Physics Reports vol.320

Physics Reports 320 (1999) 1}15 Long-range forces in the universe A.D. Dolgov* Teoretisk Astrofysik Center Juliane Ma...

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Physics Reports 320 (1999) 1}15

Long-range forces in the universe A.D. Dolgov* Teoretisk Astrofysik Center Juliane Maries Vej 30, DK-2100, Copenhagen, Denmark

Abstract Possible existence of new long range forces and their interaction strength are critically analyzed and reviewed. 1999 Elsevier Science B.V. All rights reserved. PACS: 04.90.#e

1. Introduction All forces in nature can be divided into two groups: the forces that are induced by an exchange of massless particles and fall as a power of distance, and the rest. The "rst group can be further sub-divided into Newton or Coulomb forces, which follow the celebrated inverse square decay law, F&1/r, and forces that fall faster or slower than that. The term `long-range forcesa usually relates to inverse square forces, though sometimes it is applied to all forces that fall slower than a power of distance. Among them are such extremes as the Van der Waals forces between electrically neutral atoms, which behave as 1/r, and quark con"nement forces, which presumably do not decrease with distance. The Van der Waals forces are not fundamental ones; they appear as a result of screening of the usual Coulomb interaction and we will not discuss them further. Historically the "rst long range forces that were quantitatively studied were gravitational forces, though gravitational coupling is the weakest in nature. Electric forces are stronger than gravitational ones by a factor of 10 but still the Coulomb law was not discovered until almost exactly a century after the Newton law. The reason for that is that gravity cannot be screened and large astronomical bodies possess huge `gravitational chargesa, proportional to their masses, and create strong (as we all feel) and easily measurable gravitational "elds. By contrast, macroscopic electric charges are usually screened and some e!orts are required to create a large Coulomb "eld. * Corresponding author: Tel.: #45-3532-5908; fax: #45-3532-5910. E-mail address: [email protected] (A.D. Dolgov) Also at ITEP, Bol. Cheremushkinskaya 25, Moscow 113259, Russia. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 7 0 - 8

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A.D. Dolgov / Physics Reports 320 (1999) 1}15

Macroscopically signi"cant long-range forces (behaving as 1/r) can be created through an exchange of massless bosons with di!erent spins, s"0, 1, 2. A massless bosonic "eld in the static case generically satis"es the equation *U "4po , (1) Q Q where o is the source density, i.e. charge density ( o ) for electromagnetic "eld and mass density ( o ) for gravitational "eld in non-relativistic limit. The solution to this equation is well known and can be expressed through the Green function:

o (r) P QQ , dr&& U (r)" Q Q "r!r" r

(2)

where Q is the total electric (Q ), gravitational (Q "GM M ), or possibly a scalar (Q ) charge. Q E The potential U decreases as 1/r, so that the corresponding force falls as 1/r. One more essential Q point is that the potential is proportional to the total charge, Q"dro, so that macroscopically large "elds can be created. This is typical for massless particle exchange. It is worth noting that only in three-dimensional space an exchange of a massless boson creates force that decreases as 1/r. This makes possible stable Newtonian orbits in planetary systems and Coulomb orbits in atoms. Presumably life is only possible in three dimensions. For a massive "eld Eq. (2) is modi"ed as (*#m)U"4po

(3)

and the solution is

e\KP dro(r)eKrrYP . U" r

(4)

Thus the e!ective charge of the source is determined by the region with the thickness 1/m, while the transverse size may be much larger,&(d/m, where d is the smaller of the distance to the source and its size. Though the solution (2) looks the same for all spins, it is not quite so because of the di!erent properties of the sources. There is an essential di!erence between gravitational and electric "elds. There are particles with both positive and negative charges in nature. In particular, antiparticles have charges opposite to those of particles, the electric charge of an electron is opposite to the charge of a proton. Due to that electric charge is screened and matter is usually electrically neutral. A very large electric charge cannot exist even in vacuum. If the value of the charge exceeds a critical value the corresponding electric "eld creates electron}positron pairs from vacuum. As a result the charge is radiated away. As is well known, the critical value of the charge for a point-like particle is about e/a"137e, where e is the charge of the electron. For realistic ions of a "nite size the critical charge is somewhat larger. In contrast to electricity, gravitational "elds are almost always attractive, with a possible exception for unbounded sources, and cannot be screened. To see this more formally let us consider the Lagrangian of massless scalar , vector , and tensor "elds: I IJ (5) ¸"(* )(*I )!(* )(*I ?)#(* )(*I ?@)# j# jI# jIJ , I ? I ?@ I IJ I where j, jI, and jIJ are the sources of the corresponding "elds. Note that the sign of the kinetic term of the vector "eld is di!erent from those of scalar and tensor ones. The choice of signs is connected


3

to the condition of the positive de"niteness of the energy of the "eld quanta. The sources jI and jIJ should be covariantly conserved, D jI"0 and D jIJ"0, otherwise the theory of massless "elds I I is not consistent. Scalar j and tensor jIJ sources are symmetric with respect to transition from particles to antiparticles while vector current jI is antisymmetric. This is well known from electrodynamics, where currents of particles and antiparticles are di!erent in signs. It can be proven [1,2] that there is only one conserved tensor source: the energy}momentum tensor ¹IJ. Its symmetry with respect to charge conjugation follows e.g. from the fact that ¹ is an energy density that should be positive both for particles and antiparticles. Static interaction is determined by the time components of the potentials and (in the previous notations, "U and "U ). These components enter Lagrangian (5) with di!erent signs. Writing Green functions for the corresponding "elds one can see that it gives rise to attraction of the same sign charges in gravity and to repulsion in electrostatics. The conservation of the energy}momentum tensor ¹IJ is linked to the freedom in choice of di!erent coordinate frames in the space-time. There is no other symmetry that gives rise to a conserved tensor source and so no massless spin two "elds except for the gravitational one exist in nature. As for conserved vector currents their existence is related to internal (gauge) symmetries and their number can be arbitrarily large. These symmetries ensure current conservation and the vanishing of the masses of the corresponding "elds not only on the classical but also on the quantum level, if anomalies are absent. There are several known charges in particle physics which may possibly be conserved. Among them are baryonic and di!erent leptonic (electronic, muonic, and tauonic) charges and in principle they might be coupled to massless vector "elds in the same way as the photon is coupled to electric charge. This possibility was "rst analyzed by Lee and Yang [3] and by Okun [4] who considered the couplings to baryonic and leptonic charges, respectively, and obtained very strong bounds on their magnitudes. We will discuss these bounds and some possible manifestations of new vectormediated long-range forces below. Due to chiral anomaly separate baryonic and total leptonic current conservations are broken but the combination (B!¸) remains non-anomalous and may be conserved. If this is the case, no massless vector "eld can be coupled to baryonic or leptonic current separately but the coupling to (B!¸) is permitted. The results obtained in the pioneering papers [3,4] could be applied to this case with little modi"cations. In principle, an exchange by massless fermions (e.g. by neutrinos, if they are massless) would also generate a power law force, but since one-fermion exchange changes quantum numbers of the source, the coherence would be lost and macroscopically interesting "elds could not be created. Some long-range forces can be generated by an exchange of a pair of massless fermions but such forces fall much faster than 1/r. For the case of ll-exchange the force behaves as 1/r [5]. In what follows, we will concentrate, however, on `reala long-range forces, that decrease as 1/r. 2. Equivalence principle and new long-range forces Let us forget for a while that baryonic charge, B, is not conserved and assume, following Ref. [2] that there exists a massless vector "eld coupled to B. Long-range forces generated by these baryo-photons would lead to ànti-gravitya between pieces of matter and to additional attraction

4


between matter and antimatter. One would immediately conclude that this interaction must be weaker than the gravitational one, otherwise celestial bodies could not be formed, and thus the coupling constant should be quite small, a (10\+10\ a, where a" is the electromag netic coupling constant and a is the baryonic one. This restriction can be very much improved if one invokes experiments testing equivalence principle [6}9]. The equivalence principle implies that the free-fall acceleration is the same for all bodies independently of their chemical content. This has been tested by EoK tvoK s et al. [6] for the acceleration towards the earth with the accuracy 10\ and with a somewhat better accuracy, a few;10\ by Renner [7]. Roughly a half century later this result, for the acceleration to the Sun, has been improved by Roll et al. [8] who have reached the accuracy about 10\ and by Braginsky and Panov [9] who have improved the accuracy down to 10\. Since the ratio of baryonic, B, (as well as leptonic charge, ¸) to mass is di!erent for di!erent samples, the baryonic long-range force would lead to a violation of the equivalence principle. It was shown in EoK tvoK s experiment that the ratio of the accelerations of copper and platinum are a /a "(4$2);10\. The ratios of ! . baryonic charge to mass for natural isotopes of Cu and Pt are, respectively, (B/m) "1.001 and ! (B/m) "0.999. Atomic weight here is given in terms of the weight of C. This permits to obtain . the following bound on the coupling constant of baryonic photons: a (10\

(6)

which is quite close to the original one found in Ref. [3]. This bound is based on the early experiment [6]. New more accurate tests [8,9] were made for aluminum and gold and aluminum and platinum, respectively. The variation of the ratio B/M for these elements is about 10\. It permits to improve the bound by 3}4 orders of magnitude, down to a (10\}10\

(7)

Similar arguments can be applied to leptonic long-range action [4]. The variation of the ratio of lepton number to the atomic weight for di!erent elements is about 10% and that makes the restriction two orders of magnitude stronger:

10\ from acceleration to the Earth , a ( * 10\}10\ from acceleration to the Sun .

(8)

These bounds are obtained under the assumption that neither baryonic nor leptonic charges of the Earth and the of Sun are screened by antibaryons or by antileptons. It is evidently true for baryonic charge because there are no antibaryons around to do the job. However, there are plenty of relic neutrinos and antineutrinos in the universe with the number density about 50/cm for any neutrino and antineutrino species, so that they may screen leptonic charges of celestial bodies. It was claimed in Refs. [10,11] that the screening of the Earth leptonic charge is almost complete and hence the tests of the equivalence principle could only give the bound a (10\}10\. This claim * was critically analyzed by Okun [12] and in more detail in Ref. [13]. It was shown that the screening of leptonic charge by cosmic neutrinos was negligible and the previously found limits survived. However it seems possible that the screening might be realized by a hypothetical light boson with a non-zero leptonic charge [12}14], if the latter existed. In short, the screening is impossible because the Bohr radius of neutrino bound state and/or the Debye screening length are both much larger than the normal atomic size. Therefore,


5

macroscopically large pieces of matter with a non-compensated leptonic charge should exist. If a is * not too small, such pieces would be unstable, and for a small a the screening length would be * larger than the Earth radius and even the distance form the Earth to the Sun. This is in essence the arguments of Refs. [12,13]. In what follows somewhat di!erent arguments are presented, which also permit to reach the conclusion that lepton long-range forces, if exist, would not be screened. The coupling constant of the lepto}photonic interaction is bounded from above by the precise agreement with experiment of the QED predictions for anomalous magnetic moment of electron and for the Lamb shift. From these data follows, a (10\. The data on elastic l e-scattering give possibly a better bound by 2}3 * orders of magnitude. However a much stronger restriction follows from astrophysics. If the coupling a is su$ciently large, lepto-photons would be abundantly produced in the Sun. Further, * depending on the value of the coupling, they would either propagate freely from the core to the surface and in this case they might carry away much more energy than photons, or if their mean free path is smaller than the core radius, R "10 cm, they would radiate away about the same energy as photons. Both cases are not compatible with the existence of the Sun during 10 billion years. The arguments presented below are similar to that used by Bernstein et al. [15] and Okun [4] to restrict possible electric charges of di!erent types of neutrinos. Lepto-photons would be produced in the Sun through the reaction cePc e. Its cross-section * can be found from the Thomson cross-section with an evident substitution for the coupling constants: p ,p(cePc e)"8paa /3m , (9) * * * where a" is the electromagnetic "ne structure constant and a is an unknown hypothetical * coupling constant of leptonic charge. The characteristic time of the production of lepto-photons is given by the expression (10) q ,(n */n )\"(p n n )\+3;10\ a\ s , * A * * A A where the electron number density in the core, n "10 cm\ was substituted. If q is smaller than * the c -escape time, q , thermal equilibrium with respect to c and the usual photons would be * * established and the energy #ux of lepto-photons would be the same or even larger, because of a larger mean free path, than that of the usual photons. The mean free path of c in the core of the * Sun is given by Eq. (10) and is equal to l +10\a\ cm. It is smaller than the core radius if * * a '10\. In this case c are more or less in equilibrium with the usual photons. If a (10\, * * * lepto-photons very quickly escape the Sun and their energy #ux can be estimated as follows. From Eq. (10) one "nds that n *"3;10\ ) 0.24¹a\ where ¹ +1 keV is the core temperature. The * A total lepto-photon luminosity is 4pR 3;10\ 0.72¹"3;10a ¸ (11) ¸ " * > * a 3 * where we substituted 3¹ for the average energy of lepto-photons and ¸ "4;10 erg/s is the > solar luminosity in normal photons. Demanding ¸ (¸ , we obtain * > a (3;10\ (12) * Possibly a more stringent bound can be established using the data from low background experiments. The #ux of lepto-photons from the Sun would excite atoms in the detectors and could be

6


observed by the emitted light. However we do not need that here because using the limit (12) we will prove that lepton charge in the Earth and after that in the Sun are not screened and the limits (8) are valid. If the limit (12) is true, then the neutrino Bohr radius would be huge, r "1/a m '10/(m /eV) * * I J cm. There is a very large number of electrons inside this radius in the terrestrial material, in which the number density of electrons is n +10 cm\. Thus to "nd the radius of the bound state one has to substitute the charge inside the orbit Z "4pRn /3. The corresponding poten* tial behaves as ;(R)"a Z /R&R, i.e. it is a harmonic oscillator potential. The number of states * * with the principal quantum number i in this potential is N +i/2 (for a large i). The total G number of states with i(i is N (i)"i/6. The size of the orbit corresponding to i is R &(iR , where G R "(3/(8pa n m ))+0.07 cm a\m\ , (13) * J J where a "10a and the neutrino mass is expressed in eV. * Let us estimate now R , i.e. the radius of the piece of matter where compensation of leptonic charge is incomplete, say, by 50%. In other words, N (i)"N /2. This condition means N "4pn R /3"2i/6"R /3R (14) From these line of equations one "nds R "2;10 cm/a m . J The validity of normal gravitational interactions was veri"ed in laboratory experiments for bodies with a much smaller size than R . Assuming a very mild accuracy of 100%, we conclude that the leptonic coupling must be weaker than the gravitational one, a (10\. That small * a implies that R is larger than the Earth radius, R "6;10 cm. Hence the leptonic charge of * = the Earth created by electrons cannot be screened by neutrinos and the "rst of the limits (8) remains valid. Using this limit we can see that neutrinos could not screen the leptonic charge of the Sun as well, and the second of the limits (8) is also true. One may argue that the screening is achieved not by bound neutrinos but by neutrino gas inside the Earth. In this case the screening is given by the Debye screening length, l "m /en . One may " J * J check repeating the previous chain of arguments that even for n "n the screening is too weak to J jeopardize the limits (8). Screening however may be quite essential [16] in another hypothetical case. If neutrinos are subject to a novel long-range force, their trajectories from the supernova 1987A would be bended and it would create a time dispersion dt/t&1/E [17}19], where E is neutrino energy. The energy spread of emitted neutrinos was about 10 MeV, the time of propagation t+5;10 s, and the duration of the signal dt was several seconds. The source of this hypothetical long-range force could be either electrons in the Galaxy, or protons, or dark matter particles, correspondingly with charges, q , q , or q . Depending on the nature of the source particles and their distribution in the C N "+ Galaxy, the limit

1 "q q "43;10\; J H m

for e and p ,

(15) /m for dark matter "+ N was derived [17}19]. The factor m /m appears for dark matter particles because in contrast to "+ N protons and electrons the mass density and not the number density of dark matter particles is known.


7

The e!ect of screening on this bound (15) strongly depends upon the spatial distribution of the source particles in the Galaxy. If the latter are either electrons or protons, mostly concentrated in the central part of the Milky way, then the neutrino path practically lies outside the source charge distribution. In this case the Coulomb-like "eld of the source would be e!ectively screened by background neutrinos and no interesting bound on their charge can be obtained. Indeed, if neutrinos are very light, so that they remain relativistic today with temperature ¹+1.9 K, their Debye screening length is j &1/(q ¹ )+3;10\ kpc/q (16) " J J J This for q '10\ the screening radius is smaller than 3 kpc and neutrinos from SN 1987A would J not be e!ected by long-range forces from the galactic electrons or protons. If neutrinos are heavier, so that they are non-relativistic today, the screening radius would be even smaller. However if the source of the new long-range force are dark matter particles a restrictive limit on their charge can be obtained. The distribution of the neutrino charge Q (r) obeys the equation: J dQ /dr"(Q !Q )/j , (17) J 1 J " where Q (r) is the distribution of the source (dark matter) particles. For the localized charge of the 1 source the charge would be exponentially screened, Q "Q !Q &exp(!r/j ). For the dark 1 J " matter particles, which presumably have number density inversely proportional to r, the charge inside radius r behaves as Q (r)&r and the solution of Eq. (17) gives [16] "+ (18) Q (r)"Q (r)(j /r)(1!e\PH") 1 " Hence the charge is compensated only with the accuracy (j /R ), where R is the boundary radius " where the charge of the source is cut-o!. Correspondingly the bound (15) with the account of the screening by cosmic neutrinos becomes q

(10\(m /m )(r/kpc)+10\(m /m ) (19) "+ "+ N "+ N The result does not depend on the charge of neutrino, q . J It was assumed above that the universe is neutral with respect to the charge to which massless vector bosons are coupled. Cosmological implications of possible violation of this assumptions are discussed in Section 4. The considerations presented above are valid only for electronic leptonic charge. The limits for muonic and tauonic charges are not so strong by an evident reason that neither muons nor tau-leptons are present in macroscopic quantities in any available samples of matter. Di!erent manifestations of possible long-range forces associated with muonic charge are discussed in Ref. [4]. If in addition to the usual photons, leptons are coupled to other massless vector bosons associated to di!erent leptonic charges, quantum corrections would generate mixings between these bosons. For example the usual photon would be transformed into muonic one through the muon loop. It would introduce a coupling of electrons to muonic photons or coupling of muonic neutrinos to the usual photons, or in other words, this diagram generates e!ective electric charge of muonic neutrinos. Due to gauge invariance, which is assumed to be unbroken for all the charges, the muon or other similar loops give contribution proportional to k, where k is the 4-momentum of the vector boson. So these loops do not generate photon mass. They create a mixing in the

8


kinetic part of the Lagrangian: L "!F !FJ #eF FJ , (20) ?@ ?@ ?@ ?@ where l corresponds to a certain leptonic charge, electronic, muonic or tauonic, or their combination. Since all photons remain massless, this mixing do not create photon oscillations in vacuum. However, oscillations would appear in media because of di!erent refraction indices of the `photonsa, di!erent plasma frequencies, etc. Consideration of stellar evolution permits to put rather strong limits on the strength of possible couplings to leptonic charges, other than electronic charge [15,4,20]. This is closely analogous to the derivation of the limit (12). Due to induced electric charge of neutrinos the plasmons in the Sun or in other stars may decay into l l -pair. Demanding that these neutrinos do not radiate too much I I energy from the Sun, the bound a

(10\ (21)

can be obtained. This bound is valid if l -mass is below solar plasma frequency, u &100 eV. I A similar bound but valid for neutrinos with masses up to 10}20 keV can be deduced from white dwarfs. For a recent review on the subject see Ref. [21]. A more stringent bound follows from the consideration of photon oscillations inside the Sun or of the process cePc e through muonic loop, cPkkPc . In vacuum it may be absent because one I I could rede"ne photon wave function so that the state that interacts with electrons does not have mixing with muonic charge. It would not remain valid in plasma and the limit from solar luminosity obtained this way is approximately 10 times weaker (by 1/a) than (12). Still all the bounds for muonic and tauonic charges are by far weaker than those obtained for electronic charge from equivalence principle.

