (p !p ) ) (p #p M ). Also, \ > R R (6.66) F M ,(2p M ) p #m)\, 8 M ,(2p M ) p #m )\ , RR RR F F RR RR 8 8 ,(s!m )\, ,s\, ,((p!p )!m )\ . (6.67) 8 8 A 8F F 8 and

w!,(s Q !¹)c8 $Q s c , (6.68) H 5 R R H 8 R 5 5 A where s (c ) is the sin(cos) of the weak mixing angle , Q and ¹ are the charge and the 5 5 5 D D z-component of the weak isospin of a fermion, respectively, and c8 "1/2!s , c8 "!s . \ 5 5 Since at tree-level there cannot be any absorptive phases, CP-violating asymmetries only of the ¹ -odd type are expected to occur in . Note that in the ttM Z case there is also a CP-odd, , \HD ¹ -even piece, bHcH Im(H );F> M (see Eq. (6.62)), in the DCS. However, being proportional to R \HRR8 , the absorptive part coming from the Higgs propagator, it is not a pure tree-level quantity.


125

Simple examples of observables that can trace the tree-level CP-odd e!ect in e>e\PttM h; ttM Z are [78]: p ) ( p ;p M ) R R , O (ttM h)" \HRRM F ; O (ttM Z)" \HRRM 8 . (6.69) O" \ s M M >HRRF >HRR8 Here O (ttM h; ttM Z) are optimal observables in the sense that the statistical error in the measured is involved. As asymmetry is minimized [29]. Note also that only the ¹ -odd part of , \HRRM 8 mentioned before, since the "nal state consists of three particles, using only the available momenta, there is a unique antisymmetric combination of momenta that can be formed. Thus, both observables are proportional to ( p , p , p , p M ). Furthermore, O (ttM h; ttM Z) are related to \ > R R O through a multiplication by a CP-even function. In the following, we focus only on the CP-odd e!ects coming from the optimal observables. We remark, however, that the results for the simple observable O exhibit the same behavior, though slightly smaller then those for O . The theoretical statistical signi"cance, N , in which an asymmetry can be measured in an ideal 1" experiment is N "A(¸( (" M , M for the ttM h, ttM Z "nal states, respectively), where for the RRF RR8 1" observables O and O , the CP-odd quantity A, de"ned above, is A + O/( O, A +( O . (6.70) Also, ¸ is the e!ective luminosity for fully reconstructed ttM h or ttM Z events. In particular, we take ¸" L, where L is the total yearly integrated luminosity and is the overall e$ciency for reconstruction of the ttM h or ttM Z "nal states. In the following numerical analysis, we have used set II de"ned before for the angles , i.e., , , "/2, /4, 0. Figs. 30(a) and (b) show the expected asymmetry and statistical signi"cance in the unpolarized case, corresponding to O in Model II for the ttM h and ttM Z "nal states, respectively. The asymmetry is plotted as a function of the mass of the light Higgs (m ) where F again, m "750 GeV in the ttM Z case. We plot N /(¸, thus scaling out the luminosity factor from & 1" the theoretical prediction. We remark that set II corresponds to the largest CP-e!ect, though not unique since we are dealing with angles, i.e., , which may be rotated by or /2 leaving the relevant combinations of angles with the same value (e.g., bFJsin sin ). In the ttM h case tan "0.5 is favored, however, R the e!ect mildly depends on tan in the range 0.3:tan :1 [78,274]. In the ttM Z case, the e!ect is practically insensitive to and is roughly proportional to 1/tan , it therefore drops as tan is increased. Nonetheless, we "nd that N /(¸'0.1, even in the unpolarized case for tan :0.6 1" [273,274]; note that N /(¸ here is dimensionless if ¸ is in fb\. 1" From Fig. 30(a) we see that, in the ttM h case, as m grows the asymmetry increases while the F statistical signi"cance drops, in part because of the decrease in the cross-section. Evidently, the asymmetry can become quite large; it ranges from &15%, for m :100 GeV, to &35% for F m &600 GeV. Indeed, the CP-e!ect is more signi"cant for smaller masses of h, wherein A is F smaller. In contrast, Fig. 30(b) shows that, in the ttM Z case, A stays roughly "xed at around 7}8% Recall that for the ttM h "nal state we choose tan "0.5 while for the ttM Z "nal state we take tan "0.3. As a reference value, we note that for ¸"100 fb\, N /(¸"0.1 will correspond to a 1! e!ect. 1"

126


Fig. 30. The asymmetry, A , and scaled statistical signi"cance, N /(¸, for the optimal observable O for: 1" (a) the reaction e>e\PttM h with tan "0.5 and (b) the reaction e>e\PttM Z with tan "0.3, as a function of the light Higgs mass m , for (s"1 TeV and 1.5 TeV. All graphs are with set II of the parameters, as in Fig. 29. Figure taken F from [274].

for m :2m , and then drops till it totally vanishes at m "m "750 GeV, due to the `GIMF R F & likea mechanism discussed above. The scaled statistical signi"cance N /(¸ behaves roughly as 1" A . That is, N /(¸+0.1}0.2 in the mass range 50 GeV:m :350 GeV, for both (s"1 and 1" F 1.5 TeV. Figs. 31(a) and (b) show the dependence of A and N /(¸ on the c.m. energy, (s, for the ttM h 1" and ttM Z cases, respectively. We see that, in the case of ttM h, the CP-e!ect peaks at (s+1.1(1.5) TeV for m "100(360) GeV and stays roughly the same as (s is further increased to 2 TeV. In the case F of ttM Z, the statistical signi"cance is maximal at around (s+1 TeV and then decreases slowly as (s grows for both m "100 and 360 GeV. Contrary to the ttM h case, where a light h is favored, in F the ttM Z case, the e!ect is best for m 92m . In that range, on-shell Z and h are produced followed by F R the h decay hPttM , thus, the Higgs exchange diagram becomes more dominant. In Tables 7 and 8 we present N for O , for the ttM h and ttM Z cases, respectively, in Model II with 1" set II, and we also compare the e!ect of beam polarization with the unpolarized case. As before, we take tan "0.5 and tan "0.3 for the ttM h and ttM Z cases, respectively, where for the ttM Z case we present numbers for both m "750 GeV (shown in the parentheses) and m "1 TeV, to demon& & strate the sensitivity of the CP-e!ect to the mass of the heavy Higgs. For illustrative purposes, we choose m "100, 160 and 360 GeV and show the numbers for (s"1 TeV with L"200 (fb)\ F


127

Fig. 31. The asymmetry, A , and scaled statistical signi"cance, N /(¸, for the optimal observable O for: (a) the 1" reaction e>e\PttM h with tan "0.5 and (b) the reaction e>e\PttM Z with tan "0.3, as a function of the c.m. energy (s, for m "100 GeV and m "360 GeV. All graphs are with set II of the parameters, as in Fig. 29. Figure taken F F from [274].

and for (s"1.5 TeV with L"500 (fb)\ [283,284] (see also [253}255]). In both cases we take "0.5 assuming that there is no loss of luminosity when the electrons are polarized. Evidently, for both reactions, left polarized incoming electrons can probe CP violation slightly better than unpolarized ones. We see that in the ttM h case the CP-violating e!ect drops as the mass of the light Higgs (h) grows, while in the ttM Z case it grows with m . In particular, we "nd that with F (s"1.5 TeV and for m 92m the e!ect is comparable for both the ttM h and the ttM Z cases where it F R reaches above 3- for negatively polarized electrons. With a light Higgs mass in the range 100 GeV:m :160 GeV, the ttM h case is more sensitive to O and the CP-violating e!ect can reach F &4- for left polarized electrons. In that light Higgs mass range, the CP-violating e!ect reaches slightly below 2.5- for the ttM Z case. For a c.m. energy of (s"1 TeV and m "360 GeV, the ttM Z F case is much more sensitive to O and the e!ect can reach 2.2- for left polarized electron beam. However, with that c.m. energy, the ttM h mode gives a larger CP-odd e!ect in the range m &100}160 GeV. F Let us now summarize the above results and add some concluding remarks. We have shown that an extremely interesting CP-odd signal may arise at tree-level in the reactions e>e\PttM h and

Clearly if the e$ciency for ttM h and/or ttM Z reconstruction is "0.25, then our numbers would correspondingly require 2 yr of running.

128


Table 7 The statistical signi"cance, N , in which the CP-nonconserving e!ects in e>e\PttM h can be detected in one year of 1" running of a future high-energy collider with either unpolarized or polarized incoming electron beam. We have used tan "0.5, a yearly integrated luminosity of L"200 and 500 (fb)\ for (s"1 and 1.5 TeV, respectively, and an e$ciency reconstruction factor of "0.5 for both energies. Recall that j"1(!1) stands for right(left) polarized electrons. Set II means , , ,/2, /4, 0. Table taken from [274] (s (TeV) P

1

1.5

j (GeV) N

m "100 F

!1 unpol 1 !1 unpol 1

2.2 2.0 1.8 4.0 3.6 3.2

e>e\PttM h (Model II with Set II) O m "160 F 2.0 1.9 1.7 3.9 3.5 3.1

m "360 F 1.1 1.0 0.9 3.2 2.9 2.6

Table 8 The same as Table 7 but for e>e\PttM Z, with tan "0.3. In this reaction, e!ects of the heavy Higgs, H, are included and N is given for both m "750 GeV (in parentheses) and m "1 TeV. Table taken from [274] 1" & & (s (TeV) P

1

1.5

j (GeV) N

m "100 F

!1 unpol 1 !1 unpol 1

(1.8) (1.6) (1.5) (2.3) (2.1) (1.8)

1.7 1.6 1.5 2.9 2.6 2.3

e>e\PttM Z (Model II with Set II) O m "160 m "360 F F (1.8) (1.7) (1.5) (2.4) (2.1) (1.8)

1.8 1.6 1.5 3.0 2.7 2.3

(2.2) (2.0) (1.8) (2.8) (2.5) (2.1)

2.2 2.0 1.8 3.3 3.0 2.6

e>e\PttM Z. The asymmetries that were found are &15}35% in the ttM h case and &5}10% for the ttM Z "nal state. These asymmetries may give rise in the best cases, i.e., for a favorable set of the relevant 2HDM parameters, to &3}4-, CP-odd, signals in a future e>e\ collider running with c.m. energies in the range 1 TeV:(s:2 TeV. Note, however, that the simple observable, O, as well as the optimal one, O , require the identi"cation of the t and tM and the knowledge of the transverse components of their momenta in each ttM h or ttM Z event. Thus, for the main top decay, tPb=, the most suitable scenario is when either the t or the tM decays leptonically and the other decays hadronically. Distinguishing between t and tM in the double-hadronic decay case will require more e!ort and still remains an experimental challenge. If, for example, the identi"cation of the charge of the b-jets coming from the t and the tM is possible, then the di$culty in reconstructing the transverse components of the t and tM momenta


129

may be surmountable by using the momenta of the decay products in the processes e>e\PttM hPb=>bM =\h and e>e\PttM ZPb=>bM =\Z. For example, the observable (p , p , p , p M ) O " \ > @ @ @ s

(6.71)

may then be used. We have considered this observable for the reaction e>e\PttM hPb=>bM =\h in [78]. We found there that, close to threshold, this observable is not very e!ective. However, at higher energies, O is about as sensitive as the simple triple product correlation O de"ned in @ Eq. (6.69) and, therefore, only slightly less sensitive than O . Note also that for the light Higgs mass, m "100 GeV, the most suitable way to detect the Higgs F in e>e\PttM hPb=>bM =\h is via hPbbM with branching ratio &1. For m 92m , and speci"cally F R with set II used above, there are two competing Higgs decays, hPttM and hP=>=\, depending on the value of tan . For example, for tan "0.5, as was chosen above, one has Br(hPttM )+0.77 and Br(hP=>=\)+0.17, thus, the hPttM mode is more suitable. Of course, hPttM will dominate more for smaller values of tan and less if tan '0.5. In particular, for tan "0.3(1) one has Br(hPttM )+0.89(0.57) and Br(hP=>=\)+0.08(0.32). Finally, as emphasized before, the "nal states ttM h and ttM Z, in particular the ttM h, are expected to be the center of considerable attention at a linear collider. Extensive studies of these reactions are expected to teach us about the details of the couplings of the neutral Higgs to the top quark [285,286]. Thus, it is gratifying that the same "nal states promise to exhibit interesting e!ects of CP violation. It would be very instructive to examine the e!ects in other extended models. Numbers emerging from the 2HDM that was used, especially with the speci"c value of the parameters, should be viewed as illustrative examples. The important point is that the reactions e>e\PttM hP b=>bM =\h and e>e\PttM ZPb=>bM =\Z appear to be very powerful and very clean tools for extracting valuable information on the parameters of the underlying model for CP violation. 6.2.2. Generalized optimization technique and extraction of various Higgs couplings An optimization technique was employed by Gunion et al. in [73] for the process e>e\PttM h. It was shown that this reaction may provide a powerful tool for extracting the ttM h Yukawa couplings and the ZZh couplings. A similar analysis for the reaction e>e\PttM Z was likewise considered in [275]. This technique is outlined in Section 2.6. The basic idea is that the nature of the Higgs particle, i.e., whether it is CP-odd or CP-even, may well be distinguishable through studies of momentum correlations in e>e\PttM h. In particular, greater information on the detailed dependence of M ( ) on the variable is extracted to deduce limits that can be obtained on the di!erent HRRF Higgs couplings combinations in Eqs. (6.58) and (6.61). As described before, the di!erential e>e\ P ttM h cross-section contains "ve distinct terms which are explicitly written in Eqs. (6.58) and (6.61). The only CP-violating component is bFcF, while the others enter into the total cross-section R as in Eq. (6.63). Gunion et al. have investigated two issues. For a given c.m. energy and integrated luminosity at the NLC, they have examined: 1. The 1- error in the determination of the couplings aF, bF, cF, by "xing (ttM h)"1 for a given R R . input model with couplings gGL !RRM F 2. To what degree of statistical signi"cance can a model be ruled out, given a certain input model.

130


Let us now elaborate more on how these two points were studied in [73]. With the optimal technique outlined in [73], Gunion et al. used unique weighting functions such that the statistical error in the determination of the various gGLM in Eqs. (6.58) and (6.61) is minimized. They write !RRF GL (6.72) g M " M\I , GI I !RRF I where I , M and the appropriate weighting functions are given in [73] (see also Section 2.6). Then, I GI , one can given an input model, for which the couplings are denoted with the superscript 0 as gGL !RRM F compute the con"dence level at which parameters of choice, di!erent from the input model, can be ruled out )(gHL !gHL )e\PZh) contains one useful Higgs coupling combination, g> >RRM F fore, the sensitivity to (cF) is increased when the above technique is also applied to (e>e\PZh). Let us continue with a discussion of the work of Ref. [73] in which only the ttM h "nal state was considered. The ability to distinguish between di!erent models was furthermore investigated by using the optimization technique. This is shown in Table 10. We see from this table, for example, that if the Higgs is the SM one, then the pure CP-odd case (input model II) and the equal CP-mixture case (input model III) are ruled out at the 9.5- and 4.8- level, respectively. Note also that negative beam polarization slightly improves the results.


131

Table 9 The 1- errors in aF, bF and cF are given, for the three Higgs coupling cases I, II and III. Results are given for unpolarized R R beams and for 100% negative e\ polarization (P "!1) also (s"1 TeV, m "100 GeV, m "176 GeV and C F R ¸"50 fb\. Table taken from [73]

Case I

aF$aF R R

Unpolarized e\ bF$bF cF$cF R R

> \ 0> \ > \

0> \ > ( \ > \

(

II III

1> \ 0> \ > ( \

aF$aF R R

P "!1 C bF$bF R R

cF$cF

> \ 0> \ > \

0> \ > ( \ > \

1> \ 0> \ > ( \

(

Table 10 The number of standard deviations, (, at which a given input model (I, II or III) can be distinguished from the other two models, are tabulated, for (s, m , m and ¸ as in Table 9. Table taken from [73] F R Unpolarized e\ Trial model

P "!1 C Trial model

Input model

I

II

III

I

II

III

I II III

* 34 6.3

9.5 * 6.3

4.8 17 *

* 40 7.3

11 * 7.3

5.5 20 *

Finally, the ability for determining a non-zero CP-violating component, bFcF, was also investiR gated in [73]. They found that with m "100 GeV, ¸"100 fb\, a non-zero bFcF coupling can F R be established at a level better then 1- in a 1 TeV e>e\ collider. 6.2.3. CP asymmetries in e>e\PZh and in the subsequent Higgs decay hPttM We now consider the process e>e\PZh followed by hPttM . As we have discussed above, in general, one cannot ignore the SM-like diagrams of class (a) on the right-hand side of Fig. 28 when analyzing CP violation in the reaction e>e\PttM Z. Moreover, inclusion of those diagrams and interfering them with diagram (b) on the right-hand side of Fig. 28, gives a bonus in the appearance of tree-level CP violation in e>e\PttM Z. However, let us assume that the Higgs has already been discovered, with a mass of m '2m , by the time a high-energy e>e\ collider starts its "rst run. In F R such a scenario, one should, in principle, be able to separate the contribution of the Higgs exchange graph in e>e\PttM Z from the rest of the SM-like diagrams which lead to the same "nal state, by imposing a suitable cut on the invariant ttM mass. Taking this viewpoint, we will consider Higgs production via e>e\PhZ and CP violation in the subsequent Higgs decay hPttM . This is the value considered by us in the previous Section 6.2.1 for which the results in Tables 7 and 8 are given.

132


A general method for tracing CP-odd and CP-even t, tM spin-correlations, in hPttM , was introduced in [287]. There, it was assumed that an on-shell Higgs, with m '2m , is produced through, for F R example, e>e\PZh, l>l\h or even >\Ph, and that its rest system can be reconstructed. In [288], a helicity asymmetry in hPttM was suggested, where e>e\PZh was explicitly assumed as the Higgs production mechanism. We will describe below these two works. Other related works can be found in [82,289}295]. In the method suggested in [287], the decay hPttM stands out as an independent decay process, in which top spin-asymmetries can be formed. Consider, for example, the observables (6.75) S "kK ) (s !s M ) , R R R S "kK ) (s ;s M ) , (6.76) R R R S "s ) s M , (6.77) R R where s (s M ) is the spin operator of t(tM ) and k is the top quark 3-momentum in the ttM c.m. frame. R R R S and S are CP-odd, where S is ¹ -even and S is ¹ -odd. Therefore, a non-zero expectation , , value of S will also require absorptive parts, while S O0 can be generated already at the tree-level. S is CP-even and is also generated at the lowest order (i.e., tree-level). For a general ttM h Yukawa interaction Lagrangian as in Eq. (3.70), it was found in [287] that the spin-asymmetries, S in Eqs. (6.75)}(6.77), depend only on one combination of the couplings aF and bF R R bF R , (6.78) r" R aF#bF R R which takes values between 0 to 1, i.e., 04r 41, where the lower limit corresponds to bF"0 and R R the upper one to aF"0. R These observables can be translated to correlations between momenta of the t and tM decay products. To do so, one can de"ne decay scenarios of the t and tM , through which both the t and tM momenta and spins can be reconstructed in the most e$cient way [287]: A:

tP=>#bPl>#l #b ,

tM P=\#bM Pq#q #bM .

(6.79)

The sample A M is de"ned by the charge conjugate decay channels of the ttM pairs AM :

tP=>#bPq #q #b , tM P=\#bM Pl\#l #bM .

(6.80)

In these decay samples, either the t or tM decays leptonically while the other decays hadronically. Each of these samples has a branching fraction of about of all ttM pairs. With these decay scenarios Ref. [287] found the momentum correlations O which trace the spin correlations S , respectively, 2 (6.81) O " kK ) p( Hl> A # kK ) p( lH\ AM " S , R R 3


8 1!2x O " kK ) (p( lH> ;p( HM )A ! kK ) (p( lH\ ;p( H)AM " S , R @ R @ 9 1#2x

8 1!2x O " p( lH> ) p( HM A # p( Hl\ ) p( HAM " S , @ @ 9 1#2x

133

(6.82) (6.83)

where p( Hl> (p( lH\ ) is the #ight direction of l>(l\) in the t(tM ) quark rest system. Similarly, p( H(p( HM ) is the @ @ unit momentum of the b(bM ) in the t(tM ) rest system. Also, x,m /m and the factor 5 R (1!2x)/(1#2x)+0.41 measures the spin analyzing quality of the b(bM ) (see Section 2.8). of hPttM events that is required to establish a nonzero correlation O at The number N RRM the N (standard deviations) signi"cance level are given by 1" N 1" " , (6.84) N RRM Br(A);A where O A " , (6.85) ( O and Br(A)"Br(A M ) is the branching ratio of the decay samples A or AM . In particular, disregarding the leptons we have Br(A)" . The number of events needed for a 3- (i.e., N "3) observation 1" of the spin-correlations, S , of Eqs. (6.75)}(6.77) are given in Fig. 32, for m "400 GeV, F Br(A)" and as a function of the parameter r de"ned in Eq. (6.78). These can be simply obtained R in Eq. (6.84), by using the relations between O and S given in Eqs. (6.81)}(6.83). from N RRM It should be noted again that, while O are non-zero already at the tree-level, O , being ¹ -even, requires an absorptive phase and, therefore, its non-zero contribution "rst arises only at , the 1-loop level in perturbation theory. Thus, O is expected to be less e!ective (as can be seen from Fig. 32). In [287] the 1-loop QCD corrections to O were computed. For the operators, O , the QCD corrections were found to be of the order of a few percent compared to the leading tree-level contribution, and were therefore neglected. We see from Fig. 32 that, for example, values of r in the range, 0.18:r :0.52, would give rise to R R a 3- CP-odd e!ect in O with a data sample of NM +1500, i.e., &1500 (hPttM ) events. Note also RR that a simultaneous measurement of O and O , with these &1500 events, would have a 3- sensitivity to r from 0.18 to its maximal value of 1. Furthermore, production of a few thousands of R hPttM events through the Bjorken mechanism, e>e\PhZ, is indeed feasible. For example, if a CP-mixed neutral Higgs (with both scalar and pseudoscalar couplings aF and bF) with a mass R R m "400 GeV, has a ZZh coupling cF of a SM strength, then the cross-section for e>e\PhZ is of F the order of a few fb's for e>e\ c.m. energies in the range 500 GeV:(s:1000 GeV. Therefore, with a yearly integrated luminosity of L&10 fb\, hundreds of hZ pairs can be produced and, thus, a &2- limit on r may be achievable. R

The cross-section for e>e\PhZ is given, for example, in [91]. With m "400 GeV and for cF"1, i.e., the SM ZZh F coupling, (e>e\PhZ)&3.5(8.5) fb for (s"500(1000) GeV.

134


Fig. 32. Number of events hPttM to establish a non-zero correlation S with 3 standard deviation signi"cance, as a function of r (see Eq. (6.78)) and for a "xed Higgs mass of m "400 GeV. The dashed line represents the result for NM , R F RR the solid line is the result for NM and the dotted line is the result for NM . m "175 GeV. Figure taken from [287]. RR RR R

An interesting CP-violating helicity asymmetry in hPttM was suggested in [288]: (##)!(!!) OF " , RR (##)#(##)

(6.86)

where (##) and (!!) are the decay widths of the lightest Higgs-boson h, into a pair of ttM with the indicated helicities. Since under CP: (##) (!!), non-zero OF would be a signal of CP RR violation. OF is CP-odd but ¹ -even, therefore, it requires a CP-odd as well as a CP-even absorptive phase RR , (i.e., FSI phase). As mentioned several times before, in a 2HDM with a CP-mixed neutral Higgs, the CP-odd phase is provided by the simultaneous presence of the scalar and pseudoscalar ttM h couplings in the ttM h interaction Lagrangian. The FSI absorptive phase is generated at the 1-loop level from the diagrams in Fig. 33. The expressions for the di!erent contributions to OF correRR sponding to the di!erent diagrams in Fig. 33 are given in [288]. There, it was found that with m "180 GeV and m 92m , OF of the order of 50% is possible. Explicitly, assuming the Higgs to R F R RR be produced through the Bjorken mechanism, e>e\PhZ, the statistical signi"cance, N , with 1" which this asymmetry can potentially be detected is (6.87) N "(¸((e>e\PhZ);OF . RR 1" In [288] an e!ective integrated luminosity of ¸"85 fb\ was assumed and a scan for maximal N was performed as a function of the 2HDM parameters tan and . It was found that with 1" tan "0.5, for example, and with m "180 GeV, m 92m , up to a 7- detection of a CP-violating R F R signal from OF is feasible, if the Higgs is produced via e>e\PhZ. It should be emphasized, RR however, that there is a potential background to the ttM pairs (coming from the subsequent Higgs decay in e>e\PhZ) from the SM-like diagrams included in Fig. 28. Therefore, as mentioned before, in order for such a study to be practical one has to know the mass of the Higgs prior to the


135

Fig. 33. The 1-loop diagrams contributing to the asymmetry OF in a 2HDM. RR

actual experiment and, with a su$cient mass resolution, demand that the invariant ttM mass reconstructs the Higgs. 6.3. e>e\PttM g Given the importance of the top pair production at the NLC, it should be clear that the associated gluon emission will also receive considerable attention. Of course, the gluons will be radiated o! top quarks quite readily once the threshold for top pair production will be reached. One important advantage of the reaction e>e\PttM g is that it is rich in exhibiting several di!erent types of CP asymmetries which can be driven by 1-loop e!ects induced by extensions of the SM. For example, exchanges of neutral Higgs from MHDMs with CP violation in the scalar sector, or exchanges of SUSY particles which carry a CP-odd phase in their interaction vertices, could give rise to both ¹ -odd and ¹ -even type CP-violating dynamics. Therefore, both CP-odd, ¹ -odd , , , and CP-odd, ¹ -even type observables can be used to extract information on the real and , imaginary parts of the amplitude. With three particles in the "nal state there are enough linearly independent momenta available so that the construction of CP-odd, ¹ -odd observables is , straightforward; there is no need to involve the spins of the top. Following our work in [296], we give below the full analytical formulae of the tree-level di!erential cross-section (DCS) as well as a description for extracting the 1-loop CP-violating DCS that can be used for any given model. The SM tree-level diagrams, are depicted in Fig. 34. The incoming polarized electron}positron current can be written as JIH"v (p )IP u (p ) , C C > H C \

(6.88)

136


Fig. 34. Tree-level Feynman diagrams contributing to e>e\PqqN g (for q"t).

where P "(1#j; ) and j"!1(1) for left (right) handed incoming electrons. p (p ) are the H > \ 4-momentum of the positron (electron) and p"(p #p ) is the 4-momentum of the s-channel > \ gauge-boson (the contribution from an s-channel Higgs vanishes for m P0). C We also de"ne the following constants: (4) ¹?g CC, C "(4)¹?g Q , C " Q 8 H A Q A O 8 2c s 5 5

(6.89)

where ¹? is the appropriate SU(3) generator, g is the strong coupling constant, Q is the charge of Q O the quark and c (s ) stands for cos (sin ), respectively. Also CC"CC (CC ) for j"!1(1) with 5 5 5 5 H * 0 CD "!2ID #2Q s and CD "2Q s . and are the gauge-boson propagators given by * D 5 0 D 5 8 A 1 1 " , " . A p 8 p!m 8

(6.90)

Then the tree-level matrix element is given by M,M #M #M #M , (6.91) ? ? @ @ where M , M , M and M are obtained from diagrams (a ), (a ), (b ) and (b ) in Fig. 34, ? ? @ @ respectively, all emanating from the SM. We thus get 1 ! ¹ ]#2C [ ¹ ! ¹ ]v(p ) . M" JIHu (p )C [ ¹ O 8 O ?I O ?I A O @I O @I O 2 C

(6.92)

Here p (p ) is the 4-momentum of the outgoing quark (anti-quark), p is the gluon's 4-momentum E O O and the quark and anti-quark propagators are given by 1 1 , " . " O 2p ) p O 2p ) p O E O E Furthermore, the hadronic vector elements in Eq. (6.92) are ¹ I ". (p/ #p/ #m ) C> , E O I *0 O ? ¹ I " C> (p/ #p/ !m ). , ? E O I *0 O

(6.93)

(6.94) (6.95)


137

Fig. 35. The cross-section for the reaction e>e\PttM g (in fb) as a function of the c.m. energy (s, for unpolarized (solid line), negatively polarized (dotted line) and positively polarized (dashed line) incoming electron beam. The cuts (p #p )5(m #m ) and (p #p M )5(m #m ), m "25 GeV, are imposed. Figure taken from [296]. E R R E R R

¹ "¹ I (C> P1) , @I ? *0

(6.96)

¹ "¹ I (C> P1) , @I ? *0

(6.97)

being the polarization vector of the gluon and C> "CO ¸#CO R, where ¸(R)"P (P ). ? *0 * 0 H\ H In Fig. 35, we have plotted the tree-level cross-section as a function of the c.m. energy in an e>e\ collider for polarized and unpolarized incoming electron beam. To facilitate experimental identi"cation as well as to avoid infrared singularities we have imposed a cut on the invariant mass of the jet pairs so that (p #p ) and (p #p M )5(m #m ) where we have taken m "25 GeV. This cut, R E R E R which e!ectively cuts the gluon energy, also removes soft gluon emission from the secondary b-quarks of the top decays. We see from Fig. 35 that with an integrated luminosity of L&200 fb\, a 1 TeV (500 GeV) e>e\ collider will be able to produce about &3;10 (&1;10) ttM g events. In a given model, the CP-violating corrections for the reaction e\(p )e>(p )Pq(p )q (p )g(p ) E \ > O O

(6.98)

requires the calculation of the corresponding 1-loop diagrams. Let us write the general form of the 1-loop matrix elements that violate CP as MM . For a given underlying model, the subscript N indicates the diagram and the superscript indicates which gauge particle is exchanged in the

138


s-channel. Thus, MM "JIHu (p )HM v(p ) , (6.99) N C O NI O where HM is the `hadronic vectora corresponding to each diagram and exchanged quanta. NI Denoting the complete CP-violating 1-loop contribution by M4" MM , (6.100) N M N the M4 can be calculated within a given model, and the polarized CP-nonconserving DCS to 1-loop is then obtained from the interference terms between the 1-loop and the Born amplitudes (M4MH#MM4H) .

(6.101)

Here the sum is carried over the polarizations of e>, t, tM and g. 6.3.1. 2HDM and CP violation in e>e\PttM g In a 2HDM, CP-violating neutral Higgs exchanges, at 1-loop order, can give rise to the Feynman diagrams depicted in Fig. 36 [296]. We take the limit m P0, thus neglecting all the C diagrams that are proportional to the electron mass. This includes any diagram that involves electron coupling to the Goldstone modes, hence proportional to m . C The relevant Feynman rules for the diagrams in Fig. 36 can be extracted from parts of the Lagrangian involving the +M HI and ZZHI couplings aI , bI and cI de"ned in Eqs. (3.70) and (3.71), D D respectively. Again, for simplicity, we will consider only one light neutral Higgs, h, assuming that the remaining two are considerably heavier. Furthermore, whenever necessary we set the masses (m ) of the remaining two neutral Higgs particles to be 1 TeV, i.e., assume that they are degenerate. & All the CP-violating terms in the 1-loop amplitudes corresponding to the diagrams in Fig. 36 emerge through interference of the scalar coupling aF (for a quark q) with the pseudo-scalar O coupling bF in any exchange of a neutral Higgs. In the diagrams where the Higgs exchange is O generated at the ZZh vertex, the CP-violating terms will be proportional to bF ;cF. The CPO violating 1-loop amplitude can then be calculated (for details see [296]) and the DCS can be schematically written as (6.102) ()" ()#R()#I () . Here () is a CP-even piece and is a CP-odd piece which is further subdivided into two parts that depend on the real and imaginary components of the amplitude. Thus, CP-odd, ¹ -odd and , CP-odd, ¹ -even e!ects will emanate from R() and I (), respectively. , To estimate the CP-violating e!ects of both the ¹ -even and ¹ -odd types, the following two , , CP-odd observables were considered in [296] for the reaction e>e\PttM g p ) ( p #p M ) R R , ¹ -even: O , \ , G s

(6.103)

p ) ( p ;p M ) R R . ¹ -odd: O , \ , P s

(6.104)

Recall that HI stand for any one of the three, i.e., k"1, 2 or 3, neutral Higgs in a 2HDM.


139

Fig. 36. CP violating Feynman diagrams contributing to e>e\Pqq g to 1-loop order in a 2HDM (h is a neutral Higgs-boson). Diagrams with permuted vertices (i.e., qPq ) are not shown.

A non-vanishing expectation value of any one of these would signal CP violation so that experimental searches for them can be performed without recourse to any model. However, as was mentioned in Section 2.6, within the context of any given model one can also construct optimal observables i.e., those observables which will be the most sensitive to CP violation e!ects in that model [29], (6.105) O ,I / , O ,R/ , P G The number of events needed in order to detect a CP-odd signal at the 1- level via each of the above four CP-violating observables, is shown in Figs. 37 and 38 for m "100 and 200 GeV, F respectively. In Fig. 39 we have magni"ed the range (s"400}600 GeV using the same Higgs masses. In these "gures, we have focused on the case of left polarized incoming electrons while in Table 11 we give a brief comparison of the left, right and unpolarized electron beam cases. Also, the following assumptions are made: (1) As mentioned before a cut on the invariant mass of the jet pairs was imposed, so that (p #p ) E R and (p #p M )5(m #m ), where we have taken m "25 GeV. R E R (2) The ¹ -odd observables, being proportional to the dispersive parts of the loop integrals, are , sensitive also to the mass of the two heavier neutral Higgs particles, m . For simplicity, we have & chosen these two Higgs particles to be degenerate with a mass of 1 TeV. (3) We set the relevant Higgs couplings to unity. That is, aF"bF"cF"1 (for q"t) which serves R R our purpose of "nding the order of magnitude of the CP-odd signal that can arise in this reaction. In fact, CP violation in e>e\PttM g is found to be dominated by the terms proportional to aF;bF. With regard to that, we note that, for low values of tan , i.e., tan :0.5, the R R product aF;bF can reach values above &5 and, therefore, our choice above of aF;bF"1 is R R R R rather conservative. Summarizing brie#y the numerical results presented in Figs. 37}39 and in Table 11, we can see that for the optimal observable O , for both a 500 GeV and a 1 TeV e>e\ collider, the number of G

140


Fig. 37. Number of events (in units of 10) needed to detect CP violation via O , O , O and O to 1- G P G P level, as a function of the total beam energy, (s, for left-handed polarized incoming electron beam. m "100 GeV, F m "1 TeV and aF"bF"cF"1 are used. Also, the cuts (p #p )5(m #m ) and (p #p M )5(m #m ), R & R R E R R E R m "25 GeV, are imposed. Figure taken from [296].

Fig. 38. Same as Fig. 37 except m "200 GeV. Figure taken from [296]. F


141

Fig. 39. Number of events (in units of 10) needed to detect CP violation via O and O to 1- level as a function P P of total beam energy in the range (s"400}600 GeV for m "1 TeV, m "100 and 200 GeV. The rest of the parameters & F are as in Figs. 37 and 38. Figure taken from [296].

Table 11 The unpolarized case is compared with left polarization ( j"!1) and right polarization ( j"#1) of the e\. The number of events in units of 10 needed for detection of asymmetries, to 1- level are given. The values of (s and m are F given in GeV. The results for the ¹ -odd observables are given for m "1 TeV, where m is the mass of the two heavy , & & Higgs (see also text). Table taken from [296] O (s

j

O

G m "100 F

O

P m "200 F

m "100 F

O P

G

m "100 F

m "200 F

m "200 F

m "100 F

m "200 F

!1 400 unpol. 1

1.8 22.5 2.3

11.5 134.8 17.1

0.07 0.05 0.05

0.05 0.05 0.04

1.0 6.5 1.3

6.0 37.0 8.4

0.05 0.05 0.04

0.05 0.04 0.03

!1 700 unpol. 1

3.4 48.6 4.5

20.0 263.9 30.8

2.2 2.2 1.9

2.1 1.9 1.8

1.7 12.2 2.0

5.1 38.6 5.9

1.9 1.8 1.5

1.7 1.4 1.2

!1 1000 unpol. 1

4.0 63.4 5.0

10.5 158.2 14.0

14.4 14.3 14.1

14.1 10.3 10.3

2.6 20.5 3.1

4.5 35.3 5.4

11.6 10.8 10.7

10.8 8.4 8.3

142


needed ttM g events in order to detect a 1- CP-odd signal is comparable and is around few;10 with neutral Higgs masses in the range 100 GeV(m (200 GeV. With O the number of needed ttM g F P events at c.m. energies around 1 TeV is few;10. However, we see from Fig. 39 that, at a c.m. energy of 500 GeV and for 100 GeV(m (200 GeV, a 1- measurement of O will require F P few;10 ttM g events. From Table 11 we see that for the ¹ -even (i.e., O and O ) cases the , G G polarization makes a signi"cant di!erence and improves their e!ectiveness by about an order of magnitude or even more. For these it seems that the left-polarized case is marginally better than the right one. Bearing in mind that with an integrated luminosity of L&200 fb\, about &10 ttM g will be produced in a 500 GeV NLC, and few;10 in a 1 TeV NLC (see Fig. 35), the observability of a non-vanishing value for O , to the 1- level, in a NLC with c.m. energies of 500 GeV is marginal, P while O falls short by about an order of magnitude. Also, with a 1 TeV NLC that can produce up G to 3;10 ttM g's a year, the CP-odd signal from O falls short by almost two orders of magnitude, P while the number of events needed to detect a CP-odd e!ect through O is one order of G magnitude away from the expected number of available events in such a collider. Clearly, although the CP-violating e!ects driven by neutral Higgs exchanges that were found in [296] fall short by at least one order of magnitude for a 3- detection, this does not rule out the possibility of larger e!ects in other extensions of the SM (e.g., SUSY). Therefore, theoretical and experimental studies of CP violation in the process e>e\PttM g may still be worthwhile. 6.3.2. Model independent constraints on top dipole moments The e!ects of anomalous EDM (dA), ZEDM (d8) and CEDM (dE) couplings of the top quark to R R R a photon, Z-boson and a gluon, respectively, in e>e\PttM g were considered in [297,298]. Let us write again an e!ective top quark interaction with a neutral gauge-bosons e\PttM g can be used to obtain limits on the anomalous dipole-like couplings of the top to , g and Z through the analysis of the associated gluon energy spectrum. If the couplings of the top to the neutral gauge-bosons , Z and g are altered by the e!ective magnetic and electric-like interactions in Eq. (6.106), then by allowing one or more of the di!erent 's and 's to be non-zero, the shape of the gluon energy spectrum in the process e>e\PttM g can 4 4 be modi"ed. In [297], Monte Carlo data samples (assuming that the SM is correct) were generated and then a "t to the general expressions for the ! dependent spectrum were performed with 4 4 which a 95% CL allowed region in the ! was obtained. This procedure is done for each 4 4 gauge-boson separately. That is, in analyzing the limits that can be placed on the various dipole moment couplings, only one pair of , corresponding to one neutral gauge-boson, e\ NLC. We see that the CEDM coupling, , E E E can be bounded in a 500 GeV NLC to :0.6}0.8. For the CMDM coupling, for whatever it is E worth, the allowed values are &$few;10\, with integrated luminosities of 50}100 fb\ and E with a cut on the gluon energy of E '25 GeV. In a 1 TeV NLC, the limit on the CEDM coupling E is approximately twice as strong as what can be achieved in a 500 GeV NLC.