3. Nucleosynthesis bounds on new forces and light particles Consideration of big-bang nucleosynthesis (BBN) permits to put an upper bound on the number of particle species contributing to the energy density of the primeval plasma at temperatures ¹+1}0.1 MeV [22,23]. The limit is usually expressed in terms of e!ective massless neutrino species, k , which make the same contribution into the energy density as the particles in question. J Given the accuracy of the present day data, the safe upper bound is *k ,k !3(1. There are J J several recent papers [24}26] presenting di!erent limits, from *k (0.2 [24] to *k (2.3 [25]. J J The last result is obtained if the mass fraction of primordial He is quite high, > "0.250. We assume in what follows that *k (1, but will keep in mind possibilities of higher and lower limits. J If there are new long-range forces, there are additional massless vector bosons which should be present at nucleosynthesis and, if these bosons have vector-like coupling, right-handed neutrino states would be also excited. If they all are in thermal equilibrium, then l would contribute *k " 0 J 1 for every neutrino #avor which possesses coupling to massless vector "elds, and *k " for the J corresponding new photons. This issue was raised in Refs. [27,28]. The production of right-handed neutrinos and new photons, as considered in Ref. [27], proceeded through the reaction c #c l #l ? ? ? ?

(22)


9

where a"k, q. The coupling to electronic charge, as we have seen above, is so weak that this interaction is never in equilibrium. The cross-section of this reaction is [25] p "2pa/s(ln(s/k )!1) , (23) ?? ? " where s is the center-of-mass energy squared and k "4pa n ?/¹ is the Debye screening mo" ? J mentum with n ? being the number density of l . Thermal equilibrium would be established ? J if the rate of this reaction, C "p n ? would be bigger than the expansion rate, ?? ?? J H"(4pg /45)¹/m . With p inversely proportional to s&¹ equilibrium would be reached H . ?? at su$ciently small temperatures, ¹(¹ . Demanding that ¹ is below 1 MeV, so that primordial ? ? nucleosynthesis is not disturbed, the bound was obtained a (1.8;10\ [27]. It was shown in ? Ref. [28], however, that this limit may be invalid if either l are two-component massless particles ? or have Majorana mass. In the former case the theory may have infrared problems (see Section 5), while in the case of Majorana mass separate leptonic charges of, say, l and l would be I O non-conserved but it is possible to arrange conservation of Q !Q . A model of long-range forces J O coupled to the di!erence of muonic and tauonic charges was proposed by Okun [29] to avoid anomalous charge non-conservation. Thus if *k "1 is permitted then in the framework of the model [28,29] only one extra light J particle, c would be present at nucleosynthesis and the coupling a is not restricted by BBN. If I\O I\O however, *k (1 then the limit a (10\ should be true. We will show that a considerably J ? stronger limit for a or a can be obtained if one takes into account production of c through the I I\O ? reaction: c#kPc #k (24) ? This reaction goes in the "rst order in a and at ¹'m its probability is much larger than the ? I probability of the reaction (22) that goes in the second order in a . Making the same calculations as ? above but with the cross-section of reaction (24) p"8paa /3m we "nd that the new photon-like ? I particles c would not be in equilibrium at ¹"m if a (4;10\. However the number of ? I ? particle species at this temperature, including k! and three pions, is g (m )"17.25. It is 1.6 times H I larger than the usual number of particle species at BBN, g ,"10.75. If c dropped out of H ? equilibrium below ¹"m then their contribution into energy density at BBN is suppressed by the I factor [g ,/g (m )]"1.88, as follows from entropy conservation. Correspondingly c would H H I ? contribute at BBN as (8/7)/1.88"0.6 neutrino species. Thus a '4;10\ would be excluded if ? nucleosynthesis constraint is as strong as *k (0.6. With the present day accuracy, *k (1 the J J limit is approximately two orders of magnitude weaker. Indeed, the `photonsa c would be ? produced through the reaction (24) even below ¹"m , though the number of muons is Boltzmann I suppressed at lower ¹. At m /¹"4 the suppression factor is 0.1, so one may neglect the entropy I release by muon annihilation and decay. The contribution of c produced at this ¹ into cosmic ? energy density at BBN would be smaller than that of one neutrino species if a (2;10\ (25) ? Direct laboratory limits on a and possibility of observation of muonic photons in neutrino I experiments were considered in a recent paper [30]. The best bound follows from the measurement of the anomalous magnetic moment of muon, a (10\. As shown in Ref. [30] muonic photons I may be observed in high-energy neutrino experiments if their coupling is above 10\.

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4. Conserved and non-conserved charges, vanishing photon mass, and charge asymmetry of the universe If there is a strictly conserved charge coupled to a massless vector boson the universe must be neutral with respect to this charge, otherwise cosmology would be very much di!erent from the usual Friedman}Robertson}Walker one. In connection to baryonic photons this point was brie#y discussed in Ref. [31]. In what follows we will use the term èlectric chargea, meaning any conserved charge coupled to massless vector bosons (photons). It is known that net electric charge of a closed universe must be zero as is enforced by the Gauss theorem. If the Universe is open it may contain a nonzero electric charge but a non-vanishing homogeneous charge density results in a cosmological disaster. Indeed in the Friedman}Robertson}Walker (FRW) background electric "eld E satis"es the equation * FG"!4pJ , (26) G where J is the electric charge density and the Maxwell "eld strength is related to electric "eld as E "F . It is easy to see that this equation has only solutions linearly rising with distance, G G E&Jl , and this would destroy the homogeneity of the Universe and create quadratically rising 3 energy density of electric "eld, (27) oE"E/8p+a(l n ) 3 where n ,b n is the number density of the excessive charged particles, (e.g. the di!erence A between number densities of particles and antiparticles), b is the corresponding coe$cient of charge asymmetry, and l is the present day horizon size, l &t +10 cm. However if the 3 3 coupling constant is extremely weak, this might be not so dangerous. The ratio of oE to cosmological energy density o &Hm &t\m is . 3 . (28) rE"oE/o +an l /m 3 . In particular for baryonic charge n +3;10\ and this ratio is rE"10 a . It is negligible if a (10\}10\, as restricted by the bounds (6}7). Charge particles in such long-range cosmic "eld would be accelerated up to energies E+an l +10 eVa(b /3;10\). For baryons this energy would be smaller than 10\ eV. 3 This kind of cosmological charge asymmetry would be noticeable for a much stronger coupling. It would create in particular an asymmetry in cosmic rays and an intrinsic dipole in cosmic microwave background radiation. However the universe may be charge asymmetric only locally but globally charge symmetric. If this is the case one should substitute instead of l a smaller size of 3 domains with a certain sign of charge asymmetry. It is possible that the considerations presented above are not cosmologically relevant because if a charge is strictly conserved, no charge asymmetry in the universe could be developed. One may make an unnatural assumption that a charge asymmetry existed ab inito but such hypothesis is incompatible with in#ationary cosmology with a su$ciently long period of in#ation [32,33]. If cosmological density of a conserved charge were non-vanishing, the energy density associated with this charge could not remain constant in the course of the universe evolution and in#ation would be impossible.


11

The situation would be drastically di!erent if the photons have a small but non-vanishing mass. Cosmology of electrically charged universe both with zero and nonzero mass of the photon was considered in Refs. [33}38]. Though the photon mass is known to be very strongly bounded from above (for a review see [20,39,40]) cosmological e!ects associated with it in a charged universe might be essential. Analysis of the Jupiter magnetic "eld leads to m (6;10\ eV while magnetic A "elds in galaxies imply a much stronger limit m (3;10\ eV [41] which we will use in what A follows. With an addition of mass term the equation of motion for the vector "eld in FRW-background takes the form: D FI#mA "4pJ , (29) I J A J J where D is the covariant derivative in the external gravitational "eld. For a uniformly charged I universe this equation has a homogeneous and isotropic solution A "dR A (t) with A (t)" I I R R 4pJ /m. This solution is very much di!erent from the massless case, in particular, it does not make A any electric "eld. There is no smooth limiting transition to the case of zero mass. The energy density of this solution is o "mA AI/2"abn/2m , (30) K A I A A where b is the corresponding charge asymmetry. The pressure density is equal to the energy density, p "o ; it is the sti!est equation of state [42}44]. The energy density decreases as 1/a, K K while the scale factor behaves as a&t. The energy density o would be smaller than the K cosmological energy density if electric charge asymmetry b is smaller than (31) b(m m /(at n +10\(m /3;10\ eV)(1/137a A A A . For other possible charges with weaker couplings the limit on asymmetry is much less restrictive. One should keep in mind however that o scales as 1/a and thus it would dominate at an early K stage of the universe evolution. The condition that o did not dominate during primordial K nucleosynthesis permits to improve the bound (31) by 11}12 orders of magnitude. Another possibility to cure infrared problems of electrically charged universe, except of prescribing a small mass to the photon, is to introduce a non-minimal coupling of photons to gravitational "eld. The most general Lagrangian of this kind that contains only dimensionless coupling constants m has the form: G L "!F FIJ/4#mA AI/2#m RA AI/2#m RIJA A , (32) IJ A I I I J where RIJ is the Ricci tensor and R is the curvature scalar. Three last terms in this expression should vanish if gauge invariance is unbroken. The upper bounds on the photon mass quoted above permit the constants m to be as large as 10. It can be shown that for the case of dominant G coupling to curvature scalar the universe expands according to the same law as in non-relativistic case, a(t)&t. Cosmology of charged universe with vector boson "eld described by the Lagrangian (32) may possess quite interesting and unusual features. In particular, it may possibly induce cosmic acceleration and in this sense mimic a non-zero cosmological constant. We will postpone a more detailed consideration of these problems to a future work.

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5. Problems of vanishing mass and charge conservation As we have already mentioned in passing, a massless vector boson must interact with a conserved current, otherwise the theory would be infra-red pathological. Even if the photon mass is non-zero but small in comparison with the characteristic energy scale of a reaction, it does not help to achieve infra-red regularization [45]. Emission of longitudinal photons would be catastrophic and perturbation theory breaks down. The probability of emission of k longitudinal photons is proportional to gaI(u/m )I, where g is the coupling constant of the interaction that breaks A current conservation. As was argued in Ref. [45], renormalization of the constant g due to virtual longitudinal photons is very strong and exponentially diminishes g almost to zero, g "g exp(!aK/4pm), where K is an ultraviolet cut-o! momentum. Thus even if electric charge A were not conserved, the decay of electron would be very strongly suppressed. Similar problems may appear with baryonic long-range action. Baryonic charge is known to be non-conserved due to chiral anomaly. Thus baryonic photons, if exist, must be massive, otherwise the usual (massless) equation * FI"J? (33) I J J would be self-contradictory and one has to add the term mA , where a indicates the type of the ? J charge, e.g. it can be baryonic charge, a,B, or any other charge. If this is the case the upper bound on the coupling constant a would depend on the mass of baryo-photon, m . If m\ is smaller then the distance between the Sun and the Earth (this corresponds to m '10\ eV) the experiments testing validity of the equivalence principle by the acceleration to the Sun are not sensitive to the baryonic "eld. With rising m , when m\ becomes smaller then the radius of the Earth, R "6400 km, the bounds on a obtained from the EoK tvoK s-type experiments become less restrict= ive. An interesting possibility is that baryonic current is non-conserved, baryo-photons are massive but light, and the proton decay is suppressed because of renormalization of the baryon nonconserved coupling discussed above. Another infrared problem is related to vanishing masses of charged particles, e.g. neutrinos. It was analyzed in Refs. [46,47] in electrodynamics, where singularities in the limit of electron mass going to zero were considered. To avoid all these complications we assume that neutrinos have a small but non-vanishing mass and the theory would be well de"ned with at worst logarithmic singularities in the reaction amplitudes. Low mass neutrinos could be produced by leptonic "eld in the same way as e>e\-pairs are produced by electric "eld. The probability of production is proportional to the Schwinger exponent exp(!pm/"e E ") and with a very small m may be signi"cant and may lead to a screening of J * * J leptonic charge.

6. Other long-range forces As we have already mentioned, long-range forces can be associated with an exchange of bosons with spin 0, 1, or 2. Bosons with higher spins do not have interaction with conserved sources and the only massless boson with spin 2 is the graviton, that interacts with energy-momentum tensor


13

[1,2]. In the case of massless vector bosons and graviton their vanishing masses are supported, respectively, by gauge invariance and general covariance. Long-range action due to a scalar particle exchange is less natural theoretically than that associated to vector or tensor bosons because no unbroken symmetry is known that would support zero mass of the corresponding scalar boson. Massless scalar "elds could arise as a result of continuous global symmetry breaking in accordance to the Goldstone theorem. However, a Goldstone "eld s has the pseudo-scalar coupling to matter, tM c ts, and so the corresponding potential decreases as r\. Scalar type coupling, tM ts, may appear for pseudo-Goldstone "elds but simultaneously with this coupling the boson acquires a non-zero mass, m , and generates Yukawa type Q potential, ;&exp(!m r)/r. The only known theoretical principle which would ensure long-range Q action of a scalar "eld with Coulomb type potential, ;&r\, is conformal invariance and the corresponding scalar boson is called the dilaton. However, in realistic four-dimensional world conformal invariance is broken and massless dilaton does not exist. Even if a scalar "eld is massless at the classical level quantum corrections should generate its non-vanishing mass. By this reason the scalar companion of the graviton, , in the Brans}Dicke model, that interacts with the trace of the energy-momentum tensor of matter, would acquire mass and the scalar gravity would become short-range. The magnitude of the mass of generated by quantum e!ects cannot be unambiguously calculated. Even an order of magnitude estimate is very much uncertain, m "m/m . and ( . can give anything from m +m (m\+10\ cm) to m +10\ eV (m\+10 km) depending ( . ( ( ( on the characteristic mass scale of the theory m . The upper bound corresponds to m "m and . the lower one corresponds to m "K +0.1 GeV. Because of that the Brans}Dicke "eld should /!" not manifest itself on astronomical scales but might be observable in experiments testing gravity at small distances. Still if a massless scalar boson somehow exists and creates long-range forces these forces would not be screened and might be operative on astronomical scales. However a scalar force between a static source and relativistic fermion, e.g. neutrino, is suppressed by the Lorenz factor m /E. J There may exist free massless spin-2 bosons coupled only to gravity with a simple Lag-rangian, L"U UIJ_?. Since the coupling to matter is absent such bosons would not mediate long-range IJ_? forces but may possibly play a role in solution of the mystery of the cosmological term [48]. Very interesting and unusual long-range forces were proposed by Okun in the paper [49] where a model with macroscopic con"nement radius was considered. The theory is analogous to Quantum Chromodynamics but is based on a di!erent gauge group with a di!erent number of fundamental fermions. This permits to have the coupling constant a of the order of the electromagF netic one at microscopically small distances and simultaneously to have the con"nement radius 1/K between the size of atomic nuclei and the size of the universe, depending on the model F parameters. Such long-range forces do not decrease with distance and bearers of a non-zero h-charge would be connected by macroscopically long h-strings. Cosmological implications of this model were considered in Refs. [49,50].

7. Conclusion At the present day only two types of long-range forces are known: gravitational and electromagnetic. However it is not excluded and moreover is quite natural that the club of long-range forces is

14


not con"ned only to those two. Though the strength of possible new long-range interactions should be quite weak, still due to possible macroscopically large charges their role in nature, in particular in cosmology and astrophysics may be signi"cant. They may be important for general cosmological evolution, large scale structure formation in the universe, and may be possibly imprinted on cosmic microwave background radiation. It is an interesting challenge to search and to "nd such new forces. It is my great pleasure to write a paper on the subject where my advisor and teacher Lev Borisovich Okun made such outstanding contributions and to dedicate this paper to his 70th anniversary.

Acknowledgements This work was supported by Danmarks Grundforskningsfond through its funding of the Theoretical Astrophysical Center. I am grateful for hospitality to the Weizmann Institute of Science, where this work was completed.

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S. Weinberg, Phys. Rev. 140 (1965) B515. S. Weinberg, Quantum Theory of Fields, Vol. I, Cambridge University Press, Cambridge, 1995, p. 537. T.D. Lee, C.N. Yang, Phys. Rev. 98 (1955) 1501. L.B. Okun, Yad. Fiz. 10 (1969) 358; English translation: Sov. J. Nucl. Phys. 10 (1969) 206. G. Feinberg, J. Sucher, Phys. Rev. 166 (1968) 1638. L. EoK tvoK s, D. Pekkar, F. Fekete, Ann. Phys. 68 (1922) 11. J. Renner, Mat. eH s termeH szettudomaH nyi ertsitoK 53 (1935) 542. P.G. Roll, R. Krotkov, R.H. Dicke, Ann. Phys. (NY) 26 (1964) 442. V.B. Braginsky, V.I. Panov, ZhETF 61 (1971) 873; English translation: Sov. Phys. JETP 34 (1972) 463. G.A. Zisman, Uchenye Zapiski LGPI No. 386 (1971) 80 (in Russian). V.M. Goldman, G.A. Zisman, R.Ya. Shaulov, Tematicheskii Sbornik LGPI, 1972 (in Russian). L.B. Okun, A remark on leptonic photons, June 1972 (unpublished, presented at ITEP seminar). S.I. Blinnikov, A.D. Dolgov, L.B. Okun, M.B. Voloshin, Nucl. Phys. B 458 (1996) 52. A.K. C7 iftc7 i, S. Sultansoi, S7 . TuK rkoK z, Phys. Lett. B 355 (1995) 494. J. Bernstein, M. Ruderman, G. Feinberg, Phys. Rev. 132 (1968) 1227. A.D. Dolgov, G.G. Ra!elt, Phys. Rev. D 52 (1995) 2581. J.A. Grifols, E. Masso, S. Peris, Phys. Lett. B 207 (1988) 493. J.A. Grifols, E. Masso, S. Peris, Astropart. Phys. 2 (1994) 161. G. Fiorentini, G. Mezzorani, Phys. Lett. B 221 (1989) 353. A.D. Dolgov, Ya.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1. G.G. Ra!elt, hep-ph/9903472; Ann. Rev. Nucl. Part. Phys. 49 (1999), in preparation. V.F. Shvartsman, Pis'ma ZhETF 9 (1969) 315; English translation JETP Lett. 9 (1969) 184. G. Steigman, D.N. Schramm, J.E. Gunn, Phys. Lett. B 66 (1977) 202. S. Burles, K.M. Nollett, J.N. Truran, M.S. Turner, Phys. Rev. Lett. 82 (1999) 4176. K.A. Olive, D. Thomas, hep-ph/9811444. E. Lisi, S. Sarkar, F.L. Vilante, Phys. Rev. D 59 (1999) 123520. J.A. Grifols, E. Masso, Phys. Lett. B 396 (1997) 201. L.B. Okun, Mod. Phys. Lett. A 11 (1996) 3041.

A.D. Dolgov / Physics Reports 320 (1999) 1}15 [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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L.B. Okun, Phys. Lett. B 382 (1996) 389. V.A. Ilyin, L.B. Okun, A.N. Rozanov, Nucl. Phys. B 525 (1998) 51. R.H. Dicke, Phys. Rev. 126 (1962) 1580. A.D. Dolgov, Ya.B. Zeldovich, M.V. Sazhin, Kosmologiya Rannei Vselennoi (in Russian), MGU, 1988; Basics of Modern Cosmology, Edition Frontier, Dreux, 1990. A.D. Dolgov, Phys. Rep. 222 (1992) 309. R.A. Lyttleton, H. Bondi, Proc. Roy. Soc. (London) A 252 (1959) 313. A. Barnes, Astrophys. J. 227 (1979) 1. C. Ftaclas, J.M. Cohen, Astrophys. J. 227 (1979) 13. S. Orito, M. Yoshimura, Phys. Rev. Lett. 54 (1985) 2457. J.E. Kim, T. Lee, Mod. Phys. Lett. A 5 (1990) 2209. A.S. Goldhaber, M.M. Nieto, Rev. Mod. Phys. 43 (1971) 277. Particle Data Group, European Phys. J. C 3 (1998) 1. V. Chibisov, Sov. Phys. Uspekhi 19 (1976) 624. Ya.B. Zeldovich, ZhETF 41 (1961) 1609. Ya.B. Zeldovich, I.D. Novikov, Struktura i Evolyutsiya Vselennoi (in Russian), Nauka, Moscow, 1975; Structure and Evolution of the Universe, The University of Chicago Press, Chicago, 1983. I.Yu. Kobzarev, L.B. Okun, ZhETF 43 (1962) 1904. M.B. Voloshin, L.B. Okun, Pis'ma ZhETF 28 (1978) 156; English translation: JETP Lett. 28 (1978) 145. T.D. Lee, M. Nauenberg, Phys. Rev. 133 B (1964) 1549. A.D. Dolgov, L.B. Okun, V.I. Zakharov, Nucl. Phys. B 41 (1972) 197. A.D. Dolgov, Phys. Rev. D 55 (1997) 5881. L.B. Okun, JETP Lett. 31 (1980) 144. A.D. Dolgov, Yad. Fiz. 31 (1980) 1522.


Cooperation between scientists and the government in the US: bene"ts and problems夽 Sidney D. Drell Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94305-5080, USA

Abstract This paper discusses the importance of maintaining close working relations between scientists and their government. Several examples of this cooperation in the U.S. are presented to illustrate the bene"ts, as well as problems, that result from such cooperation, or lack thereof. These examples include government support of scienti"c research, as well as contributions by science to help governments understand both the possibilities and the limitations of science as they formulate national policy. 1999 Elsevier Science B.V. All rights reserved. PACS: 01.78.#p Keywords: Science and government; Science and public policy; Science advice

For Lev Okun, a dear friend and admired colleague, on his 70th birthday. For his distinguished contributions to research and to teaching; for his holding to the highest principles during the most dizcult times in the Soviet Union; and for his continuing dedicated and selyess ewort to preserve the best science and scientists in Russia during the present dizculties, Lev commands the awection and highest respect of his colleagues worldwide.

In World War II, American scientists } from university and industrial research labs and including many refugees from persecution in Europe } were recruited to work in large projects which focused on developing the latest scienti"c advances in support of the military e!ort of the US

夽

This paper is adapted from a talk given at the Institute of Theoretical and Experimental Physics in Moscow on October 28, 1998 as co-recipient, with Professor A. Akhiezer of Kharkov, of the "rst Ya. Pomeranchuk Award. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 5 5 - 1

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S.D. Drell / Physics Reports 320 (1999) 17}25

and its Allies. The Radiation Lab organized at MIT developed microwave radar into instruments that proved decisive in the aerial defense of England and in the ultimate defeat of German U-boat raiders against the life line of convoys crossing the Atlantic Ocean to England and up to Murmansk. Out of the latest developments in nuclear physics and the theory of "ssion, a successful nuclear chain reaction was achieved by Enrico Fermi and his collaborators at the University of Chicago's Metallurgical Labs on December 2, 1942. This led to the production of plutonium and eventually to the construction of the "rst atomic bombs at Los Alamos under J. Robert Oppenheimer's leadership. These are the two best known examples of the massive e!ort by US scientists to create new weapons of war, and they served as models for continuing close scienti"c}government cooperation; and not just in the US. Great national laboratories were created in all the major industrial powers, some devoted to basic and applied research, and some to secret military weapons projects. A major stimulus for creating these laboratories was the important role scientists had played in helping win WWII. The public stood in awe of the nuclear weapons, in particular, and was willing to spend money liberally to support strong scienti"c communities as a resource to rely on for national security, if for no other reason. Formal recognition of the broader importance of scientists for society came in the US with the publication of a very perceptive and pathbreaking report presented to President Truman in 1945 by Vannevar Bush who headed the O$ce of Scienti"c Research and Development for President Roosevelt during WWII. This report entitled `Science: The Endless Frontiera emphasized the need to build on wartime experience and laid out the requisites for a nation to develop and manage a major league scienti"c endeavor. Bush wrote as follows `Science, by itself, provides no panacea for individual, social, and economic ills. It can be e!ective in the national welfare only as a member of a team, whether the conditions be peace or war. But without scienti"c progress no amount of achievement in other directions can insure our health, prosperity, and security as a nation in the modern world.a And he went on to caution: `Scienti"c progress on a broad front results from the free play of free intellects, working on subjects of their own choice, in the manner dictated by their curiosity for exploration of the unknown. Freedom of inquiry must be preserved under any plan for Government support of science2a Furthermore, Vannevar Bush reminded Washington that research is a di$cult and often very slow voyage over uncharted seas, and therefore, for science to #ourish with governmental support, there must be funding stability over a period of years so that long-range programs may be undertaken and pursued e!ectively. This report, and the appreciation of what science had done, led to a remarkable period of support for science that lasted some 20 years after WWII in the US. I was of the generation entering graduate school after WWII that bene"ted greatly from all the opportunities opening for us at that time. Institutions such as the Air Force O$ce of Scienti"c Research and the O$ce of Naval Research supported our graduate studies and provided necessary facilities. In my "eld of highenergy physics, major national accelerator centers were created at Berkeley, Brookhaven, and Stanford, and accelerators were built at many universities including Cornell and MIT. The National Science Foundation was created. There was a veritable love a!air with science during a period of great support and cooperation. The cooperation intensi"ed in the late 1950s when the Soviet Union launched Sputnik, and the development of long-range ballistic missiles raised new threats to our security. This served to


19

increase government reliance on scientists to address national security needs, and led to a growing appreciation of the need to forge even stronger bonds between government leaders and the scienti"c community. Cooperation and understanding extended beyond the realm of national security to the broader realms that, as V. Bush emphasized, contribute to the quality of life and to the protection we aspire to against diseases. As government reliance on scientists increased, so did the reliance of science on government. This is especially true in big science. The only way to make progress on some of the most fundamental frontiers, in particular the "elds of elementary particles and astrophysics, is in partnership with a government willing to pay some big bills. Governments are essential partners in the work, and it is in the self-interest of scientists to improve the understanding of government o$cials of what, why, and how we do the research that they are paying for. It is especially in scientists' self-interest to improve their understanding of the very nature of scienti"c progress, its reliance on basic research as the fuel and the engine of progress, and the need for stability and patience in their support of basic research. The importance of developing mechanisms to help maintain a strong bond and e!ective interactions between science and government in matters of US national security was already very clear in the 1950s. National security concerns were paramount, and rapid and, in some instances, revolutionary advances in military technology created a growing gap between science and government leaders. Our leaders were faced with di$cult new choices: In an era of H-bombs of mass indiscriminate destruction, what should we do to defend ourselves? What could we do? What about e!ects of worldwide fallout from continuing atmospheric nuclear weapon tests? What could we do in practice } taking realities into account } to reduce nuclear danger? President Eisenhower understood very well the importance of closing a growing gap } especially after Sputnik in 1957 and the emergence of the ICBM threat } between what scientists could foresee as the potential of these revolutionary technologies, and what our government leaders understood based on what they were familiar with. This led him to create the position of a full-time Science Advisor in the White House, and also to establish the President's Science Advisory Committee (PSAC). This mechanism was his resource for direct, in-depth analyses and advice as to what to expect from science and technology, both current and in prospect, in establishing realistic national policy goals. Members of PSAC, and consultants who served on its hardworking panels, were selected apolitically and solely on the grounds of demonstrated achievements in science and engineering, and of a commitment to work hard. They undertook studies and o!ered advice in response to White House interest over a broad range of national concerns that extended beyond issues of national security. Two things set PSAC apart from the then existing governmental line organizations and cabinet departments with operational responsibilities, and from non-governmental organizations engaged in policy research. First of all, members of PSAC had White House backing and, for the broad range of national security issues, access to the relevant, sensitive information for their studies. Secondly, the individual scientists were independent and presumably, therefore, immune from having their judgments a!ected by operational and institutional responsibilities. Therein lay their unique value. Beyond this, there had to be good `chemistrya between the President and his science advisor. The e!ectiveness of presidential science advisory mechanisms has waxed and waned in the US; and di!erent societies may choose very di!erent mechanisms for providing important technical

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advice to their leaders. I interpret it to be a healthy development for Russia that Yuri Osipov, President of your Academy of Science, has been included in Premier Primakov's inner cabinet so that the voice of science is being heard at the highest councils. We may hope he proves e!ective } and is being listened to. However, if but one scientist is being heard as a solitary advisor at the top, there is a danger that his own biases may color his advice. This danger was recognized in the US, and avoided by creating the PSAC to help balance out the inevitable parallaxes that can a!ect one's scienti"c visions. In the US the honeymoon between science and government began to come apart after 20 years, starting in the late 1960s. For one thing there were those in Washington who resented the fact that scientists who were feeding from the government trough for support of their research were speaking out more and more critically and publicly on policy issues of what to do with some of the new technical achievements resulting from science. There were particularly strong debates in the US on the dangers of atmospheric testing of nuclear weapons. The criticisms became very intense in the late 1960s as many scientists, and academics more broadly, became disillusioned with Washington and opposed strongly our involvement in Vietnam; and also became leading spokesmen in the debate about whether it was practical or desirable to employ ballistic missile defenses against intercontinental range missiles. No longer were scientists being perceived as major contributors to solving di$cult issues; more often than not, they were being viewed as part of a sustained opposition to government policies. Therein one sees a price for close cooperation with government. Some feel that science advisors on government committees must become captive to political policy decisions and quietly join in step with them. But such advisers are not members of the government, and most are academics and independent. Much of this work is pro bono and done out of a sense of duty to contribute to better policy decisions by the governments. Science advisers are not policy makers, and are listened to only on technical issues where they have some expertise to o!er. But their independence was not always welcome nor were they willing to be silent on other issues. There were also strains with many colleagues, especially during the Vietnam era, which were created by the barrier of secrecy between them and some of the advisers' activities. Nevertheless, whatever the di$culties, the need for governments to get the best counsel } not only in national security, but also in energy, environmental, and health care policy issues } is still of crucial importance today and will remain so for the future. The cost of making policy decisions, based on inadequate scienti"c understanding of what is possible, and what is pure fancy, can be catastrophic, whether we are considering the attack on disease or national security issues. This highlights the need to preserve an important emphasis on research and training } two key ingredients that are the foundations for progress in advancing our understanding } an area that requires close cooperation between science and government. It is tempting for government o$cials to "nd the strongest motivation for their support of science from the widely celebrated successes of speci"c scienti"c and technical projects that achieved the strategic goals for which they were created. It is much rarer to "nd an understanding in the political process of the long tortuous path of basic research that made it possible to accomplish those goals. Certainly, in the United States the legacy of the atom bomb project during World War II, and, more recently, of other similar major focused projects like the Apollo moon landing, have created a distorted picture of what science really is, and unrealistic expectations of what one can expect from it. This has led to frequent calls by society and governments


21

to conquer diseases such as cancer and AIDS, and to overblown expectations that the victory will be timely and total. Anything less is viewed to be a failure. Often forgotten in evaluating the atomic bomb project are the decades of basic scienti"c achievements by the Chadwicks, Fermis, Hahns, Bohrs, Strassmans, Meitners, Frischs, Flerovs, Kharitons, Zel'dovichs, Semenovs, Kurchatovs that made it all possible. Political leaders must be reminded, and helped to appreciate, that a sound understanding of fundamental cellular processes underlies progress in combating disease. Several years ago the Nobel Laureate discoverer of the Hepatitis B virus, Baruch Blumberg, wrote in the Financial Times of London that the discovery of the Hepatitis B virus, which has led to life saving vaccines, would not have happened as fast, if at all, had he been assigned the task of "nding it, rather than èngaging in basic science without a speci"c application in minda. His voice echoes concerns of many biologists and medical personnel who keep reminding us that we face new dangers from the development of drug resistant organisms. Drug resistant bacterial pathogens like those which cause tuberculosis are increasing at a rapid rate, and antibiotics that have been used for the past 50 years are not working against them. Unless countered, that proliferation could result in nothing less than the end of the era of presumptive good health that we have all grown to take for granted with antibiotics at our disposal. It is true today that infectious diseases are still the leading cause of death in the world. We face a growing urgency to develop new strategies to combat widespread viral infection, like AIDS, and drug resistant bacteria, but this will not happen without a profound understanding of basic life processes with which to arm a successful counterattack. Close cooperation between science and government is needed to maintain a healthy balance of support for basic research and to avoid ill-advised political decisions to earmark funds toward speci"c goals that look popular, and are politically attractive, in the short term. Overemphasis on such projects will have the result of robbing support for the basic work that provides the seed corn for progress. There is universal agreement that basic research, that is, research not motivated by speci"c application, must largely be supported by the federal government. Private industry can hardly be expected to recapture the bene"ts of basic research on behalf of any one industry, or within a time interval which provides for a reasonable rate of return on the "nancial investment. The time interval between initial results in basic research and gainful applications has been recently in the 20}30 yr range, and of course many basic research results do not lead to applications at all. Generally, this time-span is so large that only the federal government can make the required investment. The late, lamented superconducting supercollider (SSC) is a clear example of the importance of government support and close cooperation with science, and of the loss when that relation breaks down. In 1993 the termination of the SSC by the United States Congress was a crippling blow to the United States program in this "eld. It raised major questions about the future of research in high-energy physics in our country. Ten years of intense e!ort by some of the best talent in the "eld, both the young and senior, devoted to the planning, design and beginning construction of that machine went down the drain. In the US we were faced with a disillusioned dispirited community. It had grown by perhaps 10% in numbers, in anticipation of the SSC, but with a decrease of 20% in real funding over a 5-year period for the rest of the program, and now had no coherent plan for the future. The "eld was in a widespread depression without a clear vision of where to go. If ever there was a need for scientists and the government to cooperate in setting priorities for maintaining a US high-energy physics program at the world frontier, consistent with

22


a practical funding level from the government, that was it. We needed government support for an expanded commitment to international collaboration that would support US participation in actually building a big accelerator, the large hadron collider (LHC) at CERN. This required scientists educating and convincing Washington that such a commitment not only was essential for preserving a US leadership role in high-energy physics, but it would also be a real bene"t to the country's technical base. Above all, the scientists themselves had to unite behind a plan for the broader good, and be willing to look beyond their individual pet projects. Fortunately, in our country we have the High Energy Physics Advisory Panel (HEPAP) mechanism to address such problems. This is a standing panel, with a rotating membership of active, widely-respected physicists from the labs and the university user groups who are expected to take a broad view of the national program and set priorities within budget guidelines provided by the program managers whom they advise. The government asked these scientists through this mechanism to formulate recommendations for future research priorities and organizational structures for a national program. Their e!orts were successful. The community was able to unite in creating a practical and a!ordable program that was designed to build for a strong future. Its three elements are: E A #exible diverse and dynamic ongoing research e!ort to address scienti"cally compelling questions. E Vigorous studies to develop innovative technologies for future accelerators and detectors. (One example is a linear electron collider reaching above the 1 TeV energy region to be built by an international collaboration.) E Major participation at the highest energy frontier, the best current opportunity for which was identi"ed as joining the international collaboration building the large hadron collider at CERN. This proved to be a successful example of a crucially important cooperation between scientists and government, joined together e!ectively in the e!ort to restore a badly wounded US national program. For more than 30 years, the US high-energy physics community has relied on the so-called HEPAP process to develop program priorities with funding guidelines from Washington. This process created and sustained a community-wide consensus, or at least a working convenant, with a high batting average of success, up until the demise of the SSC. The SSC failure is a singular event. I am not sure that I understand all the factors leading to it, but aside from the SSC failure, our program's success was and is based on a process that has developed and sustained a good working cooperation between scientists and government. Let me turn back once again to national security issues of high technical content. Throughout the Cold War years, the issue of how best to discourage, deter, or defend ourselves against the use of nuclear weapons was on center stage. In particular, debates about the potential value, versus the dangerous illusions, of nationwide anti-ballistic missile defenses were ongoing. Though often driven by political considerations, these were serious debates in our country about strategic policy that touched a fundamental instinct of all human beings to protect our families and homes. But was it possible to meet the requirements of an e!ective ABM defense in an era with nuclear warheads of such enormous destructive potential?


23

Scientists in the United States, both in government and in universities and think tanks, were deeply involved in the questions which were at the root of almost all national security discussions. How easy was it, and how costly, to design and deploy o!ensive countermeasures to overpower any conceivable, a!ordable defense? Most experts saw an arms buildup between competing o!enses and defenses, and their countermeasures and counter-countermeasures, as harmful to strategic stability and future prospects of reducing the nuclear threat. The history and record of these debates in my country show how important it is to have an open channel of cooperation and dialogue between scientists and government. This was lacking in 1983 when President Reagan sought to escape the limitations of massive assured destruction by calling on emerging technologies of beam weapons and advanced sensors for a space-based defense. In the absence of careful independent analyses of practical technical realities, fanciful claims preceded more measured judgments. A largely political and highly acrimonious debate ensued that, for a while, was harmful to the e!orts to improve conditions for peaceful coexistence between our two countries. Looking back over this debate, one sees compelling evidence of the importance of high level cooperation between science and government on major national security issues with high technical content. This cooperation requires participation of scientists of top quality in detailed and comprehensive analysis of what to expect from science and technology, both current and future, in establishing realistic national policy goals. After all you cannot bend the laws of nature to satisfy your policy desires! Furthermore, these scientists must be independent and objective, as well as very good. Unfortunately, there was no process functioning to provide this during the ABM debate, which became more acrimonious than substantive in our country. In this connection I remember accompanying a bipartisan delegation of US Senate leaders in national security a!airs on a visit to Moscow in March, 1988 to meet your then President Gorbachev, Foreign Minister Shevardnadze and Marshal Akhromeyev. One evening I took them to Andrei Sakharov's apartment. When one of the senators, now a high o$cial in our government, questioned Sakharov on why should one not build an ABM, he received a strong, logically compelling lecture on the very same issues that I raised before about the harmful impact of an o!ense}defense race on stability, arms control, and the dangers of nuclear war. Andrei as usual was very eloquent and e!ective. A very important contribution by physicists to the ultimate resolution of that debate was the report prepared by the American Physical Society Study Group on `Science and Technology of Directed Energy Weaponsa co-chaired by N. Bloembergen of Harvard and C.K. Patel, then of Bell Labs. This study was an important example of the essential role of government cooperation with scientists. It would have been impossible without the government providing access to, and brie"ngs on, the essential technical details of the Star Wars program and ABM technology. It was published in the Reviews of Modern Physics, July 1987, four years after President Reagan's speech. This was a de"nitive analysis of the new and prospective technologies, along with the relevant operational issues. Laser and particle beams; beam control and delivery; atmospheric e!ects; beam}material interactions and lethality; sensor technology for target acquisition, discrimination, and tracking, systems integration including computing, power needs, and testing; survivability and system deployment were all analyzed carefully, as were some countermeasures. Aspects of boost-phase, mid-course, and terminal intercepts that were all parts of the Star Wars concept of a layered `defense in deptha were included in their comprehensive analysis.

24


The sober "ndings of the APS Directed Energy Weapons Study are summarized in part as follows: Àlthough substantial progress has been made in many technologies of DEW [directed energy weapons] over the last two decades, the Study Group "nds signi"cant gaps in the scienti"c and engineering understanding of many issues associated with the development of these technologies. Successful resolution of these issues is critical for the extrapolation to performance levels that would be required in an e!ective ballistic missile defense system. At present, there is insu$cient information to decide whether the required extrapolations can or cannot be achieved. Most crucial elements required for a DEW system need improvements of several orders of magnitude. Because the elements are interrelated, the improvements must be achieved in a mutually consistent manner. We estimate that even in the best of circumstances, a decade or more of intensive research would be required to provide the technical knowledge needed for an informed decision about the potential e!ectiveness and survivability of directed energy weapon systems. In addition, the important issues of overall system integration and e!ectiveness depend critically upon information that, to our knowledge, does not yet exista. These questions are still central in 1998, although now, happy to say, it is no longer a question of US versus Russian missiles, but concern about proliferating missile technology as exhibited recently in North Korea and Iran; and the threat that is posed by small numbers of missiles, in contrast to the massive numbers that threatened a holocaust between the US and the former Soviet Union. Another major issue that we face today is the e!ort to end all underground nuclear explosions under a Comprehensive Test Ban Treaty (CTBT). Presently, 152 countries, including US and Russia, have signed a CTBT that bans, for all time, all nuclear explosions, anywhere, of any size. This treaty presents us with an historic opportunity. When President Clinton signed the CTBT at the United Nations on September 26, 1996, he said that it was `The longest sought, hardest fought prize in the history of arms control.a The e!ort to end all nuclear tests commenced four decades earlier. Upon leaving o$ce President Eisenhower commented that not achieving a nuclear test ban `would have to be classed as the greatest disappointment of any administration } of any decade } of any time and of any party2a. A decisive political and strategic reason for the powers with nuclear weapons to sign a ban on all nuclear testing was the importance of such a treaty for accomplishing broadly shared nonproliferation goals. This was made clear in the debate at the United Nations in May 1995 by 181 nations when they signed on to the inde"nite extension of the Non-Proliferation Treaty (NPT) at its "fth and "nal scheduled "ve-year review. A commitment by the nuclear-powers to cease testing and developing new nuclear weapons was a condition for many of the non-nuclear nations when they signed on to the Treaty. Not only will the CTBT help limit the spread of nuclear weapons through the non-proliferation regime, particularly if current negotiations succeed in strengthening the provisions for verifying that treaty and appropriate sanctions are applied for non-compliance, it will also dampen the competition among nations who already have nuclear warheads, but who now will be unable to develop and deploy with high con"dence more advanced ones at either the high or the low end of destructive power. The CTBT would also force rogue states seeking a nuclear capability to place con"dence in untested bombs. Notwithstanding a strong case for the CTBT, the United States and Russia } and the other nuclear powers } if they are to be signatories of this treaty, must be con"dent of a positive answer to


25

the following question. Under a ban on all nuclear explosions, will it be possible to retain the currently high con"dence in the reliability of our nuclear arsenal over the long term, as the weapons age and the numbers are reduced through arms control negotiations? A group of independent scientists, working with government cooperation and support, addressed the scienti"c and technical challenge of answering this question. The "nding of this study was that the US could maintain con"dence in its enduring stockpile and meet our national security needs as currently perceived under a CTBT. To do so it was essential for the US to support a well-de"ned set of programs that are necessary to maintain the health of the enduring stockpile and to deepen our fundamental scienti"c understanding of the processes occurring during a nuclear explosion. They serve as the substitute for new nuclear test data. This conclusion helped form the technical base for President Clinton's decision for the United States to support and seek a true zero-yield CTBT announced in September, 1995. It is one more example of the importance of a close working relation between the scienti"c community and government. The CTBT must now be rati"ed by all 44 nations deemed to be nuclear capable, i.e. possessing nuclear warheads, nuclear power or research reactors, in order to enter into force. As we look to the future, we can see ominous new threats emerging which involve other weapons of indiscriminate destruction beyond the remaining 20 000 or so nuclear warheads still possessed today by at least 8 nations. Chemical and biological weapons in the hands of substate entities and terrorists are a growing concern. As the attack in the Tokyo subway system by the Aum Shinrikyo reminded us in 1995, these threats can no longer be ignored. Scientists will have to remain strongly involved in e!orts to build a safer 21st century, and this involvement means helping our government decision makers understand both the potential and the limits of science and technology.