Fig. 40. 95% CL allowed region in the ! (recall that ,(2m /g );dE) plane obtained from "tting the gluon E E E R Q R spectrum. On the left above: E "25 GeV at a 500 GeV NLC assuming an integrated luminosity of 50(solid) or E 100(dotted) fb\. On the right above: for a 1 TeV collider with E "50 GeV and luminosities of 100(solid) and E 200(dotted) fb\. Figure taken from [297].

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Fig. 41. The 95% CL allowed regions obtained for the anomalous couplings (a) , and (b) , at a 500(1000) GeV A A 8 8 NLC, assuming a luminosity of 50(100) fb\, lie within the dashed(solid) curves (recall that ,(2m /e);dA and A R R ,(2m /g /2c );d8). The gluon energy range z,2E /(s50.1 was used in the "t. Only two anomalous couplings 8 R 5 5 R E are allowed to be non-zero at a time. Figure taken from [297].

It should be noted that in [298] two other CP-even quantities were considered: the cross-section itself and a CP-even combination of the top and anti-top polarizations. The limits obtained there for the CEDM of the top, , are somewhat weaker then those shown in Fig. 40. E In analyzing the anomalous EDMs of the top to a photon and a Z-boson, the `normalizeda gluon energy distribution was used in [297]: d(e>e\PttM g) 1 dR " . dz (e>e\PttM ) dz

(6.109)

It was then found that the EDM coupling of the top quark to a photon, , can be constrained at A the NLC by studying the reaction e>e\PttM g. From Fig. 41(a) we see that at a 500 GeV NLC with integrated luminosity of 50 fb\, only long narrow bands around &!1 or 0 are allowed which A then gives :0.4}0.6. In a 1 TeV NLC with integrated luminosity of 100 fb\, one circular A narrow band between !0.4: :0 is allowed giving 0: :0.2. A A The anomalous EDM coupling of the top quark to the Z, , is much less constrained. 8 In particular, from Fig. 41(b) we see that with a 500 GeV NLC and integrated luminosity of 50 fb\, 0: :0.5 is allowed if !0.5: :0.1. However, with a 1 TeV NLC and integrated 8 8 luminosity of 100 fb\, the allowed region in the ! plane is considerably reduced. Namely, 8 8 0: :0.1 can be achieved if !0.2: :0. Also, as was shown in [297], doubling the integrated 8 8 luminosity does not increase the sensitivity of the NLC to these anomalous couplings of the top quark. To conclude, note that while the process e>e\PttM will presumably be more appropriate for the exploration of CP-odd e!ects driven by the top dipole moment couplings to and Z (see Section 6.1), the reaction e>e\PttM g might be the only place for searching for the CEDM of the top quark at the NLC. In that sense, an investigation of the e!ects of on CP-odd quantities in the E ttM g "nal state at an e>e\ collider, is worthwhile. This was done in [298] by constructing a genuinely CP-odd observable out of the top and anti-top polarizations and is described below.


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6.3.2.2. CP-odd observables. An interesting CP-odd observable was suggested in [298]. This observable involves the top polarization and is de"ned as (6.110) \"[(!)!( )#(!)!( )] , where (!), (!) refer respectively to the cross-sections for top and anti-top with a positive spin component in its direction of #ight, and ( ), ( ) are the same quantities with a corresponding negative spin component. In [298] the sensitivity of \ to the CEDM of the top, , was studied. Note that \ is E CP-odd and ¹ -even and therefore depends on the imaginary parts of the combinations of , couplings Im(H ) and Im( ) in Eq. (6.106). The 90% CL limits on the values of Im(H ) and E E E E E Im( ) were obtained from [298] E (6.111) L \( , )!\"2.15(L (!)# (!) , E E 1+ 1+ 1+ where in the above expressions, the subscript `SMa denotes the value expected in the standard model, with " "0; is the top detection e$ciency and L is the integrated luminosity. E E Eq. (6.111) gives contours in the Im(H )!Im( ) plane which are shown in Figs. 42(a) and (b) E E E for c.m. energies of (s"0.5 and 1 TeV, respectively, for an integrated luminosity of L"50 fb\ and for "0.1. Also di!erent polarizations of the incoming electron beam, P , are considered in C Figs. 42(a) and (b). The allowed regions in Figs. 42(a) and (b) are the bands lying between the upper and lower straight lines. We see that the dependence of \ on the electron beam polarization is rather mild.

Fig. 42. \ contour plots in the Im(H ) (vertical axis)}Im( ) (horizontal axis) plane, with 90% con"dence level at E E E c.m. energies (s"500 GeV (left side) and (s"1000 GeV (right side), and for di!erent values of the electron beam polarization: P "#0.5, 0, !0.5, !1. Figure taken from [298]. C

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Fig. 43. Intersecting area in the Im(H )}Im( ) plane resulting from two independent \ measurements at E E E (s"500 GeV and (s"1000 GeV. P "!1 is used. See also caption to Fig. 42. Figure taken from [298]. C

It was also suggested in [298] that a measurement of \ at two di!erent c.m. energy may improve the limits on Im(H ) and Im( ). This is demonstrated in Fig. 43 where it is shown that E E E from measurements of \ at (s"0.5 and 1 TeV, the possible limits are !0.8(Im(H )(0.8, !11(Im( )(11 . (6.112) E E E Although dealing with a genuine CP-odd observable, the limits in Eq. (6.112) are still weaker by about an order of magnitude than those obtained through the study of the gluon jet energy distribution. Note however, that those limits are placed on the imaginary part of while the limits E obtained through the study of the gluon jet energy distribution are set on the absolute value of the top CEDM. 6.4. CP violation via == fusion in e>e\PttM C C At a NLC with a very high c.m. energy, above 1 TeV, the t-channel =>=\ fusion subprocesses =>=\PttM , where the two =-bosons are emitted from the initial e>e\ beams, starts to dominate over the simple s-channel production mechanism of a pair of ttM , i.e., e>e\P, ZPttM . As it turns out [299], the reaction e>e\P=>=\ PttM (6.113) C C C C can potentially exhibit large CP-violating phenomena, driven by CP-odd phases in the neutral Higgs sector in MHDMs. To lowest order there are four Feynman graphs, shown in Fig. 44, relevant to the reaction in Eq. (6.113). Indeed, at large c.m. energies, i.e., as s/m becomes very large, the cross-section for the 5


147

Fig. 44. The Feynman diagrams that participate in the subprocess =>=\PttM . The blob in diagram (a) represents the width of the Higgs resonance and the cut across the blob is to indicate the imaginary part.

reaction in Eq. (6.113) is dominated by collisions of longitudinally polarized ='s and the subprocess =>=\PttM shown in Fig. 44, when calculated in the E!ective =-boson Approximation [300}305], serves as a good approximation to the reaction e>e\PttM . C C The key point here, as suggested in [299], is again to construct CP-odd observables utilizing the top polarization, which in turn can be traced through the top decays. Following [299], in the rest frame of the t one de"nes the basis vectors: !e J( p > #p \ ), e Jp > ;p \ and e "e ;e . 5 W 5 5 V W X X 5 For the anti-top one uses a similar set of the de"nitions in the tM rest frame related by charge conjugation: !e J( p \ #p > ), e Jp \ ;p > and e "e ;e . Now let P (for j"x, y or z) be 5 W 5 5 V W X H X 5 the polarization of t along e , e , e and similarly, PM the polarization of tM along e , e , e . One can V W X H V W X then combine information from the t and tM systems and de"ne the following asymmetries: A "(P #PM ), B "(P !PM ) , V V V V V V A "(P !PM ), B "(P #PM ) , W W W W W W (6.114) A "(P #PM ), B "(P !PM ) , X X X X X X where it is easy to verify that, within the above coordinate systems, the A's are CP-odd and the B's are CP-even. Moreover, A , B , A are CP¹ -odd whereas B , A , B are CP¹ -even. V W X , V W X , We note here that the CP-even spin observable B , being proportional to the imaginary part of W the Higgs propagator in Fig. 44(a), is also useful for experimentally measuring the Higgs width [299]. However, since here we are only interested in CP non-conservation e!ects in the reaction e>e\PttM , we will focus below on results obtained for the CP-odd observables, i.e., the A's in C C Eq. (6.114). Let us consider a 2HDM with the ttM HI and =>=\HI Lagrangian pieces of Eqs. (3.70) and (3.71), respectively. Here also, the simultaneous presence of the scalar, aI, and pseudoscalar R couplings, bI, in Eq. (3.70) is required for a non-zero expectation value of the CP-violating R asymmetries A , A , A . Therefore, since only two out of the three neutral Higgs particles, i.e., V W X k"1, 2 or 3, (say, h,H and H,H for the lighter and heavier ones, respectively) have a simultaneous scalar and pseudoscalar couplings to ttM (see Section 3.2.3), the third neutral Higgs need not be considered. A is expected to receive signi"cant contributions from loop corrections. X Therefore, we focus below on A and A only (see discussion in [299]). V W The two asymmetries A , A are shown in Fig. 45, for a NLC with a c.m. energy of (s"1.5 TeV, V W as a function of the lighter Higgs mass m . The heavier Higgs mass is "xed to m "1 TeV. Also, for F & illustration, we use tan "0.5 and choose , , "!/2, ,!/2, where , and

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Fig. 45. The asymmetries A (solid) and A (dashes) integrated over s( as a function of m for (s"1.5 TeV and V W F m "1 TeV. The coupling parameters are for tan "0.5 and , , "!/2, ,!/2 as described in the text. & Figure taken from [299].

are the three Euler angles that specify the 3;3 orthogonal mixing matrix of the three neutral Higgs-bosons (see Eq. (3.73)). With this set of parameters the ttM h, ttM H, =>=\h and =>=\H couplings are "xed according to Eqs. (3.70}3.73) in Section 3.2.3. We observe from Fig. 45 that for a wide range of the lighter Higgs mass the asymmetries are appreciable. In particular, A is about 10% for m &400}800 GeV whereas A is around 30% for V F W m &100}300 GeV. Although not shown in Fig. 45, the asymmetries vanish when m "m due to F F & a GIM-like cancellation as explained in Section 3.2.3. As mentioned above, in order to measure those top polarization asymmetries, one needs the momentum of the t and tM decay products in a given decay scenario. In [299] two such decay scenarios useful for top polarimetry were considered: 1. The decay tP=>b followed by =>Pl>, where l"e, ; in this case only the hadronic decays of tM are included. This case occurs with a branching ratio of B +()()" . 2. The decay tP=>b followed by =>Phadrons. Now the decay of tM to a \ is excluded. This case occurs with a branching ratio of B +()()". In the case of the leptonic decay of t (or equivalently tM ), the angular distribution of the lepton is J(1#R P cos l ), where P is its polarization, l is the angle between the polarization axis and the momentum of the lepton in the top rest frame and R "1 in the SM. Thus, the optimal method to obtain the value of P is to use P"3 cos l /R (see Section 2.8). Similarly, in the case of the hadronic t decay (or equivalently tM ), one uses the distribution of the = momentum in the top frame which is J(1#R P cos ) to extract the top polarization, where R "(m!2m )/(m#2m ). 5 R 5 R 5 Therefore, in this case P"3 cos /R . 5 Hence, bearing in mind that the leptonic decay of the top is self polarizing, the number of events needed to obtain a 3- signal in the t, tM decays of case 1 above is [299]: "()(R B #R B )\;a\ , NN RRM

(6.115)


149

where a is the asymmetry in question (either A or A ). Therefore, given the above numbers for V W R , R , B and B , numerically NN M +52a\, thus requiring some 5200 events for an asymmetry of RR 10%. In fact, this can be further improved in the case of the hadronic decays of t (case 2 above) by observing that [82] the less energetic of the two jets from the decay of the = is more likely to be the dM -type quark as noted in Section 2.8. In particular, in case 2 one obtains NNM +32a\, thus RR reducing the requirement to 3200 events for an asymmetry of 10% [299]. For an asymmetry of about 30%, which was found possible in the case of A (see Fig. 45), only a few hundred events will W be needed. Indeed, the cross-section for the reaction in Eq. (6.113) was calculated in [299] for the case of a 2HDM with the couplings described above and it was found to be at the level of a few fb, reaching 910 fb for 350 GeV:m :550 GeV. Thus, given that at (s"1.5 TeV the projected luminosity F could be about 5;10 cm\ s\ [283,284] (see also [253}255]), a cross-section of 10 fb would yield about 5000 events rendering it feasible to detect asymmetries 910%. The conclusion is therefore that the top polarization asymmetries for the reaction e>e\PttM are accessible to the C C NLC and can serve as a powerful probe of CP violation driven by the neutral Higgs sector of a 2HDM. However, this last statement must be taken with some caution since, the two neutrinos in the "nal state, carry a substantial amount of missing energy and may therefore pose a problem in reconstructing the t and tM rest frames, as required for measuring the polarization asymmetries in question when the t or tM decays leptonically. No such problem arises if both the t and the tM decay purely hadronically but, in that case, it remains to be seen if it will be possible to distinguish t from tM which is also required for measuring A and A . V W An interesting generalization of this work [299] is to consider instead the reaction e>e\PttM e>e\. Now the fusion takes place via neutral gauge-bosons (, Z). Although there may be some loss of the cross-section, to compensate that, there is also the advantage that the di$culties in reconstructing the rest frames of t, tM may be far less formidable. 7. CP violation in pp collider experiments The LHC is a pp collider at CERN, with c.m. energy of 14 TeV, scheduled to start running around 2005 (For a recent review on machine parameters see [306].) Its design luminosity is L"10 cm\ s\, corresponding to a yearly integrated luminosity of 100 fb\. A low luminosity "rst stage of 10 fb\ is usually assumed in articles discussing physics at the LHC. The issues discussed in the following section, will be relevant for the future CMS and ATLAS experiments (for a review see [307]); heavy ions and LHC-B will not be discussed in the present work. For recent reviews on the physics at LHC, see [307,308]. 7.1. ppPttM #X: general comments In hadronic collisions ttM pairs are produced through the parton level subprocesses qq PttM and ggPttM . The latter, gg fusion process, dominates over the quark}anti-quark annihilation in a multi-TeV pp collider. For example, at the LHC, (ggPttM )&90% and (qq PttM )&10% are expected. It is therefore important to investigate the expected CP violation e!ects in ppPttM #X that can arise from CP non-conservation in the subprocess ggPttM .

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Note that the simple qq fusion process is the analog of the e>e\PttM production mechanism where an s-channel gauge-boson is exchanged. In the case of qq PgPttM , the CP-odd e!ect can therefore be attributed to the CEDM (dE) of the top present at the gttM vertex. In contrast, the gg R production process gives rise to a much richer possibility of CP-violating interactions and the resulting asymmetries in ggPttM need not be related merely to the CEDM of the top quark. This fact can be readily seen in model calculations (such as 2HDM and MSSM to be discussed below), where additional CP-violating 1-loop box diagrams as well as s( -channel resonant neutral Higgs exchange become relevant. We will "rst discuss an e!ective Lagrangian approach in which all CP-violating e!ects are assumed to originate only from the CEDM of the top. We will then present model dependent analysis of CP non-conservation in ggPttM where all possible CP-violating operators are taken into account. As will be shown, the typical size of the CP-violating asymmetries in ppPttM #X is &10\. Although, naively one may expect such asymmetries to be within the experimental reach of the LHC, which is expected to produce &10}10 ttM pairs, there are at least two types of hurdles that make this objective very di$cult to attain. First there is the detector-dependent systematics which are expected to present serious limitations for asymmetries at the &10\ level. Another serious di$culty is that the initial state (pp) is not an eigenstate of CP. Therefore, one expects fake asymmetries to arise at some level even though the underlying interactions do not violate CP. These backgrounds are process dependent and the fake asymmetries that they produce need to be much smaller in comparison to the CP-violating signal that is of interest. In some cases, e.g., an s( -channel resonant Higgs exchange within a 2HDM, as will be described in Section 7.3.2, by employing clever cuts on the ttM invariant mass one can obtain asymmetries at the percent level. In these cases, the CP signal is more robust and may be within the reach of the LHC if the 2HDM parameter space turns out favorable. 7.2. ppPttM #X: general form factor approach and the CEDM of the top As already mentioned in previous sections, in close analogy to the EDM and the weak(Z)}EDM of the top, one can generalize the top quark-gluon e!ective Lagrangian to include terms of dimension 5 which can give rise to a CEDM for the top quark (see Eq. (6.106) in Section 6.3.2). In general, the CEDM coupling, dE, may be considered as a form factor. Its momentum dependence is R generated by e!ective Lagrangian operators of dimension greater than 5. In model dependent calculations, this from factor may acquire momentum dependent imaginary parts as well as real parts. In momentum space, similar to the EDM and weak-EDM cases, the CEDM modi"es the ttg interaction to read (we will not concern ourselves here with the CP-conserving chromo-magnetic dipole moment of the top) !i¹ (g I#dEIJ k ) , (7.1) ? Q R J where k"p #p M is the gluon four-momentum and p (p M ) is the t(tM ) four momentum. R R R R The subprocess ggPttM then proceeds through diagrams (a)}(d) in Fig. 46 where the heavy dots indicate the vertices modi"ed by the CEDM of the top de"ned in Eq. (7.1). Diagram (d) involves an additional dimension 5 ttgg contact term and is needed to preserve gauge invariance (see Section 2.5). Assuming that dE is small enough such that one can expand the matrix element R


151

Fig. 46. Feynman diagrams contributing to ggPttM in the presence of a top CEDM in the ttM g vertex which is denoted by the heavy dot.

squared to "rst order in dE, the di!erential cross-section for the subprocess ggPttM can be written, R similar to the e>e\PttM case, as [35] () d" () d#[Re dE(s( )R ()#Im dE(s( )I ()] d , (7.2) R R

where s( "x x s and x , x are the gluons momentum fractions. In Eq. (7.2) above, the gluon structure functions are included and thus represents the "nal state phase space including the gluon momentum fraction variables. Also, with no summation over the t and tM spins s and s M , R R respectively, the CP-odd di!erential cross-sections R () and I () are functions of s , s M .

R R Therefore, because of the correlation between the top spin and the momentum of the charged lepton from the top decay tPb=>Pbl>l , Eq. (7.2) with the s and s M dependence, gives in fact R R the di!erential cross-section for the complete process ggPttM including the subsequent leptonic decay chains of the tops. 7.2.1. Optimal observables With the e!ective Lagrangian in Eq. (6.106) in Section 6.3.2 and by ignoring operators of dimension greater than 5, only the e!ect of a constant real dE was investigated in [35]. Indeed, in R model calculations to be described below, the real part of the CEDM form factor is a constant to a good approximation. Similar to the e>e\PttM case, an optimal ¹ -odd, CP-violating observable , for ggPttM was de"ned in [35] as R (7.3) O" . In a realistic hadronic collider however, not all momenta which enter into the problem are immediately observable. For example, with leptonic decays of both t and tM , the momenta of the neutrinos and the longitudinal momenta of the initial gluons are not observed. As was shown in [35], this leads to a twofold or fourfold ambiguity (depending on the number of solutions to the kinematics which results in a quartic equation) in determining the neutrinos momenta. To bypass this di$culty an ìmproveda optimal observable, that averages over the reconstruction ambiguity, was suggested in [35]: R ( ) (7.4) O , G G , ( ) G G where the sum is over the di!erent possible reconstructions of the neutrino and anti-neutrino momenta from the leptonic t and tM decays, respectively.

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Using the optimal observables O and O , the attainable 1- limits on Re dE, assuming 10 dilepton R ttM decays, were given in [35]. Note that one can consider also leptonic}hadronic and purely hadronic decays of the ttM pairs. Due to the branching ratios of the =-boson, 10 leptonic ttM pairs implies a sample of &6;10 leptonic}hadronic ttM pairs and &9;10 hadronic ttM pairs. With m "175 GeV, for the `simplea optimal observable O, with dilepton ttM pairs, it was found that the R 1- limit is Re dE&2.8;10\g cm. For the ìmproveda optimal observable O , Re dE&3.0; R Q R 10\g cm with dilepton or hadronic ttM pairs and Re dE&2.0;10\g cm with leptonic}hadQ R Q ronic ttM pairs. Comparing the 1- limits on Re dE attainable with dileptonic ttM pairs and through the R use of the optimal observables O and O in Eqs. (7.3) and (7.4), respectively, we see that the reconstruction ambiguity does not cause any signi"cant changes. Evidently, with these optimal observables, Re dE may be measured to a precision of &10\g cm. O with the leptonic}hadronic R Q ttM channel seems to be the most sensitive to Re dE. Comparable results for Re dE were found in [309] R R by using the same type of optimal observables. Ref. [309] has also extended the analysis of [35] by including e!ects of the imaginary part of dE. They found that the attainable limit at the LHC for R Im dE is of the same order, i.e., Im dE&10\g cm, although slightly better than the one for Re dE. R R Q R This result is encouraging since, as we have discussed in Section 4 the CEDM of the top may be 910\g cm in some extensions of the SM, e.g., MHDMs and the MSSM. Q 7.2.2. Observable correlations between momenta of the top decay products It is also instructive to consider simple observables constructed exclusively out of momenta which are directly observed. With the decays tPbll and tM PbM lM l , the momenta pl , plM , p and @ p M will be directly observed and observables which involve correlations between those momenta are @ the most appropriate. Two such CP-odd, ¹ -odd correlations were considered in [35]: , M pI pJ pNp M f " IJNM C C @ @ , (7.5) (p ) p p ) p M ) C C @ @ (7.6) f "(pVpW !pW pV ) ) sgn(pX !pX )(p ) p ) , C C C C C C C C where sgn(X)"#1 for X50 and !1 for X(0. The attainable 1- limits on Re dE for the observables f and f , with m "175 GeV and R R assuming 10 dileptonic ttM pairs, were also given in [35]. The "ndings were for f : Re dE&5.3;10\g cm and for f : Re dE&3.0;10\g cm. We see that the limit that might be R Q R Q achieved with f is about an order of magnitude smaller than that from f . However, f depends only on the lepton momenta and is, therefore, easiest to determine experimentally. Also, the limit from f is about 2 times weaker then the one obtained from the ìmproveda optimal observable discussed previously. In [310] CP-odd ¹ -even observables which might be used to probe the imaginary part of the , CEDM, i.e., Im dE, were considered R (7.7) A "ElM !El , # (7.8) Q "2(pXlM #pXl )(pXlM !pXl )!( plM !pl ) . A is the energy asymmetry between l and lM and Q is an asymmetry originally suggested by # Bernreuther et al. in [249] (see Section 6.1.2). In a pp collider with (s"14 TeV, an integrated


153

luminosity of 10 fb\ and an acceptance e$ciency of "10%, taking only leptonic (l"e, ) ttM pairs and assuming m "175 GeV the following 1- limits on Im dE were obtained through the R R observables A and Q # A : Im dE"8.58;10\g cm , (7.9) # R Q Q : Im dE"2.05;10\g cm . (7.10) R Q Thus, the limits on the imaginary part of the top CEDM are weaker by about an order of magnitude than those that might be obtained on the real part of the top CEDM, using the optimal observables discussed before. 7.2.3. Polarized proton beams A very interesting CP-violating polarization rate asymmetry was originally suggested by Gunion et al. in [311], for Higgs production through gg fusion in a pp collider. This asymmetry was applied to ppPttM #X in [310]. The basic idea is that, if the gluons in a polarized proton are polarized, then the initial CP-odd gluon}gluon con"guration allows to probe CP-violating e!ects without requiring full reconstruction of the ttM "nal state. The polarization rate asymmetry is de"ned as ! \ , (7.11) A , > NP # > \ where , in the subprocess ggPttM , is the cross-section for ttM production in collisions of an ! unpolarized proton with a proton of helicity $. Clearly, A is CP-odd and ¹ -even and therefore NP , can only probe the imaginary part of the top CEDM. Of course, a crucial point for such an analysis is the degree of polarization that can be achieved for gluons in the pp collider. The amount of gluon polarization in a positively polarized proton beam is de"ned by the structure functions di!erence g(x)"g (x)!g (x). The structure functions of polarized gluons, g , are not well known and > \ ! depend on the amount of the proton's spin carried by the gluons. In [310] the following, parameterization was adopted (g is the unpolarized gluon distribution)

g(x)

(x'x ) , A (7.12) (x/x )g(x) (x(x ) , A A where x &0.2 yields a value of g&2.5 at Q"10 GeV. The above distribution was actually A evaluated at Q"100 GeV disregarding any scale evolution. The 1- attainable limits on Im dE were calculated in [310] and are given in Table 12, for R various transverse-momentum cuts and for (s"14 TeV, m "175 GeV, L"10 fb\ and an R e$ciency acceptance of 10%. Also, in Table 12 N"N #N is the total number of ttM events, > \ NK "N !N and N (N ) is the number of ttM events predicted for positively(negatively) > \ > \ polarized proton. They have included all possible t decay modes so that the net branching ratio was taken as unity. We see that, even with high p -cuts, it is possible to put a 1- limit on Im(dE) up to 2 R the order of 10\g cm in the LHC with polarized incoming protons. This limit is more stringent Q than the ones obtained in Eqs. (7.9) and (7.10) through the leptonic correlations A and Q , # respectively, and is of the same order as that obtained on the real part of the top CEDM with the optimal observables discussed before. g(x)"

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Table 12 The number of ttM events N, the ratio NK /N (see text), and the attainable 1- limits on Im dE, for various p -cuts with R 2 (s"14 TeV, m "175 GeV and L"10 fb\. Table taken from [310] R p -cuts (GeV) 2

N (;10)

NK /N

Im dE (;10\g cm) R Q

0 20 40 60 80 100

2.62 2.55 2.36 2.08 1.74 1.41

1.44 1.42 1.37 1.30 1.22 1.14

0.766 0.788 0.847 0.951 1.107 1.313

Fig. 47. Feynman diagrams for the tree-level QCD and neutral Higgs exchanges (denoted by the dashed lines) which contribute to the production density matrix for ggPttM . Diagrams with crossed gluons are not shown.

7.3. 2HDM and CP violation in ppPttM #X In a 2HDM with the CP-violating ttM H couplings in Eq. (3.70), neutral Higgs exchanges can give rise to CP violation in ggPttM and qq PttM at the 1-loop order in perturbation theory. In Fig. 47(c)}(h) all possible 1-loop CP-violating Higgs exchanges in ggPttM are drawn and in Fig. 48(b) the only CP-violating 1-loop diagram for qq PttM is shown. Interference of diagrams (c)}(h) with the SM tree-level diagrams (a) and (b) in Fig. 47 and interference of diagram (b) with diagram (a) in Fig. 48 can then give rise to CP non-conservation e!ects in gg and qq fusion, respectively. One can then identify various CP-violating correlations to trace the resulting CP-odd quantities which appear in the corresponding di!erential cross-sections. Here also we assume that two out of the three neutral Higgs particles in the 2HDM model are very heavy or have very small CP-violating couplings, such that either way their e!ects decouple. Thus, only the couplings of the lightest neutral Higgs (denoted by h) are important and there will be only one dimensionless CP-odd quantity relevant for the study of CP violation in qq , ggPttM . Using the notation in [289,291,312] in conjunction with our parameterization in Eq. (3.70), this quantity is ,!2aFbF , !. R R

(7.13)


155

Fig. 48. Born level QCD and relevant neutral Higgs exchange (denoted by the dashed line) Feynman diagrams for qq PttM .

where aF and bF are de"ned by the ttM h (say h"H) coupling in Eq. (3.70) and are functions of tan R R } the ratio between the two VEVs in the Higgs potential and of the three Euler angles which parameterize the Higgs mixing matrix (for details see Section 3.2.3). Below we present two very interesting approaches of probing CP violation in ppPttM #X. The "rst is the Schmidt and Peskin (SP) approach [33], which utilizes the distribution of the leptonic decay products of the top. The second is the Bernreuther and Brandenburg (BB) approach [289,291,312], which studies the CP-violating e!ect in the resonant s( -channel Higgs shown in Fig. 47(h). 7.3.1. Schmidt}Peskin signal Schmidt}Peskin (SP) proposed [33] a signature for CP violation in production and decays of ttM pairs for hadron colliders, namely via the reaction

P l\l P =\bM pp( pN )PttM #X . =>b P l>l P

(7.14)

Despite the complexity of the reaction and the hadronic environment, the signal for CP violation that they suggest, i.e., the lepton energy asymmetry El> ! El\ " # El> # El\

(7.15)

is very simple and robust. Such an asymmetry can only arise from non-SM sources such as an extended Higgs sector or supersymmetry. The size of the asymmetry is unfortunately rather small &10\. Since this asymmetry is CP-odd and ¹ -even, it requires an absorptive part to the Feynman , amplitude. Such an absorptive part is already present (see Figs. 47(c), (g) and (h) and 48(b)) as ttM pair production requires the kinematic threshold s( '4m , (7.16) R where s( is the square of the energy in the subprocess qq or gg c.m. frame. In particular, when a neutral Higgs exchange leads to the CP-violating phase (as in their study), then the absorptive part due to the threshold condition in Eq. (7.16) arises even if m's( . F

156


Note again that for the subprocess qq PttM , the underlying cause of CP violation in extended Higgs models is the CEDM of the top quark. Of course given the extremely short lifetime of the top quark (&10\ s) the CEDM as such (i.e., at q"0) is extremely di$cult to be seen. Consider, however, the asymmetry [33]: [N(t tM )!N(t tM )] * * 0 0 , N " *0 all ttM

(7.17)

where N(t tM ) is the number of t tM pairs produced via qq (gg)PttM , etc. Clearly, N is CP-odd * * * * *0 and ¹ -even. The qq contribution to N arises from interference of Fig. 48(b) with the , *0 lowest-order graph for qq PttM depicted in Fig. 48(a). They found 2 Re(F ) , N " *0 3!

(7.18)

where "(1!4m/s( ) and, in their notation, F (s( ) is the CEDM form factor and Re(F ) R involves the absorptive part of the Feynman integral

m s( 1 m 4m R R 1! F ln 1# . (7.19) Re(F )" s( !. s( m 8 v F Here m is the mass of the lightest neutral Higgs and is de"ned in Eq. (7.13). It is easy to F !. understand [33] the e!ect intuitively: for s( <m, the gluon will predominantly couple to t tM or R * 0 t tM . However, when P0, t tM and t tM , which are related to each other via CP, are dominantly 0 * * * 0 0 produced, which may thus lead to N O0. The resulting asymmetry at the parton level, N , *0 *0 for the subprocess qq PttM for m "175 GeV, "( and for di!erent values of m and (s( , is F R !. found to be of order 10\. For ggPttM the calculation is more involved. In particular, in addition to the tK channel h exchange, now an s( -channel Higgs exchange graph is also present (see Fig. 47(h)). There is in fact constructive interference between these two channels for m (2m . The result for F R the asymmetry in the gg fusion case, but without the s( -channel Higgs exchange (see Fig. 48(b)), was also given in [33]. Near threshold, i.e., (s( 92m , the asymmetry in the gg fusion case is R about twice as big as that of the qq fusion case. However, although larger than the qq fusion subprocess, it is again at the level of 10\. Adding the qq and gg subprocesses, then N can reach *0 optimistically &10\, for low values of m and tan . In any case, the gg initial state gives rise to F a much richer possibility of CP-violating operators and, as was mentioned before, the resulting asymmetry cannot be attributed merely to the CEDM of the top quark. Indeed, as noted in [289,291,312], the s( -channel neutral Higgs exchange that was ignored in [33], can give rise to larger asymmetries in ggPttM and may be attainable at the LHC. We will return to this e!ect in the next section. As has been emphasized at several places in this review, the fact that top decays are a powerful spin analyzer comes in extremely handy here too in leading to a detectable signature. The CP Note that in this notation, tM means an anti-top quark with momentum, for instance, along #z and spin along !z. *


157

violation in the production process causes the polarization asymmetry above, which leads to an asymmetry in the energies of the charged leptons emerging from t and tM decays. The distribution of the charged lepton in the t-rest frame is given by d 1#cos d " , 2 dEl dcos dEl

(7.20)

where is the angle between the top spin and the lepton momentum. When the top quark is boosted to the qq (gg) c.m. frame, Eq. (7.20) provides the correlation between the helicity of the top and the energy of the decay lepton. The resulting energy spectrum for t (tM ) and t (tM ) is * 0 0 * signi"cantly di!erent from each other as was shown in [33]. Clearly, their "ndings indicate that the energy spectrum of the leptons, serves as a useful spin analyzer. The asymmetry in pp collision is calculated by folding in as usual the parton distributions. For this purpose SP used the parton density functions proposed in [313]. The e!ects of the longitudinal boost of the parton}parton collision are eliminated by considering the transverse energy (E ) of the 2 leptons. The resulting asymmetry is [33]: d/dE l> !d/dE l\ 2 2 (7.21) N(E )" 2 d/dE l> #d/dE l\ 2 2 and it was calculated in [33] for m "100 GeV, m "150 GeV and "( . Unfortunately, F R !. numerically it is again only of order 10\. Let us brie#y discuss the background for these type of CP-violating asymmetries in ppPttM #X. As was mentioned at the beginning of this section, the initial state (pp) at hadron Supercolliders, such as the LHC, is not an eigenstate of CP. Consequently energy asymmetry in the decay lepton spectrum are not necessarily due to CP violation. The point is that the protons in the initial state produce more energetic quarks than anti-quarks. Also the reaction qq PttM has a small forward}backward asymmetry induced by corrections. Thus, the top quarks produced by this Q reaction tend to have a slightly higher energy than tM , leading to an asymmetry in the energy of the decay lepton. Such an e!ect, originating from higher order QCD corrections, causes an irreducible background. Fortunately this background is very small. First of all, qq annihilation is subdominant at such pp collider energies and the leading reaction ggPttM is free from such a forward}backward asymmetry. Also, as mentioned before, the background to the asymmetry arises from higher order (QCD) radiative corrections. Furthermore, since the forward}backward asymmetry mainly a!ects longitudinal variables, its e!ect on the transverse energy asymmetry in Eq. (7.21) would cancel if there were no lepton acceptance cuts. This background can be crudely estimated from the electromagnetic analog of the forward}backward asymmetry for e>e\P>\. The analogous asymmetry is crudely estimated by the replacement P[(d?@A)/32] "(5/12) . SP in [33] used the approximate formula in [314] Q Q which allows them to get an estimate for massless ttM pairs. This approximation tends to overestimate this background. For numerical estimates SP also impose a cuto! on the gluon energy of E/E"0.3. The resulting background was found to be of the order of 10\. Therefore, it is much smaller than the desired CP-violating e!ect and also it is essentially independent of the lepton energy.