From Alexander of Aphrodisias to Young and Airy J.D. Jackson University of California, Berkeley, CA, USA and Lawrence Berkeley National Laboratory, University of Berkeley, Berkeley, CA, 94720, USA

Abstract A didactic discussion of the physics of rainbows is presented, with some emphasis on the history, especially the contributions of Thomas Young nearly 200 years ago. We begin with the simple geometrical optics of Descartes and Newton, including the reasons for Alexander's dark band between the main and secondary bows. We then show how dispersion produces the familiar colorful spectacle. Interference between waves emerging at the same angle, but traveling di!erent optical paths within the water drops, accounts for the existence of distinct supernumerary rainbows under the right conditions (small drops, uniform in size). Young's and Airy's contributions are given their due. 1999 Elsevier Science B.V. All rights reserved. PACS: 01.30.Rr; 42.15.Dp; 42.25.Fx; 42.68.Ge

*This pedagogical piece on rainbows is dedicated to Lev B. Okun, colleague and friend, on his 70th birthday. On an extended visit to Berkeley in 1990, Lev saw on my ozce wall a picture of a double rainbow with at least three supernumerary bows visible inside the main bow. As part of my `lecturea on the photograph, I showed Lev a copy of these 1987 handwritten notes prepared for a class. He said, Àre these published somewhere?a My answer was no, but now they are, in augmented form. Lev is an amazing man, a physicist-mensch } a brilliant researcher, mentor, and warm human being. I have a vivid memory of a wonderful trip to Yosemite National Park with an allegedly ailing Lev. In the early morning hours, we found Lev outside our tent in Curry Village perched on a sloping rock doing vigorous calisthenics! Lev, may you have Many Happy Returns!+ The rainbow has fascinated since ancient times. Aristotle o!ered an explanation (not correct), as did clerics and scholars through the ages. Newton and Descartes established the elementary theory, according to what e now know as geometrical optics. But long before Newton and Descartes, as early as the 13th century, the puzzling occasional phenomenon of supernumerary rainbows was noted. These àberrationsa were inexplicable in terms of geometrical optics. It was not until the beginning of the 19th century that Thomas Young, promoting the wave theory of light against acolytes of Newton, o!ered the correct explanation of the supernumeraries as results of interference. Airy put the theory on a "rm mathematical footing in 1836. A scholarly treatment of the 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 8 8 - 5

28

J.D. Jackson / Physics Reports 320 (1999) 27}36

history of the attempts to understand the rainbow by Boyer [1] contains much of interest, including striking paintings and photographs with the rainbow as subject. A semi-popular account of the theory of rainbows is presented by Nussenzveig [2]. The discussion that follows traces the theory of the rainbow from the simple Cartesian} Newtonian description to the interference-di!raction-caustic treatment of Airy.

1. Geometrical optics, no dispersion A light ray is incident on a water drop of radius a at impact parameter b, as shown in Fig. 1. The index of refraction of water at the wavelength of the sodium D lines (j"5890, 5896 As ) and at 203C is closely n". The ray has an angle of incidence i whose sine is sin i"b/a,x. The angle of refraction r is given by Snell's law as r"sin\(x/n). The scattering angle h for the emerging light ray (de"ned here as the angle of emergence of the ray relative to the incident direction) can be computed by adding up the angular bends made by the ray: The entering bend is (i!r) Each internal re#ection bend is (p!2r) The exiting bend is (i!r) For m internal re#ections, the scattering angle is thus h ""2(i!r)#m(p!2r)" [Modulo 2p] . (1) K The primary rainbow has m"1, the secondary, m"2, and so on. Fig. 2 shows the scattering angle as a function of sin i"b/a for m"1 and m"2. At the extremes, the angle is either 0 or p,

Fig. 1. Geometrical optics of a primary rainbow.


29

but for intermediate b/a values, the light is scattered at various angles. Note the gap between 1293 and 1383. This is a region of negligible scattering (from higher orders) and appears as a dark space between the primary and secondary rainbows (known as Alexander's dark space, after Alexander of Aphrodisias, a follower of Aristotle and head of the Lyceum in Athens around 200 AD). The feature that causes the rainbow is the extremum in angle as a function of impact parameter. For the primary rainbow (upper curve in Fig. 2), the minimum angle is h "1383 at x "0.86066 for n". Classically, the scattering cross section is dp/dX""b db/(sinh dh)" (2) At the extremum, db/dh is in"nite, corresponding to a (classically) in"nite cross section. Wave aspects prevent the in"nity, of course, but it is indicative of a large cross section. The singular behavior is an example of a caustic. To examine the vicinity of the extremum and see its dependence on the index of refraction, we make a Taylor series expansion around the minimum. For the primary bow (m"1), we have h"p#2 sin\x!4 sin\(x/n) ,

(3)

where x"b/a. The "rst two derivatives are 2 4 dh " ! , dx (1!x (n!x

(4)

2x 4x dh " ! . dx (1!x) (n!x)

(5)

The extremum occurs for dh/dx"0, i.e., (n!x "2(1!x , or x "((4!n)/3 and (1!x "((n!1)/3.

Fig. 2. Rainbow scattering angles according to geometrical optics for index of refraction n"1.34. As indicated by the dotted lines, the dark band is somewhat wider in violet light, for which n"1.345. Typical m"1 and m"2 rays are shown in Fig. 4(a).

30


The second derivative at x"x is of interest: dh 9 (4!n h, " . (6) dx 2 (n!1) VV For n", h"9.780, and h "137.973. For x near x , we have hKh #h(x!x )/2. In passing, we note that near h"h , the classical scattering cross section is

2 x dp Ka ; . (7) h(h!h ) sin h dX As sketched in Fig. 4(b), scattering is concentrated at h"h , but also occurs for h'h . This is what causes the white appearance ìnsidea the primary bow (and òutsidea the secondary bow).

2. Colors of the rainbow, dispersion The beautiful colors of the rainbow are a consequence of the variation of the index of refraction of water with wavelength of the light. This dispersion, as it is called, is shown quantitatively in Fig. 3. If we arbitrarily de"ne the visible range of wavelength to be from 400 nm (violet) to 700 nm (red), we "nd that the index of refraction di!ers by *n"1.3;10\ from one end of the range to the other. Now consider the e!ect of a change in n on h: R 4x dh "!4 [sin\(x/n)]" . Rn dn n(n!x

(8)

Fig. 3. Index of refraction of water as a function of wavelength. The visible light interval is between 400 and 700 nm.


31

Fig. 4. Sketches to accompany the text.

At the rainbow angle,

dh dn

2 " n

4!n n!1

(9) V For n"4/3, dh/dn" "2.536. With *n"1.3;10\, we "nd *h "3.3;10\ radians "1.893. V The colors of the rainbow are spread over about 23 out of the 423 away from the anti-solar point (1803!1383). Since dn/dj(0, the red light emerges at a smaller angle than the violet.

32


The viewer thus sees the rainbow with the red at the outer side of the arc and the violet on the inner side, as indicated in Fig. 4(c). For the secondary bow, the order of the colors is opposite.

3. Consequences of the wave nature of light, supernumerary rainbows For rays incident at impact parameters close to b "x a, the scattering angle is equal to h , correct to "rst order inclusive. In fact, because of the quadratic dependence of h!h on *x"x!x , two rays incident at impact parameters greater and less than b by an amount "*x" will emerge with the same scattering angle. In the wave picture, as observed by Young [3] in 1803, these two waves emerging in the same direction can interfere. Whether the interference is constructive or destructive depends on the di!erent in optical path length of the two rays. This varies as a function of *x and so provides the potential for interference e!ects in addition to dispersion in rainbows. Referring to Fig. 1, we see as the solid line the critical ray, which emerges at h"h . On either side are shown neighboring rays with small "*x" that emerge at angles di!ering from h only in O(*x). The surfaces AA and BB are convenient ones for de"ning the optical path of a ray in the neighborhood of the critical ray. The optical path, or more appropriately, the phase accumulated along the ray, is given by

(x)"2ka(1!cos i#2n cos r) ,

(10)

where the 2(1!cos i) represents the sum of the distances from AA to the drop's surface and similarly for the exit leg, while 4n cos r is the length (times n) of the path interior to the drop. The free-space wave number is k"u/c"2p/j. In terms of x"b/a, the phase is

(x)"2ka[1!(1!x#2(n!x] .

(11)

We are interested in the behavior of (x) near x!x . Consider the derivative, 2x x d

"2ka ! . dx (1!x (n!x

(12)

Comparison with dh/dx in part(a) shows the relation, d /dx"kax dh/dx .

(13)

Writing x"x #m, we can put this equation in the form, d /dm"ka[x dh/dm#m dh/dm] . Integration on both sides from 0 to m yields

K dh m dm dm K "ka x (h!h )#mh! h(m) dm .

(m)! "ka x (h!h )#

(14)

(15) (16)


33

Inserting h(m)"h #hm/2#O(m) in the integral, we "nd

(m)" #ka[x (h!h )#hm/3#O(m)] .

(17)

For two rays, a and b, as shown in Fig. 1, with equal and opposite small m values, the phase di!erence is d" (m)! (!m)"2kahm/3 .

(18)

If we equate this phase di!erence to 2pN and express m in terms of h!h , we "nd the angles of constructive interference to be h !h K[(h)/2](3pN/ka) . ,

(19)

(Actually, a more correct procedure has N# replacing N. See Ref. [4, p. 243] and Section 3.21.) The angles of constructive interference mark the positions of additional rainbows, called supernumerary rainbows. They lie at larger angles than h and so fall ìnsidea the main bow. Their colors are in the same order as in the primary bow. They are rarely seen because conditions must be optimized for them to appear unobscured or not washed out. The angle (h !h ) depends on the , droplet size, varying as (ka)\. For large drops, the angle becomes very small and the supernumerary bows fall inside the various colors of the primary bow. Using N"5/4, h"9.780, and (h!h ) "3;10\ radians (corresponding to the spread caused by dispersion), we "nd (ka) K2.5;10. With k appropriate to the sodium D lines, we obtain a K0.28 mm. Larger

drops will cause the supernumeraries to be obscured by the e!ects of dispersion. Variation in drop size, even if the drops are small enough, also causes the maxima of the supernumerary bows to be washed out in angle. Thus, one needs small drops, uniform in size, in order to see clearly the supernumeraries. All this was understood by Young [3]. For very small drop size, a(50 lm, the whole pattern of primary peak (N") and supernumer ary peaks for a given wavelength is so spread in angle that dispersion e!ects are unimportant. All the colors have broad primary peaks lying almost on top of each other in angle. The result is a `white rainbowa or `fog bowa.

4. Huygens: construction for the rainbow, Airy integral The scalar di!raction theory of Huygens, Young, Fresnel and Kirchho! [5, Section 10.5] can be used to obtain an approximate description of the rainbow in wave theory, as was "rst done by George B. Airy (1836). Consider the line BB in Fig. 1, where we have evaluated the expression for the phase (m). A wave along this line will have the form tJexp[ik z#ik ) r #i (m)] , (20) , , , where we choose our axes so that z is in the direction of scattering at h and r is measured along , BB, with value ax at the critical ray. If the wave is propagating in the direction h, then k ) r "!ka(h!h )x , (21) , ,

34


where the negative sign comes from the fact that k and r are antiparallel (h'h ). Since z is , , constant on BB, the relevant parts of the wave's overall phase are k ) r # (m)"ka[(h!h )(x !x)#(h/3)m]#

(22) , , "ka[!m(h!h )#(h/3)m]# . (23) With the approximation, h!h "hm/2, we "nd k ) r # (m)! "!(ka/6)hm#O(m) . (24) , , Along the line BB, the wave amplitude in the neighborhood of x"x or m"0 has the form t(m)"exp(!ikahm/6) (25) assuming the slowly varying amplitude function is a constant. We can now use the simplest version of the Kirchho! integral for di!raction,

k e I0 t " t(x) da . 1 2pi R

(26)

With kRKkr!k ) x in the usual way, we "nd a scattering amplitude,

t J 1

e I?F\F Ke\ I?FK dm .

(27) \ This can be put in the form of the Airy integral Ai(!g), as de"ned by Abramowitz and Stegun [6, p. 447]:

1 Ai(!g)" cos(t/3!gt) dt , (28) p where g"(2ka/h)(h!h ). This function is shown in Fig. 4(d). For positive g, Ai(!g) oscillates with an amplitude that decreases as g\. For negative g, Ai(!g) is exponential in character, falling rapidly to zero for !g'1. The maxima and minima occur successively at g"1.0188 (1.1155), 3.2482 (3.2616), 4.8201 (4.8263), 6.1633 (6.1671), 7.3722 (7.3748) . The numbers in parentheses are values of [(3p/2)(N#)] for N"0, 1, 2, 2, from our previous discussion of the angles for constructive interference. For larger N values, the agreement is excellent. It is of interest to compare the angular positions of the supernumerary rainbows implicit in the tabulated g values with the examples quoted by Young [3]. Notorious for not giving details of his calculations, he only quotes answers. He states that for drops inches in diameter, the reds of the "rst and fourth supernumerary bows are approximately 23 and 43 inside the red of the primary (the "rst just clearing the violet of the primary). With h"9.912 and j"700 nm for red light, we "nd ka"1.50;10 and g"1.34 (h!h ), with the angles measured in degrees. With *g"2.23 and 6.35 for the "rst and fourth supernumeraries, we obtain *h"1.7 and 4.7 degrees, in rough agreement with Young. Incidentally, the fact that gO0 for N"0 (primary bow) explains the


35

long-standing puzzle that the angular positions of some rainbows were observed to vary appreciably away from the Cartesian}Newtonian angle h (evidently dependent on drop size). The peak intensities of the supernumerary bows, relative to the primary bow, are 0.612, 0.504, 0.446, 0.408, 2, falling o! only as g\ or (N#)\. Note that the g\ behavior is just what our classical cross section gave. The wave aspect rounds the corners and gives interference. The intensity pattern for red and violet light is sketched in Fig. 4(e) for a"64 lm (ka+10 for violet light). The "rst supernumerary bow would be visible, but subsequent ones would not. Add some variation in drop size and everything except the primary bow will wash out. An approximate cross section for a given ka can be written in terms of Airy's integral by normalizing the average intensity at large g to the classical cross section. From Abramowitz and Stegun [6], one "nds that for large g the leading term in an asymptotic series is (29) Ai(!g)K(1/(p)(1/g)sin(g#p/4) . The average value of its square is 1"Ai(!g)"2"1/(2p(g). With the expression for g in terms of h!h , this becomes 2 1 h 1 . (30) 1"Ai(!g)"2" ka h(h!h ) 2p 2 Comparison with the classical cross section, near h"h , dp x 2 Ka , (31) dX sin h h(h!h ) leads to the cross section in the Airy approximation,

dp 2 x (ka)"Ai(!g)" . (32) K2pa dX sin h h (See Fig. 4(f ).) For n", h "137.973, sin h "0.66952, x "0.86066, and h"9.780. Ignoring the loss of intensity from the refractions and re#ection, the cross section for a given component of the rainbow of "xed ka is thus dp K2.80(ka)"Ai(!g)"a . dX

(33)

At the peak of the rainbow, dp/dX"0.803(ka)a. For ka"10, this cross section is 32 times as great as an isotropic cross section, dp/dX"a/4.

5. Comment on polarizations and loss of intensity After Young, but before Airy, David Brewster showed in 1812 that the scattered rainbow light was almost completely polarized, con"rming earlier observations of Biot (of the Biot}Savart law in magnetism). The polarization comes about because the refracted and re#ected intensities at each of the interfaces are di!erent for di!erent polarizations. The formulas of pp. 305}306 of Jackson [5]

36


can be used to show that at h"h , the ratios of scattered amplitude to incident amplitude are i 8/27 g E 1 "j 2!n E g 2 2n k n#2 2#n

for E plane of incidence , ,

(34) for E plane of incidence . ,

For n", the intensity of perpendicular polarization is 8.78;10\ of the incident, while the intensity of the parallel polarization relative to the perpendicular is 3.9;10\. The cross section quoted above must therefore be multiplied by approximately ;1.039;8.78;10\ for un polarized light incident.

6. Note on notation Van de Hulst [4] de"nes his Airy integral to be

f (z)"

cos

p (zt!t) dt . 2

(35)

His z and our g are related by z"(12/p)g. His function f (z) is f (z)"(2p/3)Ai(!g). Note that other notations are used for the Airy integral. For example, see [7, Section 59].

Acknowledgements Special thanks are owed to Gail Harper and Betty Armstrong, who TE X'ed the equation-rich handwritten manuscript, and to Don Groom for generating postscript "les from my hand-drawn "gures.

References [1] C.B. Boyer, The Rainbow, Princeton University Press, Princeton, 1959, 1987. [2] H.M. Nussenzveig, Scienti"c American, April 1977, p. 116. [3] T. Young, Experiments and Calculations Relative to Physical Optics, Bakerian Lecture, November 24, 1803, publ. Phil. Trans. 1804, in: G. Peacock (Ed.), Miscellaneous Works of Thomas Young, Vol. I, John Murray, London, 1855, p. 179, esp. pp. 185}187. [4] H.C. Van de Hulst, Light Scattering by Small Particles, Wiley, New York, 1957, Chapter 13; also Dover reprint. [5] J.D. Jackson, Classical Electrodynamics, 3rd Edition, Wiley, New York, 1998. [6] M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1972. [7] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Pergamon, Oxford, 1962.


Particle creation by charged black holes I.B. Khriplovich Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia Novosibirsk University, Russia Dedicated to Lev Borisovich Okun on his birthday

Abstract A simple derivation is given for the leading term (n"1) in the Schwinger formula for the pair creation by a constant electric "eld. The same approach is applied then to the charged particle production by a charged black hole. In this case, as distinct from that of a constant electric "eld, the probability of the charged particle production depends essentially on the particle energy. The production rate by black holes is found in the nonrelativistic and ultrarelativistic limits. The range of values for the mass and charge of a black hole is indicated where the discussed mechanism of radiation dominates the Hawking one. 1999 Elsevier Science B.V. All rights reserved. PACS: 12.20.Ds; 04.70.Dy Keywords: External "elds; Black holes

The problem of particle production by the electric "eld of a black hole has been discussed repeatedly [1}6]. The probability of this process was estimated in these papers using in some way or another the result obtained previously [7}9] for the case of an electric "eld constant all over the space. This approximation might look quite natural with regard to su$ciently large black holes, for which the gravitational radius exceeds essentially the Compton wave length of the particle j"1/m. (We use the units with "1, c"1; the Newton gravitational constant k is written down explicitly.) However, in fact, as will be demonstrated below, the constant-"eld approximation, generally speaking, is inadequate to the present problem, and does not re#ect a number of its essential peculiarities. It is convenient to start the discussion just from the problem of particle creation by a constant electric "eld. In this paper we restrict ourselves to the consideration of the production of electrons E-mail address: [email protected] (I.B. Khriplovich) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 7 8 - 2

38

I.B. Khriplovich / Physics Reports 320 (1999) 37}49

and positrons, primarily because the probability of emitting these lightest charged particles is the maximum one. Besides, the picture of the Dirac sea allows one in the case of fermions to manage without the second-quantization formalism, thus making the consideration most transparent. To calculate the main, exponential dependence of the e!ect, it is su$cient to restrict to a simple approach due to [7] (see also Refs. [10,11]). In the potential !eEz of a constant electric "eld E the usual Dirac gap (Fig. 1) tilts (see Fig. 2). As a result, a particle with a negative energy in the absence of the "eld, can now tunnel through the gap (see one of the horizontal dashed lines in Fig. 2) and go to in"nity as a usual particle. The hole created in this way is nothing but an antiparticle. The exponential factor in the probability of the particle creation depends obviously on the action only inside the barrier. This action does not change under a shift of the dashed line in Fig. 2 up or down, i.e., under a shift by *E of the energy E of the created particle. Being obviously an integral of motion, E is also the energy of the initial particle of the Dirac sea. If we put for instance E"!m,

Fig. 1.

Fig. 2.


39

so that the particle enters the barrier at z"0, the squared four-dimensional momentum (E!e )!p"m becomes (!m#eEz)!p"m . For the time being we assume that the transverse part of the particle momentum p "(p ,p ), which , V W is also an integral of motion, is equal to zero. Inside the barrier the modulus of the momentum p(z)"p (z) is X "p (z)""(m!(m!eEz) . The action inside the barrier equals:

KC# pm dz" p(z)"" . 2eE Finally, the exponential factor in the probability = is [7]: S"

(1)

=&exp(!2S)&exp(!pm/eE) .

(2)

One can easily take into account in the exponent (2) the transverse momentum p . This integral , of motion will, clearly, enter all the previous formulae in the only combination m#p . So, , expression (2) demands in this case the substitution mPm#p , , changing thus to

p(m#p ) , . =&exp ! eE

(3)

Let us calculate now the pre-exponential factor in the probability of particle creation. The obtained exponential (3) is the probability that a particle of the Dirac sea approaching the potential barrier from the left (see Fig. 2), will tunnel through it to the right, thus becoming a real electron. To obtain the total number of pairs created per unit volume per unit time, the exponential (3) should be multiplied by the current density of the particles of the Dirac sea j "ov . X X For the velocity we use the common relationship

(4)

v "RE/Rp (5) X (the subscript z of the longitudinal momentum p is again omitted here and below). The particle density is as usual o"2dp dp/(2p) , , the factor 2 being due to two possible orientations of the electron spin.

(6)

40


For a "xed coordinate z and "xed p the identity holds: , (RE/Rp) dp"dE .

(7)

On the other hand, it is obvious that the interval of energies dE of the tunneling particles is directly related to the interval dz of longitudinal coordinates of the points where the particles enter the barrier: dE"eE dz (up to an inessential sign). Being interested in the probability per unit volume in general, and per unit longitudinal distance in particular, we should delete thus the arising factor dz when calculating the e!ect. So, the total number of pairs created per unit volume per unit time is

dp p(m#p ) , exp ! , . = "2eE (2p) eE

(8)

Now the trivial integration over the transverse momenta gives the "nal result = "(eE/4p)exp(!pm/eE) .

(9)

The probability = in the above formulae is supplied with the subscript 1/2 to indicate that the result refers to particles of spin one half. Obviously, the notion of the Dirac sea, and hence the above derivation by itself, do not apply to boson pair creation. However, in the semiclassical approximation, the creation rate for particles of spin zero is almost the same. The only di!erence is that since these particles do not have two polarization states, the rate is two times smaller than (9): = "(eE/8p)exp(!pm/eE) .

(10)

The corresponding exact results for a constant electric "eld are [9]

eE 1 pm = " exp !n , (11) 4p eE n L eE (!1)L\ pm =" exp !n . (12) n 8p eE L Obviously, the account for higher terms, with n52, in the sums (11), (12) makes sense only for very strong electric "elds, for eE9m. For smaller "elds, when eE;m, simple formulae (9) and (10) are correct quantitatively. The above straightforward derivation explains clearly some important properties of the phenomenon. First of all, the action inside the barrier does not change under a shift of the dashed line in Fig. 2 up or down. Owing to this property alone expressions (2) and (9) are independent of the energy of created particles. Then, for the external "eld to be considered as a constant one, it should change weakly along the path inside the barrier. Obviously, the length of this path l&m/eE di!ers essentially from the Compton wave length j"1/m of the particle. The ratio l/j is of the same order of magnitude as the action S inside the barrier, and therefore should be large for the semiclassical approximation to be applicable at all. Thus, one may expect that, generally speaking, the constant-"eld approximation is not applicable to the problem of a charged black hole radiation, and that the probability of particle


41

production in this problem is strongly energy-dependent. The explicit form of this dependence will be found below. We restrict ourselves in the present work to the case of a non-rotating black hole. We start the solution of the problem by calculating the action inside the barrier. The metric of a charged black hole is well-known: ds"f dt!f\ dr!r(dh#sinh d ) ,

(13)

where f"1!2kM/r#kQ/r ,

(14)

M and Q being the mass and charge of the black hole, respectively. The equation for a particle 4-momentum in these coordinates is f \(e!eQ/r)!fp!l/r"m .

(15)

Here e and p are the energy and radial momentum respectively, of the particle. We assume that the particle charge e is of the same sign as the charge of the hole Q, ascribing the charge !e to the antiparticle. Clearly, the action inside the barrier is minimum for the vanishing orbital angular momentum l. It is rather evident therefore (and will be demonstrated below explicitly) that after the summation over l just the s-state de"nes the exponential in the total probability of the process. So, we restrict for the moment to the case of a purely radial motion. The equation for the Dirac gap for l"0 is e (r)"eQ/r$m(f , !

(16)

which is presented in Fig. 3. It is known [12] that at the horizon of a black hole, for r"r "kM#(kM!kQ, the gap vanishes. Then, with the increase of r the lower boundary > e (r) of the gap decreases monotonically, tending asymptotically to !m. The upper branch e (r) at \ > "rst, in general, increases, and then decreases, tending asymptotically to m. It is clear from Fig. 3 that those particles of the Dirac sea whose coordinate r exceeds the gravitational radius r and whose energy e belongs to the interval e (r)'e'm, tunnel through > \ the gap to in"nity. In other words, a black hole loses its charge due to the discussed e!ect, by emitting particles with the same sign of the charge e, as the sign of Q. Clearly, the phenomenon takes place only under the condition eQ/r 'm . >

(17)

For an extreme black hole, with Q"kM, the Dirac gap looks somewhat di!erent (see Fig. 4): when Q tends to kM, the location of the maximum of the curve e (r) tends to r , and the value of > > the maximum tends to eQ/r . It is obvious however that the situation does not change qualitatively > due to it. Thus, though an extreme black hole has zero Hawking temperature and, correspondingly, gives no thermal radiation, it still creates charged particles due to the discussed e!ect.

42


Fig. 3.

In the general case Q4kM the doubled action inside the barrier entering the exponential for the radiation probability is

2S"2

P

dr"p(r,e)"

P P dr r (!pr#2(eeQ!kmM)r!(e!km)Q . "2 r!2kMr#kQ P

(18)

Here p "(e!m is the momentum of the emitted particle at in"nity, and the turning points r are as usual the roots of the quadratic polynomial under the radical; we are interested in the energy interval m4e4eQ/r . Of course, the integral can be found explicitly, though it demands > somewhat tedious calculations. However, the result is su$ciently simple: 2S"2p[m/(e#p )p ][eQ!(e!p ) k M] . (19) Certainly, this expression, as distinct from the exponent in formula (2), depends on the energy quite essentially. Let us note that the action inside the barrier does not vanish even for the limiting value of the energy e "eQ/r . For a nonextreme black hole it is clear already from Fig. 3. For an extreme K > black hole this fact is not as obvious. However, due to the singularity of "p(r,e)", the action inside the


43

Fig. 4.

barrier is "nite for e"e "eQ/r for an extreme black hole as well. In this case the exponential K > factor in the probability is exp(!p((km/e)kmM) .

(20)

Due to the extreme smallness of the ratio (km/e&10\ ,

(21)

the exponent here is large only for a very heavy black hole, with a mass M exceeding that of the Sun by more than "ve orders of magnitude. And since the total probability, integrated over energy, is dominated by the energy region e&e , the semiclassical approach is applicable in the case of K extreme black holes only for these very heavy objects. Let us note also that for the particles emitted by an extreme black hole, the typical values of the ratio e/m are very large: e/m&e /m"eQ/kmM"e/(km&10 . K In other words, an extreme black hole in any case radiates highly ultrarelativistic particles mainly. Let us come back to nonextreme holes. In the nonrelativistic limit, when eQ/r Pm and, > correspondingly, the particle velocity vP0, the exponential is of course very small: exp(!2pkmM/v) .

(22)

44


Therefore, we will consider mainly the opposite, ultrarelativistic limit where the exponential is exp(!p(m/e) eQ) .

(23)

Of course, here also the energies e&e &eQ/kM are essential, so that the ultrarelativistic limit K corresponds to the condition eQ space 2

dp dp dp dx dy dz V W X (2p)

(30)


45

is a scalar. On the other hand, the number of particles in the elementary cell dx dy dz equals (see [13], A 90) o(cdx dy dz ,

(31)

where c is the determinant of the space metric tensor. Since all the states of the Dirac sea are occupied, we obtain by comparing formulae (30) and (31) that the following expression o

2 dp dp dp 2 dp dp dp " V W X" V W X (2p) (2p) (g c (!g (g should be plugged in formula (28) for the current density (the summation here and below is performed with "xed e and l, see (28)). In our case the determinant g of the four-dimensional metric tensor does not di!er from the #at one, so that the radial current density of the particles of the Dirac sea is dp Re jP(e,l )"2 . (2p) Rp

(32)

The summation on the right-hand-side reduces in fact to the multiplication by the number 2l#1 of possible projections of the orbital angular momentum l onto the z-axis and to the integration over the azimuth angle of the vector l, which gives 2p. Using identity (7), we obtain in the result 2p(2l#1) jP(e,l )"2 . (33) (2p)r(e,l ) Finally, the pre-exponential factor in the probability, di!erential in energy and orbital angular momentum, is 2(2l#1) . p

(34)

Correspondingly, the number of particles emitted per unit time is

dN 2 " de (2l#1)exp[!2S(e,l)] . (35) dt p J In the most interesting, ultrarelativistic case dN/dt can be calculated explicitly. Let us consider the expression for the momentum in the region inside the barrier for lO0:

"p(e,l,r)""f \

l m# r

eQ f! e! . r

(36)

The main contribution to the integral over energies in formula (35) is given by the region ePe . In K this region the functions f (r) and e!eQ/r, entering expression (36), are small and change rapidly. As to the quantity k(r,l )"m#(l/r) ,

(37)

one can substitute in it for r its average value, which lies between the turning points r and r . Obviously, in the discussed limit ePe the near turning point coincides with the horizon radius, K

46


r "r . And the expression for the distant turning point is in this limit >

2k (kM!kQ . (38) r "r 1# > e !k r K > Assuming that for estimates one can put in formula (37) r&r , one can easily show that the > correction to 1 in the square bracket is bounded by the ratio l/(eQ). Assuming that this ratio is small (we will see below that this assumption is self-consistent), we arrive at the conclusion that r +r , and hence k can be considered independent of r: k(r,l)+m#l/r . As a result, we > > obtain 2S(e,l)"peQ(m/e#l/r e) . > Now we "nd easily

(39)

dN/dt"m(eQ/pmr )exp(!pmr /eQ) . (40) > > Let us note that the range of orbital angular momenta, contributing to the total probability (40), is e!ectively bounded by the condition l:eQ. Since eQ


47

of inequality (42), which guarantees the localization of the initial wave packet in the region of a strong "eld, means in particular that eQ"Za'1

(43)

(we have introduced here Z"Q/e). It is well-known (see, e.g. [14,15]) that the vacuum for a point-like charge with Za'1 is unstable, so that such an object loses its charge by emitting charged particles. It is quite natural that for a black hole whose gravitational radius is smaller than the Compton wave length of the electron, the condition of emitting a charge is the same as in the pure quantum electrodynamics. (Let us note that the unity in all these conditions should not be taken too literally: even in quantum electrodynamics, where the instability condition for the vacuum of particles of spin 1/2 is for a point-like nucleus just Za'1, for a "nite-size nucleus it changes [14,15] to Za'1.24. On the other hand, for the vacuum of scalar particles in the "eld of a point-like nucleus the instability condition is [16,17]: Za'1/2.) As has been mentioned already, for a light black hole, with kmM(1, the discussed condition eQ'1 leads to a small action inside the barrier and to the inapplicability of the semiclassical approximation used in the present article. The problem of the radiation of a charged black hole with kmM(1 was investigated numerically in [18]. The exponential exp(!pmr /eQ) > in our formula (40) coincides with the expression arising from formula (2), which refers to a constant electric "eld E, if one plugs in for this "eld its value Q/r at the black hole horizon. As > has been mentioned already, an approach based on formulae for a constant electric "eld was used previously in Refs. [1}6]. Thus, our result for the main, exponential dependence of the probability integrated over energies, coincides with the corresponding result of these papers. Moreover, our "nal formula (40) agrees with the corresponding result of Ref. [6] up to an overall factor 1/2. (This di!erence is of no interest by itself: as has been noted above, for a non-extreme black hole the semiclassical approximation cannot guarantee an exact value of the overall numerical factor.) Nevertheless, we believe that the analysis of the phenomenon performed in the present work, which demonstrates its essential distinctions from the particle production by a constant external "eld, is useful. First of all, it follows from this analysis that the probability of the particle production by a charged black hole has absolutely nontrivial energy spectrum. Then, in no way are real particles produced by a charged black hole all over the whole space: for a given energy e they are radiated by a spherical surface of the radius r (e), this surface being close to the horizon for the maximum energy. (It follows from this, for instance, that the derivation of the mentioned result of Ref. [6] for dN/dt has no physical grounds: this derivation reduces to plugging E"Q/r to the Schwinger formula (9), obtained for a constant "eld, with subsequent integrating all over the space outside the horizon.) Let us compare now the radiation intensity I due to the e!ect discussed, with the intensity I of & the Hawking thermal radiation. Introducing additional weight e in the integrand of formula (35), we obtain I"pm(eQ/pmr ) exp(!pmr /eQ) . > >

(44)

48


As to the Hawking intensity, the simplest way to estimate it, is to use dimensional arguments, just to divide the Hawking temperature ¹ "(r !r )/4pr & > \ > by a typical classical time of the problem r (in our units c"1). Thus, > I &1/4pr . (45) & > More accurate answer for I di!ers from this estimate by a small numerical factor &2;10\, but & for qualitative estimates one can neglect this distinction. The intensities (44) and (45) get equal for p (mr ) p (kmM) > & eQ& . (46) 6 ln(mr ) 6 ln(kmM) > (One cannot agree with the condition eQ&1/(4p) for the equality of these intensities, derived in Ref. [6] from the comparison of e "eQ/r with ¹ "(r !r )/(4pr ).) K > & > \ > Let us consider in conclusion the change of the horizon surface of a black hole, and hence of its entropy, due to the discussed non-thermal radiation. To this end, it is convenient to introduce, following Refs. [19,20], the so-called irreducible mass M of a black hole: (47) 2M "M#(M!Q ; here and below we put k"1. This relationship can be conveniently rewritten also as M"M #Q/4M . (48) Obviously, r "2M , so that the horizon surface and the black hole entropy are proportional to > M. When a charged particle is emitted, the charge of a black hole changes by *Q"!e, and its mass by *M"!eQ/r #m, where m is the deviation of the particle energy from the maximum one. > Using the relationship (48), one can easily see that as a result of the radiation, the irreducible mass M , and hence the horizon surface and entropy of a non-extreme black hole do not change if the particle energy is the maximum, eQ/r . In other words, such a process, which is the most probable > one, is adiabatic. For m'0, the irreducible mass, horizon surface, and entropy increase. As usual, an extreme black hole, with M"Q"2M , is a special case. Here for the maximum energy of an emitted particle e "e, we have *M"*Q"!e, so that the black hole remains K extreme after the radiation. In this case *M "!e/2, i.e., the irreducible mass and the horizon surface decrease. In a more general case, *M"!e#m, the irreducible mass changes as follows:

e!m # *M "! 2

e m M ! # m. 2 4

(49)

Clearly, in the case of an extreme black hole of a large mass, already for a small deviation m of the emitted energy from the maximum one, the square root is dominating in this expression, so that the horizon surface increases. I am grateful to I.V. Kolokolov, A.I. Milstein, V.V. Sokolov, and O.V. Zhirov for their interest in this work and useful comments. The work was supported by the Russian Foundation for Basic


49

Research through Grant No. 98-02-17797, by Grant No. 96-15-96317 Leading Science Schools, and by the Federal Program Integration-1998 through Project No. 274.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

M.A. Markov, V.P. Frolov, Teor. Mat. Fiz. 3 (1970) 3. W.T. Zaumen, Nature 247 (1974) 531. B. Carter, Phys. Rev. Lett. 33 (1974) 558. G.W. Gibbons, Comm. Math. Phys. 44 (1975) 245. T. Damour, R. Ru$ni, Phys. Rev. Lett. 35 (1975) 463. I.D. Novikov, A.A. Starobinsky, Zh. Eksp. Teor. Fiz. 78 (1980) 3 [Sov. Phys. JETP 51 (1980) 3]. F. Sauter, Z. Phys. 69 (1931) 742. W. Heisenberg, H. Euler, Z. Phys. 98 (1936) 714. J. Schwinger, Phys. Rev. 82 (1951) 664. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum Electrodynamics, Pergamon Press, Oxford, 1989. W. Greiner, J. Reinhardt, Quantum Electrodynamics, Springer, Berlin, 1994. N. Deruelle, R. Ru$ni, Phys. Lett. 52 B (1974) 437. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, London, 1995. Ya.B. Zel'dovich, V.S. Popov, Uspekhi Fiz. Nauk 105 (1971) 403 [Sov. Phys. Uspekhi 14 (1972) 673]. A.B. Migdal, Uspekhi Fiz. Nauk 123 (1977) 369 [Sov. Phys. Uspekhi 20 (1972) 879]. A. Sommerfeld, Wave Mechanics, Dutton, New York, 1930. H. Bethe, Intermediate Quantum Mechanics, Benjamin, New York, 1964. D.N. Page, Phys. Rev. D 16 (1977) 2402. D. Christodoulou, Phys. Rev. Lett. 25 (1970) 1596. D. Christodoulou, R. Ru$ni, Phys. Rev. D 4 (1971) 3552.


Decoherence}#uctuation relation and measurement noise L. Stodolsky Max-Planck-Institut fu( r Physik (Werner-Heisenberg-Institut), Fo( hringer Ring 6, 80805 Mu( nchen, Germany

Abstract We discuss #uctuations in the measurement process and how these #uctuations are related to the dissipational parameter characterizing quantum damping or decoherence. On the example of the measuring current of the variable-barrier or QPC problem we discuss the extra noise or #uctuation connected with the di!erent possible outcomes of a measurement. This noise has an enhanced short time component which could be interpreted as due to `telegraph noisea or `wavefunction collapsesa. Furthermore, the parameter giving the strength of this component is related to the parameter giving the rate of damping or decoherence. 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz Keywords: Decoherence; Mesoscopic; Measurement

1. Introduction I have always shared Lev Okun's interest in the fundamentals of quantum mechanics, and it is a pleasure to share the following thoughts with him for his 70th birthday. Our topic has to do with `quantum dampinga or `decoherencea, the description of how a quantum system loses its coherence when in contact with an external system or environment. This is an interesting and amusing subject with many aspects. In studying the loss of coherence one may say we are seeing how a quantum system `gets classicala. On the other hand, the environment in question may be a `measuring apparatusa and so we get involved with the `measurement problema. Finally, in a more pedestrian vein, many of the problems and equations are those of ordinary kinetic theory. We have studied these issues over the years and applied the results in many contexts, ranging from optically active molecules [1}3] to neutrinos [4] to gravity [5] and E-mail address: [email protected] (L. Stodolsky) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 6 5 - 4

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L. Stodolsky / Physics Reports 320 (1999) 51}58

quantum dots [6]. In the particularly simple case of the two-state system, such as the two states of a handed molecule, two mixing neutrino #avors, or an electron tunneling between two quantum dots, it is possible to give a fairly complete phenomenological treatment of the problem. Here we would like to discuss a further idea, that there are certain #uctuations in the measuring system and these are connected with the parameter characterizing the damping or decoherence of the observed system.