158


7.3.2. s( -channel resonance Higgs ewects } Bernreuther}Brandenberg approach For m '2m , as noted before, there is an interesting s( -channel Higgs contribution to ggPttM F R shown by diagram (h) in Fig. 47. This was explored in some detail by Bernreuther and Brandenberg [289,291], who recently improved their analysis in [312]. For the simple on-shell decay hPttM , a large ttM spin}spin correlation can be induced already at the tree-level if h is not a CP eigenstate, as happens in a class of 2HDMs. In a 2HDM, with the ttM h coupling of Eq. (3.70), this spin}spin correlation is given by [289] !. R , (7.22) kK ) (s ;s M )" R R R (bF)#(aF) R R R where "(1!4m/m), s , s M are the spin operators of t and tM , respectively, kK is the unit vector R R R F R R of the momentum of the top quark and is de"ned in Eq. (7.13). It is remarkable that this !. CP-violating spin}spin correlation can, in principle, be as large as 0.5. In practice, though, this decay has to be coupled to some particular production process and the "nal asymmetry can vary signi"cantly between di!erent processes. Moreover, for pp collisions, there is an interference between the continuum and the resonant ttM production which tends to diminish the spin}spin correlation. For the gluon}gluon fusion, the CP-violating expectation value of kK ) (s ;s M ) was R R calculated in [291]. The resulting asymmetry was found to be at best only a few percent and falls signi"cantly short compared to the value of 0.5 mentioned above. In fact, when this is translated to an asymmetry that utilizes the t and tM decay products, as was done in [289,291], the signal-to-noise ratio for such an asymmetry was found to be at best &10\. The same non-vanishing spin}spin correlation of Eq. (7.22) can arise in qq PttM . The asymmetry for the qq fusion subprocess was also calculated in [291] for the same set of parameters as in the gg fusion case. As expected, in the case of qq fusion, the asymmetry is about one order of magnitude smaller than gg fusion, since in this channel the resonant Higgs graph is absent. Furthermore, the asymmetry gets smaller with growing Higgs-boson masses as opposed to the gg fusion case which we now discuss in some detail. Let us now focus on an improved analysis of the results mentioned above. This was recently suggested by Bernreuther, Brandenburg and Flesch (BBF) in [312]. In their analysis the basic idea was to include new cuts on the ttM invariant mass which signi"cantly improved their previous results in [291]. For the case when both t and tM decay leptonically, consider the CP-violating observables [312]: Q "kK ) q( !kK M ) q( R > R \

(7.23)

and Q "(kK !kK M ) ) (q( ;q( )/2 , (7.24) R R \ > where kK , kK M are the t, tM momentum directions in the parton c.m. system and q( , q( are the l>, l\ R R \ > momentum directions (from tPbl>l and tM PbM l\l ) in the t and tM rest frames, respectively. Note that the decay channels to l>, l\ (disregarding the lepton) may include all di!erent combinations A more detailed analysis of the possible spin}spin correlations in hPttM is given in Section 6.2.3. Recall that the lightest neutral Higgs is assumed to be h"H, i.e., k"1 in Eq. (3.70).


159

of e and and, in [312], all possible combinations were summed over. It is useful to note that Q in Eq. (7.24) is, in fact, a transcription of the spin}spin correlation kK ) (s ;s M ) de"ned in Eq. (7.22) R R R above, and is a ¹ -odd quantity. Also, Q traces the spin}spin correlation kK ) (s !s M ) which , R R R corresponds to the CP asymmetry N de"ned in Eq. (7.17) and was suggested originally in [33]. *0 Clearly, it is ¹ -even, thus requiring an absorptive phase. , For the leptonic}hadronic decay channel of the ttM pairs, it is useful to consider the two possible decay scenarios: sample A in which the t decays leptonically and tM decays hadronically, and sample A M in which the t decays hadronically and tM decays leptonically (see Eqs. (6.79) and (6.80), respectively, in Section 6.2.3). One can then de"ne the following CP-violating quantities with respect to samples A and AM [312]: E " O A ! OM AM , E " O A # OM AM , where

(7.25) (7.26)

O ,kK ) q( , OM ,kK M ) q( , R > R \ O ,kK ) (q( ;q( M ), OM ,kK M ) (q( ;q( ) . (7.27) R > @ R \ @ Here q( and q( M denote the momentum direction of the b and bM jets in the t and tM rest frames, @ @ respectively. Again, E e!ectively corresponds to the spin}spin correlation kK ) (s !s M ) and is R R R a ¹ -even observable, and E traces the spin}spin correlation kK ) (s ;s M ) and is therefore a ¹ -odd , , R R R quantity. In [312] it was shown that, in the region m '2m , the magnitude of the asymmetry increases F R signi"cantly and the dominant contribution comes from the interference of the CP-violating terms in the amplitude of the neutral Higgs resonant diagram in Fig. 47(h) with the Born amplitude. They have evaluated the dependence of the di!erential expectation values of Q and Q on the ttM invariant mass, M M . An example of such a dependence for the ¹ -odd observable Q and in the , RR dilepton decay channels of the ttM pairs is shown in Fig. 49. In this "gure the resonant contribution in ggPhPttM is compared with the resonant#the remaining h contribution (i.e., all the nonresonant graphs), for di!erent values of m , for (s"14 TeV and setting the CP-violating quantity, F , from the ttM h vertex to be equal to 1. Also, since the CP-violating e!ect is sensitive to the neutral !. Higgs total width, it is therefore sensitive to the ZZh, ==h `reduceda coupling cF (de"ned in Eq. (3.71) for h"H, i.e., k"1), which determines the decay rates (hPZZ) and (hP==) [312]. In Fig. 49, cF"0 was chosen, in which case the above decay channels of a neutral Higgs to the massive gauge-bosons are forbidden at tree-level. Clearly, looking at Fig. 49, the non-resonant contributions are negligible with respect to the s( -channel h contribution which in turn gives rise to a CP asymmetry at the level of a percent when M M is in the vicinity of m . F RR The sharp peaks observed in the range M M &m give an extra handle in an attempt to enhance F RR the CP signal. Indeed, Figs. 49(b)}(d) show that, in the case of m '2m , Q changes sign as one F R goes from M M : m to M M 9 m , such that integrating over M M will diminish the CP-violating RR F RR F RR e!ect. Therefore, choosing appropriate M M mass bins below or above m , allows for a signi"cant F RR enhancement of the CP-odd signal. This is demonstrated in Table 13, where the three values cF"0, 0.2 and 0.4 were considered (see discussion above). Also, in Table 13, the left column gives

160


Fig. 49. d Q /dM as a function of M (Q is de"ned in Eq. (7.24)) at (s"14 TeV, for reduced Yukawa couplings RR RR cF"0, aF"( , bF"!( , i.e., "1 (see text), and Higgs}boson masses: (a) m "320 GeV, (b) 350 GeV, (c) 400 GeV, R !. F R and (d) 500 GeV, in the dilepton channel. The dashed line represents the resonant and the solid line the sum of the resonant and non-resonant h contributions. Figure taken from [312].

the expectation value of Q in percent and the right column shows the statistical signi"cance in which this CP e!ect can be measured at the LHC with an integrated luminosity of 100 fb\. The M M intervals (i.e., mass bins) in Table 13 where chosen below m (see caption of Table 13). Also the F RR rows in Table 13 correspond in descending order to "1, 0.3, 0.09. !. The numbers for the statistical signi"cance of the CP-violating signal that were found in [312] and are shown in Table 13 are quite remarkable. In most cases, the CP-violating signal is well above the 3- level, perhaps even a lot better than 5- in the best case. As an example, note that with m "370 GeV and as low as 0.09, the observable Q can yield a 7- e!ect with an F !. appropriately chosen interval for M M . Recall that values as large as &4, corresponding to !. RR tan :0.5, are still allowed by present experimental data (see Section 3.2.3). A few additional remarks are in order regarding the analysis in [312]: 1. The expected statistical signi"cance for a CP-odd signal from the observable E in Eq. (7.26) corresponding to the leptonic}hadronic channel (i.e., lepton # jet from the ttM pairs) was found to be comparable to that of Q discussed above. Moreover, for the ¹ -even observables ,


161

Table 13 The expectation value of Q and its sensitivity at the LHC with (s"14 TeV and an integrated luminosity of 100 fb\, for the dilepton ttM decay channels. The M M intervals are chosen below m such that: for m "370, 400, 500 GeV, Q was F F RR integrated over M M "15, 40, 80 GeV, in the M M ranges 355}370, 360}400 and 420}500 GeV, respectively. For each pair RR RR (m , cF) the "rst column is Q in percent and the second column is the sensitivity in standard deviations. The rows F correspond, in descending order, to (aF, bF)"(1,!1)/(2, (1,!0.3)/(2 and (0.3,!0.3)/(2, i.e., to "1, 0.3 and 0.09, R R !. respectively. Numbers for m are in GeV. The non-resonant h contributions have been neglected for these values of m . F F Table taken from [312]

m F

cF 0.2

0.0

0.4

370

4.4 3.9 1.2

29.8 23.4 6.6

4.1 2.9 0.75

27.4 16.7 4.1

3.3 1.6 0.39

20.9 9.0 2.1

400

2.3 1.3 0.49

24.4 13.4 4.9

2.1 1.1 0.35

22.8 11.1 3.5

1.8 0.75 0.21

18.7 7.5 2.1

500

0.65 0.31 0.14

8.6 4.1 1.9

0.59 0.26 0.10

7.9 3.5 1.4

0.46 0.18 0.06

6.0 2.4 0.77

Q (see Eq. (7.23)) and the corresponding one for the leptonic}hadronic channel E (see Eq. (7.25)) the CP-violating signal, although somewhat smaller, may yield more than a 3- e!ect as long as 90.3. !. 2. Apart from the cuts on the M M invariant mass, i.e., the chosen intervals/mass bins, Ref. [312] RR employed additional cuts on the rapidities of the t and tM and on the transverse momenta of the "nal state charged leptons and quarks in both the dilepton channel and leptonic}hadronic channel samples. 3. It is important to note that the ¹ -odd observables Q and E are insensitive to CP violation , from the subsequent t, tM decays to leading order in the CP-violating couplings. This is ensured by CPT invariance [312]. Moreover, the ¹ -even observables Q and E , although may acquire , contributions from CP-violating absorptive parts in the t, tM decays, at least in the 2HDM case, these absorptive parts are absent in the limit of vanishing b-quark mass (see also related discussion in Section 5.1.2). Therefore, for the 2HDM case, both the ¹ -even and the ¹ -odd , , quantities in Eqs. (7.23)}(7.26) are `cleana probes of CP violation in the production mechanism of ttM at the LHC. Refs. [289,291] considered possible contaminations to an asymmetry of the type Q (or equivalently the spin}spin correlation in Eq. (7.22)). Again, the key point is that the dominant gg fusion subprocess is free from undesired CP-conserving background to Q . Therefore, background considerations are relevant only for the case of qq PttM . Refs. [289,291] estimated the CPconserving background to Q to be of order 10\ which is about 3 orders of magnitude smaller than the actual asymmetry. The reason for that is that ¹ -odd observables such as Q do not ,

162


receive contributions from CP-invariant interactions at the Born level but only from absorptive parts. Thus, in the case of qq PttM , the main background comes from order and Q Q absorptive parts [289,291]. Finally, let us note that the optimization technique (with no additional cuts), i.e., the use of optimal observables, employed in [309] for CP violation in ggPttM , yield roughly the same results as those obtained in [312]. That is, optimal observables can be sensitive to values down to &0.1 with no cuts on the ttM invariant mass. !. 7.4. SUSY and CP violation in ppPttM #X As we have discussed in previous sections, 1-loop exchanges of SUSY particles may give rise to CP-violating phenomena in top systems which are driven by SUSY CP-odd phases in the supersymmetric vertices. Such SUSY CP-violating 1-loop e!ects in ggPttM were investigated by Schmidt [231]. In the case of ggPttM , only exchanges of gluinos and stops are relevant and are shown in Fig. 50. The only CP-odd phase arises then from the o! diagonal elements of the tI !tI * 0 mixing matrix. Writing again (see also Sections 3.3.2 and 4.5) the top squarks of di!erent handedness in terms of their mass eigenstates, tI , tI , as > \ tI "cos tI !e\ @R sin tI , R > * R \ (7.28) tI "e @R sin tI # cos tI , R \ R > 0 the asymmetry will then be proportional to the quantity R "sin(2 ) sin( ) . (7.29) !. R R Schmidt neglected possible CP-non-conserving e!ects in the subdominant process qq PttM and considered the asymmetry N de"ned in Eq. (7.17) only for ggPttM . As mentioned before, N *0 *0 being CP-odd and ¹ -even, requires both a CP-odd phase and an absorptive phase. Such , absorptive phases are present in diagrams (a), (b) of Fig. 50 if the c.m. energy is large enough to produce on-shell gluino (g ) pairs and in diagrams (c)}(e) in Fig. 50 if the c.m. energy of the colliding gluons is su$cient to produce on-shell top squark (tI ) pairs. Obviously, if SUSY particles have masses of O(1 TeV), then this condition is satis"ed at the LHC in which the c.m. energy of the colliding protons is 14 TeV. Thus, N is a sum of the two contributions *0 d cos [AE (cos )#ARI (cos )] , (7.30) N , *0 (all ttM )

Fig. 50. 1-loop Feynman diagrams contributing to CP violation in ggPttM in supersymmetry. Diagrams with crossed gluons are not shown.


163

where is the production angle of the top quark in the gg c.m. frame and AE (ARI ) is the contribution from on-shell gluino (stop) pairs. Schmidt found 9 Q R (!1)N ((s( !2m ) AE (cos )" E 64s( N 1 R (1! )K\!(1!)K\# sin (K\!K\) ; E R R E 1! cos R E 1 R (1! )K>!(1!)K># sin (K>!K>) # E R R E 1# cos E R

, (7.31)

and 9 Q R (!1)N ((s( !2m ) ARI (cos )" N 64s( N 1/81 10/81 R (1!)KM \! sin KM \ # ; N R N 1! cos 1# cos R N R 10/81 1/81 R (1! )KM >! sin KM > , # # (7.32) N R N 1# cos 1! cos R R N where "(1!4m/s( ) and the index refers to the two mass eigenstates of the stop. Also, here m m , E R R , (7.33) Q s( !. R and the form factors K! (i"0}3) are given in [231]. The asymmetry N was calculated for several values of m I \ , m I > (the masses of the two stop *0 R R eigenstates) and m , and for m "150 GeV, R "!1 (i.e., its maximal negative value). In general, E R !. the asymmetry was found to be dominated by the amplitudes which contain the intermediate gluinos (Fig. 50(a) and (b)) even if the intermediate stops in Fig. 50(c)}(e) can go on their mass shell. For example, with m "210 GeV, m I \ "100 GeV and m I > "500 GeV, Schmidt found that the R R E asymmetry at the parton (i.e. gluons) level can reach &2% if the c.m. energy of the colliding gluons is around 450 GeV. Note that, in this c.m. energy, and with the above stops masses, N receives *0 contributions only from diagrams (a) and (b) in Fig. 50 since there is no absorptive cut along the stops lines in diagrams (c)}(e) in Fig. 50. These results for N are about an order of magnitude *0 larger than what was found in the 2HDM case [33] and it is roughly comparable to the s( -channel Higgs resonant e!ect [289,291,312]. However, in a more realistic study, one will have to integrate over the structure functions of the incoming gluons and present asymmetries in the overall reaction ppPttM #X. Unfortunately, folding in the gluons structure functions, the CP-violating asymmetry drops again to the level of 10\. For this purpose Schmidt again considered the asymmetry in the transverse energy of the leptons de"ned in Eq. (7.21). He [231] found that, similar to the 2HDM case when no additional cuts are made (like the ones suggested in [312] and were described in the previous section), at the

164


LHC with (s"14 TeV and with typical SUSY masses of a few hundreds GeV, the asymmetry N(E ) of Eq. (7.21) is again typically of the order of &few;10\. Schmidt also examined the 2 non-CP-violating background, and again found it to be negligible compared to the CP-violating e!ect. Clearly, the Schmidt SUSY CP-violating e!ect in ggPttM as it stands, is smaller than that of Ref. [312], i.e. the signal caused by the resonant Higgs contribution. This is mainly due to the appreciable improvement that can be achieved with appropriate cuts on the ttM invariant mass, as was discussed in the previous section. Note however, that by using optimal observables for extracting information on CP violation in ggPttM , it was shown in [309] that the signal to noise ratio for the CP-violating e!ect in ggPttM (driven by the diagrams in Fig. 50 that were considered by Schmidt) can reach the percent level after folding in the gluons structure functions. This allows a 1- detection of the CP-odd e!ect even for smaller values of R &0.1. !. 8. CP violation in pp collider experiments Shortly after the demise of the SSC, it was suggested to upgrade the energy of the Tevatron. More recently, substantial, two stage, upgrades in luminosity without a factor of two or so increase in energy remain as viable options. For a review see [315]. In the previous run at the Tevatron, at c.m. energy of 1.8 TeV, the D and CDF experiments accumulate more than 0.2 fb\ of integrated luminosity. In the "rst upgrade, called Run II, L will be increased from its current peak value to 2;10}10 cm\ s\ (or even to twice this value). In the second stage, so called Run III (or TeV-33), the luminosity will be further increased to L+10 cm\ s\. The working hypotheses are that in Runs II and III the integrated luminosity will be 2 and 30 fb\, respectively, with a modest increase of c.m. energy to 2 TeV. In addition, the D and CDF detectors are also being upgraded. 8.1. pp PttM #X Contrary to the LHC pp collider, where ttM pairs are produced predominantly through the gg fusion process ggPgPttM , in the Tevatron pp collider with c.m. energy of (sK2 TeV the main production mechanism of ttM pairs is the qq fusion, qq PgPttM . In particular, the qq fusion process is responsible for about 90% of the cross-section pp PttM #X. Being so, the processes pp PttM , ttM g#X, where g stands for an extra gluon jet in ttM production, will presumably be sensitive to the CEDM and CMDM of the top quark, which can be incorporated as e!ective interactions at the ttg vertex. As already mentioned in previous sections, being a CP-odd quantity, a nonvanishing CEDM coupling might give rise to observable CP violation in top systems in such a hadron collider. If so, this will be a clue for new physics, as in the SM the CP-non-conserving e!ects in the reactions pp PttM , ttM g#X are extremely small. In principle, CP-non-conserving e!ects due to the CEDM can be searched for through a study of either CP-even or CP-odd correlations in the reactions pp PttM , ttM g#X. Of course, CP-odd correlations are expected to be more sensitive to the CEDM than CP-even observables as the former are linear in CEDM whereas the latter are proportional to the square of the CEDM form factor.


165

We will "rst present a study of the sensitivity of some CP-even observables to the CEDM, and then describe some interesting CP-odd correlations that may be applied to pp PttM #X. 8.1.1. CP-even observables in pp PttM #X and pp PttM #jet#X The e!ective Lagrangian for the interaction between the top quark and a gluon, that includes the CEDM and CMDM form factors of the top, is given in Eq. (6.106) ( for l\X ,

(8.5)

while in the second scenario only one of the t or tM decays leptonically (see also Eqs. (6.79) and (6.80) in Section 6.2.3) pp PttM Pl>X, pp PttM Pl\X .

(8.6)

The processes in Eq. (8.6) have better statistics than the one in Eq. (8.5) and give the best signature for the top quark identi"cation. Within these decay scenarios two possible CP-odd observables were considered in [319] which we will describe below. 8.1.3. Transverse energy asymmetry of charged leptons The transverse energy asymmetry of the charged leptons was originally suggested by Peskin and Schmidt in [33] for ttM production in a pp collider (see Eq. (7.21)) and was discussed in detail in Sections 7.3 and 7.4. Recall that in the case where both t and tM decay leptonically it can be de"ned as (using here the notation in [319]) (E\'E>)!(E>'E\) 2 2 2 . A " 2 2 (E\'E>)#(E>'E\) 2 2 2 2

(8.7)

Recall that this is not necessarily true in model calculations, e.g., in the MSSM } see discussion in Section 7.4.


169

The result for the expected CP-violating asymmetry in the transverse energy of the muons (AI ) was 2 given in [319]. They considered AI as a function of the imaginary part of the top CEDM, Im dE 2 R (recall that A is ¹ -even thus requiring an absorptive phase), and of Re( f 0!fM * ). Also, the usual 2 , CDF cuts were applied. They found that CP violation from the production mechanism, i.e., JIm dE, is larger then that arising from the decay process, i.e., JRe( f 0!fM * ). For example, they R found that with Im dE&10\g cm and Re( f 0!fM * )&0.2 one can obtain an asymmetry around R Q the &10% level. In order to understand the feasibility of extracting such values for the CP-violating couplings in production and decays of the ttM , it is useful to decompose AI as follows [319]: 2 AI ,c A #c A , (8.8) 2 . . " " where the dimensionless couplings c and c are . " 1 m (8.9) c , R Im dE, c , Re( f 0!fM * ) . R " 2 . g Q Then, in terms of c and c , the statistical signi"cance for AI determination is given by . " 2 N2 ,c A #c A (Nll , (8.10) 1" . . " " where Nll is the number of dilepton events expected to be &80 and &1200 at an integrated luminosity ¸"2 and 30 fb\, respectively [320]. A and A can be calculated and, thus, it was . " found in [319] that a 3- e!ect will require the following relations to be satis"ed 2.5c #0.9c 51 for ¸"2 fb\ , (8.11) . " 9.8c #3.3c 51 for ¸"30 fb\ . (8.12) . " Clearly, AI is more sensitive to CP violation in the production mechanism than in the ttM decays. So, 2 for example, ¸"30 fb\ allows for an observation of c "c "0.08 at (s"2 TeV. Note that . " c "0.08 corresponds to Im dE"8.8;10\ which, again, is more than an order of magnitude . R larger than what is expected for the CEDM in beyond the SM scenarios such as MHDMs and SUSY (see Section 4). Similarly, the resulting 3- limit Re( f 0!fM * )"0.16 (corresponding to c "0.08) falls short by about one order of magnitude from model predictions for this quantity (see " discussion in Section 5). A related asymmetry which can be used in the case when only one top decays leptonically was also suggested in [319]: $E\'E )!>(E>'E ) 2 2 2 2 . AI (E )" 2

(8.13)

The in Eq. (8.13) denotes the integrated cross-section with no cuts except for the standard experimental cuts. We note that, with the CP-violating coupling c and c of the order of 0.1, which . " tends to be somewhat optimistic, this asymmetry in Eq. (8.13) can also reach the &10% level. 8.1.4. Optimal observables It is useful to be able to experimentally separate CP violation in the production from that in the decay. The optimization method outlined in Section 2.6 can provide for such a detection and was also used in [319].

170


Consider the transverse lepton energy spectrum in the single leptonic (say l>) and the dileptonic "nal states 1 d "f>(E>)#c f>(E>)#c f>(E>) , (8.14) 2 . . 2 " " 2 dE> 2 1 d "f!(E>, E$#c f!(E>, E\)#c f!(E>, E\) , (8.15) 2 2 . . 2 2 " " 2 2 dE> dE\ 2 2 where in Eqs. (8.14) and (8.15) denotes the cross-section for the process pp Pl>#jets and pp Pl>l\#jets, respectively. Also, c , c are de"ned in Eq. (8.9) and f ! are known functions of . " E> and E\. 2 2 As can be seen from Eqs. (8.14) and (8.15), the transverse lepton energy spectrums, in both the single- and double-leptonic channels, are sensitive to c and c . Using the above transverse lepton . " energy spectrum, the optimal weighting functions can be obtained. This was done in [319] where in both cases the statistical signi"cances for the experimental determination of c and c , i.e., . " N.",c /c , were calculated. For the single-leptonic events they obtained 1" ." ." c c (8.16) N. " . (Nl , N" " " (Nl , 1" 18.43 1" 2.37 and for the dileptonic events c c N. " . (Nll , N" " " (Nll , 1" 1.17 1" 5.76

(8.17)

where Nl and Nll are the expected number of single- and double-leptonic events, respectively. Nl &1300(20,000) and Nll &80(1200) for ¸"2(30) fb\, respectively [320]. The minimal values of c and c necessary to observe a 3- CP-violating e!ect with the optimization technique are . " listed in Tables 14 and 15 for the single- and double-leptonic channels, respectively. We see that the single-leptonic modes are more sensitive to the non-standard couplings. Thus, for example, the Tevatron upgrade Run III with ¸"30 fb\ will be able to probe Im dE down to values of R &5;10\g cm in the single-leptonic channel. Note that, as expected, this result is somewhat Q better than what can be achieved with `naivea observables such as AI in Eq. (8.13). We note in passing that comparable limits for Im dE but also for Re dE, i.e., Im, R R Re dE&few;10\g cm, were found also in [309] using optimal observables for the reaction R Q pp PttM #X and with the Tevatron upgrade parameters. Table 14 The minimal values of c and c necessary to observe CP violation in the single-lepton mode at the 3- level for . " ¸"2, 30 fb\. As a reference value, recall that c "1 corresponds to Im dE"1.1;10\g cm. Table taken from [319] . R Q ¸(fb\)

2

30

c . c "

0.20 1.50

0.05 0.40


171

Table 15 The minimal values of c and c necessary to observe CP violation in the dilepton mode at the 3- level for . " ¸"2, 30 fb\. See also caption to Table 14. Table taken from [319] ¸(fb\)

2

30

c . c "

0.39 1.93

0.10 0.50

To summarize, as was noted in [319], it is not inconceivable that non-standard CP-violating couplings of the top quark to a gluon may be discovered at the Tevatron before precision measurements at the LHC are done. 8.2. pp PtbM #X In spite of the fact that at the Tevatron pp collider, top quarks are mainly produced as ttM pairs via an s-channel gluon exchange [321], the subleading electroweak production mechanism of a single top forms a signi"cant fraction of the ttM pair production. It will therefore be closely scrutinized in the next runs of the Tevatron [322}329] (see also [315]). The production rate of tbM through an s-channel o!-shell =-boson, pp P=HPtbM #X, (the corresponding partonic reaction, udM P=HPtbM , is depicted in Fig. 54) is expected to yield about 10% of the ttM production rate [322}329]. In this section, we examine CP violation asymmetries in top quark production and its subsequent decay via the basic quark level reactions [215,258,330]: udM PtbM Pb =>bM ,

u dPtM bPbM =\b .

(8.18)

Indeed this reaction is rich for CP violation studies as it exhibits many di!erent types of asymmetries. Some of these, which we consider below, involve the top spin. Therefore, the ability to track the top spin through its decays becomes important and top decays have to be examined as well (see e.g. [331]). The asymmetries in tbM production can be appreciably larger than those in ttM pair production wherein they tend to be about a few tenths of percent (see Sections 6 and 7). Moreover, while in the SM, CP-odd e!ects in pp P=HPtbM #X are expected to be extremely small since they are severely suppressed by the GIM mechanism (see e.g., [64,65,235]), it is shown below that, in extensions of the SM, CP asymmetries can be sizable } in some cases at the level of a few percent. Therefore, since the number of events needed to observe an asymmetry scales as (asymmetry)\, the enhanced CP-violating e!ects in tbM (tM b) may make up for the reduced production rates for tbM compared to ttM . In fact larger asymmetries could be essential as detector systematim cs can be a serious limitation for asymmetries less than about 1%. Let us discuss now the asymmetries in the udM (u d) subprocess. We consider four types of asymmetries that may be present [258]. First, is the CP-violating asymmetry in the cross-section (8.19) A "( M ! M )/( M # M ) , R@ R@ R@ R@ where M and M are the cross-sections for udM PtbM and u dPtM b, respectively, at s( "(p #p M ). R @ R@ R@

172


Fig. 54. The tree-level Feynman diagram contributing to udM PtbM .

The spin of the top allows us to de"ne three additional types of CP-violating polarization asymmetries. For these it is convenient to introduce the coordinate system in the top quark (or anti-top) rest frame where the unit vectors are e J!p , e Jp ;p and e "e ;e . Here X @ W S @ V W X p and p are the 3-momenta of the bM -quark and the initial u-quark in that frame. We denote the @ S longitudinal polarization or helicity asymmetry as A(z( )"(N !N !NM #NM )/(N #N #NM #NM ) , (8.20) 0 * 0 * 0 * 0 * where N is the number of left-handed top quarks produced in udM PtbM and NM is the number of * 0 right-handed tM produced in u dPtM b, etc. Therefore, in the frame introduced above right-handed tops have spin up along the z-axis and left-handed ones spin down. We further de"ne the CP-violating spin asymmetries in the x and y directions as follows: !N #NM !NM )/(N #N #NM #NM ), V> V\ V> V\ V> V\ V> V\ (8.21) A(y( )"(N !N !NM #NM )/(N #N #NM #NM ) , W> W\ W> W\ W> W\ W> W\ where, for example, N (NM ) represent the number of t(tM ) with spin up with respect to jK -axis, for H> H> j"x, y, etc. While all these four asymmetries are manifestly CP-violating, A , A(z( ) and A(x( ) are even under naive time reversal (¹ ) whereas A(y( ) is ¹ -odd. So the "rst three, unlike A(y( ), require a complex , , amplitude, i.e., absorptive phases. These asymmetries are related to form factors arising from radiative corrections of the =HPtb production vertex due to non-standard physics. To see this, it is useful to parameterize the 1-loop tbM and tM b currents of the production amplitude as follows [215]: A(x( )"(N

P. pI g @ #P. I Pv , (8.22) JIR@M ,i 5 u @ (2 .*0 R mR PM . pI g @ #PM . I Pv , (8.23) JIRM @,i 5 u R (2 .*0 @ mR where P"¸ or R and ¸(R),(1!(#) )/2. P*0 and P*0, de"ned in Eqs. (8.22) and (8.23), contain the necessary absorptive phases as well as the CP-violating phases in a given model. It is easy to show that if one de"nes P* &e BQ ;e BU , P* &e BQ ;e BU ,

(8.24) (8.25)


173

where , are the CP-even absorptive phases (i.e., FSI phases) and , are the CP-odd phases, Q Q U U then (8.26) PM 0&!e BQ ;e\ BU (8.27) PM * &e BQ ;e\ BU . In terms of these form factors, the two ¹ -even asymmetries A and A(z( ) are given by [258]: , (x!1) A " Re(P* #PM 0)!Re(P* !PM * ) , (8.28) 2(x#2) x!2 (x!1) Re(P* #PM 0)! Re(P* !PM * ) , A(z( )" x#2 2(x#2)

(8.29)

where x,m/s( . Notice that the three quantities A , A(z( ), A(x( ) are linear combinations of only R two form factors. Thus, one can show that A(x( )"!3x\[(2#x)A #(2!x)A(z( )]/32 (8.30) which will therefore hold if the new CP violating physics takes place through such a vertex correction of =HPtb. Similarly, the asymmetry A(y( ) is proportional to the real parts of the 1-loop integrals and may therefore be obtained from the corresponding imaginary parts through the use of dispersion relations. In particular, since the ¹ -even asymmetries are proportional to the absorptive phases in , the above form factors, one can express A(y( ) in terms of A and A(z( ), 1!x 2#x 3 Re A(y( )[s( ]"! (!x)(!1#i ) 32 (2#x)(x

;[(2!x)A [s( ]#(2#x)A(z( )[s( ]] d .

(8.31)

Note that the integrand is 0 if s( is below the threshold to produce an imaginary part since then A and A(z( ) will vanish. Let us now evaluate the form factors de"ned in Eqs. (8.22) and (8.23) in two extensions of the SM: the 2HDM of type II and the MSSM. As was pointed above, once these form factors are calculated in a given model, the asymmetries A , A(z( ), A(x( ) and A(y( ) can be readily obtained. 8.2.1. 2HDM and the CP-violating asymmetries As emphasized throughout this article, in the 2HDM of type II, a CP-odd phase can reside in neutral Higgs exchanges and there is only one Feynman diagram that contributes to CP violation in udM P=HPtbM to 1-loop order. This diagram is shown in Fig. 55. The relevant Feynman rules for this diagram, required for calculating the asymmetries of interest, can be extracted from the parts of the 2HDM Lagrangian involving the ttM HI and ==HI couplings in Eqs. (3.70) and (3.71), with k"1, 2, 3 for the three neutral Higgs "elds. Recall again that the coupling constants, aI, bI and R R cI are functions of tan , which is the ratio between the two VEVs in this model, and of the three mixing angles which diagonalize the 3;3 Higgs mass matrix (see Section 3.2.3). As usual, for simplicity, we have assumed that two of the three neutral Higgs-bosons are much heavier

174


Fig. 55. The CP-violating 1-loop graph in 2HDMs with CP violation from neutral Higgs exchanges; h denotes the lightest neutral Higgs.

compared to the third one which we denote by h. Thus, the CP-violating e!ect will be dominated by the lightest neutral Higgs, h, and is proportional to bFcF. Choosing the angles " "/2 and R "0 which give maximal e!ects [258], one obtains bFcFJ cos cot so the asymmetries are now R functions of tan and m only. F Using the Lagrangian in Eqs. (3.70) and (3.71), the form factors Re(P* #PM 0) and Re(P* !PM * ) can be readily calculated and, we get

bFcF m R R Im[2m (C !C ) Re(P* #PM 0)"! 5 (2 2 sin 5 m5 !m(C #C )!CI !CI ] , (8.32) R bFcF m R R Im[2m (C #C )!2C ] , Re(P* !PM * )" (8.33) 5 4 sin m (2 5 5 where the C above, x30, 11, 12, 21, 22, 23, 24, are the three-point form factor associated with the V 1-loop diagram in Fig. 55, and are given via [258,330]:

C "C (m, m , m, m, s( , m) , (8.34) V V R 5 F @ R where s( "(p #p M ) and C (m , m , m , p , p , p ) is de"ned in Appendix A. V R @ The quark level asymmetries of interest can be converted to the hadron (i.e., pp ) level by folding in the structure functions in the standard manner [91]. The results for the 2HDM case are shown in Fig. 56, for tan "0.3 and as a function of the lightest Higgs-boson mass, m . For the asymmetry F A(y( ) we apply a cut of s( '(m #m ). We can see that A and A(x( ) can reach above the percent F 5 level for m :200 GeV. The measurable consequences for such an asymmetry are discussed in F Section 8.2.3 below. It is interesting to note that in the 2HDM (to the 1-loop order) the ¹ -even asymmetries A , A(z( ) , and A(x( ) do not receive any contribution from the decay vertex in tPb=. The only diagram that can potentially drive CP violation in tPb= is the same one as shown in Fig. 55 with the momenta of the t and the = reversed. Thus, an important necessary condition for A , A(z( ), A(x( )O0, that there is an imaginary part in the decay amplitude, is not satis"ed. Moreover, as it turns out, the observed value of A(y( ) is not a!ected by CP violation in the decay process. The key point is that the measurement of A(y( ) through the decay chain u(p ) dM (p )PbM (p ) t(p ) followed by S B @ R t(p )Pb(p ) e>(p ) (p ) is equivalent to a measurement of a term proportional to (p , p , p , p ) R @ C J C B R @ ( being here the Levi-Civita tensor). On the other hand, CP violation arising from the decay process is proportional to (p , p , p , p ). It is easy to see that an observable related to the "rst of C B R @


175

Fig. 56. The CP-violating asymmetries in pp PtbM #X for the 2HDM case and in the pp c.m. frame for (s"2 TeV, as a function of m (horizontal axis); A (solid), A(z( ) (dashed), A(x( ) (dotted) and A(y( ) (dot-dashed). Figure taken from [258]. F

these will be insensitive to the second. So, in this way asymmetries in the production can be separated from those in the decay. 8.2.2. SUSY and the CP-violating cross-section asymmetry The MSSM possesses several CP-odd phases that can give rise to CP violation in pp PtbM #X and in the subsequent top decay tPb= (see Section 5.1.4). In [215] we have constructed a plausible low-energy MSSM framework in which there are only "ve relevant free parameters needed to evaluate the cross-section asymmetry A (see also Sections 3.3.4 and 5.1.4). These are: M a typical SUSY mass scale that characterizes the mass of the heavy squarks, m the mass of the 1 % gluino, the Higgs mass parameter in the superpotential, m the mass of the lighter stop and tan J the ratio between the VEVs of the two Higgs "elds in the theory. In this framework there are two sources of CP violation that can potentially contribute to A [215]. The "rst may arise from the Higgs mass parameter which may be complex in general. The second CP-violating phase arises from tI !tI mixing (see e.g., Eq. (5.60)) and may be parameterized by a single quantity R de"ned * 0 !. in Eq. (5.61). In this scenario, when no further assumptions are made, there are 12 Feynman diagrams that can give rise to CP violation at the parton level process udM PtbM which are depicted in Fig. 57. However, if we assume that arg()"0 as implied from the existing experimental limit on the NEDM (see discussion in Section 3.3.4) and take m "m "m "0, then only diagrams (a), (b), (d), (e) and (g) S B @

176


Fig. 57. CP-violating, SUSY induced, 1-loop diagrams for the processes udM PtbM . g is the gluino, is a chargino, is a neutralino and tI and bI are top and bottom squarks, respectively.

have the necessary CP-violating phase, being proportional to one quantity } R . In [215] we have !. shown that with M and m of about several hundred GeV, diagram (d) is, in fact, responsible for 1 % &90% of the total CP violation e!ect. Let us, therefore, present the results for the asymmetry A corresponding only to diagram (d) in Fig. 57. The Feynman rules needed to calculate A can be derived from the following parts of the SUSY Lagrangian [185]:

L K "u >dM S BQ

(2m (2m S ZGZ>H R# B ZGZ\ ¸ <SBA #h.c. , !g ZGZ>H# 5 S K S K S K K v v

L L "u \ S SQ L

g (2m S ZGHZL ¸ ! 5 ZGH¸>! S S , v (2

(8.35)

(2m 2(2 S ZGHZLH R u#h.c. , g tan ZGHZLH! (8.36) , , S S 5 5 3 v (8.37) L K L "g I(K\¸#K>R)=>#h.c. , L I 5Q Q 5 K where ¸(R)"(1!(#) ), u and u (dI and d) stand for up squark and up quark (down squark H and down quark), respectively, (m"1, 2) and (n"1!4) are the charginos and neutralinos, K L respectively. Also, in Eqs. (8.37) and (8.36) we have de"ned #