2. Damping}decoherence parameter In the description of the loss of coherence a certain parameter D arises, which can be thought of as the quantum damping or decoherence rate. We "rst describe how this parameter arises. Our description of the two-state system is in terms of the density matrix, which is characterized by a three-component `polarization vectora P, via (1) o"(I#P ) p) , where the p are the pauli matrices. P gives the probability for "nding the system in one of the two X states (l or l , electron on the left or right dot and so on) via P "Prob(¸)!Prob(R). (We shall C I X refer to our two states as L and R). Hence P gives the amount of the `qualitya in question. The X other components, P , contain information on the nature of the coherence. "P""1 means the system is in a pure state, while "P""0 means the system is completely randomized or `decohereda. P will both rotate in time due to the real energies in the problem and shrink in length due to the damping or decoherence. The time development of P is given by a `Bloch-likea equation [1}3] PQ "V;P!DP . (2) The three real energies V give the evolution of the system in the absence of damping or decoherence, representing for example the neutrino mass matrix or the energies for level splitting and tunneling on the quantum dots. The second term of Eq. (2), our main interest here, describes the damping or decoherence. D gives the rate at which correlations are being created between the `systema (the neutrinos, the external electron on the dots) and the environment or detector, and this is the rate of damping or decoherence. The label `tra means `transversea to the z axis. The damping only a!ects P because the `dampinga or òbservinga process does not induce jumps from one state to another: the neutrino interaction with the medium conserves neutrino #avor, the read-out process for the quantum dots leaves the electron being observed on the same dot; the observing process conserves P "1p 2. X X A formula for D can be given [1}3] in terms of the S matrices for the interaction of the environment or measuring apparatus with the two states of our system. There is a certain complex quantity K whose imaginary part gives the damping D and whose real part gives an energy shift to the system being measured. K is given by K"i( -ux)1i"1!S SR "i2 . (3) * 0 The factor -ux is the #ux or probing rate, where in the QPC application to be discussed one can use the Landauer formula -ux"e< /p , with < the voltage in the detector circuit [7]). The label


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i refers to the initial or incoming state of the probe and the S's are the S matrices corresponding to the di!erent states of the observed system. In our problem with two barriers (¸ and R) we may express the barrier penetration problem in S matrix form, calculate K according to the above formula and take the imaginary part to "nally obtain D [6]. In doing so, we "nd that D, as might have been expected, involves the di!erence in transmission by the two barriers. However, because of the various phases which are in general present in S, there are some further and more subtle phase-dependent e!ects which can contribute to D. These e!ects are quite interesting, (and not uncontroversial) but we shall not discuss them here and simply con"ne ourselves to the most straightforward situation where the only contributions to D are due to the di!erence in transmission by the two barriers. Thus the `measurementa consists solely in the fact that each barrier passes a di!erent current. With the S matrix parameterized such that the transmission coe$cients for the two barriers are called cos h ,cos h , * 0 corresponding to transmission probabilities p "cos h , p "cos h , we "nd [6] with this * * 0 0 neglect of phases D"( -ux)+1!cos *h, ,

(4)

where *h"h !h . Hence D is maximal for very di!erent transmission probabilities or large *h, * 0 while for the case of *h small: (*h) . D+( -ux) 2

(5)

D is a phenomenological parameter representing a kind of dissipation, resembling in some ways, say, the electrical resistance. Now for resistance and similar dissipative quantities there is the famous #uctuation}dissipation theorem [8] which relates the resistance or similar parameter to #uctuations in the system, as in the relation between resistance and Johnson noise. Should there be such a relation here? Of course here it is not energy that is being dissipated. Rather it is `coherencea that is being lost or perhaps entropy that is being produced. Nevertheless we should expect such a relation. For an interesting treatment of the #uctuation}dissipation theorem not based on energetic considerations see Ref. [9]. Indeed, looking at the Bloch equation Eq. (2) in its original context as the description of the polarization in nuclear magnetic resonance, it is quite natural to see the decay of the polarization as due to #uctuating magnetic "elds in the sample. Here, however, we wish to consider not "elds in a sample but something being observed by a `measuring apparatusa. We will nevertheless reach a similar conclusion, in that a `measuring apparatusa is something that reacts di!erently according to the state of the thing being observed. If it does not react di!erently, it does not `measurea, obviously. Hence we expect a measuring apparatus to show #uctuations related not only to the state of the system being observed, but also to how strongly it reacts to di!erences in that system. Furthermore since D, according to Eq. (4), is determined by these same di!erences, we expect some relation between the damping or decoherence rate and the #uctuations. In the following we would like to show how such expectations are realized in the variable barrier or QPC (quantum point contact) measuring process, where the `measuring apparatusa is a current determined by a variable tunneling barrier.

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3. The current in the variable-barrier problem Brie#y, the measurement process using a quantum point contact (QPC) detector [10] can be described as the modi"cation of a barrier whose transmission varies [11] according to whether an external electron is nearby or farther away. When the external electron is close by there is a certain higher barrier, and when it is farther away, there is a reduced barrier. Given an incident or probing #ux on the barrier (in practice also electrons), the modi"cation of the resulting current through the barrier, thus `measuresa where the external electron is located. Thus we have a quantum system, the external electron, which is being òbserveda by the detector current. The state of this external electron, in the case where it represented as a two-state system is given by the density matrix, evolving as in Eq. (2). The density matrix elements o and o , give the probabilities of the system ** 00 being observed being found in the state ¸ or R (o #o "1 and o !o "P ). ** 00 ** 00 X Experiments of this type give a fundamental insight into the nature of measurement, and in an elegant experiment Buks et al. [10] * stimulated by the work of Gurvitz [12] * saw the expected loss of fringe contrast in an electron interference arrangement when one of the interferometer paths was ùnder observationa by a QPC. Here, however, we wish to focus not so much on the object being measured but rather on the behavior of the `measuring apparatusa * the current through the variable barrier or QPC. We wish to examine certain èxtraa #uctuations in this current due to the measurement process. We thus turn to the current through the two-barrier system. Consider the probability for a given sequence of transmissions and re#ections through the barriers. Let 1 represent a transmission and 0 a re#ection for the probing electrons. Also let p be the probability of transmission and q that for a re#ection, (q#p"1), where each of these quantities has a label ¸ or R. We write the probability that in N probings the "rst probing electron was transmitted, the second re#ected, 2 the (N!1)th transmitted and the Nth transmitted, as Prob[11201]. Now it is relatively easy to write down a formula for this probability in the situation where the time in which the N probings take place is small compared to the time in which the observed system changes its state. Taking there to many probings in this time, we "nd [6] Prob[11201]"o (p p 2q p )#o (p p 2q p ), N;N , ** * * * * 00 0 0 0 0

(6)

where N is the number of probings in the time it takes the observed system to change states. The

main point about this formula is that each sequence contains factors of only the ¸ or R type. We need the restriction N(N because for longer times the observed system can change states and

the string 2q p will get `contaminateda with factors with R labels. For short times, where this is * * not a problem, the formula applies and may be arrived at either by thinking about the amplitude for the whole multi-electron process or by repeated wavefunction `collapsesa. Using it, one can understand the continuum from àlmost no measurementa to `practically reduction of the wavefunctiona by varying the parameters p and p . The former occurs for approximate equality of * 0 p and p and the latter in the opposite limit where, say, p is one and p is zero. One can also * 0 * 0 understand the origin of `telegraphica signals resembling a `collapse of the wavefunctiona in this latter case: for p and p close to zero and one, respectively, we will have predominantly sequences * 0 of either transmissions or re#ections with high probability, while mixed sequences are improbable [6]. Here we would like to use Eq. (6) to examine the #uctuations in the current.


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4. Measurement noise Eq. (6) says that for short times N;N we have a simple combination of two processes, each

one consisting of a sequence characterized by statistical independence. Naturally, a distribution consisting of the (normalized) sum of two such distinct distributions will have a variance greater than the average of the variances from individual distributions. For example, in the classical limit we would have two distinct peaks for the transmitted current. Hence we expect greater #uctuations than we would have with just a single distribution. To quantify this we calculate the variance
(14)

Above we considered two regimes, N(N for times short compared to the relaxation time for

the two-level system controlling the current, and N'N for long times compared to this

relaxation time. We can now rewrite Eq. (8) as a relation for the variance of the current for short averaging times

p #p * 0#(p !p ) (e -ux), N;N * 0

N

( j )!( j )" R R

(15)

while for long averaging times Eq. (9) becomes

( j)!( j )" R

(p !p ) p #p 0 N * 0# * (e -ux), NK4 > E(r)" dz o do+[(1/o)R (oA)#R]#[R A] M X e \ # [R f ]#[R f ]#f A#i( f !1), M X

(37)

1 r dm d z!m . R" d(o) ) o 2 \

(38)

In the limit rP0 the Coulombic contribution becomes singular. The easiest way to separate the singular piece is to change the variables A"A #a, where A is the solution in the absence of the B B Higgs "eld: A "(1/2o)[z /r !z /r ], z "z$r/2, r "[o#z ] . B \ \ > > ! ! !

(39)

Upon this change of variables the energy functional and the classical equation of motion take the form E(r)"E

!p/r#EI (r) ,

> p > EI (r)" dz o do e \

1 R (oa) #[R a] X M o

#[R f ]#[R f ]#f (a#A )#i( f !1) , M X B

(40)

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V.I. Zakharov / Physics Reports 320 (1999) 59}78

R

1 R (oa) #Ra"f (a#A ) , M o M X B

1 1 R [oR f ]#R f"f (a#A )# if ( f !1) . X B 2 o M M

(41)

(42)

The energy functional has been minimized numerically. The numerical results [4] for various i values clearly demonstrate that there is a linear piece in the potential even in the limit r;1. The slope p at rP0 was de"ned by the "tting the numerical data to:

EI (r)"C

1!e\P !1 #(p #C )r"p r#O(r) , r

(43)

The resulting slope p depends smoothly on the value of i, For the purpose of orientation let us note that for i"1 the slope of the potential at rP0 is the same as at rPR. That is, within error bars: p +p

(44)

where p determines the value of the potential at large r. Thus, existence of short strings is proven in the classical approximation to the Abelian Higgs model [4]. The linear piece in the potential at small distances re#ects the boundary condition that U"0 along the straight line connecting the monopoles.

6. Lessons for QCD Similarity between QCD and the Abelian Higgs model becomes transparent in the ;(1) projection of QCD which is a certain way to "x gauge in a non-Abelian theory [7]. The interest in the ;(1) projection has been related mostly to the con"nement mechanism. There exist detailed numerical simulations on the lattice which con"rm the dual-superconductor picture of the con"nement (for review and references see [16]). Moreover, condensation of a scalar "eld U with

magnetic charge is con"rmed within the ;(1) projection as well. We will concern ourselves here with implications of short strings discussed above for QCD. To begin with, existence of a ;(1) gauge invariant operator "U " of dimension d"2 at "rst sight,

causes problems. Indeed, because of the condensation of U in the vacuum, this operator has

a non-vanishing vacuum-to-vacuum matrix element. On the other hand, in QCD the lowest dimension of operator which may have a non-vanishing matrix element over the vacuum is d"4 [17]. It is (G? ) where G? is the gluon "eld strength tensor. The dimensions of local gauge IJ IJ invariant operators play a crucial role in applications of the operator product expansion (OPE) and we come to a disturbing conclusion that applying the OPE in the Abelian projection would be inconsistent with QCD. The paradox is resolved through the observation [4] that the OPE breaks in the Abelian Higgs model, as is explained below. Since the ;(1) projection of QCD is similar to the AHM, we expect, therefore, that the OPE breaks down in this projection as well. Moreover, the


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OPE results in certain predictions for observables and the breaking of these predictions may not depend on the use of the ;(1) projection. The breaking of the OPE in the AHM at the level of 1/Q corrections re#ects the presence of the topological strings discussed above. Indeed, validity of the OPE is equivalent to the assumption that at short distances the e!ects of particle exchanges are calculable explicitly, while the e!ects of large distances are included into matrix elements of various operators constructed from the "elds propagating at short distances. It is clear, on the other hand, that short strings cannot be reconstructed from particle exchanges and should be added as new objects. The simplest objects to apply the OPE are propagators (for review and references see, e.g., [19]). The standard logic can be illustrated by an example of the photon propagator connecting two electric currents which can be fully reconstructed within the AHM from the OPE:

1 1 1 1 1 1 d IJ . ! e1U2 # e1U2 e1U2 !2 " (45) D (Q)"d IJ IJ Q Q Q Q Q Q Q#m 4 Thus, one uses "rst the general OPE assuming "Q"<eU, then substitutes the vacuum expectation of the Higgs "eld U and upon summation of the whole series of the power corrections reproduces the propagator of a massive particle. The latter can also be obtained by solving directly the classical equations of motion. This approach fails, however, if there are both magnetic and electric charges present. Indeed, as is shown in Section 3 the propagator constructed along these lines contains a singular piece m 4 (d !n n ) , I J (Q#m )(Qn) IJ 4 see Eq. (18). The presence of such a term immediately implies that the standard OPE does not work at the level of Q\ corrections. Indeed, choosing Q large does not guarantee now that the m correction is small since the factor (Qn) in the denominator may become zero. 4 The di$culties with the propagator stem from the fact that one cannot avoid overlap of the Dirac strings and trajectories of charged particles in case of vacuum condensation of a charged "eld U. In other words, the singularities in the (Qn) variable signal presence of the Dirac string. The issue cannot be settled by formal manipulations with these stringy singularities. To the contrary, the Dirac string acquires a physical meaning in terms of energy and controls the 1/Q corrections to the potential, see the preceding section. As a result, instead of a 1/Q correction associated with "U" and large distances which is expected on the basis of the OPE, there is a 1/Q correction associated with small distances. We expect similar transmutation to happen within QCD. Then emergence of the operator "U " signals rather 1/Q corrections associated with small distances than violation

of the OPE with respect to the infrared corrections.

Appearance of the 1/Q corrections which go beyond the standard OPE is not in con#ict with QCD because of the ultraviolet renormalon. The ultraviolet renormalon signals that the perturbative QCD is inconsistent at the level of 1/Q terms and short strings may be thought of a non-perturbative counterpart which is needed to make the theory unique. There were many attempts to build up a phenomenology based on the ultraviolet renormalon (see, e.g., [18]). Phenomenological consequences of the short strings appear to be di!erent from the schemes considered so far.

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There is a common question in QCD, how the con"nement a!ects the structure of singularities of the gluon propagator (see, e.g., [2]). The prevailing viewpoint is that the con"nement is manifested through a 1/Q singularity in the infrared (see, e.g., [21]). There are also arguments that because of the Gribov's copies the propagator, to the contrary, vanishes in the infrared limit [22]. The AHM allows for a fresh look at the problem. Namely, one argues [20] that it is the same stringy singularities in the (Qn) variable that signal con"nement. Indeed, on one hand the AHM is a con"ning theory. On the other hand, in the London limit the propagator of the photon connecting two (con"ned) magnetic charges can be found explicitly [20]. The result is that, apart from the term (18), one should account only the e!ect of the scattering of the "eld A o! closed strings world sheets: I D (Q)"[1/(Q#M )]X (Q)#D(Q) , IJ 4 IJ IJ

(46)

1 dp 1 nn m e\ ? @ 1R (Q)R (!p)2R . D(Q)"! 4 J@ IJ Q#M (Q ) n) (2p) p#M (p ) n) I? 4 4

(47)

Note that the expression (46) for the gluon propagator is exact in the London limit (there are no loop contributions to Eq. (46))! Moreover, the string}string correlation function in Eq. (46) is de"ned as follows: 1 1O2R" Z

DR e\1RO, . R

Z "

DR e\1(R) .

(48)

R

and the string action S(R) is the same as in Eq. (22). The string}string correlator is parametrized in terms of a single function DR(Q): 1R (Q)R (!p)2R"(2pe)d(Q!p)e e Q Q DR(Q) . I? J@ I?KM J@DM K D

(49)

The behavior of the function DR(Q) in the infrared region, QP0, can be estimated as follows: C , DR(Q)" Q#m

(50)

where m is the mass of the lightest glueball with quantum numbers of the photon, J."1\. Collecting all the factors we get for the propagator in the axial gauge:

Q n #Q n QQ 1 I J! I J F(Q)! (d !n n )G(Q) D " d # I J IJ I J IJ (Q ) n) (Q ) n) (Q ) n) IJ

(51)

where

1 Cm m 4 F(Q)" 1# , Q#m (Q#m )(Q#m ) 4 4

(52)

m Cm m 4 4 G(Q)"! 1! . Q#m (Q#m )(Q#m ) 4 4

(53)


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Thus, we do not reproduce any special behavior of the propagator at QP0. In particular, there is neither Q\ singularity nor vanishing of the propagator as QP0. The only unusual feature is the singularities in (Q ) n). To summarize, the Abelian Higgs model demonstrates breaking of the OPE at large Q at the level of 1/Q corrections due to the topological strings [4]. Also, one can derive an expression for the photon propagator which is exact in the London limit [22]. The expression demonstrates that the con"nement is manifested in the propagator rather through stringy singularities than through a particular behavior at small Q.