¸!, tan ZL$ZL , 5 , ,

(8.38)


177

1 K>,ZLHZ\ # ZLHZ\ , , K (2 , K

(8.39)

1 K\,ZLZ>H! ZLZ>H , , K (2 , K

(8.40)

and the mixing matrices Z , Z , Z , Z\ and Z> were given in [215] (also given in Section 3.3.2). S B , Using the Lagrangian in Eqs. (8.35)}(8.37), the form factors Re(P* #PM 0) and Re(P* !PM * ) (corresponding to diagram (d) in Fig. 57) can be evaluated within the MSSM and are given by m [m O Im(CB #CB )#m L O Im CB Re(P* #PM 0)"! R R B Q B sin 5 (8.41) ! m K O Im(CB !CB )] , Q B 1 [O((s( !m) Im CB !s( Im CB Re(P* !PM * )" B R 2 sin 5 ! 2 Im CB !m Im CB ) R ! m m L O Im CB #m m K O Im(CB #CB ) R Q B R Q B (8.42) # m K m L O Im CB ] , Q Q B where s( "(p #p M ) and O contain the SUSY CP-weak phases associated with diagram (d) in R @ B Fig. 57. In fact, the same CP-violating phases occur also in the decay tPb= (see Section 5.1.4) and, therefore, the O above are the same as the ones given in Eqs. (5.50)}(5.53). The Im CB , B V x30, 11, 12, 21, 22, 23, 24, in Eqs. (8.41) and (8.42) are the imaginary parts of the three-point form factors associated with diagram (d) in Fig. 57. Thus, CB are given via [215]: V (8.43) CB "C (mI , m K , m L , m, s( , m) , Q @ R V V R Q and C (m , m , m , p , p , p ) is de"ned in Appendix A. V In [215], instead of calculating the cross-section asymmetry A , we considered a partially integrated cross-section asymmetry, A.'!, in which we have imposed a cut on the tb invariant mass, m (350 GeV. Such a cut on m may help to remove the ttM `backgrounda from a measureR@ R@ ment of a cross-section asymmetry in pp PtbM #X. The results in the SUSY case are shown in Figs. 58 and 59 as a function of and m , respectively, for M "400 GeV, m "50 GeV (m is the % 1 J J mass of the lighter stop particle) and for tan "1.5, 35. Maximal CP violation was chosen in the sense that R "1, thus the asymmetry plotted in Figs. 58 and 59 is, in fact, given in units of R . !. !. Evidently, for some values of around 100 GeV and with m &450 GeV the asymmetry can % almost reach the 3% level. The asymmetry is above the 1% level for several other choices of . It was also shown in [215] that, in general, in order for the asymmetry to be above the percent level the mass of the lighter stop is required to be below &75 GeV. Furthermore, the asymmetry tends to drop as tan is increased in the range 1:tan :10, and it is almost insensitive to tan for tan 910. In the MSSM, 1-loop radiative corrections to the amplitude of top decay tPb= that can violate CP are also present. In fact, disregarding the incoming u and d lines, diagrams (a)}(d) in Fig. 57 with

178


Fig. 58. The SUSY-induced partially integrated cross-section asymmetry A.'! de"ned in the text, as a function of , for M "400 GeV, m "50 GeV and for (s"2 TeV. With (a) tan "1.5 and (b) tan "35. Figure taken from [215]. 1 J

the t and = momenta reversed, can give rise to a CP-violating tb= decay vertex. This was discussed in some detail in Section 5.1.4. To 1-loop order in perturbation theory, where the CP-violating virtual corrections enter only either the production or the decay vertices of the top in the overall reaction pp PtbM #XP=>bbM #X, and in the narrow width approximation for the decaying top, an overall CP asymmetry, A, can be broken into A"A #A , . "

(8.44)

In Eq. (8.44) A and A are the CP asymmetries emanating from the production and decay of the . " top, respectively, and are de"ned by (pp PtbM #X)!(pp PtM b#X) , A , . (pp PtbM #X)#(pp PtM b#X)

(8.45)

(tP=>b)!M (tM P=\bM ) A , . " (tP=>b)#M (tM P=\bM )

(8.46)

The PRA A , de"ned in Eq. (8.46), does not depend on the speci"c production mechanism of the " top and was calculated in Section 5.1.4. We have shown there that, with the low-energy MSSM parameters described above, one gets A (0.3% where, in most instances, it tends to be even "


179

Fig. 59. The SUSY-induced partially integrated cross-section asymmetry A.'! de"ned in the text, as a function of m , % for several values of , M "400 GeV, m "50 GeV and for (s"2 TeV. With (a) tan "1.5 and (b) tan "35. Figure 1 J taken from [215].

smaller } A (0.1%. Therefore, it is about one order of magnitude smaller than the asymmetry in " the production of tb and its relevance to the overall asymmetry in pp PtbM #XP=>bbM #X is negligible. As a "nal remark here, let us recall that in the 2HDM case discussed before, the PRA A in " Eq. (8.46) is forbidden at 1-loop order because of CPT invariance, i.e., no rescattering of "nal states as shown in Section 2.3. This was also discussed in Section 5.1.2. 8.2.3. Feasibility of extraction from experiment To summarize the results of Sections 8.2.1 and 8.2.2, CP-violating asymmetries in single top production and decay at the Tevatron through pp PtbM #XP=>bbM #X may optimistically reach a few percent in extensions of the SM such as SUSY and 2HDMs. In future upgrades of the Tevatron to (s"2 TeV, the cross-section for pp PtbM #X is expected to be about 300 fb, if a cut of m (350 GeV is applied on the invariant mass of the tbM [215]. Therefore, with an integrated R@ luminosity of L"30 fb\ [315,322}329], an asymmetry of &3% can be naively detected with a statistical signi"cance of 3-. Therefore, a percent level CP-violating signal in the reaction pp PtbM #XP=>bbM #X is especially notable as it may become accessible at the near future 2 TeV pp collider. In particular, based on the results presented in this section, such a measurement

180


may impose limits on the CP-violating parameters arg(A ) and bFcF of the MSSM and the 2HDM, R R respectively. However, it should be noted that such a detection at the Tevatron will require the identi"cation of all tbM pairs, which, in principle, can be achieved only if the top can be reconstructed even when the = decays hadronically. It will also be useful to explore SUSY or 2HDM mediated CP-violating e!ects that can originate from the =-gluon fusion subprocess which contributes to the same "nal state (i.e., =gPtbM d) and which has a comparable production rate to that of the simple udM PtbM in the 2 TeV Tevatron. While in the MSSM various 1-loop triangle and box corrections can give rise to CP nonconservation in the =-gluon fusion subprocess, in the 2HDM CP-violating radiative corrections to =g fusion, at the 1-loop order, do not yield absorptive parts in the limit m "0. Therefore, @ it will not contribute to CP asymmetries of the ¹ -even type in single top production. Note, , however, that the =-gluon fusion subprocess has its own characteristics, e.g., the extra light jet in the "nal state, which may be used in order to experimentally distinguish it from the `simplea ud fusion process (see e.g., [324]). 8.3. pp PtbM h#X, a case of tree-level CP violation Motivated by the large, tree-level, CP-violating e!ects found in the reaction e>e\PttM h (see Section 6.2), we were led to consider an analogous reaction in the Tevatron pp collider with a tbM h "nal state [330]. Thus, in this section we focus on CP violation, driven by 2HDM in the process pp PtbM h#X, where h is the lightest neutral Higgs in the 2HDM of type II. From the outset we remark that a statistically signi"cant CP study in the reaction pp PttM h#X in a future Tevatron upgrade with c.m. energy of 2 TeV and even 4 TeV, is unlikely due to the low tbM h event rate. As in the case of e>e\PttM h, a very interesting feature of the reaction udM PtbM h (at the parton level) is that it exhibits a CP asymmetry at the tree graph level. Such an e!ect arises from interference of the Higgs emission from the t (but not from the bM in the limit m P0) @ with the Higgs emission from the =-boson. Being a tree-level e!ect the resulting asymmetry is quite large. This asymmetry may be measurable, in principle, through a CP-odd, ¹ -odd , observable. Let us now discuss the tree-level cross-section and CP violation e!ects in our reactions, u(p )dM (p M )Pt(p )bM (p M )h(p ), u (p )d(p )PtM (p M )b(p )h(p ) . R @ F S B R @ F S B

(8.47)

In the limit m "0 (and also m "m "0), and to lowest order, the only two diagrams that can @ S B contribute to CP violation in the reactions of Eq. (8.47) are depicted in Fig. 60. The relevant Feynman rules needed to calculate the tree-level CP asymmetry are extracted from the Lagrangian in Eqs. (3.70) and (3.71). Here also, we assume that two of the three neutral Higgs particles are much heavier than the remaining one, i.e. h. We, therefore, omit the index k in Eqs. (3.70) and (3.71), and denote the couplings for the lightest neutral Higgs as: aF, bF and cF. The tree-level di!erential R R cross-section K at the parton level, is a sum of two terms: the CP-even and CP-odd terms K and > K , respectively, \ K ,K #K , > \

(8.48)


181

Fig. 60. Tree-level Feynman diagrams contributing to udM PtbM h in the limit m "m "m "0; h stands for the lightest B S @ neutral Higgs in a 2HDM.

where K are calculated from the tree-level diagrams in Fig. 60. The expression for the CP-even ! part, i.e., K , can be parameterized as >

2 m 5 R ; (aF)A#(bF)B K " > R R R R sin 2 m 5 5

m # 2(cF) 5 C#2aFcF D , R R 5F m 5F R

(8.49)

where the terms A, B, C and D are quite involved and were calculated in [330]. is the 5 =-boson propagator, and together with and are given by R 5F 1 , , 5 s( !m 5

1 , R 2p ) p #m R F F

1 , . 5F m!m #2p ) p M R 5 R @

(8.50)

Furthermore, p,p #p M is the s( -channel 4-momentum at the quark level, and s( is de"ned to be S B s( "p. The CP-violating piece of the tree-level di!erential cross-section is [330]:

2 m R bFcF; (p M , p M , p , p );( f!s #w) , K "2 \ @ B R S R sin m 5 5F R R 5 5

(8.51)

where s ,(p #p M ), f,(p !p M ) ) (p #p M ), w,(p #p M ) ) (p #p M ) and is the Levi-Civita S B R @ S B R @ R R @ tensor. For illustration, we adopt here also the value tan "0.3. We fold in the structure functions of the u and the dM inside the p and p , respectively, and plot in Fig. 61 the tree-level cross-section for pp PtbM h#X, with c.m. energies of (s"2 and (s"4 TeV. Four possible sets of the Higgs coupling constants aF, bF and cF were chosen. For illustrative purposes, the "rst two sets of R R parameters which are chosen for a 2 TeV collider are: set I with tan "0.3, , , " /4, /2, 3/4 and set II with tan "0.3, , , "/2, /2, 0. The other two, chosen for a 4 TeV collider are: set III with tan "0.3, , , "/4, /2, /2 and set IV with tan "0.3, , , "/2, 3/4, 3/4. The general feature of these sets is that sets I and III give rise to a large CP asymmetry but `smalla cross-section, while sets II and IV increase the event

182


Fig. 61. The cross-section for the reaction pp PtbM h#X (in fb), for (a): (s"2 TeV and for sets I (solid line) and II (dashed line), (b): (s"4 TeV and for sets III (solid line) and IV (dashed line). For the de"nition of the sets I}IV, see text.

rate but decrease the asymmetry. Also, note that each set by itself is not unique, as there are other values of the angles , and for each set which lead to the same e!ect. As in the reaction e>e\PttM h discussed in Section 6.2, we are dealing here with a tree-level CP-violating e!ect. Thus, the CP-violating term K can probe only CP asymmetries of the ¹ -odd \ , type. In this case, the "nal state is not a CP eigenstate. Therefore, one has to construct a ¹ -odd, , triple correlation product (or equivalently, a Levi-Civita tensor) which takes into account the conjugate reaction as well (u dPtM bh), thus endowing the observable with de"nite CP properties. This led us to consider the following CP-odd, ¹ -odd observable: , O"( (p , p M , p , p )# (p , p , p M , p ))/s . S @ R F S @ R F

(8.52)

Fig. 62 shows the results for the signal-to-noise ratio, i.e., the asymmetry A , O/( O, for (s"2 TeV with sets I and II and for (s"4 TeV with sets III and IV. Evidently, the asymmetry A is of the order of 10}15% for a light Higgs particle in the mass range 50 GeV(m (100 GeV F and of the order of 20}30% for a heavy Higgs particle with mass in the range 200 GeV(m (250 GeV, for both set II (which corresponds to a 2 TeV collider) and set IV (which F we chose for the 4 TeV collider). Sets I and III give asymmetries of the order of a few percent. Although with these sets, i.e., sets I and III, the cross-section can be 10 times larger than that corresponding to sets II and IV, the statistical signi"cance of the CP-violating e!ect that can be achieved when the free parameters of the 2HDM are chosen according to sets I and III is much smaller than that with sets II and IV. This is simply due to the fact that the number of events needed to detect the CP-violating e!ect scale as (asymmetry)\. Therefore, the enhanced e!ect for sets II and IV makes up for the reduced production rate in those scenarios.


183

Fig. 62. The asymmetry, A (see text) as a function of m for (a): (s"2 TeV with sets I (solid line) and II (dashed line), F (b): (s"4 TeV with sets III (solid line) and IV (dashed line). For the de"nition of the sets I}IV, see text.

Let us proceed by analyzing the two scenarios which give the large asymmetries. For the reaction at hand, the statistical signi"cance N of the CP-odd signal in the collider is 1" N "(L((pp PtbM h#X);A , 1" -

(8.53)

where L is the collider luminosity. From Fig. 61 we see that (pp PtbM h#X)&0.1}10 fb, depending on the parameters aF, bF and cF and the neutral Higgs mass. Thus, since A &0.1}0.3 in R R the best cases, it is evident from Eq. (8.53) that, typically, an integrated luminosity of about &100 fb\ will be required to be able to observe a statistically signi"cant e!ect in this reaction. Therefore, although the CP asymmetry in this process could reach the 10}30% level, it is, unfortunately, not likely to be able to detect a CP-violating signal in the next runs of the Tevatron with L"2 fb\ and even with 30 fb\.

9. CP violation in collider experiments Future electron}positron colliders include the attractive option of a linear collider, where each beam of photons is produced by Compton backscattering of laser light on an electron or positron beam. The peak energies and luminosities of the are expected to be slightly smaller than those of the corresponding e>e\ collider. The idea was originally suggested in [332}335]; for recent reviews see [336,337]. The attractive option of obtaining polarized photon beams is also being considered. Note also that collisions have been discussed in the context of heavy ion colliders (for a review see [338]) as well. Unfortunately, the invariant mass reach for the LHC (running in its heavy ion

184


mode), will only be about 100 GeV, with lower values attainable at RHIC [338]. Therefore, this option will not be discussed here. 9.1. PX: general comments In a collider there are two distinguishable modes: unpolarized and polarized incoming photons. In the unpolarized case, in order to be able to detect CP violation in the reaction PttM , one needs information on the spins of the t and the tM , or equivalently, one needs to construct asymmetries involving the decay products of the top quark. In this case, including the subsequent decays of the t and tM , one can break the di!erential cross-section for the process (k , ) (k , )Pt(p , s )tM (p M , s M ) , R R R R to its CP-odd and CP-even parts as d,d#d .

(9.1)

(9.2)

In Eq. (9.1) , denote the helicities of the incoming photons , , respectively, and s , s M are R R the covariant spins of t, tM , respectively. In general, the CP-odd terms in Eq. (9.2) has the form (9.3) dJ q ) (s ;s M )# q ) (s !s M ) , \ R R > R R where q is a three momentum of any of the particles in the "nal or the initial state (i.e., q"k , k , p or p M ) and s (s M ) is the spin three vectors of the t(tM ). and are non-zero only if R R R R \ > there is CP violation in the underlying dynamics of the process PttM . Furthermore, is \ proportional to the dispersive, CP-odd, ¹ -odd contributions, while gets its contribution from , > absorptive, CP-odd, ¹ -even terms. Of course, as mentioned above, the CP-odd polarization , correlations of the top and the anti-top in Eq. (9.3) will lead to CP-odd correlations among the momenta of the decay products of the t and the tM . Asymmetries which involve the top decay products in the case of the unpolarized photons were investigated in [261,339]. If the incoming photons are polarized, then one can construct CP-odd correlations by using linearly polarized photons where no information on the momenta and polarization of the top quark decay products are needed. The amplitude squared for a general "nal state X, i.e., PX, in the case where the two photons are linearly polarized is given by [257] ![ cos (#)# cos(!)] Re( ) # [ sin(#)! sin(!)] Im( ) ! [ cos(#)! cos(!)] Re( ) # [ sin(#)# sin(!)] Im( ) # cos(2) Re( )# sin(2) Im( ) # cos(2) Re( )n# sin(2) Im( ) .

d(, ; , )"

(9.4)


185

Here , are the degrees of linear polarization of the two initial photons, and are the azimuthal angle di!erence and sum, respectively, and the invariant functions are de"ned as [257] 1 " [M #M #M #M ] , 4 >> >\ \> \\ 6

(9.5)

1 " [M (MH #MH )#(M #M )MH ] , 2 >> >\ \> >\ \> \\ 6

(9.6)

1 " [M (MH !MH )!(M !M )MH ] , 2 >> >\ \> >\ \> \\ 6

(9.7)

1 " (M MH ) , >\ \> 2 6

(9.8)

1 " (M MH ) . 2 >> \\ 6

(9.9)

The subscripts 0 and 2 in Eqs. (9.6)}(9.9) represent the magnitude of the sum of the initial photon helicities and the notation for the helicity amplitudes for the reaction ( ) ( )PX using Eqs. (9.5)}(9.9) is M

H H

" XM .

(9.10)

Furthermore, the event rate of any "nal state production through fusion can be written, in general, as [340] dN"dL GH dGH , (9.11) AA GH where G(H) are the so called Stokes polarization parameters for ( ) with ""1. In particular, and are the mean helicities of and , respectively, and (()#() are their degrees of linear polarization. Also, dL is the luminosity of the two photons and GH are the AA corresponding cross-sections. There are only three CP-odd functions out of the nine invariant functions in Eqs. (9.5)}(9.9): Im( ), Im( ) and Re( ). While Im( ) and Im( ) are ¹ -odd, Re( ) is ¹ -even. , , A CP-odd asymmetry can be formed at a collider if, for example, the J"0 amplitudes of two photons in the CP-even and CP-odd states are both non-vanishing: M[PX(CP"#)]J ) , M[PX(CP"!)]J( ; ) ) k ,

(9.12)

where and are the polarizations of the two colliding photons and k is the momentum vector of one photon in the c.m. frame. Such an asymmetry can be constructed, for example, by taking

186


Fig. 63. Lowest-order Feynman diagrams for the process I (k )J (k )Pt(p )tM (p M ). and are Lorenz indices. R R

the di!erence of distributions at "$/4 [257,341]:

d d ! d Im( ) d d " Q Q\ . A , (9.13) d d # d d d Q Q\ Alternatively, in terms of event rates which correspond to the (0, 0) and the unpolarized initial photon}photon states this reads N A " . N

(9.14)

9.2. PttM and the top EDM Recall that a top EDM, i.e., dA, modi"es the SM ttM coupling to read R "ie #idA (p #p M )J , (9.15) I I R IJ R R and CP-violation arising due to this EDM of the top can be studied in the reaction PttM of Eq. (9.1). The relevant lowest-order Feynman diagrams for PttM are shown in Fig. 63 wherein the top EDM can be folded into any of the ttM vertices in those two diagrams. We note however again that, in general, the CP-violating e!ects in PttM cannot necessarily be all attributed to the top quark EDM. For example, in a 2HDM (see next section) additional box diagrams can give rise to CP-non-conserving terms in the amplitude of the reaction PttM . With the notation M( , , , M ) for the amplitude, where , , and M correspond to the R R R R helicities of the two incoming photons, the top quark and the top anti-quark, respectively, the non-vanishing helicity amplitudes for the process in Eq. (9.1), obtained for combinations such as " " or "! " and M " or M "! , are given by [257,341,342] R R R A A R m M( , , , )"!4C R ( # ) A A R R R (s A R R

s ! idA 2m 2# ( ! ) sin R R R A R 4m R R R s 4m R # ( ! ) sin # (dA) A R 2 R R A R R s

,

(9.16)


187

M( , , ,! )"!4m C A A R R R R ; sin cos [ idA!m (dA)] , R R R A R R R m M( ,! , , )"4C R A A R R R (s

(9.17)

s s ; #idA !(dA) sin , R R R 2m R R 2 R R R R

(9.18)

M( ,! , ,! )"2C sin ( # cos ) A A R R R R R A R R

s 4m R cos # (1! cos ) ! (dA) R A R R R R 2 s

,

(9.19)

where is the scattering angle in the c.m. frame and is the top quark velocity and R R C "eQ/(1! cos ). R R R R The CP-odd ¹ -odd distribution Im( ) de"ned in Eq. (9.9) depends linearly on dA and is given , R by [257] Im( )"24(1! cos ) Re(dA) , (9.20) R R R and the asymmetry A de"ned in Eq. (9.14) can then be calculated [257,341]. After extracting the top EDM from A and de"ning A ,Re( )AI , where ,2m dA/e is a dimensionless EDM A A R R form factor, as in Eq. (6.108), one obtains the allowed sensitivity (i.e., N " number of standard 1" deviations) to the dispersive part, Re( ), in the case that no asymmetry is found A (2N 1" Max(Re( ))" , (9.21) A AI ( N where is the detection e$ciency which was taken to be 10% in [257]. The kinematics of the Compton backscattering process at hand is characterized in part by the dimensionless parameter x,2p ) p /m. Larger x values are favored to produce highly energetic photons but the degree of C A C linear polarization is larger for smaller x values (for more details see [257]). In particular, the denominator on the r.h.s. of Eq. (9.21) depends on x, which for a given c.m. energy squared, s, is bounded by 2m R 4x42(1#(2) , (9.22) (s!2m R for the process PttM where the upper bound is required to prevent e>e\ pair production in the scattering of the photon while the lower bound is required to have photons energetic enough to produce top pairs. In [257] the x dependence of the Re( ) upper bound, i.e. Max(Re( )), was given which is A A shown in Fig. 64 for two c.m. energies (s"0.5 and 1 TeV. We see that Max(Re( )) can reach A below 0.1 where the optimal sensitivities are obtained with x"3.43 and x"0.85 for (s"0.5 and

188


Fig. 64. The x dependence of the Re( ) upper bound, i.e., Max(Re( )), at (s"0.5 TeV (solid line) and 1 TeV (dashed A A line), from the asymmetry A . Figure taken from [257].

1 TeV, respectively. For these x values, and to 1-, the upper bounds that can be achieved are: Re( )"0.16 and Re( )"0.02 for (s"0.5 and 1 TeV, respectively. This corresponds to A A Re(dA)+0.9;10\ and 0.1;10\e cm for (s"0.5 and 1 TeV, respectively. R Di!erent type of asymmetries which involve the polarization of both the initial photons beams and the decay products of the t and tM (e.g., tPbll ) in the reaction PttM , were suggested in [342]. The "rst one is a charge asymmetry, A , which measures the di!erence between the number of leptons and anti-leptons produced as decay products of the top and anti-top, respectively. The second, A , is $ a sum of the forward}backward asymmetries of the leptons and anti-leptons and requires polarized laser beams. In terms of the di!erential cross-sections these asymmetries are given by \F dl (d>/dl !d\/dl ) A ( )" F . \F dl (d>/dl #d\/dl ) F

(9.23)

dl (d>/dl #d\/dl )! \F dl (d>/dl #d\/dl ) , A ( )" F $ \F dl (d>/dl #d\/dl ) F

(9.24)

and

where d>/dl and d\/dl refer to the l> and l\ distributions in the c.m. frame, respectively, and is a cuto! on the polar angle of the lepton. It is important to note that if there is no CP violation in the top decays, then the charge asymmetry is zero in the absence of the cuto! . Both A and A are ¹ -even asymmetries, thus they probe the imaginary part of the top EDM, $ , Im(dA). In [342], the case where one of the t or tM decays leptonically and the other decays R hadronically, was studied. Also, it was assumed that no CP-violation enters these top decays. The asymmetries A and A were then evaluated in the c.m. frame and 90% C.L. limits on the top $ Then the asymmetries correspond to samples A and A M de"ned in Eqs. (6.79) and (6.80) in Section 6.2.3.


189

EDM, in the case that no asymmetry is found in the experiment, were obtained. The 90% C.L. limits were evaluated for di!erent electron and laser beam energies as well as for di!erent cuto! angles. Also, di!erent helicity combinations of the initial beam and di!erent values of the dimensionless parameter x de"ned before (see Eq. (9.22) and the discussion above), were analyzed. They found that for an electron beam energy of 250 GeV, and for a suitable choice of circular polarizations of the laser photons and longitudinal polarizations for the electron beams, and assuming a luminosity of 20 fb\ for the electron beam, in the best cases and in an ideal experiment, it is possible to obtain limits on the imaginary part of the top EDM, again, of the order of 10\e cm. However, an order of magnitude improvement may be possible if the beam energy is increased to 500 GeV [342]. To conclude this section, the reaction PttM can serve to limit both the real and the imaginary parts of the top EDM. However, it is worth mentioning that the limits that may be placed on the top EDM through a CP study in PttM are roughly comparable to those which might be obtained through a study of the reaction e>e\PttM at the NLC (for comparison see Section 6.1). Therefore, the motivation for going to a collider in order to study e!ects of the top EDM is somewhat arguable. On the other hand, model calculations of CP violation in PttM , such as the one described below, i.e., a 2HDM, show that CP-non-conserving signals in this reaction, which are not necessarily associated with the top EDM, may be sizable; i.e., at the detectable level in a future photon collider. 9.3. PttM and s-channel Higgs exchange in a 2HDM A collider can also provide an interesting possibility for producing an s-channel neutral Higgs-boson, via Ph, which can then decay to a pair of fermions, hP+M (recall that a related process was considered in the context of a pp collider in Section 7.3.2). Once again, h stands for the lightest neutral Higgs-boson in a MHDM and the other neutral Higgs particles are assumed to be much heavier, thus, neglecting their contribution in what follows. The decay of a neutral Higgs, for m '2m , to a pair of ttM will inevitably dominate the other F R fermionic decays of the Higgs due to the largeness of the top mass. CP violation in the reaction PttM was investigated within a MHDM in [339] for unpolarized incoming photons, where the e!ects of the s-channel Higgs were included. In [340] polarized laser beams were considered for the s-channel neutral Higgs production, Ph. In [339], the complete set of CP-non-conserving contributions to PttM , at the 1-loop order, were considered within a 2HDM of type II. Recall that, in the SM, CP violation cannot occur in this process at least to 2-loop order. This set of 1-loop Feynman diagrams is depicted in Fig. 65(b)}(h) and Fig. 65(a) represents the only tree-level diagram for this process; this tree-level diagram and its permuted one are also shown in Fig. 63. We de"ne M to be the amplitude for a diagram i in Fig. 65 (i.e., i"a,2, h) where we further G decompose M to its CP-odd part M and CP-even part M: M "M#M. Then, to G G G G G G leading order, the CP-odd (d) and the CP-even (d) parts of the di!erential cross-section are given by F d"2 Re M M R , ? G A G@

(9.25)

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Fig. 65. Feynman diagrams for PttM . In (a) the Born diagram is shown (see also Fig. 63), and in (b)}(h) the complete set of 1-loop diagrams that can violate CP are depicted. Diagrams with crossed lines are not shown.

F d" M MR# 2 Re M M R . ? ? ? G A GD

(9.26)

Note that the CP-even contribution from the interference of the s-channel Higgs graphs Fig. 65(f)}(h) with the Born amplitude of Fig. 65(a), which is explicitly included in Eq. (9.26), can become important and was taken into account in [339] because of the non-negligible width of the Higgs. Also, the CP-odd interference in Eq. (9.25) will give rise to the simple form of d in Eq. (9.3). Again, to e$ciently trace the CP-odd spin correlations in Eq. (9.3), one de"nes a ttM decay scenario where the t decays leptonically and the tM decays hadronically and vice versa. As in Eqs. (6.79) and (6.80) in Section 6.2.3, we denote by A the decay sample in the case that the top decays leptonically and the anti-top decays hadronically, and by A M the charged conjugate decay sample [261,339]. With these decay scenarios one can evaluate a few CP-odd asymmetries of both the ¹ -odd and , ¹ -even type which may acquire a non-vanishing value only if O0 and O0, respectively (see , \ > Eq. (9.3)). To do so, let us de"ne for sample A, i.e., tP=>bPl>l b and tM P=\bM Pqq bM , the following operators: O "(q( l> ;q( H \ ) ) p( M , 5 R

(9.27)

O "El> ,

(9.28)

O "q( l> ) p( M , R

(9.29)

where the asterisk denotes the t(tM ) rest frame. The corresponding ones for the sample A M are OM "(q( l\ ;q( H > ) ) p( , 5 R

(9.30)

OM "El\ ,

(9.31)

OM "q( l\ ) p( , R

(9.32)


191

Thus, the CP-odd asymmetries are constructed as [261,339] ¹ !odd: " O # OM , (9.33) , ¹ !even: " O ! OM . (9.34) , To calculate the above asymmetries one has to fold in the distribution functions of the backscattered laser photons (for more details see [261,339]). Also, one has to choose a de"nite scheme for type II 2HDM couplings aF and bF of a neutral Higgs particles to a pair of ttM in Eq. (3.70). Recall R R that the scalar aF and pseudoscalar bF couplings are functions of the neutral Higgs mixing matrix R R R in Eq. (3.73) and of the ratio between the two VEVs, tan (see Section 3.2.3). In particular, as in HG [261,339], assuming that the other two neutral Higgs of the model are much heavier than h such that their mass lies above the c.m. energy, their contribution is neglected. Also, the mass of the charged Higgs-boson was taken as m ! "500 GeV and R "1/(3 for j"1, 2, 3 was assumed H & (see Eq. (3.73)). Let us present a sample of the results that were obtained in [261,339]. The ¹ -odd asymmetry, , i.e., the signal-to-noise ratio / , is shown in Fig. 66 and the ¹ -even asymmetry ratio / is , depicted in Fig. 67. The asymmetries are plotted for various values of tan . In Fig. 66 tan "0.5 (dashed line) and tan "1 (solid line) were used, while in Fig. 67 tan "0.3 (dashed line) and tan "0.627 (solid line) were chosen. In those "gures an e>e\ collider energy of (s"500 GeV was taken, and the asymmetries were plotted for the above values of the 2HDM free parameters and as a function of the lightest Higgs mass. We see from Figs. 66 and 67 that both the CP-odd ¹ -odd asymmetry, , and the CP-odd , ¹ -even asymmetry, , peak twice. First when the mass of the Higgs is close to the ttM threshold and , then when it is close to the maximal energy. With m +350 or 400 GeV these asymmetries can F

Fig. 66. The ratio / as a function of the lightest Higgs-boson mass, m , at (s"0.5 TeV. The dashed line F corresponds to tan "0.5 and the solid line to tan "1; m "175 GeV. Figure taken from [339]. R

192


Fig. 67. The ratio / as a function of the lightest Higgs-boson mass, m , at (s"0.5 TeV. The dashed line F corresponds to tan "0.3 and the solid line to tan "0.627; m "180 GeV. Figure taken from [261]. R

reach above 10%. In particular, would lead to a somewhat higher CP-violating signal and we see from Fig. 67 that, for tan "0.3, / is above the 10% level in almost the entire mass range 100 GeV(m (500 GeV, and peaks around m +2m at 50%. Recall that the statistical F F R signi"cance NG of the CP-violating signal that can be measured with a CP-odd asymmetry is 1" given by (9.35) NG " (N , 1" where N "RA AM ;L is the number of expected events, with RA AM the branching ratios for the decay scenarios A, A M , respectively, and assuming a reconstruction e$ciency of 1. Furthermore, L is the collider integrated luminosity and is the cross-section which, in the leading order, is calculated from d in Eq. (9.26). With L"O(10) fb\ the expected number of PttM events is of the order of few;10 for collider c.m. energies of 500}700 GeV. Thus, for example, if we take RA AM "4/27 such that only leptonic top decays into electrons and muons are considered, then an asymmetry larger than 10% will correspond to a signal-to-noise ratio above the 3- level [261]. As previously discussed, if the polarization of the backscattered photons is adjustable then CP asymmetries involving these polarizations can be constructed and they, in turn, can serve as an e$cient tool for investigating the CP properties of the neutral Higgs-boson. Three such polarization asymmetries were suggested in [340]: M !M \\ , P , >> M #M >> \\

(9.36)


2 Im(MH M ) \\ >> , P , M #M >> \\ 2 Re(MH M ) \\ >> . P , M #M >> \\

193

(9.37)

(9.38)

In the helicity basis, one can choose the polarization of (moving in the #z direction) as: !"G2\(0, 1,$i, 0), and that of (moving in the !z direction) as: !"G2\ (0,!1,$i, 0). To understand how the above polarization asymmetries (P ) trace the CP properties of the neutral Higgs, note that a CP-even (CP-odd) scalar couples to two photons via F FIJ(F FI IJ) (see [340] and references therein). Therefore, as implied from Eq. (9.12), in the c.m. IJ IJ of the two photons this will yield a coupling proportional to ) (( ; ) ) for a CP-even X (CP-odd) neutral Higgs to a pair. For the above convention of the polarizations, one "nds i 1 ) "! (1# ), ( ; ) " (1# ) , X 2 2

(9.39)

where , "$1 are the helicities of , , respectively. Now, for a mixed CP state, the general amplitude to couple to will have both the CP-even and the CP-odd pieces in Eq. (9.12) and it can be written as M" ) # ( ; ) , X

(9.40)

where ( ) is the CP-even (CP-odd) coupling strength of the neutral Higgs to the two photons. Using Eq. (9.39), the squares of the helicity amplitudes which appear in P can be readily calculated [340]: M #M "2( # ) , >> \\ 2 Re(MH M )"2( ! ) , \\ >> M !M "!4 Im( H ) >> \\ 2 Im(MH M )"!4Re( H ) , \\ >>

(9.41) (9.42) (9.43) (9.44)

where and are given in [340] for a 2HDM with scalar and pseudoscalar couplings of a neutral Higgs to a pair of fermions. It is then evident that P , P O0 and P (1 only if both , O0. That is, only if both the CP-even and the CP-odd couplings are present. Using Eq. (9.11), for the Higgs-boson production of our interest, one gets [340] dN"dL d(M #M ) >> \\ AA ;[(1# )#( # )P # ( # )P # ( ! )P ] ,

(9.45)

194


where d is the appropriate element of the "nal state phase space including the initial state #ux factor. Note that the properties of dL and of the various 's (appearing in Eq. (9.45)) as a function AA of the c.m. energy of the two photons are very important for this discussion as they depend strongly on the polarization of the incoming electrons and associated photons. Instead of presenting a detailed analysis of those parameters and the numerical results, we refer the reader to [340]. We will only give their summary for a general 2HDM in which the CP properties of a single neutral Higgs have to be determined. In particular, it was found in [340] that out of the three polarization asymmetries de"ned in Eqs. (9.36)}(9.38), P provides the best statistical signi"cance for the task at hand. A non-zero value for P requires that the h coupling has an imaginary part, as well as both CP-even and CP-odd contributions. For a mixed CP Higgs-boson with m :2m , a measurement F 5 of P will be easiest if tan is large since the b-quark loop, which makes the only large contribution to the imaginary part for such m values, will be enhanced. For m '2m , the required imaginary F F 5 part is dominated by the =-boson loop (or t-quark loop if m is also '2m ); large tan makes F R detection more di$cult since the dominant CP-odd contribution originates from the t-quark loop, which will be suppressed. To summarize, the production of a neutral Higgs-boson by fusion of backscattered laser beams can provide a systematic analysis of the CP properties of the Higgs particle. In particular, PhPttM would be a promising channel for exploring CP-violating e!ects that can arise from an extended Higgs sector, as for quite a large range of the 2HDM parameter space this reaction can exhibit statistically signi"cant CP-nonconserving signals in a high-energy collider running at c.m. energy of (sK500 GeV. Moreover, if the polarizations of the incoming photons are controlled, then detailed information on both the scalar and the pseudoscalar couplings of the neutral Higgs to a pair of fermions may be extracted by considering polarization asymmetries of the two colliding photons. If the neutral Higgs is a pure CP eigenstate, the polarization asymmetries P and P in Eqs. (9.36) and (9.37) will vanish, while, in Eq. (9.38) P "1 (!1) for a CP-even (odd) neutral Higgs. Therefore, a non-vanishing value for P and P and P (1 will imply the existence of an extended Higgs sector beyond the SM and of CP violation in the scalar potential.

10. CP violation in >\ collider experiments The idea to build a high-energy >\ collider is more than 25 years old [343,344]. It has recently gained interest in part due to the interesting possibility of using it as an `s-channel Higgs factorya. Of course, it may also be suitable for tackling some other physics issues, e.g., SUSY. For recent reviews see [345}350]. The c.m. energies considered range from 100 GeV to 4 TeV or even more, with luminosity comparable or higher than in linear e>e\ colliders. The subject is still in its infancy compared with the more established technologies of linear e>e\ colliders, and of hadronic colliders such as the Tevatron or the LHC. 10.1. >\PttM If there exists a Higgs-boson with mass of a few hundred GeV, a muon collider running at the Higgs resonance can provide the fascinating and unique possibility of an in-depth study of the


195

Higgs particle in the s-channel. In particular, the CP-violating properties of its coupling to ttM may be studied via the reaction >\PHPttM ,

(10.1)

where we have generically denoted the neutral Higgs resonance under study by H. It is perhaps surprising that Higgs-bosons can be produced at an appreciable rate at a muon collider. Indeed, the H coupling is very small since it is proportional to the mass of the muon, m . On the other hand, if the c.m. energy of the accelerator can be tuned to be at s"mH , then the I cross-section receives appreciable enhancement due to the resonant production of the Higgs. To see how this works, consider a collider tuned precisely at the Higgs resonance, s"mH , then the cross-section H ,(>\PH) for neutral Higgs-boson production is given by 4 H " B , mH I

(10.2)

where B is the branching ratio of HP>\. It is useful to compare this with the point I cross-section "(>\PHPe>e\) .

(10.3)

Thus H 3 R(H)" " B , (10.4) I where is the "ne-structure constant. Therefore, H and R(H) are enhanced if the neutral Higgs has a narrow width, i.e., a relatively large B "(HP>\)/H . I One simple way to study CP-violation at a muon collider is via the decays HPttM . CP-violating correlations can be studied in the decays of the produced ttM pair. Again, this is possible due to the fact that the weak decays of the top quark are very e!ective in analyzing the top spin (see Section 2.8). 10.1.1. A general model for the Higgs couplings To keep the discussion completely general, we will assume that a single neutral Higgs-boson, H, is under study although the underlying model may contain several Higgs doublets. In practice, of course, the muon collider will only be tuned to one Higgs resonance at a time. In Section 3.2.3, we have written an example of a useful parameterization for the H+M ( f"fermion) interaction (see Eq. (3.70)), taking into account possible CP violation in this vertex due to an extended Higgs sector. It is, however, also instructive to introduce a di!erent notation } somewhat more compact } which is useful for the investigation at hand. Let us therefore parameterize the coupling of H to fermions with the Feynman rule [351]: (10.5) CH "iC e A HD , DD DD D where C "!(g /2)(m /m ) is the coupling in the SM and for each fermion f ( f"l, u, d DD 5 D 5 D where l"charged lepton) is a real constant which gives the magnitude of the coupling in relation to the standard model. The CP nature of the coupling is determined by the value of . In D particular, which is not a multiple of /2 is indicative of CP violation since the coupling will then D

196


contain both scalar and pseudoscalar components. CP violation is thus essential in any scalar coupling which is not either pure scalar or pseudoscalar and so learning about is equivalent to D investigating CP violation in HP+M . Moreover, in models with an extended Higgs sector the coupling of H to the boson sector of the theory may be characterized as either scalar, H, or pseudoscalar, A. If H"A then it cannot couple to gauge-bosons while if H"H we can parameterize its coupling to two vector-bosons as (10.6) CH "C cos , 44 4 where, again, C is the coupling in the SM, \PttM . Note that in this reaction both initial and "nal states are CP eigenstates. Furthermore, there is no CP-odd observable that one can construct out of the total cross-section (such as a partial rate asymmetry); if one considers angular distributions of the "nal t-quark, such distributions depend only on the angle , i.e., the angle between the \ and the t-quark momenta IR in the c.m. frame. Since cos is a C-even P-even quantity we clearly need more information if we IR are to observe CP violation. Indeed, if the dominant amplitude is mediated by scalar exchange the angular distribution will be isotropic in any case. To construct CP-violating observables we therefore need information about the polarization of the fermions: either the "nal state ttM or the initial state >\. In Section 10.1.2, we consider the use of correlations in the top polarization in >\PHPttM to measure the CP-violating parameter of HttM coupling, i.e., sin 2 , where is the angle in Eq. (10.5). R R In Section 10.1.3, we consider measurement of CP violation in the same reaction (>\P HPttM ) except this time the asymmetry we construct is based on polarized muons. Clearly, to perform such experiments it is necessary to have a muon collider capable of producing muons with a signi"cant polarization. Finally in Section 10.2 we consider the possibility of #avor changing neutral Higgs couplings which could give rise to CP violation in the reaction >\Ptc versus >\PtM c. Large couplings of this sort may be expected in 2HDM of type III which is described in some detail in Section 3.2.2. Here again the use of top and/or muon polarization is essential to obtain CPviolating signals.

We note that in the language of the interaction Lagrangian in Eq. (3.70), O/2 corresponds to having aH, bHO0, D R R where aH(bH) is the scalar (pseudoscalar) H coupling to a ttM pair. R R


197

10.1.2. Decay correlation asymmetry Let us now consider the case of a muon collider where unpolarized muons produce ttM through >\PHPttM . Thus, in order to learn about the coupling in Eq. (10.5), we can observe the polarization of the top quarks through their decays [351], which is discussed extensively in Section 2.8. Here we will just consider the determination of top polarization by its correlation with the momentum of a particle in various decay modes. Thus, if X is a decay product of a top decay, we de"ne the ànalyzing powera: R ,3 cos , (10.8) 6 6 where is the angle between p and the spin of the top in the top rest frame. 6 6 Let us now extend this idea to study the correlations of the polarizations of the top quarks where the polarizations are indicated by the momenta of speci"c decay particles. As discussed in Section 2.8, for the case of a single polarized top, some further optimization may follow from going beyond this which we do not consider here. Following [351], we work in the rest frame of the Higgs-boson with the t momentum along the z-axis. Let each of the t-quarks undergo a decay which can analyze the top polarization. Let x and G x be the outgoing particles which we wish to correlate with the t and tM polarization respectively; for H instance, the lepton or =!. Also, let y and y denote the rest of the decay products. Thus the two G H decays are tPx y and tM Px y . We can then de"ne the azimuthal angle between p G and G G H H V p H projected on to the x, y plane as V ( p ;p H ) ) p V R . (10.9) sin( )" VG GH p G p H p V V R The azimuthal di!erential distribution of t and tM events is then given by d "1! R R cos 2 cos # R R sin 2 sin . (10.10) R GH 16 G H R R GH d 16 G H R GH The coupling is de"ned in Eq. (10.5) for f"t and R , R are the analyzing power of the decays R G H (de"ned by Eq. (10.8)). Also, , are phase space factors which approach 1 as mH \PHPe>e\ and for a "nal state X, R "(>\PX)/ . In the above R is R from SM processes only, which 6 H H needs to be included as it contributes to the background. Note that the interference between the SM and the Higgs exchange will be negligible; from helicity considerations, such an interference term will be suppressed by m /mH . The SM does I however contribute as a background, hence the term R in the numerator of Eq. (10.20). We must H


199

Fig. 68. The values of y( N (i.e., the number of years required for a 3 e!ect) for the three scenarios discussed in the text obtained in [351]. The solid line is y( N for scenario (1) using top polarization correlation. The upper dash-dot line is RRM y( N for scenario (2) using top polarization correlation while the lower dash-dot line (also in scenario (2)) is for the case M RR where the initial muon beams have a longitudinal polarization P"0.9. The short dashed line is y( N obtained in scenario G (2) using transverse polarization of the initial muon beams. The long dash line is y( N for scenario (3) using top RRM polarization correlation. Here we take L"10 cm\ s\.

also consider the e!ect of all of the other decay modes of the Higgs taken together since RH is proportional to B and hence inversely proportional to the total width (see Eq. (10.20)). To get an I idea of how large the CP-violating e!ects can be, we consider y( N as a function of mH ,(s in H Fig. 68 in a number of di!erent scenarios: (1) H"H with "1 and "453 for all fermions and "453. D D (2) H"A with "1 and "453 for all fermions. D D (3) H"A with " "5 and "1/5, "453 for all fermions. J B S D We assume that " " and " " . The signi"cance of H"H or A, as discussed S A R B Q @ above, is that only if H"H does HP==, ZZ contribute to the total width. Thus, in particular, the value of is not relevant to cases (2) and (3), since if H"A, then no boson pairs are produced by the resonance at tree-level. In Fig. 68 which shows the results from [351], we take a nominal luminosity of 10 cm\ s\ and a year of 10 s. (i.e., with running e$ciency of ). The solid line gives the result in the case of scenario (1) while the upper dot-dash line is scenario (2). In both of these cases, y( NM starts at about RR 5 yr near threshold and increases thereafter. The result for scenario (3) is shown with the long dashed line and is considerably smaller, 0.01}0.1 years, due to the narrow width of the neutral Higgs H in this case.

200


One can enhance the signal with respect to the SM through the use of longitudinally polarized beams. If both of the > and \ beams are left polarized with polarization P, then the Higgs production is multiplied by (1#P) thus enhanced while the SM backgrounds are multiplied by (1!P) and thus reduced. More generally, if the > has polarization P> and the \ beam has polarization P\ then the Higgs cross-section gets multiplied by (1#P>P\) while the SM background gets multiplied by (1!P>P\). In the lower dash-dot curve of Fig. 68, we consider the results for scenario (2) where we have taken P"0.9 which gives a reduction of nearly an order of magnitude in the number of years required to observe a 3- CP violating signal. 10.1.3. Production asymmetry in >\PttM via polarized muons As discussed above, some knowledge about fermion polarization is required if information about CP violation in >\PttM is to be obtained. Above we considered the case where we used the polarization of the top quarks to learn about sin 2 . Here we consider the case where the muons R are polarized. The initial production of muons results in a substantial longitudinal polarization since the weak decay P produces predominantly left-handed \ (and right-handed >). If one constructed a single pass colliding beam machine, it should not be too di$cult to preserve this polarization. On the other hand, in the case of muon storage rings the polarization would have to be manipulated in some way since a longitudinally polarized beam will precess at a rate proportional to g!2. In a recent paper Grza7 dkowski et al. [352] discuss the details of how the polarization of muons in a storage ring may be used to make measurements on the Higgs resonance of the type considered in the following sections. Here we will assume that it is possible to prepare muons in a given initial state of transverse or longitudinal polarization. Let us "rst consider an experiment where the muon beams are polarized transversely to the beam axis. The cross-section is then measured as a function of the angle between the polarizations. We can take the z-axis in the c.m. frame to be in the direction of the \ beam and the x-axis to be its polarization while the > beam is polarized at an angle of to the x-axis, that is in the direction I (cos , sin , 0). I I If the ! beams have polarization P then the cross-section (e.g., for >\PttM ) as a function of ! is [351] I ( )"(1!P P cos 2 cos #P P sin 2 sin ) , (10.21) I > \ I I > \ I I where is the corresponding unpolarized cross-section. We could therefore look for the presence of CP-violation by comparing ( "#903) with ( "!903). Thus, we de"ne the CP-odd I I (C-even and P-odd), ¹ -odd asymmetry , (#903)!(!903) "P P sin 2 . (10.22) A " > \ I I (#903)#(!903) Clearly if appreciable polarizations are available and sin 2 +1 these e!ects are dramatic. I In this experiment, we are simply observing a change in Higgs production as a function of , so I in the approximation that the Higgs resonance is dominant, it would not matter in fact what the "nal state is. In practice, the SM e!ects will also produce the same "nal states. Again using this asymmetry we can quantify the amount of run time required to see a signal through Eq. (10.20). In Fig. 68, we


201

show with the short dashed line the value from [351] of y( N, which is the number of years required G to obtain a 3 signal using the initial polarizations for this asymmetry in scenario 2 taking P "P "1. > \ It is also useful in some cases to consider asymmetries which make use of longitudinally polarized muons in the initial state. Such an asymmetry which is ¹ -even, C-even and P-odd was , considered in [353]: (\>PttM )!(\>PttM ) 0 0 . (10.23) A " * * !. (\>PttM )#(\>PttM ) * * 0 0 This asymmetry in the pair production cross-section clearly requires longitudinally polarized muon beams. In principle, a similar asymmetry is possible for other fermions as well. Since this asymmetry is ¹ -even, some absorptive phase is required. If the collider is running near the Higgs , resonance, this will be naturally provided by the complex phase in the Higgs propagator. Thus, a mechanism for generating this asymmetry is the CP violation originating from the mixing between H and Z and/or between the scalar (H) and pseudoscalar (A) Higgs that can occur in extended models. In particular, the CP invariance of the Higgs sector may be broken by the presence of heavy Majorana fermions. Such a scenario can occur in the minimal SUSY model, in which heavy neutralinos are Majorana fermions. E inspired theoretical scenarios o!er another possibility for heavy Majorana neutrinos at the TeV mass scale. The most interesting situation occurs when a CP-even H mixes with a CP-odd Higgs scalar, A, and, as is natural in SUSY models, for M <M , the two states are roughly 8 degenerate, M KM . In particular, if M '2M the broadening of the H due to the two vector & & 8 decay channel can allow signi"cant mixing between the two states. Consequently, Pilaftsis [353] "nds !2K &[ImK &&(m )!ImK (m )] & & , (10.24) A & !. (m !m )#[ImK (m )]#[ImK &&(m )] & & & where GH are coupled channel propagators derived in that paper. Note, in particular, the proportionality to the imaginary parts as expected since the asymmetry is CP¹ -even. , Fig. 69 shows the results from [353] in a model where & is generated by heavy Majorana neutrinos with masses M "0.5, 1.0 and 1.5 TeV. It is assumed that (s"m . Two scenarios for , & the masses and couplings of the Higgs-bosons are considered: (a) M "170 GeV and cos "1, "2"1/ and the asymmetry is observed with a bbM "nal B S state. (b) M "400 GeV and cos "0.1, "2"1/ and the asymmetry is observed with a ttM "nal B S state. The cross-section is shown with solid curves while the asymmetry is shown with dotted curves. Scenario (a) is shown with the curves in the region around (s"170 GeV where the "nal state is bbM while scenario (b) corresponds to the curves in the region (s"400 GeV, with a ttM "nal state. The enhancement of the asymmetry from the imaginary part of the scalar propagators is apparent in the case where the A and H masses are close together, within about 10% of each other.

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Fig. 69. This "gure shows the cross-section (solid curves) and A in Eq. (10.24) (dotted curves) as a function of (s"m !. & in a model with A}H mixing induced by heavy Majorana neutrinos with masses M "0.5, 1.0 and 1.5 TeV and with , "2"1/ . The two curves at the left are for case (a) where M "170 GeV and A is observed in the bbM channel (see B S !. text). The curves at the right are for case (b) where M "400 GeV and A is observed in the ttM channel (see text). Note !. that in both cases the curves are shown in the vicinity of M &M where the mixing e!ects are likely to be most & prominent. Figure taken from [353].

10.2. CP-violation in the yavor changing reaction >\Ptc As mentioned before, one of the unique properties of a muon collider is that, under favorable conditions, it may produce neutral Higgs states in the s-channel. If the Higgs sector contains two or more doublets, then the Higgs couplings may be #avor changing (FC), see e.g., [128,354,355] (see also Section 3.2.2). This can lead to a dramatic tree-level signature of >\Ptc (or ctM ) due to the neutral Higgs resonance [356]. At the same time FC processes do occur at the loop level in the SM and in practically all of its extensions, even if they are forbidden at the tree-level. Thus, a continuum of FCNC reactions of the form >\PZH, HPtc , tM c are expected. Indeed such couplings with CP-violating phases may also naturally arise in R-parity violating SUSY models [357]. Of course such processes are GIM suppressed in the SM but for the purpose of this discussion we are assuming that there is a FC Higgs sector as in Section 3.2.2, thus for the reactions ZH, HPtc , tM c, rates appreciably larger than the SM may be expected [128,358]. Since many such extensions of the SM contain a large number of unconstrained Yukawa couplings, they will, in general, also contain CP-violating phases. Therefore, the interference between the resonant and the continuum processes can lead to CP-odd observables; it is this possibility which we wish to study in this section. Consider now the two processes shown in Fig. 70(a) and (b). Since the Higgs #ips the helicity of the while the Z does not, for unpolarized or longitudinally polarized beams the interference will be proportional to the mass of the muon and consequently exceedingly small and uninteresting. Such a suppression will not occur if the beams are transversely polarized whence a large interference signal may be produced, especially if the resonant (Fig. 70(a)) and the continuum (Fig. 70(b)) processes are of similar strength.


203

Fig. 70. (a) Feynman diagram for >\Ptc through s-channel neutral Higgs exchange. (b) Feynman diagram for >\Ptc through virtual Z and exchanges, where the circle indicates a vertex correction. (c) An example of a vertex correction contributing to >\Ptc, where H is a neutral Higgs with #avor changing interactions to fermions and H! is a charged Higgs.

Bearing all this in mind, we will thus proceed as follows: "rst, we will consider the general case of the resonance production of tc interfering with the continuum and then we will consider, more speci"cally, what signals are produced in models similar to the ones discussed by Atwood et al. [128]. The process which produces tc via s-channel Higgs exchange is controlled by the terms in the Lagrangian (for more details see Section 3.2.2) L "[ #tM c#2#h.c.]H , & I RA where with the parameterization de"ned in Eq. (10.5),

(10.25)

"C e A HI , # . (10.26) I II I I I Similarly, for we can write RA , # . (10.27) RA RA RA Note that unitarity implies that is real while is imaginary. I I CP-violation will then occur if there exists another mechanism for producing tc which the Higgs may interfere with. Here we take this process to be >\PHPtc and/or >\PZHPtc with the amplitude M "e ( ) ) (tM M c) , 8A 8A M I RA where , , tM and c above are Dirac spinors and

(10.28)

"s[(s!m )s c ]\, "1 , 8 8 5 5 A ,A8A#B8A , ,A8A#B8A . (10.29) I I I RA RA RA Here s "sin , c "cos , where is the weak mixing angle. A8A, B8A are real and they, of 5 5 5 5 5 I I course, occur in the tree-level SM Lagrangian. A8A, B8A are form factors which are induced at the RA RA loop level. As mentioned previously, although small in the SM, they may be generated at reasonable levels in some extensions of the SM, for instance, in multi Higgs scenarios which give HPtc , see e.g. [354}356]. Since we are interferring a vector continuum with a scalar resonance, this interference is naturally suppressed by m . This suppression, however, does not apply if at least one of the beams I

204


is transversely polarized. To construct a quantity which is CP odd, we consider the case where the \ beam is polarized in the #x direction and add it to the result where the > is also polarized in the #x direction. We will also consider the case where the continuum is dominated by the Z exchange. Combining the transversely polarized > and transversely polarized \ data as suggested above, we now consider some angular distributions of this combined data which have speci"c properties under CP and ¹ . Let us de"ne the polar coordinates (, ) of p ; in particular, is the azimuthal , A separation between the beam polarization and p . For each event of the form >\Ptc or tM c, let us A also de"ne, ¸ to be #1 for the tc "nal state and !1 for the tM c "nal state. It is natural, therefore, to R consider the following possible asymmetries x " (cos ), x " ¸ (cos) , R x " (sin), x " ¸ (sin) , (10.30) R where (x)"#1 if x50 and !1 if x(0. These expectation values can be characterized in terms of their symmetry properties. Thus x , x are CP-odd; x is ¹ -even and x is ¹ -odd. x and x are CP-even; x is ¹ -even and x is , , , ¹ -odd, indicating that x and x require complex Feynman amplitudes, i.e., FSI phase(s). In the , process at hand, one source of this is the Higgs propagator (see Fig. 70(a)). In fact, since the Higgs is close to resonance, in the experiments being envisioned here, these (CP¹ -odd) observables (i.e., , x and x ) are likely to be the most prominent of the observables since they are enhanced by the resonant phase of the Higgs propagator. In order to observe the signals suggested above one "rst requires a muon collider which is able to deliver beams with a large transverse polarization as well as an energy spread for the beam which is comparable to or smaller than the Higgs peak. Clearly, theories which produce detectable signals should, of course, have fairly large FC tc couplings. As a speci"c example let us consider 2HDM of types III (see e.g., [128,354,355]), also discussed in Section 3.2.2. In these scenarios a 2HDM is considered where the second Higgs doublet has arbitrary Yukawa couplings. The popular Cheng-Sher Ansatz [126]: (m m R A , (10.31) Kg RA 5 m 5 for in Eq. (10.25) is then imposed where is a parameter that needs to be extracted from RA experiment. It is perhaps natural to expect to be of O(1). It is clear that the "rst obstacle to a large signal is having the ZH-exchange continuum in Fig. 70(c), generated by loop corrections, to be sizable. Let us de"ne R "(>\PHPtc , tM c)/(>\PHPe>e\) , (10.32) & R "(>\PZHPtc , tM c)/(>\PHPe>e\) . (10.33) 8 Clearly then, a necessary condition for there to be large O(1) asymmetries is that R +R . For & 8 m &150}350 GeV, typically R &10\}10\ [358]. Such a signal would have a marginal 8 &


205

chance at a L"10 cm\ s\ luminosity machine. The Higgs signal in this case can also be of order R &10\ } 10\ in the scenario where the Higgs decay to two vector-bosons is allowed. In & that case, &O(1 GeV), so the spread in beam energy needed to be in the resonance region of the & Higgs should be achievable at a muon collider. This could allow asymmetries of a few tens of percents provided that the CP-odd weak phase di!erence were large. Since R &R &10\, & 8 observing the asymmetry would still require a 10 cm\ s\ collider. The situation, of course, would improve considerably if the continuum were larger; for example, this happens if '1 in Eq. (10.31). Since the continuum scales as the integrated luminosity required to observe these asymmetries scales as \.

11. Summary and outlook There are only two known systems which have been shown to violate the CP symmetry: the neutral kaon through the parameters and and the entire universe through the dominance of matter over antimatter. It would be of great signi"cance to understand the relation between these two e!ects or trace them to a common origin. Although experiments in the near term are likely to clarify the source of CP violation in the kaon system, the mechanisms of baryogenesis remain in the realm of theoretical speculation and may not be directly tested in the lab for some time. The top quark, however, o!ers a unique system where new CP-violating e!ects could be discovered which could, in time, shed light on the processes which were important in the early universe. The immediate source of CP violation in the kaon is thought to be the CKM phase in the SM. This will be tested in detail in the next few years through the study of the B meson. Ironically, although the exchange of virtual top quarks generates the large CP violation in the B and K mesons, the CKM phase will not produce any signal in top quark systems that is large enough to be of experimental interest. Instead, if a CP-violating signal is seen in the top quark, it must be due to some inherently large, non-standard, CP-odd phase which becomes manifest only at high energy scales. Since the e!ect of the CKM phase is also thought to be too weak to explain baryogenesis, the required phase for this process is likely to show up in top quark physics as well. Thus, the observation of CP violation in top quark reactions is an unambiguous signal of physics beyond the SM which may well shed light on baryogenesis. In this review, we have considered a number of laboratory tests of CP violation in the top quark in the context of various non-standard models for physics beyond the SM. In particular, we focus on two classes of models which are described in some detail in Section 3: 1. Multi Higgs models containing at least two Higgs doublets with phases in the Yukawa couplings. Here CP violation manifests either in the neutral or in charged Higgs sectors. 2. SUSY models wherein we consider in detail the MSSM with the Yukawa couplings given by N"1 minimal SUGRA models. One manifestation of SUSY CP violation is through mixing in the sfermion sector. We chose to focus on these models because they seem to be representative of models for physics beyond the SM which could give rise to CP violation and are most often considered in the literature. It is likely that the ability of a particular signal to detect CP violation in one of these models is a good indication of its general utility.

206


In this review we highlight some notable CP-violating phenomena which follow from these models: E The transverse polarization of the in tPb which follow from CP violation in the charged Higgs sector (see Section 5). E CP-violating correlations in e>e\PttM H, ttM Z and e>e\PttM at high energy e>e\ colliders C C generated by CP violation in the neutral Higgs sector. Since these e!ects arise through the interference of two tree-level graphs the resulting correlations can be very large indeed (see Section 6). E CP-violating correlations in hadronic top pair and single top production which can originate from CP-odd phase(s) in the neutral Higgs sector or in the squark sector of SUSY models (see Sections 7 and 8). E CP-violating top polarizations may arise in top production at muon and/or photon colliders. In particular at such colliders the neutral Higgs(es) can be produced in the s-channel, giving rise to a distinct resonant enhancement which in turn may magnify the CP-violating e!ect in reactions such as >\, PttM or even in the #avor changing channels >\, Ptc #tM c (Sections 9 and 10). E CP-violating moments of the top analogous to the electric dipole moment which may be observed at an NLC from top polarimetry in the reaction e>e\PttM . Such moments can be generated in SUSY models as well as models with an extended neutral Higgs sector (Section 4). E CP-violation in the main top decay tPb=. In this case a CP-odd phase in the stop sector of the MSSM can cause a partial rate asymmetry in tPb=> at the level of a few;0.1% (Section 5). A common feature of both the SM and the models mentioned throughout this review is that CP violation is driven directly or indirectly by Yukawa couplings in the scalar sector of the theory. In the SM the CKM matrix which contains the CP-violating parameter results from the Higgsfermion coupling while with multi Higgs models additional CP violation may result from the couplings between the various Higgs "elds, either from explicit CP violation in the Higgs potential (e.g., Model II) and/or in the Yukawa interaction terms (Model III), or CP can be violated spontaneously if there are more than two Higgs doublets. In SUSY models, phases may be associated with the scalar Lagrangian as well, for instance, from squarks and sleptons mixing. As in the SM, the amount of CP violation is proportional to the non-degeneracy of the relevant mass spectrum. For instance, in MHDMs the non-degeneracy of the Higgs particles is required while in SUSY it is the non-degeneracy of squarks or sleptons of di!erent helicities. It is important to emphasize that, in many ways, the phenomena of CP violation in top quark systems strongly relies on the large mass of the top which, therefore, becomes the key property of the top as far as CP violation is concerned: E The large mass of the top quark allows its polarization to be determined by its weak decays because, unlike the other 5 quark #avors, it decays before it hadronizes and so the information carried by its spin is not diluted. As discussed in Section 2, this allows experiments to consider CP-odd observables involving polarization (i.e., top spin correlations) which is crucial since in many settings no CP-violating observables could be constructed without this information. For example, the CP-violating transverse top polarization asymmetry, suggested in Section 8, may


E

E

E

E

207

be used to probe tree-level CP violation in pp PtbM which otherwise (i.e., without the use of top spins) cannot be observed. In MHDMs, it is the enhancement in the Yukawa coupling of a neutral Higgs to the top quark that is responsible for the enhanced CP-violating e!ect. As discussed in Section 6, this clearly manifests in e.g., e>e\PttM Z, where the only CP-violating diagram present, i.e., the one with a virtual neutral Higgs exchange, is comparable in size to the CP-even SM diagrams that contribute to the same "nal state, due to the fact that the CP-odd, HttM Yukawa coupling may be as large as the gauge coupling. As mentioned above, in the CP-non-conserving e!ects associated with SUSY particles exchanges, it is the large mass splitting between the two stop mass eigenstates of the theory that may be the cause for an enhanced CP-violating e!ect, again, due the corresponding large mass of their SM partner } the top quark. As discussed in Sections 7 and 8, this is the case for example in ppPttM and pp PtbM where the e!ect arises from CP-violating loop exchanges of stop particles. Large m enables the study of CP violation in cases where the CP-odd e!ect is driven by new R thresholds (i.e., absorptive cuts across heavy particles of the underlying theory). As discussed in Section 5, this is the case in e.g., PRA in tPb= within supersymmetry, where it is only viable if m 'm I #m } still allowed by present experimental data, basically, because of the heaviness of Q R R the top. The cases where the CP-odd e!ect is enhanced to the detectable level due only to an intermediate resonance are also a clear manifestation of the important role played by the large m in CP R violation studies. Such is the case in e.g., CP violation in the decay tPb , as discussed in O Section 5, where the intermediate =-boson resonance provides the necessary enhancement, of course, since m 'm . R 5

It is therefore evident that, due to its large mass, the top is very sensitive to new e!ects from possible new short distance theories. This sensitivity of the top quark to short distance e!ects from many models leads one to consider a more general approach for such studies. For instance, by parameterizing CP violation in a model independent way using CP-violating form factors. Such form factors which contain the information of the dynamics of some new physics scenarios at higher energy scales are expected to be more pronounced in top quark interactions. This technique is a useful prescription for extracting limits on various CP-violating couplings that may arise in new physics. Examples of such e!ective form factors are the top dipole moments and the CP-violating form factors in the top decays which were separately discussed in Sections 4 and 5, respectively. In Section 4, we "nd that in models with extra Higgs doublets as well as in SUSY models, one can expect an EDM and ZEDM of the top on the order of &10\ e cm and, likewise, a CEDM of &10\g cm. These values are many orders of magnitude larger than the SM prediction for these Q quantities. Thus, a discovery of such an e!ect in ttM production in leptonic or hadronic colliders and perhaps also in photon and muon colliders, would be a clear signal of beyond the SM dynamics. In Section 6 we discuss the sensitivity of an e>e\ NLC collider to these EDM and ZEDM. We "nd that optimal observable techniques seem to indicate that high-energy e>e\ colliders will be sensitive, at best, to a top dipole moment at the level of &10\}10\ e cm, about one to two orders of magnitude larger than what is expected in the models mentioned above.

208


In Section 9, we "nd that a similar statement is true at colliders based on the backscattered laser light from an NLC. On the other hand, hadron colliders are expected to be more sensitive to the CEDM, in particular, in Section 7, we "nd that a CEDM at the level of 10\}10\g cm Q might be observable at the LHC. Although this sensitivity of a NLC to the top electric dipole moment may seem a little discouraging, there is still very strong motivation to look for this e!ect; the observation of a top dipole moment with this magnitude (i.e., &10\ e cm), will clearly be a surprise, since such a large dipole moment cannot be accounted for in the popular models such as SUSY and MHDMs. Thus, in spite of the very large enhancements expected in such beyond the SM scenarios for the top dipole moments, it is evident that this type of signal is useful only if the dipole moments are on the very large side of the theoretical range. That being the case, one would like to search for other alternatives for the observation of CP violation in top quark systems. It may, for example, be more promising to look for speci"c signals of CP violation in the production or decay of top quarks which are not related to the dipole moments. In Section 5, we consider the CP-violating e!ects which might be present in the decay of top quarks. The simplest kind of signal is a PRA in the decay tPb= (mentioned above) which in SUSY can have an asymmetry of &10\ and thus may be detectable at the LHC. Another promising signal which is particularly applicable to 3HDM or other models with charged scalars is polarization asymmetries in the decay tPb which arises from the interference of the W pole with the charged Higgs propagator and could result in asymmetries on the order of a few tens of percents. While CP-violating e!ects in the decay of top quarks may be searched for at any experiment where top quarks are produced, there are a broader range of signals where the CP violation occurs in the production of the top quark. In this case, one must consider each kind of top quark production mechanism separately. CP violation in the production mechanism of the top was discussed in the context of an e>e\ collider (in Section 6), hadronic colliders (in Sections 7 and 8), photon collider (in Section 9) and muon collider (in Section 10). Each of these machines has its own characteristics and so special attention needs to be given in constructing appropriate CP-violating observables. For example, lepton and photon colliders have the advantage of their relative cleanliness as far as background is concerned; it should be easier to reconstruct the top quark in such colliders even when it decays via purely hadronic modes. On the other hand, it may be quite challenging for such colliders, e.g., the NLC, to posses the necessary luminosity for studying rare phenomena in top physics such as loop induced CP-violating e!ects. Hadron colliders such as the LHC may have an advantage in this context, since top quarks will be more readily produced there. However, the hadronic environment requires more e!ort in disentangling the CP-violating signal both from the experimental and the theoretical points of view. In addition for hadron colliders, it should be noted that one would prefer, in principle, to always use a pp collider (such as the Tevatron) for CP studies since then the initial state is a CP eigenstate. Unfortunately, the LHC which is expected to produce a very large number of top pairs is a pp collider. It turns out, however, that the initial state at the LHC may not e!ect the CP studies considered here greatly, primarily because the dominating ttM production mechanism there is in fact through a CP eigenstate, i.e., gluon}gluon fusion. Nonetheless, in such colliders it is important to use clean CP-violating observables that can reduce the backgrounds. One such useful observable for the LHC, that was suggested by Schmidt


209

and Peskin was discussed in Section 7. Their CP-odd signal uses the di!erence between the energy of the positrons from tPbe> and electrons from tM PbM e\ in the overall rather complicated C C reaction, ggPttM Pb=>bM =\Pbe> bM e\ . Unfortunately, the expected asymmetry is unlikely C C to be larger than a few times 10\. However, Bernreuther, Brandenburg and Flesch have shown that a considerable improvement may be achieved in this reaction by employing clever cuts on the ttM invariant mass. By doing that they were able to isolate the possible CP-violating contribution from an s-channel Higgs exchange in ggPttM . Thus, in their analysis, asymmetries at the level of a few percent may arise leading to a CP-odd signal well above the 3! level in ppPttM #X at the LHC. Another useful CP-violating signal designed for the Tevatron setting was discussed in Section 8. Speci"cally, it uses an apparent advantage of pp colliders: that there should be a high rate of virtual = production via udM annihilation. In this case a number of asymmetries involving the transverse and longitudinal components of the top spin may be constructed. In both MHDMs and SUSY models we "nd that asymmetries around 1% may thus occur in single top production at the Tevatron. Loop induced CP-violating e!ects such as that of Schmidt and Peskin as well as dipole moments tend to give asymmetries at the level of &0.1!1%. Thus experimental detection of rare CP-violation e!ects in top physics, both in hadronic and leptonic colliders, leads to at least two important challenges. (1) Can detector systematics be controlled to the point that a CP asymmetry of O (0.1%) can be observed? (2) Can CP violation be studied with purely hadronic decay modes of the ttM pair. That is, to what extent will the experimentalists be able to distinguish between the top and the anti-top via purely hadronic modes; if that can be done to a signi"cant level, then the increased statistics will improve the prospects for the observability of such rare CP-odd signals. The small CP-asymmetries which arise from phenomena that occur at 1-loop may make most of those signals too small to be of great use in putting bounds on models of new physics. On the other hand, signals which arise from the interference of tree graphs only are likely to give rise to larger asymmetries. In Section 6, we discuss some candidate signals of this type such as e>e\PttM H, ttM Z, and ttM . In addition, the decay discussed in Section 5, tPb C C O falls into this category. In these cases one "nds that CP asymmetries at the level of tens of percents are possible in models with CP-odd phase(s) in the Higgs sector. This makes that type of CP-violating mechanism quite robust, requiring about a few thousands ttM events per year in order to be detected; such a number may well be within the reach of the future colliders presently under consideration. The two other exotic technologies which may be used in the future in this context (i.e., tree-level CP violation) are colliders and muon storage rings. In Section 9 we discuss reactions which can take place at a collider constructed from an e>e\ NLC collider by backscattering laser light from the e! beams. As mentioned above, these machines can be used to produce ttM pairs through an intermediate Higgs state and interfere it with the born cross-section for top pair production. In this case, observables constructed by considering the top polarization can give asymmetries of up to &10% in 2HDM. In Section 10 we further discuss experiments at muon colliders. Clearly any experiment which can be performed at an e>e\ collider may also be performed at a muon collider. In addition, however, the larger mass of the muon allows us to contemplate the production of Higgs bosons in the s-channel. In such scenarios one can analyze the scalar versus pseudoscalar

210


couplings of the Higgs to ttM by studying the polarization correlations of the tops produced which, again, could give &10% asymmetries. Now, since a large portion of the experimental e!ort in these future colliders will be devoted to the search of supersymmetry, it is particularly gratifying that for such studies of tree-level CP violation, supersymmetry may play an important role in our understanding of the underlying mechanism for CP violation. In particular, once SUSY is discovered and SUSY particles are readily produced in high energy collider experiments, the next step would clearly be to start scrutinizing the basic ingredients of the SUSY Lagrangian, e.g., its CP-violating sector. Indeed, due to the potential richness of CP-odd phases in SUSY theories, tree-level CP violation can easily occur in production and decay of SM#SUSY particles. A promising venue to investigate such tree-level SUSY CP violation may be to search for reactions involving associated top production in "nal states which contain additional SUSY particles and to probe the CPviolating e!ect through top polarimetry, i.e., bypassing the missing energy limitation (typical to SUSY signatures) by using the top spins. A simple example may be CP violation in e>e\Pt#X#missing energy versus e>e\PtM #XM #missing energy, where X is some nonSUSY hadronic "nal state. Another interesting related venue in the context of tree-level CP violation within SUSY models is to search for CP-odd signals in reactions where, although involving SUSY particles, only SM particles are produced in the "nal state. Indeed, if SUSY theories posses R-parity violating interactions, CP may be violated at tree-level even in 2P2 processes in which the initial and "nal states consist of SM particles only. In particular, through SUSY scalar exchanges in which the CP-odd phases are carried by the R-parity violating couplings in the interaction vertices of a pair of SM particles to squarks and/or sleptons. Again, such tree-level CP violation may be probed even in a 2P2 process if one uses top spin asymmetries. Consider, for example, single top production at the Tevatron, pp PtbM #tM b. As was discussed in Section 8, a transverse top polarization asymmetry can potentially probe tree-level CP violation in this process. Indeed, since s-channel exchanges of charged sleptons can mediate pp PtbM #tM b in R-parity violating SUSY, this transverse top polarization asymmetry can potentially lead to large tree-level SUSY CP-violating signal in this reaction. These types of tree-level CP violation in SUSY models were not discussed in this review or anywhere else in the literature to date and could be useful to examine in the future, especially once SUSY is directly observed. More generally, the subject of tree-level CP violation seems promising and requires additional e!ort from the theoretical point of view. In parting, the study of CP violation in top quark physics deserves to be one of the main issues on the agenda of the future high-energy colliders. The expected high production rate of top quarks in these colliders turns these machines into practically top factories enabling the examination of what is presently considered rare phenomena in top physics. In particular, these colliders provide a unique opportunity for the study of CP violation } a phenomenon that till now seems to be essentially con"ned only to the kaon system } and its relation to top quark dynamics. The manifestation of CP violation in heavy particles systems in general and in the top quark system in particular, can shed light on new aspects of this phenomena due to the high-energy scales involved, possibly on new physics related to the dynamics of our universe in its very early stages. One, of course, should not forget the importance of the up coming CP measurements in the B system. On the other hand, it is also important to note that there is a very interesting interplay


211

Table 16 The underlying source of CP-odd phase and the mechanism responsible for CP violation in the processes indicated. / means CP / from tI !tI mixing, HCP / means CP / from scalar}pseudo-scalar mixing in the Htt vertex Note that: tI CP * 0 / means a CP / phase in the H>tb and/or H> vertices. See also Section 3 and H>CP O Process

CP source

Mechanism

tPb=

/) MSSM (tI CP

1-loop

tPb O

/) MHDM (H>CP

=>-resonance in tree-level =>!H> interference

pp PtbM

/) MSSM (tI CP & /) MHDM (HCP

1-loop

pp PttM


Top } CEDM (1-loop)

ppPttM

/) N MSSM (tI CP & /) N MHDM (HCP

1-loop s-channel H & 1-loop

e>e\PttM


Top } EDM, ZEDM (1-loop)

e>e\PttM H, ttM Z

/) MHDM (HCP

Tree-level interference

e>e\PttM C C

/) MHDM (HCP

s-channel H in Tree-level interference

>\PttM

/) MHDM (HCP

H-resonance

PttM

/) MHDM (HCP

s-channel H & 1-loop

between CP violation in b physics and in t physics. In b physics, one expects large CP-violating signals due to the CKM phase alone. Therefore, non-observation of CP violation in B decays would, in fact, stand out as a signal of new physics. This is, of course, in complete contrast to the situation in the top system in which one does not expect any CP-odd signal with the CKM phase of the SM. Therefore, any signal of CP violation in top reactions will unambiguously prove the existence of new physics. Moreover, in order to disentangle e!ects of new physics in the B system, one will need precision measurements and cross-checking of the di!erent available CP-violating B decay channels. In top systems the advantage is that no signi"cant e!ort is needed in order to establish the existence of new physics phenomena in CP-odd top correlations } any measured CP-non-conserving e!ect in top systems will su$ce. Finally, in Tables 16 and 17, we summarize the main features of some of the most interesting CP-violating signals that were discussed in this review.