7. Phenomenological applications The existence of short strings in the AHM goes back to the analysis, how unique is the gauge "xing in this model [7]. Similarly, it was argued [7] that the ;(1) projection can be introduced uniquely everywhere except for the world trajectories of magnetic charges. One can also trace strings connected to the monopoles. These observations are crucial to conjecture that short strings exist in the ;(1) projection of QCD as well [4]. However, there is no direct way to evaluate the e!ective tension for the Dirac string embedded into the physical vacuum of QCD. The reason is that the topology of the gauge "xing is described now not directly in terms of the "elds appearing in the Lagrangian but rather in terms of foreign objects like eigenvalues of some operators. There exists convincing evidence that the dual Abelian Higgs model does describe the lattice gluodynamics in the infrared limit [23]. Moreover, the structure of the observed string which determines the QM Q potential at large distances is well described by the classical Landau}Ginzburg equations [24]. In particular, results of the measurements of the QQM potential on the lattice has been compared with the predictions of the (dual) AHM since a long time (see, e.g., [25] and references therein). What we can add to this analysis is the prediction for the behavior of the QQM potential at small distances [4]. Indeed, the de"nitions of the monopoles and strings relevant to the ;(1) projection apply in fact at small distances. Thus, we predict a non-vanishing slope p of the potential at rP0. Numerically, the prediction for p depends on the ratio of m and m . It is known [23] that the 4 1 realistic case of quantum gluodynamics is close to the Bogomolny limit, m "m . In this case the 4 1 prediction for the potential at short distances is especially simple (see Eq. (44)): p "p , i.e. the slope does not depend on the distance r. It is amusing therefore that the lattice simulation [26] does not indicate indeed any change in the slope of the QM Q potential at all the distances available, r50.1 fm. Also, the data on the bound states of heavy quarks are much better described if one assumes that the linear piece in the potential persists at short distances as well [27]. The short strings is the "rst theoretical explanation of such a behavior of the potential at rP0 since according to the standard picture the correction to the Coulombic potential is of order r at small distances [10]. The linear piece in the potential is an example of a 1/Q correction where Q is a generic large mass parameter. In this case it is a correction to the leading, Coulomb-like piece in the potential and Q+1/r. It would be of course very important to see, whether there are other detectable

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corrections due to the short strings in QCD. Let us emphasize, however, that not all the 1/Q corrections in QCD are associated with short distances. For example, in case of DIS there are 1/Q corrections coming from the infrared region and this is perfectly consistent with the OPE. Thus, the class of theoretical objects for which an observation of the 1/Q corrections would signify going beyond the OPE is limited. One example was the potential l\ dileptons. In particular, as a measure of D}D mixing they proposed the measurement of the ratio of the same sign to the inclusive dilepton events, N>>#N\ \ 1 (C !C )#4(DM ) 1 * 1* , R , " (1) " N>\#N\>#N>>#N\ \ 2 (C #C )#4(DM ) 1 * 1* where C and C are the widths of the (short-lived) D and (long-lived) D mesons, respectively, and 1 * 1 * DM is their mass di!erence. They also suggested the measurement of the charge asymmetry 1* d ,(N>>!N\ \)/(N>>#N\ \)K4Re e , (2) " " as a measure of CP violation. Here, e is the CP-violating parameter in the wave functions of " D and D mesons, analogous to the corresponding parameter e in the K-system [2] 1 * ) D &D #e D , D &D #e D , (3) 1 " * " where D and D are the pure CP states. So far, neither R nor d have been measured [2]. In fact, " " in the Cabibbo}Kobayashi}Maskawa (CKM) theory of quark #avour mixing [3], which is now an integral part of the SM, no measurable e!ects are foreseen for either of the ratios R and d , due to " " the experimentally established hierarchies in the quark mass spectrum and the CKM matrix elements. Typically, one has in the SM [4], DM /(C #C )KO(10\), (C !C )/(C #C );1 , (4) 1* 1 * 1 * 1 * with d completely negligible. By virtue of this, the quantities R and d have come to be " " " recognized as useful tools to search for physics beyond the SM [5,6]. The OPZ formulae also apply to the time-integrated e!ects of mixing and CP violation in the B}B and B}B systems, and they were used in the analysis [7] of the UA1 data on inclusive Q Q B B dilepton production [8]. Calling the corresponding mixing measures R B and R Q, respectively, present experiments yield R BK0.17 and R QK1/2 [2]. These measurements are consistent with the more precise time-dependent measurements, yielding DM B"0.471$0.016 (ps)\ [9] and the 95% CL upper limit DM Q'12.4 (ps)\ [10]. However, the corresponding CP-violating charge asymmetries d B and d Q in the two neutral B-meson systems have not been measured, with the present best experimental limit being d B"0.002$0.007$0.003 from the OPAL collaboration [2] and no useful limit for the quantity d Q. These charge asymmetries are expected to be very small in the SM, re#ecting essentially that the width and mass di!erences DC and DM in the B}B and B B B}B complexes are relatively real. Typical estimates in the SM are in the range d B"O(10\) Q Q and d Q"O(10\). Hence, like d , they are of interest in the context of physics beyond the SM " [11,12]. With the advent of B factories and HERA-B, one expects that a large number of CP asymmetries in partial decay rates of B hadrons and rare B decays will become accessible to experimental and

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theoretical studies. Of particular interest in this context are the #avour-changing neutral-current (FCNC) processes which at the quark level can be thought of as taking place through induced bPd and bPs transitions. In terms of actual laboratory measurements, these FCNC processes will lead to DB"1, DQ"0 decays such as BP(X ,X )l>l\ and BP(X ,X )c, where X (X ) Q B Q B Q B represents an inclusive hadronic state with an overall quantum number S"$1(0), as well as their exclusive decay counterparts, such as BP(K, KH, p, o,2)l>l\ and BP(KH, o, u,2)c. Of these, the decays BPX c and BPKHc have already been measured [2]. The DB"2, Q DQ"0 transitions lead to B}B and B}B mixings, brie#y discussed above. Likewise, non-trivial Q Q B B bounds have been put on the CP-violating phase sin 2b from the time-dependent rate asymmetry in the decays B/BPJ/tK [13]. In K decays, the long sought after e!ect involving direct CP Q violation has been "nally established through the measurement of the ratio e/e [14,15]. This and the measurement of the CP-violating quantity "e" in K Ppp decays [2] represent the sPd FCNC * transitions. Likewise, there exists great interest in the studies of FCNC rare K decays such as K>Pp>ll and K Ppll [16], of which a single event has been measured in the former decay * mode [17]. The FCNC processes and CP asymmetries in K and B decays provide stringent tests of the SM. The short-distance contributions to these transitions are dominated by the top quark, and hence these decays and asymmetries provide information on the weak mixing angles and phases in the matrix elements < , < and < of the CKM matrix. Some information on the last of these matrix RB RQ R@ elements is also available from the direct production and decay of the top quarks at the Fermilab Tevatron [18]. The measurement of < will become quite precise at the LHC and linear colliders. R@ Moreover, with advances in determining the (quark) #avour of a hadronic jet, one also anticipates being able to measure the matrix element < (and possibly also < ). RQ RB We shall concentrate here on the analysis of the data at hand and in forthcoming experiments which will enable us to test precisely the unitarity of the CKM matrix. These tests will be carried out in the context of the Unitarity Triangles (UT). The sides of UTs will be measured in K and B decays and the angles of these UTs will be measured by CP asymmetries. Consistency of a theory, such as the SM, requires that the two sets of independent measurements yield the same values of the CKM parameters, or, equivalently the CP-violating phases a, b and c. We are tacitly assuming that there is only one CP-violating phase in weak interactions. This is the case in the SM but also in a number of variants of Supersymmetric Models, which, however, do have additional contributions to the FCNC amplitudes. In fact, it is the possible e!ect of these additional contributions which will be tested. In this case, quantitative predictions can be made which, in principle, allow experiments to discriminate among these theories [19]. As we shall see, the case for distinguishing the SM and the MSSM rests on the experimental and theoretical precision that can achieved in various input quantities. Of course, there are many other theoretical scenarios in which deviations from the pattern of #avour violation in the SM are not minimal. For example, in the context of supersymmetric models, one may have non-diagonal quark}squark}gluino couplings, which also contain additional phases. These can contribute signi"cantly to the magnitude and phase of bPd, bPs and sPd transitions, which would then violate the SM #avour-violation pattern rather drastically. In this case it is easier to proclaim large deviations from the SM but harder to make quantitative predictions. This paper is organized as follows. In Section 2, we discuss the pro"le of the unitarity triangle within the SM. We describe the input data used in the "ts and present the allowed region in o}g

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space, as well as the presently allowed ranges for the CP angles a, b and c. We also discuss the "ts in the superweak scenario, which di!ers from the SM "ts in that we no longer use the constraint from the CP-violating quantity "e". The superweak "ts are not favoured by the data and we quantify this in terms of the 95% CL exclusion contours. We turn to supersymmetric models in Section 3. We review several variants of the MSSM, in which the new phases are essentially zero. Restricting ourselves to #avour violation in charged-current transitions, we include the e!ects of charged Higgses H!, a light scalar top quark (assumed here right handed as suggested by the precision electroweak "ts) and chargino s!. In this scenario, which covers an important part of the SUSY parameter space, the SUSY contributions to K}K, B}B and B}B mixing are of the same form B B Q Q and can be characterized by a single parameter f. Including the NLO corrections in such models, we compare the pro"le of the unitarity triangle in SUSY models, for various values of f, with that of the SM. We conclude in Section 4.

2. Unitarity triangle: SM pro5le Within the SM, CP violation is due to the presence of a nonzero complex phase in the quark mixing matrix < [3]. A particularly useful parametrization of the CKM matrix, due to Wolfenstein [20], follows from the observation that the elements of this matrix exhibit a hierarchy in terms of j, the Cabibbo angle. In this parametrization the CKM matrix can be written approximately as

1!j j Aj(o!ig) . G G> 4,0. This theory has a string theory realization. It is the theory of threebranes of Type IIB at the singularity of the form 1/9 where 9 re#ects all the coordinates. It can be also viewed as 9 orbifold of the Type 0 theory. The space}time aspects as well as its AdS ;1/ dual realization will be discussed elsewhere [14]. Note added. After completion of the manuscript, a paper [15] appeared which studies further aspects of self-dual threebranes in type 0 string theory.

Acknowledgements We are grateful to I. Klebanov for discussions. The research of N. N. is supported by Harvard Society of Fellows, partly by NSF under grant PHY-98-02-709, partly by P''N under grant 98-01-00327 and partly by grant 96-15-96455 for scienti"c schools. The research of S. S. is supported by DOE grant DE-FG02-92ER40704, by NSF CAREER award, by OJI award from DOE and by Alfred P. Sloan foundation.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

A.M. Polyakov, Nucl. Phys. Proc. Suppl. 68 (1998) 1; Int. J. Mod. Phys. A 14 (1999) 645. I. Klebanov, A. Tseytlin, Nucl. Phys. B 546 (1999) 155; B 547 (1999) 143. I. Klebanov, A. Tseytlin, JHEP 9903 (1999) 015. J. Maldacena, Mod. Phys. Lett. A 13 (1998) 219. M. Bershadsky, Z. Kakushadze, C. Vafa, Nucl. Phys. B 523 (1998) 59. M. Bershadsky, A. Johansen, Nucl. Phys. B 536 (1998) 141. M.R. Douglas, G. Moore, hep-th/9603167. S. Kachru, E. Silverstein, Phys. Rev. Lett. 80 (1998) 4855. A. Lawrence, N. Nekrasov, C. Vafa, Nucl. Phys. B 533 (1998) 199. L. Dixon, J. Harvey, Nucl. Phys. B 274 (1986) 93. N. Seiberg, E. Witten, Nucl. Phys. B 276 (1986) 272. O. Bergman, M. Gaberdiel, Nucl. Phys. B 499 (1997) 183. J. Minahan, hep-th/9811156, JHEP 9904 (1999) 007. I. Klebanov, N. Nekrasov, S. Shatashvili, in preparation. A. Tseytlin, K. Zarembo, Phys. Lett. B 457 (1999) 77.


Nonperturbative QCD vacuum and colour superconductivity N.O. Agasian*, B.O. Kerbikov, V.I. Shevchenko Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia

Abstract We discuss the possibility of existence of colour superconducting state in real QCD vacuum with nonzero 1a GG2. We argue, that nonperturbative gluonic "elds might play a crucial role in colour superconductivity scenario. 1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.Lg; 12.38.!t Keywords: Colour superconductivity; Gluon condensate

We wish to start with a few introductory words. We formulate the problem in the paper which we cannot solve, at least now, without resorting to simple, sometimes naive estimates. Not everybody accepts such an approach and believes in it. Probably, the success or failure depends here on the insight and physical intuition. Inspiring example of solving complicated problems by surprisingly simple tools can be found in many works of Lev Okun. But sure the fact that we are lucky to work near Lev Borisovich at ITEP does not mean that his favourite tool would work in our hands. It is the reader of the paper who will judge.

1. Introduction The behaviour of QCD at high density has become recently a compelling subject due to (re)discovery of colour superconductivity [1,2]. The essence of the phenomenon is the formation

* Corresponding author. E-mail addresses: [email protected] (N.O. Agasian), [email protected] (B.O. Kerbikov), shevchen@ heron.itep.ru (V.I. Shevchenko) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 8 0 - 0

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N.O. Agasian et al. / Physics Reports 320 (1999) 131}138

of the BCS-type diquark condensate at densities exceeding the normal nuclear density [3] by a factor (2}3). Colour superconductivity has been studied within di!erent versions of the Nambu}Jona}Lasinio-type model [1,4,3] or the instanton model [2,5]. To our knowledge, however, no attention has been paid to the fact, that as compared to real QCD, both approaches miss important nonperturbative gluonic content of the theory. The NJL model contains the gluon degrees of freedom in a very implicit way: it is argued, that high-frequency mode (one-gluon exchange) after being integrated out gives rise to the e!ective four quark interaction (see, e.g. [6,7]). The instanton model deals only with speci"c "eld con"gurations * instantons and antiinstantons. The NJL model with gluon condensate included has been considered in [8] while con"ning background superimposed on instantons has been treated in [9]. The role played by the gluon degrees of freedom in the problem under consideration is essentially twofold. First, it is assumed, that they are responsible for producing the quark}quark attractive interaction leading at small enough temperatures to the Cooper pairing (thus gluons play the role of phonons over ion lattice speaking in condensed matter terms). On the other hand, the vacuum gluon "eld #uctuations should be a!ected by colour superconducting state itself in a way analogous to the Meissner e!ect in ordinary superconductor. The crucial point is which force will win, i.e. whether superconductivity will survive or will be destroyed by gluonic "elds as it happens in the standard BCS theory in the presence of strong enough external magnetic "eld. The studies of idealized QCD performed by several authors have shown, that depending on the chemical potential k and the temperature ¹ the system displays three possible phases: (1) chiral symmetry breaking without diquark condensation; (2) mixed phase with nonzero values of both chiral and diquark condensates; (3) diquark condensation without symmetry breaking. The system may be described by the thermodynamic potential X( ,D;k,¹), where and D are order parameters, related to the chiral and diquark condensates, respectively (explicit de"nitions will be given below). The potential X is expressed in terms of quantum e!ective action C as X"C¹/< . Suppose, that the potential X has been calculated within the framework of some NJL-type model, i.e. with the gluon sector excluded (except for contribution of high-frequency modes, giving the necessary attraction). Consider for simplicity the phase (3) of the system with "0, DO0 and ¹"0. Let X( "0,D ;k,0) be the stationary value of the thermodynamic potential, where D is the solution of the gap equation *X/*D"0 (see below). Now we superimpose the nonperturative vacuum gluon "elds on the above picture. The detailed knowledge of the nonabelian Meissner e!ect is unfortunately absent. Anyway, it is obvious that the corresponding microscopic picture is far from being trivial. The nonlinear character of the equations of motion for gluon "elds is of prime importance here, while usual Meissner e!ect for abelian "elds is essentially linear phenomenon. On the contrary, general symmetry arguments tell us, that part of gluon degrees of freedom becomes massive if the colour gauge invariance is spontaneously broken. It means e!ective screening of low-frequency modes and therefore it is reasonable to assume, that the formation of the colour gauge invariance * breaking diquark condensate should lead to the decrease of the gluon condensate by some factor, which we assume to be about a few units (but, presumably, not to exactly zero value, as it happens in abelian case). Then it will be energetically favourable for the system to remain in the colour superconducting state with DO0 only if the quantity e(i)"!(1!1/i)[b(a )/16a ]1G G 2 IJ IJ

(1)


133

is less than X( "D"0;k,0)!X( "0,D ;k,0). The factor i in (1) represents the unknown rate of decrease of the gluon condensate due to superconducting diquark state formation. Note, that e(iPR)"!e . In what follows we will show, that these two energy gaps have the same order of magnitude unless the i is close to 1. It should be noted, that the analogy with the BCS superconductor in external "eld is somewhat loose here for at least two reasons. First, strong nonperturbative gluonic "elds are inherent for the QCD vacuum. The consistent way of analysis should include a set of gap equations for the free energy of the system depending on gluon and di!erent types of quark condensates, determining energetically best values for all of them simultaneously. The second point is the relation between scales, characterizing colour superconductivity and nonperturbative gluon #uctuations. In particular, only modes with the wavelengths larger than the Cooper pair radius are responsible for the supercurrent while the rest do not admit simple interpretation in terms of Ginzburg}Landau theory. We leave the analysis of these complicated problems for the future. Another important remark is in order. It might be naively assumed, that if the system under study is in the decon"nement region, the vacuum gluonic content may be taken as purely perturbative. There are several reasons, however, why it is not the case. The most important one is the following. Finite density breaks Euclidean O(4) rotational invariance and hence chromoelectric and chromomagnetic components of the correlator 1a G2 enter on the di!erent footing. In I$ particular, decon"nement, i.e. zero string tension is associated with the vanishing of the electric components, while it is energetically favourable for the magnetic ones to stay nonzero (the same phenomenon takes place for the temperature phase transition [10]). At the same time it is just strong magnetic "eld, which is able to destroy the superconductivity.

2. General formalism We start with the QCD Euclidean partition function

Z" DA DtM Dt exp(!S) ,

(2)

where

1 S" F F dx# tM (!ic D !im#ikc )t dx . I I 4g IJ IJ

(3)

We supress colour and #avour indicies and also introduce chemical potential k (only the case N "2; N "3 is considered in this paper). Performing integration over the gauge "elds one gets e!ective fermion action in terms of cluster expansion

Z" DtM Dt exp ! dx ¸ !S

(4)

with ¸ "tM (!ic * !im#ikc )t and e!ective action S " 11hL22/n! where h" I I L dx tM (x)c A (x)t t(x) and double brackets denote irreducible cumulants. I I

134


To proceed, one is to make considerable simpli"cation of Eq. (4). First, only the lowest, four-quark interaction term is usually kept in S . Second, it is instructive to consider instead of the original nonlocal kernel some idealized local one, respecting the given set of symmetries. In the problem under study it is common to choose either instanton-induced four-fermion vertex (the choice, adopted in [2,5]) or di!erent versions of the NJL model [1,4,3] including the one, motivated by one gluon exchange [4]. In the latter case, one gets

S " dx dy(tM (x)c t t(x))(tM (y)c t t(y))D(x!y) I I

(5)

with D(x!y)"Gd(x!y) and where the coupling constant G with the dimension [m]\ is introduced. We assume, as it has already been mentioned, that this localized form of the kernel does not mimic all gluonic content of the original theory, in other words only some part of gluon degrees of freedom participates in the condensate formation. One way to analyse the role, played by other ones, would be to consider more realistic nonlocal functions D(x!y) (which, in principle, encode all necessary information if we keep only four-quark interaction). This will be done elsewhere, while in the present paper we work with the local form of the action. Performing colour, #avour and Lorentz Fierz [7] transformations and keeping only scalar terms in both tM t and tt channels, we arrive at ¸ "G[(tM (x)Kt(x))(tM (y)Kt(y))!(tM (x)U t!(y))(tM !(y)U t(x))] , ? ?

(6)

where K"(i/(6)1) q , $

U "(1/(12) e c q , ? ?@A $

and t!"CtM 2"c c tM 2. We note, that with only scalar terms kept, Lagrangian (6) is no more chiral invariant. The attraction in scalar colour antitriplet channel (which also exists if one starts from the instanton-induced interaction) could lead to the formation of the condensate, breaking colour S;(3). In close anology with [3] we replace the common coupling constant G by two independent constants G and G corresponding to the two terms in (6). Next step is to write down the partition function and to perform its bosonization. We adopt the standard Hubbard}Stratonovich trick and get

Z" DtM Dt exp ! dx(¸ #¸ )

" D DD DDR exp

# Tr Ln

dx+![ /4G#DDR/G ]

i*K #i(m# )!ic k i*K 2!i(m# )#ic k 2DRC\UR 2UCD

.

(7)


135

For the system of the massless quarks at the phase (3) and ¹"0 the thermodynamic potential reads (we remind, that Z&exp(!S)&exp(#C))

2UCD i*K !ic k "D" 1 C ! Tr Ln . X" " g < < i* K 2#ic k 2DRC\UR The value of the diquark condensate is determined by the gap equation

(8)

*X "0 . (9) *D D D By going in (8) to wave number-frequency space and making use of the gap Eq. (9), we arrive at the following expression for the thermodynamic potential at its minimum X( "D"0; k, 0)!X( "0, D ; k, 0)KD/[G ln(M/D)] , (10) where M is the NJL cuto! which is typically about 0.8 GeV [1,3]. Alternatively, the cuto! may enter via the formfactor or the instanton zero-mode. In the NJL-type calculations the values of the cuto! M and the coupling constant G are "tted simultaneously, but no unique `standarda "t exists so far [11}13]. To estimate the r.h.s. of (10) we have taken for cuto! M"0.8 GeV, for coupling G"12G"(15}40) GeV\ and the value of the gap in the diquark scalar sector D "(0.1}0.15) GeV. With these parameters, one gets X( "D"0; k, 0)!X( "0, D ; k, 0)&(1}5) ) 10\ GeV . (11) To be on a robust quantitative footing and to consider "nite temperatures one can replace estimate (11) by the result of direct numerical calculations of the thermodynamic potential performed in [3]. Our result (11) is larger than the corresponding value, presented in [3]. The discrepancy may be due to di!erent value of the coupling constant g adopted in [3]. Needless to say, that the larger is the estimate of (10) the larger is the critical "eld extinguishing superconductivity. Now, let us estimate expression (1). We use di!erent sets of data from [14}16] and take for the gluon condensate

a G " G G "(0.014}0.026) GeV. p IJ IJ Then for two #avours one gets in one loop

1 1 4 e(i)" 1! ) 11! ) 32 i 3

a G G p IJ IJ

1 & 1! ) (4}8);10\ GeV . i

(12)

It is seen, that (10) and (12) have the same order of magnitude for i52. It should be noted, that estimate (11) given above is rather optimistic in the following sense. If one naively assume the colour superconductor to be the BCS one and take its typical parameters, for example, from [2], then one has at ¹"0 for the value of the critical magnetic "eld H "0.64(D )(m p ) , $

(13)

136


where p"k!m, m is the constituent quark mass. We take k"(0.4}0.5) GeV and corre$ sponding D "(0.04}0.10) GeV from the paper [2] which are smaller, than the typical values we have analysed before. The maximum of (13) with respect to p at the "xed k is reached for $ k"(2p and it gives $ H "(0.8}8);10\ GeV .