212


Table 17 The asymmetries that can probe CP violation in the processes indicated and their expected size. Also indicated is the place in which each asymmetry was discussed in the review, i.e., its equation number. The size of the asymmetry given tends to be optimistic, i.e., on the large side of its theoretical range Process

Type of asymmetry

tPb=

PRA } A (Eq. (5.4)) pol. } e.g., (transverse) A (Eq. (5.85)) X Cross-section } A (Eq. (8.19))

tPb O pp PtbM pp PttM ppPttM

e>e\PttM

Size (%) 0.1 10

1 Top pol. } e.g., (transverse) A(y( ) (Eq. (8.21)) Lepton energy } e.g., (transverse) A (Eq. (8.7)) 2 Optimal observable } e.g., O (Eq. (7.4)) Top pol./lepton momenta } e.g., N (Eq. (7.17)) *0 Lepton energy } e.g., (transverse) N(E ) (Eq. (7.21)) 2 Optimal observable } e.g., O (Eq. (6.8)) 0 Top pol./lepton momenta } e.g., ¹K (Eq. (6.24)) GH

0.1 0.1}1

0.1

Angular distributions } e.g., A () (Eq. (6.43)) SB Energy distributions } e.g., Al l (Eq. (6.50)) e>e\PttM H, ttM Z e>e\PttM C C

Top momenta, optimal observable } O, O (Eq. (6.69)) Top pol./lepton momenta } e.g., A (Eq. (6.114)) W Top pol./lepton momenta } AR (Eq. (10.13))

>\PttM

PttM

10 10 10

Muon beam pol. } e.g., A (Eq. (10.22)) I Top pol./lepton momenta } e.g., (Eq. (9.33))

10

Photon pol. } e.g., P (Eq. (9.36))

12. Note on literature survey The literature survey for this review was primarily completed in Dec. 1999.

Acknowledgements Two of us (G.E. and A.S.) are most grateful to the US}Israel Binational Science Foundation for its support that proved very valuable during the long period that took to write this review. GE would also like to thank the Israel Science Foundation and the Fund for Promotion of Research at the Technion for partial support. This work was also supported in part by US DOE Contract Nos. DE-FG02-94ER40817 (ISU), DE-AC02-98CH10886 (BNL) and DE-FG03-94ER40837 (UCR).


213

Appendix A. One-loop C functions In this appendix, we de"ne the coe$cients C , (x"0, 11, 12, 21, 22, 23, 24, see below) correspondV ing to 1-loop integrals with three internal propagators in the loop, i.e. `triangle-likea 1-loop diagrams. In the review they appear in Section 4.3 (Eqs. (4.16) and (4.17)), in Section 4.4 (Eqs. (4.35) and (4.36)), in Section 4.5 (Eqs. (4.46), (4.53), (4.54), (4.62) and (4.63)), in Section 5.1.4 (Eqs. (5.38)}(5.45)), in Section 8.2.1 (Eqs. (8.32) and (8.33)) and in Section 8.2.2 (Eqs. (8.41) and (8.42)). These three-point loop from factors which are functions of masses and momenta are de"ned by the 1-loop momentum integrals as follows [359,360]:

dk 1; k ; kk ; k k I I I J , C ; C ; CI ; C (m , m , m , p , p , p ), I I IJ i D D D where

(A.1)

D ,k!m , (A.2) D ,(k#p )!m , (A.3) D ,(k!p )!m , (A.4) and p "0, i"1!3, is to be understood above. G G The three-point loop from factors are then given through the following relations [361]: C "p C #p C , (A.5) I I I CI "p CI #p CI , (A.6) I I I C "p p C #p p C #p p C #g C , (A.7) IJ I J I J IJ IJ where ab ,a b #a b . The numerical evaluation of the above form factors can be performed IJ I J J I using the algorithm developed in [359,360].

Appendix B. Abbreviations We list in this appendix all the abbreviations used throughout this review: SM CKM GIM NLC LHC PRA FSI PIRA MHDM

standard model Cabibbo}Kobayashi}Maskawa Glashow}Iliopoulos}Maiani next linear collider large hadron collider partial rate asymmetry "nal state interactions partially integrated rate asymmetry multi Higgs doublet models

214


SUSY SSB VEV NLO 2HDM 3HDM MSSM FCNC FC NFC REWSB NEDM EDM RGE SUGRA EW TDM ZEDM CEDM FF LSP DCS

supersymmetry or supersymmetric spontaneous symmetry breaking vacuum expectation value next-to-leading order two Higgs doublet model three Higgs doublet model minimal supersymmetric standard model #avor changing neutral currents #avor changing natural #avor conservation radiative electroweak symmetry breaking neutron electric dipole moment electric dipole moment renormalization group equations supergravity electroweak top dipole moment weak(Z)-dipole moment chromo-electric dipole moment form factor lightest supersymmetric particle di!erential cross-section

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V.K.B. Kota / Physics Reports 347 (2001) 223}288

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EMBEDDED RANDOM MATRIX ENSEMBLES FOR COMPLEXITY AND CHAOS IN FINITE INTERACTING PARTICLE SYSTEMS

V.K.B. KOTAa,b a

b

Physical Research Laboratory, Ahmedabad 380 009, India Max-Planck-Institut fuK r Kernphysik, Postfach 10 39 80, D-69029 Heidelberg, Germany

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 347 (2001) 223}288

Embedded random matrix ensembles for complexity and chaos in "nite interacting particle systems V.K.B. Kota * Physical Research Laboratory, Ahmedabad 380 009, India Max-Planck-Institut fu( r Kernphysik, Postfach 10 39 80, D-69029 Heidelberg, Germany Received September 2000; editor: W. Weise

Contents 1. Introduction 2. Embedded ensembles: EGOE(k) 2.1. De"nition 2.2. Basic results 2.3. Statistical spectroscopy 3. Deformed embedded ensembles 3.1. Onset of chaos in "nite interacting many-particle systems 3.2. Strength functions: transition from Breit}Wigner to Gaussian form 3.3. Statistical mechanics for "nite systems of interacting particles via smoothed strength functions 3.4. Other deformed EGOE 4. Interaction-driven thermalization and Fockspace localization 4.1. Chaos and interaction-driven thermalization

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4.2. Embedded ensembles and Fock space localization 5. Conclusions Acknowledgements Appendix A. Some basic results for classical random matrix ensembles Appendix B. EGOE(2) for Boson systems Appendix C. Edgeworth expansions Appendix D. EGOE results for NPC and S Appendix E. Unitary decomposition of the Hamiltonian and trace propagation E.1. Unitary decomposition of operators E.2. Trace propagation Appendix F. Convolution forms in statistical spectroscopy References

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Abstract Universal properties of simple quantum systems whose classical counter parts are chaotic, are modeled by the classical random matrix ensembles and their interpolations/deformations. However for "nite interacting

* Corresponding author. Physical Research Laboratory, Ahmedabad 380 009, India. Tel.: 91-79-6302129; fax: 91-796301502. E-mail address: [email protected] (V.K.B. Kota). 0370-1573/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 1 3 - 7


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many-particle systems such as atoms, molecules, nuclei and mesoscopic systems (atomic clusters, helium droplets, quantum dots, etc.) for wider range of phenomena, it is essential to include information such as particle number, number of single-particle orbits, lower particle rank of the interaction, etc. These considerations led to resurgence of interest in investigating in detail the so-called embedded random matrix ensembles and their various deformed versions. Besides giving a overview of the basic results of embedded ensembles for the smoothed state densities and transition matrix elements, recent progress in investigating these ensembles with various deformations, for deriving a statistical mechanics (with relationships between quantum chaos, thermalization, phase transitions and Fock space localization, etc.) for isolated "nite systems with few particles is brie#y discussed. These results constitute new progress in deriving a basis for statistical spectroscopy (introduced and applied in nuclear structure physics and more recently in atomic physics) and its domains of applicability. 2001 Elsevier Science B.V. All rights reserved. PACS: 02.50.Ey; 05.45.Mt; 21.10.!k; 21.60.Cs; 24.60.Lz Keywords: Chaos; Shell model; Random matrix ensembles; GOE; EGOE; Information entropy; Bivariate strength distributions; Strength functions; Statistical spectroscopy; Statistical mechanics; Finite interacting many-particle systems; Fock space localization

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1. Introduction The slow neutron resonances mark a region of chaos in heavy nuclei [1,2]. The level and strength #uctuations in this region are described by the Gaussian orthogonal ensemble of random matrices which are "rst introduced and studied by Wigner in the 1950s. The classical (Gaussian orthogonal (GOE), unitary (GUE) and sympletic (GSE)) ensembles are classi"ed by Dyson and studied in detail by Wigner, Dyson, Mehta, Gaudin, Porter, Rosenzweig, French, Pandey, Bohigas and others in the 1960s and 1970s. Signi"cant results of GOE are: (i) the nearest-neighbor spacing (S) distribution P(S) dS (of unfolded spectra) well represented by the Wigner's surmise P(S) dS&Se\1 dS (but not by the Poisson law e\1 dS); (ii) the Dyson}Mehta spectral statistic showing spectral rigidity; (iii) locally renormalized transition strengths (x) obey the Porter}Thomas law P(x) dx&x\ e\V dx (Appendix A). These results are well found in the slow neutron resonance data produced by the Columbia group in the 1960s and early 1970s, however a decisive positive test of the GOE model via these data came from the much improved data analysis by Bohigas et al. [3]. This and the seminal paper by Bohigas et al. in 1984 [4] on the analysis of level #uctuations of the quantum Sinai's billiard whose classical counter part is known to be completely chaotic has established that the #uctuation properties of classical random matrices are generic and therefore applicable for local spectral statistics of a wide variety of quantal systems. In fact, as Berry states [5] if the system is classically integrable corresponds to that of Poisson systems, if the system is classically chaotic and has no symmetry corresponds to that of GUE and if the system is chaotic and has time reversal symmetry corresponds to that of GOE. Study of various billiard systems, kicked rotor, kicked top, quartic oscillator, quantum maps, polynomial potentials in two dimensions, experiments with hydrogen atom in a strong magnetic "eld, microwaves in metal cavities (superconducting microwave billiards) and acoustic resonances in aluminum blocks, etc. by large number of authors in post Bohigas era (i.e. from 1984 onwards) established random matrix physics to be one of the central themes of quantum chaotic systems. In the study of these systems and applications to real physical systems (atoms, molecules, nuclei, mesoscopic systems, etc.), it became clear that one has to consider various interpolations and deformations of the Gaussian ensembles with or without the regular Poisson and uniform spectra. To this end studied are: (i) GOE}GUE, GOE}GSE, GSE}GUE interpolations, with, for example GOE}GUE giving bounds on the time reversal non-invariant (TRNI) part of the nucleon}nucleon interaction, statistics of levels in a metallic ring when pierced by a magnetic "eld, etc.; (ii) Poisson to GOE, GUE and GSE and uniform to GOE, GUE and GSE, with, for example Poisson to GOE being important for order}chaos transitions in the low-lying and near-yrast levels in nuclei, Poisson to GUE for metal}insulator transitions, etc.; (iii) partitioned random matrix ensembles with the 2;2

The word chaos (or quantum chaos) refers to quantum systems which in the classical limit show chaotic dynamics. However, the usage of this phrase even for systems like atomic nuclei, which do not possess a classical limit, is now quite wide spread. At present, it should be understood that complex quantum states are referred as chaotic. In fact, most of the studies presented in this article attempt to characterize and quantify complexity and/or chaos in interacting many particle systems. The dynamics of a classical Hamiltonian system is called regular if the orbits of the system are stable, to in"nitesimal variations of initial conditions. It is called chaotic if the orbits are unstable to in"nitesimal variations of initial conditions. Useful quantities to calculate this behavior are the Lyapunov exponents.


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ensembles giving bounds on isospin breaking in Al, the 5;5 ensembles being important for two coupled quartic oscillators, mixing of several GOE's being important for ¸S to JJ coupling transitions in atoms, etc.; (iv) Dyson's circular ensembles which do not need unfolding of energy levels; (v) banded random matrices which are, for example, used to describe one-dimensional disordered models in solid-state physics (for example quasi-1D disordered wires); (vi) sparse random matrices and quasi-random matrices for localization; (vii) random matrices related to orthogonal polynomials and non-Gaussian random matrix ensembles; (viii) Verbaarschot's chiral Gaussian ensembles (chGOE, chGUE, chGSE) in the study of the Dirac operator of QCD; (ix) parametric random matrix theory (Gaussian process corresponding to GOE, GUE, GSE) for universal statistical correlations; (x) the C, D, CI, DIII class of random matrices introduced by Zirnbauer with applications in mesoscopic systems; (xi) Ginibre's non-Hermitian random matrices with applications for example in dissipative quantum chaos; (xii) embedded random matrix ensembles (they are de"ned and described in detail ahead) for interacting many-particle systems, etc. It is useful to mention that many diversi"ed methods are used to derive results for the ensembles mentioned above and they are: (i) in some limited situations via 2;2 matrices; (ii) Monte-Carlo methods using powerful computers; (iii) exact matrix methods (largely due to Mehta); (iv) Dyson's Brownian motion method which is, for example, used recently in deriving analytical results for banded random matrices; (v) supersymmetry method (nonlinear model) of Efetov which was later extended by Verbaarschot, WeidenmuK ller and Zirnbauer (using this they developed, for example, the theory for cross-section #uctuations in the chaotic compound-nucleus resonances); (vi) the so-called binary correlation approximation due to Wigner which was later used extensively by French for studying embedded ensembles and also the classical ensembles; (vii) in some situations perturbation theory, etc. Finally, we mention that, in the nuclear context nuclear models such as the shell model, cranked shell model for high-spin states, quasipaticlephonon nuclear model (QPNM) of Soloviev, particle-rotor model, etc. on the one hand and group theoretical models such as the interacting boson model (IBM), interacting boson}fermion and fermion}fermion models (IBFM, IBFFM), Ginocchio's SO(8) model, Feng's FDSM model and the simple Lipkin}Meshkov}Glick SU(3) model, etc. on the other are used to study relationships between level and strength #uctuations in quantum systems, random matrix predictions and classical chaos, onset of chaos, order}chaos transitions, etc. Group theoretical models for interacting particle systems in "nite-dimensional Hilbert spaces allow one to obtain their classical analogues via coherent states. Other advantage of these models is that they are semi-realistic and in these models the Hamiltonian matrix dimensions are usually not very large. As the literature for all these is quite vast, we refer the readers to some representative reviews and papers [1}41]; large bibliography is given in the latest review article by Guhr et al. [26]. Besides the analysis of the slow neutron resonance data for testing the GOE results for level and strength #uctuations [3], low-energy data (complete level schemes with respect to JL) of nuclei with 244A4244 by Von Egidy, Mitchell and Shriner, energy levels up to &4 MeV in Sn by Raman et al., near vibrational nuclei by Abul-Magd and rotational bands in deformed nuclei with Z"62}75, A"155}185 by Garrett et al. are analyzed for identifying the region of onset of chaos in nuclei and for order-chaos transitions interpreted mainly in terms of the Brody parameter There are several phenomena in nuclear spectra which can be termed as precusors to chaos and some examples are backbending, signature splitting in B(M1)'s, ¸"4 staggering in superdeformed bands, etc. [42].

228


(Appendix A). Chaotic nature of the slow neutron resonance domain is exploited: (i) to derive a bound on the TRNI part of the nucleon}nucleon interaction by French and collaborators via level and strength #uctuations and statistical spectroscopy; (ii) bounds on parity breaking meason exchange coupling constants in the nucleon}nucleon interaction via experiments with polarized neutrons by the TRIPLE collaboration. Similarly, French and collaborators and WeidenmuK ller and collaborators derived bounds on the TRNI part of the nucleon}nucleon interaction from cross-section #uctuations in detailed balance pair of reactions proceeding through compound nucleus, the major example till to date is Mg# Al#p with intermediate Si compound nucleus. Finally, Mitchell, Shriner and collaborators carried out analysis of the ¹"0 and 1 levels in 0}8 MeV excitation (and the electromagnetic transition strengths among them) in Al for order}chaos transitions due to possible isospin breaking and more recently for the same in P and Cl. See Refs. [3,19,23,43}45] for details of all these data analysis. In addition to all these, the relationship between quantum chaos and random matrix results for #uctuations is being applied to the problem of decay from superdeformed band to normaldeformed (low-lying) states [46] via chaos assisted tunneling [17] and to problems in dissipative collective dynamics as occurs in the decay of giant resonances, "ssion and in collisions between two heavy nuclei, etc. [47]. At this stage, it is appropriate to recall the purpose, as stated by the organizers Altshuler, Bohigas and WeidenmuK ller, of a recent workshop (held at ECTH, Trento in February 1997) on chaotic dynamics of many-body systems: &The study of quantum manifestations of classical chaos has known important developments, particularly for systems with few degrees of freedom. Now, we understand much better how the universal and system-specixc properties of &simple chaotic systems' are connected with the underlying classical dynamics. The time has come to extend, from this perspective, our understanding to objects with many degrees of freedom, such as interacting many-body systems. Problems of nuclear, atomic, and molecular theory as well as the theory of mesoscopic systems will be discussed at the workshop'. Working in this direction, several research groups recently recognized the importance of investigating the embedded ensembles (EE) in detail. With the development of nuclear shell model codes in late 1960s French recognized very early that the statistical properties of nuclear levels and strengths (produced by various transition operators) have their origin in EE (with a shell model code one can construct EE) which primarily take into account the fact that the interaction rank of the nuclear force is much smaller compared to the number of valence nucleons and the many particle states, in the shell model, are direct products of single-particle states. French and collaborators have, in fact, investigated in considerable detail both spectral averages and #uctuations (the later coinciding with the results of the classical ensembles). With this one has statistical spectroscopy (SS) [48}67]. These developments are brie#y reviewed in Section 2. However in the last three years, shell model results for Si [22], spectroscopic calculations for the Ce atom [68}70] and the description of shell model results for the measures, number of principal components (NPC) and information entropy (S ) which depend both on averages and #uctuations, of complexity and chaos [66] substantiating many results of EE and SS, new interest is generated in EE with deeper connections to chaos giving simplicities and possible applications to many other interacting particle systems such as atomic clusters, isolated quantum dots, etc. Nuclear shell model codes, the binary correlation approximation and Monte-Carlo methods, etc. are employed recently in investigating: (i) critical energy and temperature for onset of chaos in "nite interacting many-particle systems; (ii) change in the shape of strength functions as the strength of


229

the interaction is varied, (iii) statistical mechanics for "nite system of interacting particles via smoothed strength functions or partial densities with examples of occupancies, Gamow} Teller (GT) strength sums and transition matrix elements of one-body operators; (iv) 2;2 partitioned EE for mixing between distant con"gurations and EE with additional symmetries generating order out of chaos; (v) chaos and interaction driven thermalization in isolated "nite interacting many-particle systems. Topics (i)}(iv) are discussed brie#y in Sections 3.1}3.4, respectively, with special emphasis on their relevance in future nuclear spectroscopy. Topic (v) and a brief discussion of EE and Fock-space localization form Section 4. These results give a new basis, in terms of chaos and complexity, for SS and enlarge its scope. Finally Section 5 gives concluding remarks.

2. Embedded ensembles: EGOE(k) 2.1. Dexnition Embedded ensembles, in particular, the embedded Gaussian orthogonal enemble of random matrices with k-body interactions (EGOE(k)), are introduced by French and Wong [71] and Bohigas and Flores [72]. Early studies used, for analyzing EGOE(k), the nuclear shell model codes along with Monte-Carlo methods. However, good insight into EGOE(k) is obtained by using the binary correlation approximation [49,57]. The EGOE(k) for many fermion (boson) systems assumes at the outset that the many particle spaces are direct product spaces, of single-particle states, as in the nuclear shell model [73] (in the interacting boson model [74]). Before going further let us de"ne EGOE(k) for m (m'k) particle systems (bosons or fermions) with the particles distributed say in N single-particle states. The EGOE(k) is generated by dexning the Hamiltonian H, which is k-body, to be GOE in the k-particle spaces and then propagating it to the m-particle spaces by using the geometry (direct product structure) of the m-particle spaces. To make clear this de"nition, let us consider EGOE(2) for fermions which is appropriate for atomic nuclei when studied using the shell model (note that most of the discussion in this article is restricted to fermion systems). Given the single-particle states , i"1, 2,2, N, the two-particle G Hamiltonian H(2) is de"ned by H aRJ aRI a G a H , H(2)" I J G H ? J J J J JG JH JI JJ

(1)

where aRJ creates a fermion in the state and similarly a J destroys a fermion in the state J J J . The symmetries for the antisymmetrized two-body matrix elements (TBME) H J I J G H ? being, H "! H , I J H G ? I J G H ? H " H . I J G H ? G H I J ?

(2)

230


The Hamiltonian H(m) in m-particle spaces is de"ned in terms of the TBME via the direct product structure. The non-zero matrix elements of H(m) are of three types, 2 H 2 " H , K K ? G H G H ? JG JH XJK JK 2 H 2 " H , N K K ? N G G ? JG J 2 H 2 " H , N O K K ? N O ?

(3)

all other 2 H 2 "0 due to the two-body selection rules. ? The EGOE(2) is de"ned by (1)}(3) with GOE representation for H in the two-particle spaces, i.e. H I J G H ?

are independent Gaussian random variables ,

H "0 , I J G H ? H "v(1# ). I J G H ? GHIJ

(4)

In (4) bar denotes ensemble average and v is a constant. Note that the H(m) matrix dimension d is d(N, m)"(,) and the number of independent matrix elements ime are ime(N)"d (d #1)/2 K where the two-particle space dimension d "N(N!1)/2. For example, d(11, 4)"330, d(12, 5)"792, d(12, 6)"924, d(14, 6)"3003, d(14, 7)"3432, d(40, 6)"3838380, d(80, 4)"15815 80, etc. and similarly ime(11)"1540, ime(12)"2211, ime(14)"4186. It should be mentioned that the EGOE(2) is also called two-body random ensemble (TBRE). Using (1)}(4), construction of EGOE(2) on a machine is straightforward. In general for fermions, in the dilute limit (de"ned ahead) EGOE(k) is more tractable and this limit is considered in detail in Section 2.2. With EGOE(k) operating in the m-particle spaces, as the corresponding Hamiltonian matrix in the k-particle spaces is represented by a GOE, the m-particle Fock space can be referred as strongly interacting (in the situations that the interaction is not strong, as discussed ahead in Sections 3 and 4, modi"cations of the GOE structure of H in the k-particle spaces are called for). Let us mention that extension of (1)}(4) for interacting boson systems is straightforward; see Appendix B and Refs. [65,75,76]. Similarly de"ning EGOE for mixed particle rank Hamiltonians (in nuclear case H is (1#2)-body) and for Hamiltonians with other extra information is direct and they are considered in Section 3. 2.2. Basic results Investigating EGOE(k) numerically (using shell model with realistic interactions and shell model#Monte-Carlo) and analytically (using the so-called binary correlation approximation; see for example [7]), some generic results that are essentially valid in the dilute limit, which corresponds to (N, m, k)PR, m/NP0 and k/mP0, are derived. These basic results form Sections 2.2.1}2.2.3.


231

2.2.1. State density The state density (eigenvalue density) I(E) or its normalized version (E) takes Gaussian form and it is de"ned by [49] I(E)"[(H!E)\"d(H!E)"d(E) ,

1 1 E! #%-# . exp! (E) P (E)"G (E)" 2 (2

(5)

In (5) [2\ denotes trace (similarly 2 denotes average), the , and d are centroid, width ( is variance) and dimensionality, respectively, of the space over which I(E) is de"ned. Note that "H, "(H! ), &G' stands for Gaussian and the bar over (E) indicates ensemble average (smoothing) with respect to EGOE. The binary correlation approximation, originally used by Wigner for deriving the semi-circle state density for GOE (k"m in EGOE(k)) was employed by Mon and French [49] to derive (5) via the m-particle space moments (H(k))NK of I(E). Keeping technical details to minimum, the dilute limit results are brie#y described here. Firstly, it is seen that by de"nition all the odd moments of I(E) will vanish. Given the k-body Hamiltonian H(k)" = PR(k)Q(k) where PR(k) creates the ./ ./ k-particle state (k) and Q(k) destroys (k), the m-particle trace of H is generated by the trace . / equivalent HH" = = PR(k)Q(k)QR(k)P(k). Under EGOE(k) ensemble average and in the ./ ./ /. dilute limit, using the normalization HK"1, the binary correlated pair HHP(L); n is the I number operator. Then HK"(K). More generally, under ensemble average and in the dilute I limit,

H(k) O(t) H(k)N

n!t k

O(t) .

(6)

In (6) O(t) is a t-body operator and the binary link denotes averaging over the pair of H's below which the link is drawn. Using (6) and that in the trace HNK binary associations dominate, formulas for the moments are derived by writing down all the possible binary associations in HN. Denoting the binary linked pairs as A, B, 2 (A, B, etc. are independent), it is easily seen that H contains three patterns (diagrams), (H(k))K"AABBABBAABABKN2AABB ABABK

"2

m m!k # k k

m k

.

(7)

The values of the irreducible diagrams AABB and ABBA are simple while that of ABAB follows from (6). Then the 4th reduced moment and cumulant k are,

k " !3"HK \HK!3"

m!k k

; m \ !1IPK !k/m . k

(8)

232


Similarly for the 6th moment (H(k)) there are 15 binary association diagrams and they are ACCBABACBCABACBACBACBABCABCCAB ABACCBABACBCABABCCACCABBAACCBB AABCCBAACBCBACCBBAABCCBAACBCBA N5AABBCC 6ABABCC ABCABC 3ACBCAB .

(9)

Here the second row gives the irreducible diagrams. Evaluating them (as in (15) ahead) and taking the dilute limit gives for the 6th cumulant and similarly for the 8th cumulant [57,59] k(6k!1) !4k(23k!9) k " #O(1/m), k " #O(1/m) . m m

(10)

Further details of (6}10) can be found in [49,77] (let us mention that the special case of EGOE(2) was also discussed by Gervois [78]). From (8) and (10) one recovers, in the dilute limit, the Gaussian form for the state densities (note that we need in fact not k/mP0 but k/mP0). Thus, for a two-body interaction m&12 gives a good Gaussian. Note that for m"4 one has k "!1 implying semi-circle shape as seen in many numerical calculations. In practice, one has to apply Edgeworth (mostly 3rd and 4th moment/cumulant) corrections [79] to the Gaussian form; Appendix C gives the Edgeworth form. The EGOE derives its signi"cance from the important result that local #uctuations in energy levels are of GOE type [7]. In fact, it is seen that there is a natural separation of information into long and short wavelengths with damping of intermediate wavelengths. The long wavelength parts correspond to the smoothed Gaussian form (5) 䉴 Fig. 1. (a) State density I(EK ) vs. EK "(E! )/ for a EGOE(1#2) ensemble H "h(1)# atoms [68,70,92]; (iv) occupancies in a symmetrized coupled two-rotor model [115]; (v) occupancies and GT strength sums in Mg using two di!erent order}chaos generating Hamiltonians [88,90]; (vi) occupancies, GT and EM strength sums for several ( fp) shell nuclei using the modern shell model code NATHAN [116]. Most signi"cant conclusion of these studies is that the transition strength sums show quite di!erent behavior in regular and chaotic domains of the spectrum and the agreement shown in Fig. 3 between EGOE and shell model, for the strength sums, is a consequence of the chaoticity of the shell model spectrum. Let us "rst consider the shell model results shown in Fig. 8 for GT strength sums in Mg. They are studied using two di!erent interpolating Hamiltonians. First set of calculations use the spherical shell model mean-"eld (MF) Hamiltonian h(1) as the unperturbed Hamiltonian H and in this case the occupation number operators commute with H , H (MF)"h(1)# to 5> (15}25 levels for each J) with 75 ¹"0 and 25 ¹"1 levels; the ¹"0 and ¹"1 levels coexist. Thus there are 2 GOE's for this system (one for ¹"1 and other for ¹"0) with possible mixing between the two due to, say, coulomb (c) force. Then the appropriate random matrix model is a 2;2 p-GOE with 2GOE to 1GOE transition. The ensemble H then is de"ned by H #< where H is a 2;2 block matrix with dimension d"d #d (d is dimension of the upper block and d of the lower block) and < is a GOE with variance v. The o!-diagonal block of H is zero, upper block H with dimension d is _ a GOE(v ) where v "v(d #d )/d and similarly the lower block H with d is a GOE(v ) _ with v "v(d #d )/d (a slightly di!erent 2;2 p-GOE was considered in [149]). Thus "0 corresponds to a superposition of two GOE's and PRgives GOE. With the transition parameter "v/DM , the number variance in the so-called binary correlation approximation is [35],

r 1 . (r, )"(r,R)# ln 1# " 4(#)

(A.21)

The cut-o! parameter is determined using the result (r, 0)" ([d /d]r)# ([d /d]r) " %-# %-# and formula (A.21) is good for r'2. Note that (r,R) is the GOE value. As discussed before, (r) formula gives the expression for M (r) (Leitner in [37]), 2 1 1 1 ! ! ln(1#Xr) M (r, )"M (r,R)# " 2 Xr 2Xr

#

4 1 9 tan\(Xr)# ! ; X" . Xr 2Xr 4 2(#)

(A.22)

A direct and good test of (A.21) came recently from experiments with two-coupled #at superconducting microwave billiards [150]. It is not out of place to mention that in [150] Eq. (A.21) is ascribed to Leitner [37] although the equation is derived much earlier in [35]. Appendix B. EGOE(2) for Boson systems Let us consider a system of interacting bosons occupying N single-particle states , i"1, 2,2, N and the Hamiltonian is say two-body. Then H(2) is G H I J G H Q bR bR b b H(2)" (B.1) ((1# )(1# ) JI JJ JG JH JG XJH JI XJJ GH IJ

274


with the symmetries for the symmetrized TBME H being, I J G H Q H " H , I J H G Q I J G H Q H " H . (B.2) I J G H Q G H I J Q The Hamiltonian H(m) in m-particle spaces is de"ned in terms of the TBME via the direct product structure of the m-particle states in occupation number representation. The non-zero matrix elements are of three types,

n (n ! ) GH H , ( )LP H ( )LP " G H P P G H G H Q (1# ) 2 2 Q GYH GH PGH PGH

( )LG \( )LH > ( )LPY H ( )LP G H PY P Q PYIJ2 PGH2 n (n #1)(n ! ) IY IYG " G H H , IY H IY G Q (1# )(1# ) IYG IYH IY

( )LG >( )LH >( )LI \( )LJ \ ( )LP H ( )LP G H I J P P Q PKL2 PGH2 n (n ! )(n #1)(n #1# ) IJ G H GH " I J H . (B.3) G H I J Q (1# )(1# ) GH IJ In the second equation in (B.3) iOj and in the third equation four combinations are possible: (i) k"l, i"j, kOi; (ii) k"l, iOj, kOi, kOj; (iii) kOl, i"j, iOk, iOl; (iv) iOjOkOl. EGOE(2) for bosons is de"ned by (B.1), (B.2) and (B.3) with the two-particle H being GOE. Note that the H(m) matrix dimension d is

d(N, m)"

N#m!1 m

and the number of independent matrix elements ime are ime(N)"d (d #1)/2 where the two particle space dimension d "N(N#1)/2. For example, d(4, 11)"364, d(5, 10)"1001, d(6, 12)"6188. Similarly, ime(4)"55, ime(5)"120 and ime(6)"231. For interacting bosons, in general, the dense limit (mPR, NPRand m/NPR) is more interesting as this limit does not exist for fermion systems. Nature of energy level #uctuations in dense boson systems is being studied using EGOE(2) [76]. In this article we will not consider any further the EGOE(2) for boson systems.

Appendix C. Edgeworth expansions Given the standardized variable x( "(x! )/ and the corresponding Gaussian density 1 x( exp! , G (x( )" 2 (2

(C.1)


275

the Edgeworth expansion to fourth-order is [79]

#"

(x( )"G (x( ) 1# He (x( ) # He (x( )# He (x( ) 6 24 72

,

He (x( )"x( !3x( , He (x( )"x( !6x( #3 , He (x( )"x( !15x( #45x( !15 ,

(C.2)

where He (x( ) are Hermite polynomials. It should be noted that the centroid and the width of (x) P and G (x) that correspond to (x( ) and G (x( ), respectively, are identical in the above Edgeworth #" expansion. It is worth noting that several alternatives to Edgeworth expansion (C.2) are suggested in the literature [151]. Given the bivariate Gaussian

1 x( !2x( x( #x( G (x( , x( )" exp ! 2(1!) 2((1!)

,

(C.3)

the bivariate Edgeworth expansion including bivariate cumulants k corrections with r#s44 is PQ [62,79]

(x( , x( )" 1# \#"

k k He (x( , x( )# He (x( , x( ) 2 6

k k # He (x( , x( )# He (x( , x( ) # 6 2

k k He (x( , x( )# He (x( , x( ) 6 24

k k k # He (x( , x( )# He (x( , x( )# He (x( , x( ) 6 24 4

#

k k k k k k He (x( , x( )# He (x( , x( )# # He (x( , x( ) 12 12 72 8

#

k k k k k k k # He (x( , x( )# # He (x( , x( ) 4 12 36 8

k k k # He (x( , x( )# He (x( , x( ) 12 72

G (x( , x( ) .

The bivariate Hermite polynomials He (x( , x( ) in (C.4) are generated by KK RK RK He (x( , x( )G (x( , x( )"(!1)K >K G (x( , x( ) KK Rx( K Rx( K

(C.4)

276


and they satisfy the recursion relation (1!)He (x( , x( )"(x( !x( )He (x( , x( ) K >K K K ! m He (x( , x( )#m He (x( , x( ) , K \K K K \ He (x( , x( )"1 , He (x( , x( )"(x( !x( )/(1!) , He (x( , x( )"(x( !x( )/(1!) , (x( !x( )(x( !x( ) # He (x( , x( )" . (1!) 1!

(C.5)

Appendix D. EGOE results for NPC and Sinfo Information entropy (S ) and number of principal components (NPC) are measures of complexity and chaos in many-body systems. GOE gives d/3 for NPC and ln(0.48d) for S where d is matrix dimension. Recently, the corresponding EGOE formulas are derived [66] and these results are brie#y described here. Let us introduce normalized strength R, average (smoothed) normalized strength RM and locally renormalized strength RK (for a transition operator O), where R(E, E )"EOROE \E OE , D D R(E, E )"EOROE \E OE , D D RK (E, E )"E OE \E OE . D D D

(D.1)

Then the measures NPC and S for strength distributions are

(NPC) " R(E, E ) D # #D

\

,

(S ) "! R(E, E ) ln R(E, E ) . D D # #D

(D.2)

The EGOE expression for NPC is derived by "rst writing (NPC) in terms of (RK ) and RM # and then using the bivariate Gaussian form (11) for the smoothed transition strength densities plus the fact that RK (E, E ) is Porter}Thomas; i.e. the locally renormalized amplitudes D E OE/E OE are Gaussian distributed with zero center and unit variance. The "nal D D


277

formulas are [66],

(NPC) #%-# P RK (E, E ) R(E, E ) # D D #D

"(dD) ( (1!X exp!

\ \ " 3 R(E, E ) D #D

( EK #K d (E) " , X 3

(S ) #%-# P ! R(E, E ) RK (E, E ) ln RK (E, E ) # D D D #D ! (RK (E, E )) R(E, E ) ln R(E, E ) D D D #D

"ln 0.48dD ( (1! exp

1!( (1!) (( EK #K ) exp! 2 2

;

( " /, K "( ! )/, EK "(E! )/ , X"[2!(( )(1!)] . D D G G

(D.3)

In (D.3) and are the centroid and width of ID(E ) and (11) de"nes the other parameters. The D EGOE formulas for NPC and S in shell model transition strength distributions are tested by performing shell model calculations in (ds)K(2 space using a two-body transition operator. Results shown in Fig. 2b con"rm that the EGOE (but not GOE) describes shell model results; see [66] for details. The NPC and S in shell model wavefunctions are de"ned by expanding the eigenfunctions

in terms of the shell model (mean-"eld) basis states (de"ned basically by h) , # I " C# , # I I I

(NPC) " C# I # I

\

,

(S ) "! C# ln C# . # I I I

(D.4)

The EGOE formulas (D.3) are applicable to the NPC and S in wavefunctions de"ned by (D.4). Here ( "1, K "0, ( , ) are same as the centroid and width of the eigenvalue distriG G bution and "(1!( / ). Note that is the average width of the basis states I I # ("d\ O H ) and " . Shell model wavefunctions for the same Hamiltonian I G H G H # G used in Fig. 2b are analyzed for (NPC) and (S ) and the results are shown in Fig. 2c. For this # # example, "10.24 MeV and "0.68. Just as in Fig. 2c, the EGOE formulas (D.3) also explain # the shell model results for NPC and S in wavefunctions given in [22]. See [152] for a very recent discussion on NPC in EGOE(2) wavefunctions.

278


Appendix E. Unitary decomposition of the Hamiltonian and trace propagation Let us consider m identical fermions distributed over r spherical orbits each carrying angular momentum j . Then there are N"P (2j #1) single-particle states and one can recognize the ? ? ? appearance of the ;(N) group which is generated by the N operators aR? ?Ya @ @Y ; , "1, 2,2, r, HK HK !j 4m 4j , !j 4m 4j . The (,) antisymmetric states of the m-fermions form an irrep of ? ?Y ? @ @Y @ K the group ;(N), usually denoted by the Young shape 1K . With this the single-particle creation operators aR? ? belong to 1 and the destruction operators a @ @ belong to 1,\ . The only scalar HK HK operator in the m-particle spaces is the number operator n as it remains invariant under the transformations produced by the generators of the ;(N) group. Partitioning of the shell model ;(N) space can be carried out in many di!erent ways and the partitioning de"ned by irreps of groups that can be realized in shell model spaces are in fact the most signi"cant ones. The signi"cance derives from the fact that the moments de"ned over the irreps in fact propagate from the few particle spaces to the many particle spaces as discussed ahead. Simplest and signi"cant (from shell model mean-"eld point of view) partitioning of m-particle spaces is according to spherical and unitary con"gurations [62,64,94]. The spherical con"gurations m"(m , m 2) ? @ where m is the number of particles in the orbit and m" m ; mP m. Note that the ? ? ? dimensionality d(m) of the con"guration m is

N ? . d(m)" m ? ? For example, for identical particles, denoting 1d , 2s and 1d orbits as C1, C2 and C3 orbits, the (ds) spherical con"gurations are m"(m , m , m )"(3,0,0)(2,1,0)(2,0,1)(1,2,0) (1,1,1)(1,0,2)(0,2,1)(0,1,2)(0,0,3). The unitary group ;(N ) acting in each spherical orbit ? generates m of the spherical con"gurations m; i.e. m behaves as 1K? 1K@ 2 with respect ? to the direct sum group ;(N ) ;(N ) 2 . Thus in the spherical con"gurations space, the ? @ scalar operators are n 's. A unitary orbit is de"ned as a set of spherical orbits. With this, ? decomposition of the m-particle spaces into unitary con"gurations [m] is possible; mP [m]. As above, a unitary con"guration [m]"(m , m ,2); m" m . Using the convention that the spherical orbits belong to unitary orbit and similarly the orbits belong to etc., the number of single-particle states in a unitary orbit is N " N . The dimensionality of the con"guration ?Z ? [m] is d([m])" (, ). For example in the above ds-shell case one can choose (1d , 2s ) K and (1d ) to be two unitary orbits (C1, C2 orbits) and then (ds)P[m]"(m , m )" (3,0)(2,1)(1,2)(0,3). It is important to recognize that [m]P m and the set of spherical con"gurations m that belong to a given unitary con"guration is easy to enumerate. For example in the (ds) case, (3,0)P(3,0,0)(2,1,0)(1,2,0); (2,1)P(2,0,1)(1,1,1)(0,2,1); (12)P(1,0,2)(0,1,2); (03)P(003). The unitary group ;(N ) acting in each unitary orbit generates m of a unitary con"guration [m]; i.e. [m] behaves as 1K 1K 2 with respect to the direct sum group ;(N );(N )2 . Thus in the unitary con"gurations space, the scalar operators are n 's. With spherical and unitary con"gurations, decomposition of the m-particle space is mP [m], [m]P m .

(E.1)


279

It is worth remarking that this decomposition automatically produces "xed-parity subspaces by choosing the unitary orbits to be set of spherical orbits with all of them having same parity. Before turning to unitary decomposition and trace propagation, it is useful to introduce some symbols. With m particles in N single-particle states, the number of holes m""N!m. Similarly for m particles ? in N states and m particles in N states, m""N !m and m" "N !m . Correspondingly, ? ? ? ? n""N !n and n" "N !n . Finally, the symbols [X] , X 2 and X2 are P ?@ ? ? ? [X] "X(X!1)(X!2)2(X!r#1) , P X 2 "X (X ! )(X ! ! )2 , ?@A ? @ @? A A? A@ X2 "X (X ! )(X ! ! )2 , X"N, m, m", n, n" .

(E.2)

E.1. Unitary decomposition of operators One can in general seek a decomposition of a given operator into tensor operators with respect to the ;(N) group similar to what one does with respect to O(3) in angular momentum algebra. Let us consider the tensor decomposition of shell model Hamiltonian H"h(1)#

<M "[n] [N] \ N /(1# ) < , ?@ ?@ ?@ ?Y@

2

(E.5)

One real signi"cance of the ;(N) tensor decomposition is that it is orthogonal with respect to the ;(N) trace, HJHJY" (HJ). Thus there is a ;(N) geometry [48]. JJY With respect to the spherical con"guration group ;(N );(N )2, the non-interacting ? @ particle Hamiltonian h is a scalar and therefore, h" n "h . (E.6) ? ? In (E.6) h "h 2 denotes that it is a scalar with respect to each spherical orbit. The scalar part < "< 2 of < with respect to spherical con"gurations follows by recognizing that it must be a second-order polynomial in n 's. Therefore < "a # b n # C n and then taking ? ? ? ?Y@ ?@ ?@ ? traces on both sides in zero-, one- and two-particle spaces, one has the result < " < n /(1# ) . (E.7) ?@ ?@ ?@ ?Y@ As a one-body Hamiltonian will be a scalar with respect to spherical con"gurations group (see [48] for exceptions), there cannot be an e!ective one-body part of < with respect to spherical con"gurations group. Thus V"
and A"A 0 #A 1 #A 2 ; A 0 "B 0 #C 0 , A 1 "B 1 #C 1 and A 2 "C 2 . Explicit expressions for B , C and A are derived using the methods that gave (E.3)}(E.8) and the "nal results are [62],

, B 0 " [E (A)]n , E (A)" E N N\ ? ? ?Z B 1 "[E 1 (A)]n , E 1 (A)"E ![E (A)] , ? ? ? ?

C 0 0 C (A)"

N C [N ]\ , ?@ ?@ ?Z @Z


C 1 0 E 1 _(A)" ?

281

(N ! )C !(N ! )C ;N !2 \ , @ ?@ ?@ @Z C 2 0 C 2 (A)"C ![C (A)]![E 1 _(A)]#[E 1 _(A)] , ? @ ?@ ?@ A 0 " [E (A)]n # [C (A)]n /(1# ) , Y

A 1 " [E 1 (A)]# (n ! )[E 1 _(A)] n , ? ? ? ? P m ([m]:A)n , ? ? ? ([m]: A)"E 1 (A)# 1 ([m]: A) , ? ? ? 1 ([m]: A)" (m ! )E 1 _(A) , ? ? @ A 2 0 C 2 (A) . (E.9) ?@ Just as the case with (E.4) and (E.5), ([m]) are the induced SPE and 1 ([m]) are the renormalized ? ? SPE (for a "xed unitary con"guration [m]). E.2. Trace propagation Given the (,)-dimensional m-particle space generated by distributing the particles in N singleK particle states, the ;(N) (or scalar) average (or trace)

N \ mOm m ?ZK of an operator O (de"ned, for fermions, by the antisymmetric irrep 1K of ;(N)) propagates from the corresponding averages in few particle spaces, i.e. the ;(N) average is a polynomial in the particle number m (the only ;(N) scalar) with the expansion coe$cients determined by the basic few particle (or input) traces. For example, for a k-body operator F(k) one has the elementary result F(k)K"(K)F(k)I and hence h(1)K"m and #HPPs#p for energies below the threshold for three-body breakup [70] and in three-body nuclear scattering processes [71]. Extension to energies above the three-body threshold, also including Coulomb interactions, employs Faddeev type of components each expanded on hyperspherical harmonics modi"ed with factors describing two-body correlations [72,73]. The e!ort is shifted from a large number of relatively simple basis states to fewer complicated basis states designed to describe parts of the correlations, e.g. the atomic helium trimer ground state [74] and nuclear scattering n#d and p#d [71,74]. The hyperspherical adiabatic expansion was originally introduced to describe the autoionizing states in the helium atom [75]. Although these states are sharp resonances in the continuum they appear as bound states or as shape resonances in the individual adiabatic potentials. The couplings between these states are responsible for the decays. The method is speci"cally powerful for low-energy scattering and bound states. It provides physical insight, e.g. H\ [76], but is less accurate than variational methods, which on the other hand cannot predict resonances such as the autoionizing states in helium. In atomic and chemical physics this method has been successfully applied to many reactions, e.g. scattering e#He> [77], e!#Ps [78], H#H [79], and reactions e\#HPPs#p [80,81], p#\H>, H #FPHF#H [26] and D #FPDF#D [27]. 1.3. The philosophy and structure of the report This report employs the hyperspherical adiabatic expansion allowing thorough, yet transparent, general analyses of a number of physical systems within essentially all sub"elds of physics. The analytic formulations of the method are directly suited for numerical implementations. The distinct advantages are accurate treatment of both large and small distances, in contrast to the generally accepted belief, and identical procedures for bound and continuum states. The method, "rst developed for s-states [21], proved its power by supplying numerical properties of the E"mov

E. Nielsen et al. / Physics Reports 347 (2001) 373}459

379

states [20]. Detailed knowledge of the basic three-body method is used in applications to study few-body correlations within a number of genuine many-body systems [82}85]. The report is intended to describe the method in details and as complete as possible, on a level allowing all physics researcher to follow and judge the arguments. We have generalized the formulation to d dimensions and focused on the mathematical structure, especially at large distances, and extracted as many model independent results as possible. These are often concerned with weakly bound and spatially extended systems, where the details of the interactions are less important. The numerical examples are molecular and nuclear halos. Interesting e!ects are halo properties and E"mov occurrence conditions in d (not necessarily integer) dimensions. Comparison between properties in two and three dimensions are particularly interesting [39,86]. We believe the report will be useful due to the detailed basic descriptions, the general conclusions, the physical insight and the topical systems used as illustrations. We shall "rst in Section 2 sketch a derivation of the angular and radial equations of motion. In Section 3 we explain analytical and numerical details of our method including improvements to deal with a strong repulsive core. In Section 4 we pay special attention to the (sometimes crucial [13,43,39,87,88]) large-distance structure of the Faddeev equations and derive analytically as far as possible the asymptotic behaviour of the corresponding adiabatic potentials and the couplings between them. The exotic E"mov e!ect is discussed in Section 5 for d (perhaps non-integer) dimensions and for arbitrary total angular momentum [11,39,46,86,89]. The practical examples begin with a brief sketch of the properties of two-dimensional systems in Section 6. Then we discuss in Section 7 the bound state energies and structures of the atomic helium trimers [17,30,67]. They provide severe tests of both the numerical method and the possible model independence as well as information about properties and occurrence of E"mov states [30,89]. In Section 8 we study the nuclear halo structures exempli"ed by Li (Li and two neutrons) [54], He (alpha particle and two neutrons) [90] and the hypertriton (proton, neutron and -particle) [91]. Other examples could have been isospin and beta decay [92], the solar neutrino problem [93], the dense helium plasma changing the properties of the three- system with strong consequences for the triple -rate [94] or few-body correlations producing new structures on, or even outside, the neutron dripline [95,96]. Application to the continuum structure and scattering involving three particles in both initial and "nal states are discussed in Section 9. The examples are the continuum structure of nuclear halo systems [97}101]. Other examples could have been the high-energy fragmentation processes of He and Li [40,54,90,102}104]. We shall also discuss the atomic helium trimers. First subject to an external electric "eld as a way of controlling the E"mov e!ect [105]. Second by computing recombination rates of three helium atoms into a dimer and a third helium atom by employing hidden crossing theory [106] combined with the hyperspherical adiabatic expansion [107]. Finally, we included three appendices containing independent results used in the derivations while Section 10 contains a brief summary and the conclusion.

2. Hyperspherical description of three-body systems Three points in a d-dimensional space always de"ne a plane. A three-body problem is therefore basically a planar problem independent of the number of dimensions d of the space, provided

380


of course that d52. For d(2 the internal con"guration of the system is restricted. For d52 the number of available dimensions only in#uences the centrifugal barrier terms. This has profound e!ects on both structure and dynamics of the three-body system. These observations are valid for all types of interactions. The total number of degrees of freedom is 3d including the centre of mass. As we shall see the three-body problem can then be formulated in general for any dimension d52. We shall use Jacobi and hyperspherical coordinates and exploit the hyperspherical adiabatic expansion. We shall assume that d is an integer in all the derivations, even though this restriction is unnecessary. In fact, it is possible to carry out analytic continuations of some of the "nal results and thereby provide the generalization to non-integer dimensions. Similar techniques are used other places, e.g. dimensional 1/d expansion in chemical physics [108,109], analytical continuations in the dimension parameter and dimensional regularization in mathematical and particle physics [110,111]. We shall restrict ourselves to consider systems with short-range two-body interactions as de"ned by Eq. (C.5). It is tedious, perhaps technically di$cult, but straightforward to add three-body short-range forces as well as the spin-dependent spin}orbit, spin}spin and tensor forces. The present formulation may also be useful for long-range interactions. 2.1. Hyperspherical coordinates Let us consider 3 particles in d-dimensions, i.e. 3d degrees of freedom of which d and 2d are related to centre of mass and relative motion, respectively. The masses, coordinates and momenta of the particles are m , r and p , i"1, 2, 3. The total mass and momentum are M " G G G R m #m #m and P "p #p #p . The Hamiltonian is, after subtraction of the centre of mass R energy, given by P p (1) H" G ! R # < (r !r )#< (r , r , r ) , G H I @ 2M 2m R G G G where i, j, k is as an even permutation of 1, 2, 3 such that i is associated with the particle pair ( j, k). The two-body interaction between the pair ( j, k) is then denoted < , which here is assumed to G be central and short range as de"ned in Eq. (C.5). The three-body interaction < is added for @ later use. 2.1.1. Dexnition of the coordinates Let us for each i"1, 2, 3 de"ne the ith set of Jacobi coordinates (x , y ) as G G mm H I , x " (r !r ), " G HI H I HI m(m #m ) H I

m r #m r m (m #m ) I I , G H I y " r! H H " , G GHI G GHI m #m (2) m(m #m #m ) H I G H I where m is a normalization mass. Each of the sets, i, j, k"1, 2, 3, 2, 3, 1, 3, 1, 2, combined with the centre of mass coordinate, describes the system. The space-"xed hyperspherical coordinates


381

(, , , ) are [112] G VG WG x " sin , y " cos , (3) G G G G where is the hyperradius and is the hyperangle con"ned by 04 4/2. The angular parts of G G x and y describing their directions are denoted and , each therefore representing (d!1) G G VG WG angles. The total number of coordinates is thus 2d of which only one carries the dimension length. The total set of the 2d!1 angular coordinates ( , , ) is denoted by or simply . G VG WG G The volume element corresponding to the relative motion is given by dBx dBy "B\d sinB\ cosB\ d d d ,B\dd . G G G G G VG WG The kinetic energy operator in the centre of mass system is now given by

R 2d!1 R K

! ! # , ¹" R R 2m

(4)

(5)

R lK lK R !2(d!1) cot(2 ) # VG # WG , (6) K "! G R sin cos R G G G G and lK and lK are the angular momentum operators corresponding to x and y respectively, see VG WG G G Appendix B for de"nitions when dO3. Here K is the square of the grand angular momentum operator in 2d dimensions, see Eq. (B.3). 2.1.2. The kinematic rotation The connection between di!erent sets of Jacobi coordinates are [113,114] x "!x cos #y sin , y "!x sin !y cos , (7) H G GH G GH H G GH G GH where the rotation angle is con"ned by !/24 4/2 and given by GH m (m #m #m ) I (8) "arctan i, j, k GH mm G H and i, j, k is the sign of the permutation i, j, k. This transformation is usually called the kinematic rotation [115]. The six possible rotations corresponding to iPj, iOj, and the identity operation form a group. Successive rotations of 1P2, 2P3 and 3P1 then return all vectors back to their initial positions and therefore # # ". The hyperradius is independent of the choice of Jacobi coordinates whereas the di!erent hyperangles are related by G sin "sin cos #cos sin !2 cos sin cos sin cos , (9) H G GH G GH G G GH GH G where is the angle between x and y . For a "xed value of , can assume any value in the G G G G G interval between 0 and 2. Then we obtain the constraints

! 4 4 ! ! ! . G GH GH G H 2 2

(10)

382


We may use a body-xxed set of coordinates, e.g. (, , ), where the angles describing the G G orientation of the plane of the particles, the three Euler angles for d"3, are removed. A number of coupled equations increasing with angular momentum and particle spins appears with coordinate and frame singularities related to the Coriolis force for non-vanishing angular momentum [115]. We shall use the laboratory system and the hyperspherical coordinates. Restrictions on the total angular momenta and its partial wave decomposition arise instead from the choice of basis functions. The total number of basis functions needed presumably turn out to be roughly the same in both these procedures. 2.2. Hyperspherical adiabatic expansion We shall use the adiabatic hyperspherical expansion [75], where we "rst solve the angular part of the SchroK dinger equation and then expand the full wave function on the complete set of these angular basis functions. For "xed the set of eigenvalues and eigenfunctions are then obtained as solutions to

2m K ! ()# < (\ sin ) (, )" () (, ) , (11) G HI G L L L

G where we assumed that < only depends on . We also introduced () as an arbitrary function of @ to be chosen later for numerical e$ciency. Then we expand the total wave function on this complete set of solutions " \B\f () (, ) , (12) L L L where we included the radial phase-space factor \B\. The spectrum arising from Eq. (11) is discrete due to the "nite intervals con"ning the angular variables or, alternatively, their periodic nature. The corresponding set of solutions is complete for each value of as shown directly for d"3 in [116]. Inserting Eq. (12) into the SchroK dinger equation with the Hamiltonian in Eq. (1) we obtain by use of Eqs. (5) and (11) the coupled set of hyperradial equations

R 1 (2d!3) (2d!1) 2m(E!< () ) @ ! #

()# ()# !Q ! f () L LL L R 4

R #Q f () , " 2P LLY R LLY LY LY$L where E is the three-body energy and the functions P and Q are de"ned by

R (, ) P (), (, ) L LLY R LY

,

(13)

(14)

R (, ) Q (), (, ) L LLY R LY

,

(15)


383

where these angular matrix elements for an operator O are de"ned by

OI , d H()OI () ,

(16)

with d from Eq. (4). All () are normalized and P "0. The expansion in Eq. (12) is e$cient L LL when the e!ective diagonal potentials in Eq. (13),

()# ()#(2d!3) (2d!1)/4 L !Q #< () , < , LL @ L 2m

(17)

dominate over the coupling terms P and Q , where nOn . LLY LLY The equations in Eq. (13) decouple completely in the adiabatic limit where R R " "0 . L R L R

(18)

Therefore < can provide substantial insight [75] in close analogy to the corresponding L potentials obtained in the Born}Oppenheimer approach [25]. Di!erentiating Eq. (11) with respect to give P and Q as LLY LLY P "0 , (19) LL R< L R LY for nOn , (20) P "! LLY

!

L LY

Q " P P , (21) LL LK KL K$L

!

L KP P Q "2 LLY

! LK KLY LY K$LK$LY L R< 1 R( ! ) L LY #2P for nOn , (22) ! L LY LLY R

!

R L LY 2m 1 (23) V ( )> W ( ) ] G G G G J VG J WG *+

(30)

JV JW ( )"NJV JW sinJV cosJW P B\>JV B\>JW (cos 2 ) , G G L G G L G

(31)

"K(K#2d!2),

(32)

K"2n#l #l , V W

where n is a non-negative integer, P B\>JV B\>JW are Jacobi polynomials and the quantum L numbers n (or K), l and l are non-negative integers limited by l #l 4K. The normalization V W V W constants NJV JW are derived from Eq. (A.4): L

NJV JW " L

(2n#d!1#l #l )(n#1)(n#d!1#l #l ) V W V W . 2(n#d/2#l )(n#d/2#l ) V W

(33)

The free solutions in Eq. (30) form a convenient basis. They can be ordered into degenerate subspaces corresponding to each value of K"0, 1, 2, 2 with the parity (!1)JV >JW "(!1)), see Eq. (32) and Appendix B. The spectrum in Eq. (32) is the same as for the angular momentum operator for one particle in a space of dimension 2d, see Appendix B. This is because the number of degrees of freedom determines the strength of the centrifugal barrier term as seen by transforming the kinetic energy operator from Cartesian to hyperspherical coordinates. The result is then obvious when the degrees of freedom are reduced by d e!ectively removing the centre of mass motion. The familiar example is the spectrum K(K#4) for three particles in three dimensions. 3.1.1. Degeneracy of the free solutions The eigenvalues are determined by the non-negative integer values of K obtained from n, l and V l as in Eq. (32). The eigenstates related to each K-value are separated into states of given total W angular momentum ¸. The remaining degeneracy related to each of the Faddeev components is denoted D(d, ¸, K), where we do not include the trivial degeneracy of 2¸#1 due to angular momentum projection. From Eq. (32) we have in general that l #l 4K. For d"3 each ¸-value, V W limited by l !l 4¸4l #l , corresponds to one state, where the di!erent angular momentum V W V W projections still are not counted. For d"2 the restrictions are instead ¸"l !l or ¸"l #l , V W V W see Appendix B. For d'3 the triangular inequalities and consequently the restrictions on K and ¸ remain the same as for d"3 [120].

388


Fig. 2. The possible combinations of (l , l ) for "xed values of the quantum numbers K and ¸ are shown as every second V W of the points with integer coordinates within the rectangle de"ned by l !l 4¸4l #l 4K. Only every second V W V W point is allowed since l #l must have the same parity as K. For d"2 we have the restriction ¸"l !l or V W V W ¸"l #l implying that only points on three sides of the rectangle are possible. The number of states for d53 is thus V W (¸#1) (K!¸#2)/2 when K#¸ is even and ¸(K!¸#1)/2 when K#¸ is odd. For d"2 we get instead the number of states 2(K!¸)#(¸#1)"K#1.

In Fig. 2 we sketch and compute the allowed sets of (l , l )-values for "xed K and ¸ arriving at the V W number of states D(d"2, ¸, K)"K#1 ,

D(d, ¸, K)"

(34)

(¸#1) (K!¸#2)/2 for K#¸ even ,

¸(K!¸#1)/2

for K#¸ odd .

(35)

3.1.2. The kinematic rotation of the free solutions The free solutions in Eq. (30) can be expressed in any of the three sets of Jacobi coordinates. The operator RK JV JW J V J W describing this kinematic rotation from system j to system i is given in Eq. (28). GH The transformation relating the corresponding free solutions can be de"ned as an overlap matrix RI : GH RI *+)JVG JWG *+)JVH JWH ,*+)JVG JWG ( ) *+)JVH JWH ( ) , (36) GH G G H H which is diagonal in K, ¸ and M due to (kinetic) energy and angular momentum conservation. The matrix elements are also independent of M. The remaining part, RI JVG JWG JVH JWH , is a square matrix of GH dimension D(d, ¸, K). The free solutions constitute a complete orthonormal basis for each of the Faddeev components as well as for the total wave function, i.e. the sum of the three Faddeev components. Each choice of Jacobi coordinate system selects the set of basis functions in Eq. (30). The transformation between these di!erent sets must therefore be unitary and two subsequent transformations must be identical to one connecting the same initial and "nal basis sets, i.e. (37) RI JVG JWG JVH JWH RI JVH JWH JYVG JYWG " VG VG WG WG . GH HG J J J J JVH JWH The matrix elements RI JVG JWG JVH JWH for d"3 are called the Raynal}Revai coe$cients [113,121]. For GH d"2 and ¸"0 they can be found in [39]. RI RI "RI , GH HI GI

RI \"RI R , GH GH


389

The total wave function may be obtained from the three Faddeev components, expressed in any set of Jacobi coordinate, by use of the Hermitian matrix RI of dimension 3D(d, ¸, K) de"ned in terms of the matrices RI : GH RI RI RI . (38) RI " RI RI RI RI RI RI Using the group properties in Eq. (37) we then get RI "3RI from which it follows that the eigenvalues of RI must be 0 or 3. Let us now for given (K, ¸, M) expand each Faddeev component of a wave function " # # on the free solutions in Eq. (30) with expansion G coe$cients CJV JW . The inner product in Eq. (29) between and I is then G I " CJV JW HRI JV JW J V J W CI J V J W ,CRI CI , (39) GGY GY G GGY JV JW J V J W where RI is de"ned in Eq. (38) and C and CI are the states described by the set of coe$cients CJV JW and CI JV JW , respectively corresponding to and I . G G Let us assume that C is a non-trivial eigenstate of RI . Then its norm is zero according to Eq. (39) if the eigenvalue is 0 and non-zero if the eigenvalue is 3. Therefore the space spanned by the eigenstates corresponding to the eigenvalue 3 must be identical to the space of normalizable physical states. This space consisting of the free solutions to the SchroK dinger equation has dimension D(d, ¸, K). Since the full space has the dimension 3D(d, ¸, K) the remaining space spanned by eigenstates corresponding to the eigenvalue 0 must have the dimension 2D(d, ¸, K). This is then the space of non-normalizable states, identically vanishing functions named spurious solutions in Section 2.3. The simplest example corresponds to K"¸"0, where the dimension is D(d, 0, 0)"1 and the free solutions are constants independent of all angular coordinates. Expressing these constants in other coordinate systems give the same constants and the transformation matrix in Eq. (38) must then be

1 1 1

RI " 1 1 1 .

(40)

1 1 1

Thus, the unnormalized eigenfunctions of RI corresponding to the eigenvalue 3 is ( , , )" (1, 1, 1) and a complete set corresponding to the eigenvalue 0 is for instance ( , , )"(1,!1, 0) and (, ,!1). Therefore, the physical space has dimension 1 and the space of spurious solutions has dimension 2. 3.2. Small distance solutions We shall now formulate a method to solve the Faddeev equations in Eq. (27) for general short-range potentials for relatively small values of , i.e. from zero and roughly up to the order of the ranges of the potentials.

390


3.2.1. Expanding on the free solutions We shall expand each angular Faddeev component *+( ) on the complete set of free G G solutions in Eq. (30), i.e. *+( )" CLJV JW *+)JV JW ( ) , G G G G G LJV JW

(41)

where ¸ and M are the conserved angular momentum, CLJV JW are the expansion coe$cients. G The dependence and subsequent summation over n is needed, since non-vanishing interactions imply that K"2n#l #l no longer, in contrast to ¸ and M, is a conserved quantum V W number. Let C denote the column vector consisting of the expansion coe$cients in Eq. (41) of all three Faddeev components sequentially ordered from i"1 to 3. The angular Faddeev equations in Eq. (27) can then be written as [¹I #J V B\>J W (cos 2 ) . G HI G LY G

3

(43)

In JV B\>JW (cos 2 ) , GH G G J G

(60)


397

where , and are related by Eqs. (55) and (56) through energy conservation (same ). The coe$cients RJV JW J V J W ( ) may depend on , but not on . Using the de"nition in Eq. (28) of the GH G operator RK JV JW J V J W we "nd GH RJV JW J V J W ( ) GH sinJ V cosJ W ( #1)PB\>J W B\>J V (!cos 2 ) H H JY H " d d VG WG sinJV cosJW (#1)PB\>JV B\>JW (cos 2 ) G G J G ;[> V ( )> W ( ) ]H [> V ( )> W ( ) ] , (61) J VG J WG *+ J VH J WH *+ where ( , , ) are functions of ( , , ) through Eq. (7). This result formally depends on the H VH WH G VG WG value of , but if the point of divergence "0 is excluded from the integral in Eq. (61) G H the coe$cients are independent of . This condition of excluding "0 is ful"lled for ( , G H G GH see Eq. (10). The coe$cients in Eq. (61) are related to an analytic continuation to non-integer of the matrix elements RI de"ned in Eq. (39). For integer "n and "n we get the precise relation from Eqs. (30) and (A.7) to be

(n#1)NJ V J W LY RI LJV JW LYJ V J W "

(!1)LY GH L>JV >JW LY>JV >JW (n #1)NJV JW L ;RJV JW J V J W ((2n#l #l ) (2n#l #l #2d!2)) (62) GH V W V W for iOj and n, n 50. The Kronecker re#ects the conservation of K. We are now ready to express the above particular solution to Eq. (27), "! ! , G H I explicitly in terms of , i.e. G

JV JW ( )"! RJV JW J V J W ( )AJ V J W G G GH H H$G J V J W ;sinJV cosJW (#1)PB\>JV B\>JW (cos 2 ) . (63) G G J G The complete solution to Eq. (27) for 4 is this particular solution added to the complete G G solution to the corresponding homogeneous equation, i.e.

R R l (l #d!2) l (l #d!2) ! !2(d!1) cot(2 ) #V V #W W G R R sin cos G G G G 2m # < (\ sin )! JV JW ( )"0 . G HI G G G

A change of variable from to r,\ changes Eq. (64) into G HI G R cot(2r /) R l (l #d!2) HI ! !2(d!1) HI # V V Rr (/ )sin (r /) Rr HI HI r l (l #d!2) 2m

W W # # < \ sin HI ! HI uJV JW (r)"0 , G HI (/ ) cos (r /) G HI HI HI JV JW ( )"BJV JW uJV JW (\ ) , G G G G GH G

(64)

(65) (66)

398


where the arbitrary constant BJV JW expresses the relative weight of the homogeneous solution in the G ith Faddeev component. Eq. (65) reduces to the two-body radial equation in Eq. (C.3) when JV B\>JW (cos 2 ) and Q ,Q B\>JV B\>JW (cos 2 ). J J G J J G Continuity of the derivative gives a similar equation obtained by deriving with respect to on G both sides of Eq. (70). Eliminating BJV JW from Eq. (70) gives a matrix equation for the coe$cients G AJV JW valid for all i"1, 2, 3, i.e. G RQ R ln uJV JW G sin() !l cot !l tan Q ! J AJV JW V G W G J G R Rr G HI R ln uJV JW RP G !l cot !l tan P ! J " V G W G J R Rr HI G

; cos()AJV JW # RJV JW J V J W ( )AJ V J W , G GH H H$G J V J W where r"\ in the logarithmic derivative of uJV JW (r). G HI G

(71)


399

This equation is linear in the coe$cients AJV JW and the corresponding determinant must be zero G to allow non-trivial solutions. This in turn de"nes a non-linear eigenvalue equation in . The information about the potentials is contained in the logarithmic derivative of uJV JW at the energy G ( /2m) \, which then in general has to be found numerically by solving Eq. (65). This has two important implications. First, solving Eq. (71) is reduced to "nding solutions to a number of two-body equations. In practice, both angular eigenvalues and eigenfunctions can be computed by this procedure, see Section 7. Secondly, when is a few times larger than r , the angular G eigenvalues are essentially model independent in the sense that they are identical for di!erent potentials provided these logarithmic derivatives are identical. This can be rephrased by saying that potentials resulting in the same two-body phase shifts produce the same angular eigenvalues at large . 3.4. Finite spins of the particles For simplicity of notation we have so far assumed that the particles either were spin zero bosons or that the spin degrees of freedom were totally uncoupled and therefore could be ignored. However, this is not possible for a number of interesting systems, perhaps especially in nuclear physics, where the spin}orbit, spin}spin and tensor forces are very important two-body interactions. The previous chapters and sections demonstrated that the orbital part of the wave functions is decisive for the large-distance asymptotical behaviour. The essential ingredients are therefore already established. On the other hand, a two-body bound state located far away from a third particle must at least asymptotically conserve the total (not orbital) angular momentum of that two-body state. Thus the spins cannot simply be factorized away by including another coupling of the total spin to the orbital part in Eq. (26). Instead it is natural to employ the ls coupling scheme for the individual pairs of particles. Incorporating the corresponding couplings at smaller distances is then rather straightforward. Details can be found in [23]. Several practical examples are discussed in the published literature [40,91,100,123].

4. Large-distance asymptotic behaviour The previous sections discussed solutions for intermediate distances roughly understood as hyperradii larger than the ranges of the two-body interactions r or as de"ned by Eq. (58). In this G section we shall discuss very large distances including the asymptotic limit of in"nite hyperradii. The behaviour of the adiabatic potentials and the coupling terms in this limit is necessary in order to understand the method of the adiabatic expansion. In numerical calculations of bound state properties this knowledge is revealing but not essential, since the wave functions are exponentially decreasing with distance. However, for scattering problems the boundary conditions at "R de"ne the process and the large-distance information is absolutely essential, see Section 9. 4.1. Expansion of eigenvalue equation The overall assumption is that the hyperradius is much larger than the range of the potentials, i.e. JV )AJV JW , G G

402


since the "rst-order pole in the -function cancels the zero point in sin(). Therefore, this coe$cient must vanish accordingly as AJV JW "O(\B\JV ) implying that AJV JW (R)"0. G G For a component with l #l 4K we have Pn50 and Eq. (73) becomes V W d d!2 d #l #l n# #l (n#1) V V W 2 2 2 bJV \J AJV JW ( aJV )B\>JV G (n#B #l )(n#d!1#l #l ) V V W HI G

(76) "AJV JW #(!1)L RJV JW J V J W ( )AJ V J W . G GH H

V W H$G J J From this equation we see that AJV JW "O(J \JV ) and AJV JW (R)"0 for l 'l . For l (l the G G V V left-hand side of Eq. (76) vanishes for PR and therefore the right-hand side must also vanish. Therefore we obtain (77) DJV JW ,cos()AJV JW # RJV JW J V J W ( )AJ V J W "O(JV \J ) . G G GH H

H$G JV JW We conclude that only components with l 4l and l #l 4K can be non-zero in the limit V V W "R. These results are summarized in Table 1. We can now "nd the constant b by solving the equation in Eq. (76) at "R with inclusion of only the contributing components of l 4l and l #l 4K. As and are dimensionless and V V W b has the dimension of length raised to the power d!2#2l we must "nd that b"aB\>J , where a is an average over the only available lengths in the problem, i.e. the scattering lengths aJ , G i"1, 2, 3. The other scattering lengths cannot enter in the eigenvalue behaviour, because Eq. (76) dictates that those with l (l disappear for large and those with l 'l require that the V V coe$cient AJV JW vanishes. G For K"¸"l "0 for three identical bosons in three dimensions a turns out to be 12/ times the two-body scattering length [20,21]. We then get

"n#

\B\J , a

(78)

\B\J . (79) a These equations exhibit the asymptotic behaviour of the angular eigenvalues. The features of a given value are the speci"c -dependence characterized by an angular momentum quantum number l and the approach towards a constant recognized from the free spectrum.

"K(K#2d!2)#2(K#d!1)

Table 1 The asymptotic large-distance behaviour the Faddeev components in the limit PR, i.e. the coe$cients AJV JW , DJV JW G G and BJV JW de"ned in Eqs. (54), (77) and (66). The eigenvalues behave as "K(K#2d!2)#2(K#d!1)b\B\J . G The integers K and l characterize how the eigenvalues approach the asymptotic value at "R

l #l 4K and l 4l V W V l #l 4K and l 'l V W V l #l 'K V W

AJV JW G

DJV JW G

BJV JW G

O() O(J \JV ) O(\B\JV )

O(JV \J ) O() O()

O(JV \J ) O(\JV ) O(\JV )


403

With this knowledge we can now use Eqs. (67), (70), (A.10), (A.11) and the results in Table 1 to calculate the wave function inside the potential regions, i.e. the coe$cients BJV JW related to the G region 4 , see Table 1. G G 4.2.1. Normalization of the asymptotic wave function The normalization of the asymptotic wave function, de"ned by Eqs. (54), (67) and (68), can now be computed when approaches an integer in the limit PR. All contributions from regions, 4 , inside the potentials vanish, because the corresponding coe$cients BJV JW vanish or G G approach a constant as seen in Table 1 and the sizes of the intervals also approach zero, i.e. P0. G The norm is therefore entirely due to the contributions from the regions, ' , outside the G G potentials. From Eqs. (54), (30), (39), (A.7), (62), (77) and Table 1 we obtain for a solution characterized by l that (n#1)RI LJV JW LYJ V J W (n #1) GGY + (!1)L>LY AJV JW HAJ V J W G H NJV JW NJ V J W

L LY GGY JV JW JV JW (n#1) " AJV JW H(!1)L G NJV JW L G JV JW

; (!1)LYAJV JW # RJV JW J V J W ( )AJ V J W G H GH H$G J V J W (n#1) (!1)LAJ JW HDJ JW , + (80) G G NJ JW W L G J where we used that AJV JW DJV JW vanishes in this limit for l Ol . The sum over l in Eq. (80) is G G V W restricted by l #l 4K, since AJV JW vanishes in the large -limit for l #l 'K. From Eq. (76) we G V W W see that if b"0 then also DJ JW "0. In this case also components with l "l do not contribute to G V the norm, which consequently vanish in this large-distance limit. Thus, solutions with b"0 have zero norm and must be spurious. We can now see that there must exist at least one non-vanishing component, AJ JW , with G l #l 4K. Otherwise no non-zero terms would be left in the sum in Eq. (80). The triangular W inequalities l !l 4¸4l #l imply that l 5l !¸, where ¸ is the angular momentum W W W obtained by coupling of l and l . Therefore we also have K5l #l 5l #l !¸ and W W l 4(K#¸). If we now consider a component with l #l 'K then the triangular inequality V W gives us that l 5l !¸'K!l !¸ and therefore l '(K!¸)5(l #l !¸!¸). Thus, V W V V for ¸"0 all components with l #l 'K also have l 'l . V W V For ¸"0 states we can also conclude that inside the potential regions the Faddeev components with l "l fall o! slower with than all other components. This is consistent with the fact that the V scattering lengths aJ , i"1, 2, 3, determine the low-energy properties of any given system at large G distances. This conclusion does not hold in general for ¸'0, where a component with minimum l (l and l #l 'K could fall o! even slower inside the potential regions. For such an l -value V V W V we have l 4¸#l and l "(K!¸)#1(l , which corresponds to a slower fall o! than for W V V that of l "l , see Table 1. V

404


4.2.2. The asymptotic degeneracy The asymptotic states in the limit of "R are identical to the free states obtained without interactions. The angular spectrum of the free solutions are given by (R)"K(K#2d!2) and the degeneracy for each of these values is 3D(d, ¸, K), but only D(d, ¸, K) of these are physically acceptable while 2D(d, ¸, K) are spurious states, see Section 3.1.1. We shall now try to "nd how many physical solutions are there for a given K and ¸ for each l between 0 and the maximum value (K#¸). Let ¹(k) denote the number of Faddeev components with l 4k and l #l 4K, where each l -value in each Jacobi set has to be counted. Let S(k) denote V V W V the number of spurious solutions, which can be obtained from these ¹(k) Faddeev components by combining into identically vanishing total wave functions. The total number of components are ¹((K#¸) )"3D(d, ¸, K) and the total number of spurious solutions are S((K#¸) )"2D(d, ¸, K). A solution corresponding to l "k is a linear combination of the ¹(k) components with l 4k, see V Section 4.2. The total number of solutions with l "0, 1, 2, k is thus ¹(k), but S(k) of these are spurious. Therefore P(k)"¹(k)!S(k) is the number of physical solutions with l 4k and thus P(k)!P(k!1)"¹(k)!S(k)!P(k!1) is the number of physical solutions with l "k. We may understand this by stepwise increasing l : the number of physical states with l "0 is the number of components with l "0 minus the number of ways these can be combined into V spurious states with zero norm. The number of physical states with l "1 is given by the number of components with l 41 minus the number of spurious states obtained by combining components V with l 41 minus the number of physical states with l "0. We may continue in this way until we V have obtained the total number of D(d, K, ¸) physical states and 2D(d, K, ¸) spurious states, all found from l 4(K#¸). 4.2.3. Numerical accuracy and the potential cutows r G The accuracy of the numerical procedure is closely related to the choice of r , i.e. the point G outside which the potential < is assumed to be zero. Then < (\ sin )"0 for ' , see G G HI G G G Section 3.3. Using the asymptotic wave function of a state characterized by l and the approxima tion in Eq. (80) give sinB\>J cosB\>JW 4 AJ JW (n#1) G G G G ? G JW 2m ;P B\>J B\>JW (cos 2 ) < (\ sin ) DJ JW . (81) L G G HI G G

Using the asymptotic behaviour in Table 1 we obtain, after the change of integration variable to r"\ sin 4\, the simple accuracy estimate HI G HI 2m O(\B\J ) dr rB\>J < (r) for l 'l ,

G PG (82) 4 2m O(\B\J ) dr rB\>J < (r) for l 4l ,

G PG where we introduced the maximum short-range angular momentum quantum number l 50 for which the integrals in Eq. (82) are "nite, see Eq. (C.5). These estimates can be as small as desired by choosing r su$ciently large. Except for the solutions of K"0, where l "0 and "O(\B), we G can even conclude that both absolute and relative error / vanish when PR.


405

The freedom in the choice of r can be exploited to obtain a faster convergence of the G eigenvalues, i.e. faster vanishing of with increasing . For r increasing with as r "O(\C), G G where is a small positive number, we use

dr rB\>J

PG

2m 2m < (r) 4rJ \J dr rB\>J < (r) G G

G PG

(83)

and Eq. (82) to obtain the accuracy "O(\B\J >C ) for l (l , where "2(l !l ). For potentials with l "R, e.g. Gaussian, Yukawa and square well potentials, we can therefore choose r J\C causing to vanish faster than any power of . Choosing r as a function of G G does not alter the leading-order terms in Table 1 and the -dependent correction term in Eq. (79). The reason for this is that as long as P0 and Eq. (73) consequently is valid the length scale G decisive for the convergence of each solution is aJ . G The -dependent term in Eq. (79), 2(K#d!1) (/a )\B\J , vanishes faster than in Eq. (82) when l 'l . Dividing the solutions into categories according to the dominating l at large distance is therefore only meaningful for l 4l . Other solutions with l 'l and the same limiting value "K(K#2d!2) couple and all contain a piece of the slowest converging component with the same resulting large-distance behaviour "K(K#2d!2)#O(\B\J ). 4.3. Two-cluster continuum states Let us investigate solutions to Eq. (73) corresponding to large-distance con"gurations with a two-body bound state and the third particle far away. Then we have "O(!), "!(d!1#l )#it and from Eq. (56) therefore "!4t!(d!1), where tP#R. We W assume "rst that l "0 and d(4. V 4.3.1. Weakly bound two-body states The -function for zPR and arg z( may be approximated by [124]

(z)K(2 exp

1 z! log z!z [1#O(z\) ] , 2

(84)

which inserted into Eq. (84) along with the expression for leads to

d i !exp t#i (d!1#l ) cot W 2 2 2

d 2 # exp t#i (1#l ) W (d!2) 2 2

\B AJW G at HI G

1 " exp t#i (d!1#l ) AJW # RJV JW J V J W (!4t!(d!1))AJ V J W . (85) W G H GH 2 2 H$G J V J W

406


The argument in RJV JW J V J W ( ) diverges towards !R. By choosing in Eq. (61) su$ciently G GH small and using Eqs. (A.9), (A.10), (10) and (84) we get (86) RJV JW J V J W (!4t!(d!1))"O(exp( (!2 #)t) ) , GH GH where is an arbitrarily small positive number. The terms in Eq. (85) containing RJV JW J V J W (!4t!(d!1)) therefore fall o! faster than the other terms. Thus for non-zero GH coe$cients AJW we "nd to leading order that G d d!2 B\ sin 2 2 t+C . (87) , C, a d!2 HI G 2

Since t must be a positive real number also the scattering lengths must be positive, i.e. a'0. If G a(0 a bound two-body state is not present in the subsystem, see Appendix C. Using Eq. (C.9) we G now obtain

2m " E , (88) a

HI G where E is the (small) two-body energy for angular momentum 0. In this derivation we have assumed that \r ;1 to allow the use of uJV JW appearing in G G Eq. (70) as the wave function of zero energy of subsystem i. The solution in Eq. (88) is only consistent with this assumption when a?>@>A#e ?> e JA) (1#O( \) ) .

(A.9)

We need a"(d!2)/2#l , b"(d!2)/2#l , x"cos(2), where d is the spatial dimension, l and V W V l are the partial angular momenta and is the hyperangle, see Section 2. From Eqs. (A.2), (A.6) and W (A.8) we then get that (#d/2#l ) V (1#O() ) PB\>JV B\>JW (cos 2)" J (#1)(d/2#l ) V QB\>JV B\>JW (cos 2) J

(A.10)

452


1 ! (2 # (1#)# (1##l )#2 ln ) W #O( ln )

for d#2l "2 , V

d!2 d #l # #l V W 2 2 \B\JV (1#O() ) (#d!1#l #l ) V W #O(ln ) for d even and d#2l 54 , V " d!2 d #l # #l V W 2 2 \B\JV (1#O() ) (#d!1#l #l ) V W d # #l V 2 d !cot (1#O() ) else , d 2 (#1) #l V 2 (A.11)

where "! (1) is Euler's constant. A case of special interest is d"3, l "l "0 and therefore V W a"b". The corresponding P is given by J (#) sin(2(#1)) . (A.12) P(cos 2)" J (#1)() (#1)sin(2) Appendix B. Spherical coordinates in d dimensions We shall here give some pertinent details and derive key formulae for an integer dimension d52, see [108] for the details. We divide the Cartesian coordinates (x , 2, x ) into the "rst d!1 B coordinates (x , 2, x ) and the last coordinate x . We de"ne angles and radii r recursively B\ B I I /x )3[0, ]. Thus by r "(x#r "(x #2#x and "arctan(r B B B\ B B B B\ x "r cos , B B B x "r sin "r sin cos , B\ B\ B\ B B B\ x "r cos "r sin 2 sin cos , B B x "r sin 2 sin sin , B B x "r sin 2 sin cos . (B.1) B B The parity transformation is given by r C r , C ! for 34i4d and C # . The B B G G Laplace operator is RB d!1 R K " # ! B , B Rr r Rr r B B B B

(B.2)


453

where the square of the angular momentum operator for d52 is given by R 1 R !(d!2)cot # K , K "! B R B sin B\ R B B B

K "0 .

(B.3)

The angular eigenfunctions to K "!R/R can be chosen as (1/(2)exp(!il s ), where l is a non-negative integer and s"$1 correspond to the two degenerate solutions related to angular momentum projections of opposite sign. Then the eigenvalues of K are l and the parity of the eigenfunctions are (!1)J . For d'2 we now assume the recursive form of the eigenfunction to be ( , , )"f ( )>B\ ( , , ), >B B JB\ 2J Q B\ 2 JB 2J Q B 2

(B.4)

where >B\ ( , , ) is an eigenfunction of K with the eigenvalue l (l #d!3). JB\ 2J Q B\ 2 B\ B\ B\ We then get the eigenvalue equation and the solutions as

R R l (l #d!3) ! !(d!2) cot # B\ B\ f ( )" f ( ) , B B B R R sin B B B

(B.5)

f ( )"sinJB\ P B\>JB\ B\>JB\ (cos ) , B B L B

(B.6)

"(n#l

(B.7)

B\

) (n#l #d!2)"l (l #d!2) , B\ B B

where n is a non-negative integer and l ,n#l . The parity of this solution is B B\ (!1)L(!1)JB\ "(!1)JB . Thus the eigenvalues of K are l (l #d!2), where l are non-negative B B B B integers, and the parity of the eigenfunctions are (!1)JB . The complete set of (unnormalized) angular eigenstates in d dimensions are > (),> B B\ 2 ( , 2, ) J J J J Q B B G\ B\>JG\ (cos ) , (B.8) "exp(!isl ) sinJG\ PGB\>J G J \JG\ G G where l 5l 5l 50 and s"$1. The two quantum numbers of three dimensions lm B B\ 2 (l"l , m"sl ), are in d dimensions generalized to d!1 quantum numbers lm , where l"l and B m "(l ,l , l , s). B\ B\ 2 The two angular momenta lK and lK associated with the Jacobi vectors x and y couple to a total V W angular momentum ¸K "lK #lK . For d"3 the simultaneous eigenfunctions of lK , lK and ¸K are V W V W given by C*+ > ( )> ( ) , (B.9) [> V ( )> W ( ) ] , JV KV JW KW JV KV V JW KW W J W *+ J V KV >KW + are the Glebsch}Gordan coe$cients. where C*+ JV KV JW KW For d"2 the eigenfunctions of the two angular momentum operators lK and lK are V W (1/(2)exp(!is l ) and (1/(2)exp(!is l ), where l and l are non-negative integers and VV V WW W V W

454


s and s both are $1. The total angular wave function is an eigenfunction of lK , lK and V W V W ¸K "iR/R #iR/R (eigenvalue s l #s l ), i.e. V W VV WW 1 " exp(!is l !is l ) 2 VV V WW W

1 1 1 " exp !i(s l #s l ) ( # )!i (s l !s l ) ( ! ) . VV WW 2 V W WW V W 2 2 VV

(B.10)

We can now assign the quantum numbers ¸S where ¸"s l #s l and S"sign(s l #s l ). VV WW VV WW For each l 50, l 50 and ¸51 there is a degeneracy of 2, S"$1, corresponding to the same V W size of ¸. Thus, S plays the same role for d"2 as the projection quantum number M for d"3. For each ¸"0 and l , l 51 the degeneracy is still 2, namely arising from s "!s "1 and V W V W !s "s "1. These two states belong to the same eigenspace of the total angular momentum V W ¸K and even rotationally invariant two-body potentials may therefore couple them. For ¸"l "l "0 there is no degeneracy. V W We can generalize Eq. (B.9) to d'3, see [120] for details. The corresponding summations over the two sets of the d!2 projection quantum numbers, m and m , each related to a partial angular V W momentum, then produce the state of total angular momentum quantum number ¸ with d!2 projections MM . Appendix C. Basic properties of two-body systems in d dimensions The SchroK dinger equation for two particles of masses m and m described by the relative coordinates r and interacting via a central potential H GH> 4 8 16 G H G H 1 *V *W ! ( # ) GH G>H> G>H\ 16 G H 1 *V *W ( # ) # GH G>H GH> G>H> 16 G H 1 *V *W ( # ) # GH> G>H GH G>H> 16 G H 1 *V *W , (5c) ! GH G>H G>H> GH> 16 G H where (¸ , ¸ ) denote the number of lattice sites in the (x, y)-direction. Readers familiar with the V W Ising model recognize immediately that up to irrelevant constants, the area A and perimeter ; correspond to the magnetization and energy of the Ising model with nearest-neighbor

K. Michielsen, H. De Raedt / Physics Reports 347 (2001) 461}538

471

interactions. The Euler characteristic is a weighted sum of all possible correlations of up to four neighboring spins. Furthermore, it is clear that the Minkowski functionals (A, ; and ) and the two-spin correlation function (or structure factor) 1 *V *W S(k, l)" (6) GH G>IH>J ¸ ¸ V W G H perform di!erent, hence complementary measurements on the con"guration of spins (or pixels).

3. Morphological image processing In the preceding section we took for granted that the digitized images are free of noise and other artifacts that may a!ect the geometry and topology of the structures of interest. Such perfect images are easily generated by computer and are very useful for the development of theoretical concepts and models (see e.g. Sections 6 and 7). Unfortunately, as we all know, genuine pictures or patterns obtained from computer simulations (e.g. a polymer solution, see Section 8) are all but perfect. Therefore, some form of image processing may be necessary before attempting to make measurements of the features in the image. Digital image processing is very important for many industrial, medical and scienti"c applications. There is a vast amount of literature on this subject so we can only cite a few books here [3}6]. There is also a huge number of di!erent processing steps and methods. The type of measurements that will be performed on the image is an important factor in making a selection of the most appropriate processing steps. In morphological image analysis the geometric and topological content of the image are of prime importance and this should be re#ected in the operations that are used to enhance the image quality. The morphological image processing (MIP) technique reviewed below is well adapted for this purpose. This is because MIP and MIA are based on the same mathematical concepts (see below). Most importantly it is #exible, fast and easy to use. Pioneering work in this "eld was carried out by Matheron [25] and Serra [26]. We have found the book of Giardina and Dougherty [7] a very valuable source of information and inspiration. Most of the material of Sections 3.1}3.3 can be found in [7], albeit in di!erent form. We have chosen to present the material in the same order as MIP is actually performed: From a gray scale to a black-and-white image. The emphasis is on the practical application, much less on the mathematical foundations which are given in [7,25,26]. 3.1. Preliminaries In this section we introduce the basic concepts of MIP. We start by giving a more precise de"nition of an image. For simplicity, we will discuss MIP of 2D images only. Extension to 3D is trivial, also in practice. As usual a 2D image will be represented by an ¸ ;¸ array I(i, j) of gray V W values, intensities represented by integers, ¸ (¸ ) is the number of pixels in the x (y) direction. We V W will see below that some operations may refer to pixels that are out of bounds of this array, meaning that they refer to pixels that are not de"ned. It is convenient to assign the value minus

472


in"nity to such pixels [7]. Hence, we will write I(i, j)"!R whenever the pixel at (i, j) (within or outside the bounds of the array) is unde"ned. The structuring element or template is a key concept in MIP. A template is a predetermined geometrical structure, hence also an image, such as a square, a disc or star. Consistency of notation would suggest the use of the symbol ¹(i, j) to denote the image corresponding to the template ¹ but we will not do so. Instead we de"ne a template by specifying the displacement k, l relative to its 2 origin (0, 0) together with the value ¹(k, l). The size of a template is de"ned as maxk, l . A template cannot contain pixels that are unde"ned. Some examples of templates are shown in Fig. 5. Very often templates are chosen to be symmetric (with respect to the symmetry operations of a square lattice). In essence MIP is the study of how a template (or several templates) "t into an image [7,25,26]. A template represents the viewer's a priori knowledge or expectation about the morphological content of the image. Finally, we need a de"nition of an object. For reasons of consistency with the integral geometry approach discussed below an object is de"ned as a collection of pixels that satisfy the following criteria: (i) they all have the same intensity, and (ii) they are nearest neighbors or next-nearest neighbors of each other or can be connected by a chain of pixels that are nearest and/or next-nearest neighbors. It may seem strange that it is necessary to include next-nearest neighbors in counting objects but in fact it is not. This can already be seen by looking at a very simple example: A pattern that consists of two squares that touch each other at the vertex yield an Euler characteristic of one (one connected component), since n "2, n "8 and n "7 (see (1)). Clearly, there are no holes in this pattern. Hence the number of objects must be equal to the Euler characteristic (recall, for 2D patterns the Euler characteristic is equal to the number of connected components, i.e. objects, minus the number of holes, see Section 2). The only way to get a consistent procedure of counting objects and computing the Euler characteristic is to include next-nearest neighbors. 3.2. Gray-scale images We now have all the ingredients to de"ne the two basic MIP operations: Dilation and erosion of an image. Dilation D transforms an input image I(i, j) as follows: D(I, ¹)(i, j),max [I(i!k, j!l)#¹(k, l)] . 6IJ72

(7)

Fig. 5. Some examples of templates used in MIP. As in most practical image processing work we adopt the convention that the intensity is digitized in the range [0, 255]. Template (1): ¹(0, 0)"64; (2): ¹(0, 0)"64, ¹(1, 0)"128, ¹(0, 1)"128, ¹(!1, 0)"192, ¹(0,!1)"255; (3): ¹(0, 0)"64, ¹(1, 0)"128, ¹(0,!1)"192; (4): ¹(0, 0)"¹(1, 0)"¹(0, 1)" ¹(!1, 0)"¹(0,!1)"255; (5): ¹(0, 0)"2"¹(1,!1)"110; (6): ¹(0, 0)"2"¹(2,!2)"255.


473

Erosion E uses the minimum instead of the maximum: (8) E(I, ¹)(i, j), min [I(i#k, j#l)!¹(k, l)] . 6IJ72 The maximum and minimum are to be taken over all values of displacement k, l of the template 2 ¹. In general for some (i, j), (i!k, j!l) may well go out of the bounds of array I(i, j), a situation we already anticipated for by setting I(i, j)"!R whenever (i, j) is out of bounds. A similar argument applies to erosion: If one of the pixels I(i#k, j#l)"!R, in the output image the pixel at position (i, j) will be unde"ned too. Usually unde"ned pixels are displayed in background color (black on a display, white on paper). In Fig. 6 we show some illustrative examples of D and E. We used three di!erent templates to perform dilate D and erode E on a rather schematic picture of a rabbit. The original image is shown in the top left panel of Fig. 6(D and E). The top right image of Fig. 6(D) is obtained by replacing a pixel by its most intense nearest neighbor. This has the e!ect of transforming gray pixels at the boundaries of the gray objects into white pixels. The same happens to black pixels touching white and gray objects, hence the rabbit gets in#ated a little. The bottom left panel of Fig. 6(D) shows the e!ect of changing the intensity during the process of dilation. In this case we use D to remove all the gray objects of the rabbit, increase the size of the rabbit and change the background color. The bottom right panel of Fig. 6(D) shows the result of using a square (5;5) template. Apparently, this template is so large that D replaces all gray objects by white ones, except for the legs of the rabbit, which get severely distorted. If our intention was to extract certain features from the original image of the rabbit, using the large square template obviously is not the right thing to do. Indeed, as

Fig. 6. Illustration of Dilate D and Erode E of a gray-scale image (top left panel). Top right: star-shaped template of size 1; bottom left: star-shaped template of size 2; bottom right: square-shaped template of size 2. The values of the templates is zero in all cases.

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mentioned earlier, the choice of the template is directly linked to the viewers expectation about the morphological content of the image. A similar sequence of images, obtained by employing E instead of D, is shown in Fig. 6(E). Not surprisingly, the `mina operation generally reduces the number of non-black pixels, i.e. the rabbit shrinks. However E can increase the area of gray objects too. The top-right image shows that E tends to emphasize internal structures: Eyes, inner ears, legs and other features became larger. The `stara template of size 2 (see Fig. 5 (5)) reduces the gray level of all de"ned pixels (viewed on a computer screen `blacka and ùnde"neda are synonymous). Also, it reduces the number of dark-gray objects. As in the case of D, using an oversized square template (bottom right panel) yields a fairly distorted image of the rabbit. The basic morphological operations D and E can be used to construct other operations that perform more complicated "ltering operations. There are two other operations called Open (O) and Close (C) that play a central role in MIP [7]. Open and Close are de"ned as O(I, ¹),D(E(I, ¹), ¹)

(9)

and C(I, ¹),!O(!I,!¹) ,

(10)

where !I,!I(i, j) and !¹,!¹(k, l). Open and Close have all the mathematical properties that are required for MIP [7]. In particular, O and C are idempotent, i.e. O(O(I, ¹), ¹)"O(I, ¹) and C(C(I, ¹), ¹)"C(I, ¹), implying that in practice it does not help to òpena or `closea an image twice or more using the same template. In Fig. 7 we illustrate the e!ect of O and C, again using the image of the rabbit as an example. Open and Close act as "lters, the exact result of the "ltering operation depending on the template. Open O generally rounds corners from the inside of the objects (see the legs of the rabbit in Fig. 7(O)

Fig. 7. Illustration of Open O and Close C of a gray-scale image (top left panel), using the same templates as in Fig. 6.


475

for example). Close C, on the other hand, smooths from the outside. Objects that do not "t the template are removed from the image (see Fig. 7(C)). It is instructive to compare, e.g. E and C (top right panel of Figs. 6(E) and 7(C)). We see that E generally increases the size of the gray features whereas C removes the small gray features but leaves other gray objects intact. This is most clearly seen by comparing the bottom right panel of Figs. 6(E) and 7(C) that show the results of E and C using the square template. Whereas C does not change the overall image very much, E makes the rabbit look like a cat. Our experience is that in practice O and C are more useful than D and E. 3.3. Black-and-white images Obviously, a black-and-white image B=,B=(i, j) may be considered as special case of the gray-scale images treated earlier. As such a discussion of black-and-white MIP may seem super#ucious. However, in practice, it is often necessary to perform MIP on the gray-scale image, convert it to black-and-white, and carry out some further MIP on the black-and-white image before the image can be used as input for MIA. Therefore, it is worthwhile to discuss MIP on black-and-white images in more detail. On a computer display a black pixel may be considered as being unde"ned [7]. Instead of assigning unde"ned pixels the value !R, in this case it is more convenient to assign to a black pixel the traditional value of zero. A white pixel takes the value one. Hence, a black-and-white image B= is represented by an array of Boolean variables B=(i, j). In analogy with gray-scale MIP the four basic operations dilate D, erode E, open O and close C are de"ned by D(B=, ¹)(i, j), B=(i!k, j!l) , 6IJ72

(11a)

E(B=, ¹)(i, j), B=(i#k, j#l) , 6IJ72 O(B=, ¹),D(E(B=, ¹), ¹) ,

(11b)

C(B=, ¹),E(D(B=,!¹),!¹) ,

(11d)

(11c)

respectively. Operations (11a)}(11c) are Boolean versions of (7), (8) and (9), respectively, but this is not the case for pair (10) and (11d) [7]. Operations (11a) and (11b) are digital versions of set-theoretic operators known as Minkowski addition and subtraction [7]. The latter are basic concepts in point-set geometry and integral geometry [8]. This correspondence suggests that MIP and MIA are closely related and indeed they are [7]. The collection of images shown in Figs. 8 and 9 serve to illustrate the e!ect of these four operations on the thresholded image of the rabbit (top left panel). The threshold is chosen such that the gray pixels are converted to black ones. The examples shown would suggest that MIP of gray-scale images followed by thresholding yields pictures that are almost identical to the corresponding morphological imaging processed black-and-white images. As a matter of fact comparison of the bottom left panel of Figs. 6(D) and 8(D) already shows that interchanging the order in which thresholding and MIP are performed changes the output. Indeed after MIP of the black-and-white image, some of the internal features remain visible, notably legs and eyes. In

476


Fig. 8. Illustration of Dilate D and Erode E of a black-and-white image (top left panel), using the same templates as in Fig. 6.

Fig. 9. Illustration of Open O, Close C and Filter F of a black-and-white image (top left panel), using the same templates as in Fig. 6.

contrast MIP of the gray-scale image yields a completely smoothed image of the rabbit. For the input image of the rabbit used in the examples, interchanging thresholding and Erode (or Open or Close) yields the same output image. In general, this will not be the case unless the images have a very simple gray-scale structure, as the ones considered here. As a "nal example of MIP we consider a more complicated "lter F de"ned as [7] F(B=, ¹),C(O(B=, ¹), ¹) .

(12)


477

Also this "lter is e!ective in removing small noisy structures (small with respect to the template ¹), and leaves larger objects intact, whereas O has the tendency to tear objects apart (see bottom right panel of Fig. 9(O)) and C has the opposite e!ect. Filter F may give a more satisfactory output image in some cases. 3.4. Miscellaneous operations Averaging an image using a template can sometimes help to remove artifacts. In our notation this operation reads 1 I(i#k, j#l) , (13) I (i, j)" C¹ 6IJ72 where C¹ denotes the number of elements of template ¹. As before, in computing this average, we use the template to express our a priori knowledge or expectation about the shape and size of the objects in the image. Often it is useful to enhance the contrast of a gray-scale image. Again we can use a template to I(i#k, j#l) perform this task. For each pixel in the image I(i, j) we determine M(i, j),max 6IJ72 I(i#k, j#l) and then replace each pixel in the image by invoking the rule: and m(i, j),min 6IJ72 m if I(i, j)!m(M!I(i, j) , I (i, j)" (14) M if I(i, j)!m'M!I(i, j) .

Note that the value of the template does not play any role in this operation. 3.5. Mapping gray-scale to black-and-white images Excluding applications of MIA that use the threshold as a control parameter (see Section 2.4), in many situations it may be expedient to reduce the number of di!erent gray values in an image. For instance, to determine the number of objects in a gray-scale image, we will have to group pixels according to their gray value. Fluctuations in the gray values due to noise and other experimental limitations may prevent us from making the correct identi"cation if we use the full resolution of gray values (typically 256 values). Clearly, a procedure that reduces the number of gray-scale levels may be very useful. Thus, we would like to have a procedure to map the original gray-scale image onto another one with only a small number N of distinct gray levels (e.g. N"2, 4). A simple approach would be to use histogram equalization to optimize the dynamic range of the gray levels, followed by thresholding to classify pixels as either background or objects [3,4]. Clearly, it is much better to use a scheme that computes a nearly optimal distribution of the N gray levels from the original image itself. The method we will describe next performs very well in practice. It is a gray-scale version of a scheme that is used to determine nearly optimal color pallets [27]. The "rst step of the algorithm consists of making a histogram of the gray-scale image. This we can easily do at full gray-scale resolution. Let us consider the case of a reduction by a factor of two (i.e. N"128). We want to group gray levels but keep the image quality as high as possible. Which gray level should we remove "rst? A natural choice would be to select from the histogram the gray

478


level k with the lowest count, say 04k4255. Then we merge the bins 2[k/2] and 2[k/2]#1 ([k/2]"k/2 if k is even, [k/2]"(k!1)/2 if k is odd). This we do by adding the count of bin 2[k/2]#1 to the count of bin 2[k/2] and then clearing bin 2[k/2]#1. This process of merging bins is repeated until we have 128 empty bins (which could be rather exceptional) and we can stop the whole procedure or until we conclude that all posibilities to merge two bins have been exhausted. In the latter case we repeat the procedure by grouping the bins 4[k/4], 4[k/4]#2 (note that in the previous step the counts in bins 4[k/4]#1 and 4[k/4]#3 have been set to zero). Thereby care has to be taken to group the same four bins only once, a technical but crucial point. Again we repeat this process, always working with groups of four bins, until the number of bins with a count larger than zero is 128 (in which case the procedure terminates) or we keep restarting the grouping of bins using increments of 8, 16,2 and so on. Clearly, this procedure terminates as soon as the number of distinct gray levels becomes equal to the desired number of gray levels. Then it is a straightforward matter to assign new gray-scale values to the pixels of the original image. Although there is some ambiguity in chosing the strategy for grouping bins, experience has shown that the procedure outlined here yields very satisfactory gray-scale images, and can be used to automatically reduce a gray-scale image to a black-and-white picture.

4. Scanning electron microscope images of nano-ceramics As an example of an application of MIP we consider the problem of identifying objects in scanning electron microscope (SEM) images of nano-ceramic materials. These materials may exhibit physical properties such as ductility, toughness and hardness of both metals and ceramics and are useful for a number of technological applications that demand good mechanical behavior and good resistance against the degrading e!ects of high temperature, corrosive environments, etc. These materials can be manufactured by di!erent techniques, for instance by covering a surface by layers of nano-sized ceramic particles. The mechanical and other properties of these materials depend on the morphology, the microstructure and the initial stress due to the use of dissimilar materials. Di!erent preparation techniques and additional (heat) treatments often yield materials that have di!erent morphologies [28}30]. The changes in the morphology during the sintering process can be monitored by means of high-resolution low-voltage scanning electron microscopy (HRSEM) [31]. In Fig. 10 we show two SEM images of SiO particles on a substrate of fused silica, before (top panel) and after (bottom panel) a heat treatment [31,32]. From Fig. 10 it is clear that the latter causes particles to aggregate. Although their size does not seem to change much, the voids get larger. A more quantitative analysis of such images requires the identi"cation of objects (i.e. particles) in the image. MIP is well suited for this purpose. At the right-hand side of Fig. 10 we depict the images obtained by MIP. The #uctuations in the intensity (i.e. gray value) within what our eyes would consider to be one particle can be rather large. This experimental artifact can be removed from the image by means of O (open) and contrast enhancement operations, both using as a template a disc with a radius of 10 pixels. The size of the template re#ects our rough guess about the size of the objects. Then we use the algorithm described in Section 3.5 to map the gray-scale image onto a black-and-white picture. The "nal step consists of removing some minor artifacts of the size of one pixel by means of a C (close) operation. For this


479

Fig. 10. Electron microscope micrographs of silica at two stages of the sintering process before (left) and after (right) MIP.

purpose we use a single pixel as a template. Clearly the quality of these black-and-white pictures is su$ciently good for further analysis of the particle size, distribution etc. In Fig. 11 we present SEM images of another material, TiO on the same substrate. Depending on the heat treatment grains of TiO grow in size, leading to the mosaic-like coverages shown in Fig. 11 [31,33]. Also in this case gray-scale O and contrast-enhancement operations are used to remove noisy features from the image. Here the template is a 4-pixel-radius disc, smaller than in the previous example, but consistent with our expectation that the images of individual grains are smaller. Then the images are converted to black-and-white pictures, using the same procedure as the one described above. Also in this case the "nal pictures are of su$cient quality so that objects can easily be identi"ed and analyzed. As a "nal example we consider a rather di!erent type of system, namely small Mn O precipitates in Ag observed by high-resolution transmission electron microscopy (HRTEM). The top-left panel of Fig. 12 shows a high-resolution picture of a small part of the sample shown in the bottom-left panel. From the former we would like to extract information about the geometrical properties of the individual grains, from the latter we want to learn how the particles are distributed over the surface. It is somehow remarkable that the same MIP procedure can be used for both, apparently quite di!erent, tasks. The top-left image has very low contrast. Moreover, due to experimental conditions, the averaged intensity at the left-hand side of the image di!ers signi"cantly from the one at the right-hand side. After correcting for this artifact, repeated averaging and contrast enhancement operations with a 15-pixel-radius template followed by the standard of mapping to black-and-white yields the image shown in the top-right panel of Fig. 12. The bottom-left image is processed in the same manner, except that instead of a 15 pixel-size disc we use

480


Fig. 11. Electron microscope micrographs of zirconia at two stages of the sintering process before (left) and after (right) MIP.

Fig. 12. High-resolution electron microscope image of Mn O precipitates before (left) and after (right) MIP.


481

a radius of 4 pixels and we added a D (Dilate) step to remove some sharp edges from the objects. Physically, relevant information about the particle size and spatial distribution is easily extracted from these pictures. The examples discussed above illustrate that MIP is a #exible and powerful tool for enhancing image quality and object identi"cation, without destroying the morphological content of the image. Of course, depending on the type of image technique used, additional non-morphological image processing steps may be required to produce patterns that are suitable for morphological image analysis. As the emphasis of this paper is on image analysis rather than on image processing an in-depth discussion of the latter is outside the scope of the paper and we refer the reader to standard treatises on the subject [3}6]. We now review the theory that provides a rigorous framework for the quantitative characterization of the morphological properties of black-andwhite images.

5. Integral geometry In this section we present the mathematics that lies at the heart of integral-geometry-based morphological image analysis. The reader who is not interested in the mathematics can skip this section and resort to Section 2. 5.1. Preliminaries Consider the set of points of a line ¸ of length a embedded in one-dimensional (1D) Euclidian space. We take a similar line of length 2r and put the center of this line at each point of the line ¸. How does the union of all these points look like? Obviously, it is another line that is longer than ¸. The sets ¸ (black line) and ¸ (union of black and light gray lines), the result of this operation, are P shown in Fig. 13. The length l of ¸ is given by P l(¸ )"a#2r"l(¸)#2r . P

(15)

The set ¸ is called the parallel set of ¸ at a distance r. P The one-dimensional case easily extends to two and three dimensions. We consider a circular disk D of radius a, a square Q of edge length a and a equilateral triangle ¹ of side length a embedded in the 2D space. Now, we use a disc of radius r and perform the same operation as we did for the 1D case: We put the center of the disc of radius r at each point of D (or Q or ¹) and consider the union of all points. The resulting parallel sets at a distance r are shown in Fig. 13. The area A of D , Q and ¹ is given by P P P A(D )"a#2ar#r , P

(16a)

A(Q )"a#4ar#r , P

(16b)

(3 a#3ar#r . A(¹ )" P 4

(16c)

482


Fig. 13. Parallel sets K (union of black and grey area) at a distance r of the sets K (black area). Top: parallel set of a line P segment ¸ of length a embedded in one dimension; bottom: parallel set of a circular disk D of radius a, a square Q of edge length a and an equilateral triangle ¹ of side length a embedded in the two-dimensional space.

Formulae (16) suggest that there may be a general relationship between the area of the original set and its parallel set at a distance r. It is not di$cult to see that the areas of the three parallel sets can be written as A(K )"A(K)#;(K)r#r , (17) P where ;(K) denotes the boundary length (or perimeter) of the geometrical object K. The similarity between the construction of the parallel sets and the dilation of an image by means of a template of `radiusa r is not an accident: Dilation on a black-and-white image (see Section 3.3) is a digital equivalent of building the parallel set in Euclidian space [7]. This again shows that MIA and MIP have common roots. As a last example we consider a cube C of edge length a embedded in 3D space. A simple calculation shows that the volume < of the parallel set C can be written as P 4 (18)

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