(14)

Strictly speaking, we are not allowed to apply formulas like (13) to the colour "elds and interpret them in terms of Meissner e!ect, but we note, that (14) even without any numerical factors is about the order of magnitude smaller than (12).

3. Finite density e4ects The comparison made in the previous section was intuitively based on the analogy with the Meissner e!ect in ordinary superconductors. The actual value of the gluon condensate in (12) was taken to be the vacuum one. This is not quite correct, however. Even without the formation of any diquark condensate, the gluon condensate in the hadronic matter is di!erent from that in the vacuum. In order to get an idea about such dependence, let us consider e!ective dilaton Lagrangian [17,18] ¸(p)"(* p)!> in our case) mass. Low-energy dilaton physics can be used for description of the gluon condensate behaviour at "nite density and temperature [19,20]. In the chiral limit masses of nucleons in QCD are determined by the nucleon}dilaton vertex ¸ "mHq q with the e!ective mass , mH"m ) (p/p ). In isotopically symmetric system the energy density takes the form: , ,

e(p ,n)"w\Pzz)"(M)!M))

(7)

3. E4ective-Higgs propagator To obtain the e!ective-Higgs propagator we `transcribea the K-matrix model from R-gauge to U-gauge [4,5]. The heart of the matter is to "nd the contribution of the symmetry-breaking sector in U-gauge, which encodes the dynamics speci"ed in the original R-gauge formulation of the model. This is accomplished using the ET as follows. Suppose that the longitudinal gauge boson modes scatter strongly. At leading order in the weak gauge coupling g we write the amplitude =>=\PZZ as a sum of gauge-sector and Higgs-sector * * terms, M "M #M , (8) 2 % 1 where SB denotes the symmetry-breaking (i.e., Higgs) sector. Gauge invariance ensures that the contributions to M that grow like E cancel, leaving a sum that grows like E, given by % M "g(E/om )#O(E, g) (9) % 5 where o"m /(cos h m). The neglected terms of order E and of higher order in g include the 5 5 8 electroweak corrections to the leading strong amplitude. The order E term in Eq. (9) is the residual `bad high-energy behaviora that is cancelled by the Higgs mechanism. It is also precisely the low-energy theorem amplitude, M "s/ov"M #O(s, g) (10) *#2 % using m "gv/2 and s"4E. Eqs. (8) and (9) may be used to derive the low-energy theorem 5 without invoking the ET. Now consider an arbitrary strong scattering model, designated as model `Xa, formulated in the usual way in an R-gauge in terms of the unphysical Goldstone bosons, M6 (wwPzz). The % total gauge boson amplitude is gauge invariant and the ET tells us that for E<m it is 5 approximately equal to the Goldstone boson amplitude, i.e., M6 (= = )KM6 (ww) (11) 2 * * % in the same approximation as Eq. (9). Eq. (8) holds in any gauge. Specifying U-gauge we combine it with Eqs. (9)}(11) to obtain the U-gauge Higgs sector contribution for model X, M6 (= = )"M6 (ww)!M . 1 * * % *#2

(12)

If the symmetry-breaking force is strong, the quanta of the symmetry-breaking sector are heavy, m <m , and 1 5 decouple in gauge boson scattering at low energy, M ;M . Then the quadratic term in M dominates 1 % % M for m ;E;m , which establishes the low-energy theorem without using the ET [9,10]. 2 5 1

M.S. Chanowitz / Physics Reports 320 (1999) 139}146

143

The preceding result applies to any strong scattering amplitude. Now we specialize to s-wave ==PZZ scattering and use Eq. (12) to obtain an e!ective-Higgs propagator with standard `Higgsa-gauge boson couplings. Neglecting m ;s and higher orders in g as always, the e!ective 5 scalar propagator is P (s)"!(v/s)M6 (= = ) 6 1 * *

(13)

Eqs. (10) and (12) with o"1 then imply P (s)"!(v/s)M6(ww)#1/s 6 0

(14)

The term 1/s, corresponding to a massless scalar, comes from M in Eq. (12). It ensures good *#2 high-energy behavior while the other term in Eq. (14) expresses the model-dependent strong dynamics. Finally, we substitute the K-matrix amplitude, Eq. (7), into Eq. (14) to obtain the e!ective propagator for the K-matrix model as the sum of two simple poles

1 1 1 2 # , P " ) 3 s!m 2s!m

(15)

where m and m are m"!16piv

(16)

m"#32piv .

(17)

and

It is surprising to "nd such a simple expression involving only simple poles. It is not surprising that the poles are far from the real axis since they describe nonresonant scattering. Interpreted heuristically as Breit}Wigner poles they correspond to resonances with widths twice as big as their masses.

4. Oblique corrections The oblique corrections are evaluated from the vacuum polarization diagrams that in the SM include the Higgs boson [11]. In place of the SM propagator, P "1/(s!m ), we substitute 1+ & P from Eq. (15). Where the SM contribution depends on the log of the Higgs boson mass, ) ¸ "ln(m /k), we now "nd instead the combination ¸ , 1+ & )

2 1 m m m ¸ "ln & P¸ " ln # ln , ) 3 1+ 3 k k k where m are complex masses de"ned in Eqs. (16)}(17).

(18)

144


The results quoted in Eqs. (1) and (2) follow from the usual expressions for S,¹ where we use the real part of ¸ in place of ¸ , ) 1+ S"Re(¸ )/12p (19) ) and ¹"!3 Re(¸ )/16p cos h . (20) ) 5 The imaginary part of ¸ is an artifact which we discard; it results from the fact that our ) approximation neglects the = mass, as in any application of the ET. At q"0, where the oblique corrections are computed, there is no contributution to the imaginary part of the vacuum polarization from the relevant diagrams. Combining the I"0 and 2 terms in Eq. (18) we have Re(¸ )"ln(216pv/k) . (21) ) Evaluating Eq. (21) we "nd that the oblique correction from the K-matrix model is like that of a Higgs boson with mass 2.0 TeV.

5. Questions The I"2 component of the e!ective propagator has peculiar properties, perhaps due to the fact that for the I"2 channel we are representing t- and u-channel dynamics by an e!ective s-channel exchange. The minus sign in the I"2 low-energy theorem, Eq. (5b), which may be thought of as arising from the identity t#u"!s, leads to interesting di!erences between the I"0 and I"2 components of the e!ective propagator P . ) First, the I"2 component of the e!ective scalar propagator has a negative pole residue, which would correspond to a unitarity violating ghost if it described an asymptotic state (which it does not). In fact the sign is required to ensure unitarity, since it is needed to cancel the bad high-energy behavior of the gauge sector amplitude which has a negative sign in the I"2 channel. In Eq. (15) for P the I"2 pole appears with a positive sign because of a second minus sign from the isospin ) decomposition, Eq. (7). Neither pole of the e!ective propagator has a negative (ghostly) residue. In any case the amplitude is exactly unitary by construction. The sign di!erence between the pole positions, m and m in Eqs. (16) and (17), may also be traced to the phases of the low-energy theorems in Eq. (5). The position of m on the negative imaginary axis of the complex s plane corresponds to poles in the fourth and second quadrants of the complex energy plane, consistent with causal propagation as in the conventional m!ie prescription. But the position of m on the positive imaginary axis corresponds to poles in the "rst and third quadrants of the complex energy plane. This would imply acausal propagation if the poles are on the "rst sheet but not if they are on the second sheet. Working in the limit of massless external particles as we are it is not apparent on which sheet they occur.

I thank Henry Stapp for a discussion of this point.


145

I conclude that the sign of the pole residue arising from the I"2 amplitude is not problematic but that the implications of the pole position requires better understanding.

6. Physical interpretation We have used a convenient representation of the K-matrix model to estimate the low-energy radiative corrections from strong == scattering. The result that the corrections are like those of a Higgs boson with mass at the unitarity scale is plausible and agrees with an earlier estimate using the cut-o! nonlinear sigma model [13,14]. The estimate establishes a &default' radiative correction from the strongly coupled longitudinal gauge bosons in theories of dynamical symmetry breaking. In general there will be additional contributions from other quanta in the symmetry breaking sector. Those contributions are model dependent as to magnitude and sign. In computing their e!ect it is important to avoid double-counting contributions that are dual to the contribution considered here. Current SM "ts to the electroweak data prefer a light Higgs boson mass of order 100 GeV with a 95% CL upper limit that I will conservatively characterize as :300 GeV [2]. Since the corrections computed here are equivalent to those of a Higgs boson with a mass of 2 TeV, they are excluded at 4.5 standard deviations. Therefore, there must be additional, cancelling contributions to the radiative corrections from other quanta in the theory if strong == scattering occurs in nature. This would not require "ne-tuning although it would require a measure of serendipity. There are good reasons for the widespread view that a light Higgs boson is likely and for the popular designation of SUSY (supersymmetry) as The People's Choice. But SUSY also begins to require a measure of serendipity [19}21] to meet the increasing lower limits on sparticle and light Higgs boson masses. While the community of theorists has all but elected SUSY, the question is not one that can be decided by democratic processes. At the end of the day only experiments at high-energy colliders can tell us what the symmetry breaking sector contains. Collider experiments, particularily those at the LHC, should be prepared for the full range of possibilities, including the capability to measure == scattering in the TeV region.

Acknowledgements I wish to thank David Jackson, and Henry Stapp for helpful discussions. This work was supported by the Director, O$ce of Energy Research, O$ce of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contracts DE-AC0376SF00098.

References [1] M.I. Vysotsky, V.A. Novikov, L.B. Okun, A.N. Rozanov, Phys. Atom. Nucl. 61 (1998) 808}811; Yad. Fiz. 61 (1998) 894}897.

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[2] The LEP collaborations, LEP EWWG, and SLD Heavy Flavour and Electroweak Groups, CERN-EP/99-15, 1999. [3] M.S. Chanowitz, M.K. Gaillard, Nucl. Phys. B 261 (1985) 379. For a recent review including a discussion of experimental signals see M. Chanowitz, in: D. Graudenz, Proceedings of the Summer School on Hidden Symmetries and Higgs Phenomena, p. 81, (Zuoz, Switzerland, August 1998) PSI-Proceedings 98-02 and eprint hepph/9812215. [4] M.S. Chanowitz, Phys. Lett. B 373 (1996) 141, hep-ph/9512358. [5] M.S. Chanowitz, Phys. Lett. B 388 (1996) 161, hep-ph/9608324. [6] M.S. Chanowitz, M.K. Gaillard, Phys. Lett. 142 B (1984) 85. [7] G. Kane, W. Repko, B. Rolnick, Phys. Lett. B 148 (1984) 367. [8] S. Dawson, Nucl. Phys. B 29 (1985) 42. [9] M.S. Chanowitz, M. Golden, H.M. Georgi, Phys. Rev. D 36 (1987) 1490. [10] M.S. Chanowitz, M. Golden, H.M. Georgi, Phys. Rev. Lett. 57 (1986) 2344. [11] M.E. Peskin, T. Takeuchi, Phys. Rev. D 46 (1991) 381. [12] G. Altarelli, R. Barbieri, Phys. Lett. B 253 (1991) 161. [13] M.K. Gaillard, Phys. Lett. B 293 (1992) 410. [14] O. Cheyette, Nucl. Phys. B 361 (1988) 183. [15] J.M. Cornwall, D.N. Levin, G. Tiktopoulos, Phys. Rev. D 10 (1974) 1145. [16] C.E. Vayonakis, Lett. Nuovo Cimento. 17 (1976) 383. [17] B.W. Lee, C. Quigg, H. Thacker, Phys. Rev. D 16 (1977) 1519. [18] H-J. He, W. Kilgore, Phys. Rev. D 55 (1997) 1515, hep-ph/9609326. [19] P.H. Chankowski, J. Ellis, S. Pokorski, Phys. Lett. B 423 (1998) 327}336, e-Print Archive: hep-ph/9712234. [20] R. Barbieri, A. Strumia, Phys. Lett. B 433 (1998) 63-66, e-Print Archive: hep-ph/9801353. [21] P.H. Chankowski, J. Ellis, M. Olechowski, S. Pokorski, Nucl. Phys. B 544 (1999) 39}63, e-Print Archive: hep-ph/9808275.


Non-renormalization of induced charges and constraints on strongly coupled theories S.L. Dubovsky*, D.S. Gorbunov, M.V. Libanov, V.A. Rubakov Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, 117312 Moscow, Russia

Abstract It is shown that global fermionic charges induced in vacuum by slowly varying, topologically non-trivial background scalar "elds are not renormalized provided that expansion in momenta of background "elds is valid. This suggests that strongly coupled theories obey induced charge matching conditions which are analogous, but generally not equivalent, to 't Hooft anomaly matching conditions. We give a few examples of induced charge matching. In particular, the corresponding constraints in softly broken supersymmetric QCD suggest non-trivial low-energy mass pattern, in full accord with the results of direct analyses. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.15.Me

1. Introduction Four-dimensional gauge theories strongly coupled at low energies often exhibit interesting content of composite massless fermions. This property is potentially important for constructing composite models of quarks and leptons, which is long being considered as a possible `major step in our way into the nature of mattera [1]. Powerful constraints on the low-energy spectrum are provided by 't Hooft anomaly matching conditions [2]. These are extensively used, in particular, in establishing duality properties of supersymmetric gauge theories (see, e.g., Refs. [3,4] and references therein).

*Corresponding author. E-mail address: [email protected] (S.L. Dubovsky) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 7 3 - 3

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S.L. Dubovsky et al. / Physics Reports 320 (1999) 147}158

The basis for anomaly matching is provided by the Adler}Bardeen non-renormalization theorem [5]. In non-Abelian theories, the absence of radiative corrections to anomalies is intimately connected to topology: if one introduces background gauge "elds corresponding to the #avor symmetry group, the anomalies in global currents are proportional to the topological charge densities of these background "elds. Integer-valuedness of global fermionic charges, on the one hand, and integer-valuedness of topological charges of background gauge "elds, on the other, imply that anomaly equations should not get renormalized. Gauge "eld backgrounds are not the only ones that may have topological properties. Topology is inherent also in scalar background "elds of Skyrmion type. Indeed, one-loop calculations [6] show that slowly varying in space, static scalar "elds induce, in vacuum, fermionic global charges which are proportional to the topological charges of the background. By analogy to triangle anomalies, this suggests that induced charges do not receive radiative corrections, and hence may serve as constraints for low-energy spectrum of strongly coupled theories [7]. Unlike triangle anomalies, however, the one-loop expressions for the induced charges are promoted to full quantum theory only if the expansion in momenta of the background "elds is valid in the full theory. The latter property can often be established to all orders of perturbation theory (exceptions are easy to understand); in some models the validity of the expansion in momenta can be also shown non-perturbatively. We will see that induced charge matching conditions emerging in this way have a certain relation to anomaly matching. However, in some cases the two sets of matching conditions are inequivalent, so the induced charges give additional information on the properties of the low-energy theory. This information is particularly interesting in softly broken supersymmetric gauge theories. Fermionic charges in vacuum are induced due to Yukawa interactions of fermions with background scalar "elds. These interactions introduce masses to some of the fermions in the fundamental theory and hence explicitly break a subgroup of the #avor group. As a consequence, some of the fermions of the low-energy e!ective theory acquire masses. Induced charge matching conditions constrain the resulting mass pattern of the e!ective theory. We will see that these conditions are satis"ed automatically (provided the triangle anomalies match) if all composite fermions charged under explicitly broken #avor subgroup become massive. The latter situation is very appealing intuitively; however, we are not aware of any argument implying that it should be generic. This paper is organized as follows. In Section 2 we show that global charges induced in vacuum by slowly varying background scalar "elds do not get renormalized provided that the derivative expansion is valid. In Section 3 we discuss exceptional situations by presenting a model where the derivative expansion fails at the one-loop level itself. In Section 4 we give several examples of induced charge matching (ordinary QCD, supersymmetric N"1 QCD exhibiting the Seiberg duality [8], SQCD with softly broken supersymmetry). We conclude in Section 5 by discussing the relation between induced charges and triangle anomalies.

2. Non-renormalization of induced charges To be speci"c, let us consider QCD with N colors and N massless fermion #avors. Let t? and tI , a, a "1,2, N , denote left-handed quarks and anti-quarks, respectively. To probe this theory, ?


149

we introduce background scalar "elds m? (x) of the following form, @ m?(x)"m ;?(x) , @ @ where m is a constant and ; is an S;(N ) matrix at each point x. Let these "elds interact with quarks and anti-quarks, (1) L "tI m? t@#h.c. ? @ Besides the global S;(N ) ;S;(N ) symmetry, the theory exhibits non-anomalous baryon * 0 symmetry, tPe ?t, tI Pe\ ?tI , mPm. The baryonic current is conserved and obtains nonvanishing vacuum expectation value in the presence of the background scalar "elds. To the leading order in momenta, the one-loop expression for this induced current is [6] 1jI 2"(N /24p)eIJHMTr(;R ;R;R ;R;R ;R) . (2) J H M This expression can be obtained by considering a con"guration which in the vicinity of a given point x has the following form, ;(x)"1#e(x) ,

(3)

where e(x) is a small and slowly varying anti-Hermitean background "eld. To the leading order in momenta, one-loop-induced baryonic current is, then, as shown in Fig. 1 with fermions of mass m running in the loop. The complete expression (2) is reconstructed by making use of S;(N ) ;S;(N ) global symmetry. * 0 A remarkable property of Eq. (2) is that the baryonic charge induced in vacuum by slowly varying, time-independent background "eld ;(x) with ;(x)P1 as "x"PR is proportional to the topological number of the background, 1B2"N N[;] , where

(4)

1 dxeGHITr(;R ;R;R ;R;R ;R) . N[;]" G H I 24p The higher derivative terms omitted in Eq. (2) do not contribute to 1B2. Let us see that Eq. (4) does not get renormalized in the full quantum theory provided the expansion in momenta of the background "eld works. Let us consider the same theory with the gauge coupling a depending on coordinates x. The induced current is now a functional of a(y) and ;(y), 1 jI (x)2"jI[x;a(y);;(y)] . At slowly varying a(y) we expand 1 jI (x)2 in derivatives of a at the point x, 1 jI (x)2"JI[x;a(x);;(y)]#BIJ[x;a(x);;(y)]R a(x)#O[(Ra)] , J

(5)

150


Fig. 1. Leading order contribution to the induced baryonic current.

where the coe$cients on the right-hand side are now functions, rather than functionals, of a(x) (but still functionals of ;(y)). The structure analogous to the right-hand side of Eq. (2) appears in the derivative expansion of the "rst term on the right-hand side of Eq. (5), JI"( f(a)/24p)eIJHMTr(;R ;R;R ;R;R ;R)#2 J H M Our purpose is to show that f (a) is independent of a, and hence f (a)"N . Let us make use of the conservation of 1 jI (x)2, R 1 jI (x)2"0 . I If f were a non-trivial function of a, the divergence of JI would contain the term

(6)

(1/24p)eIJHMTr(;R ;R;R ;R;R ;R)(Rf/Ra)R a . J H M I The only possible source of cancellation of this term in Eq. (6) is the second term on the right-hand side of Eq. (5). The cancellation would occur i! BIJ contained the term of the following structure bIJ[;]Rf/Ra with R bIJ[;]"!(1/24p)eJNHMTr(;R ;R;R ;R;R ;R) . (7) I N H M However, the right-hand side of Eq. (7) is not a complete divergence of any tensor that is invariant under the #avor group (recall that the right-hand side of Eq. (7) is a topological current). Hence, the conservation of the baryonic current requires that Rf/Ra"0. This argument is straightforward to be generalized to the other conserved currents and to the theories other than QCD. As discussed in Section 1, it implies that induced charges should match in fundamental and low-energy theories. Examples of such a matching are given in Section 4.

3. Failure of derivative expansion: an example An important ingredient in the above argument is the derivative expansion. While it works in QCD and many other models, at least to all orders of perturbation theory, one can design models


151

Fig. 2. Dangerous diagram in the model of Section 3. Heavy and light lines correspond to massless and massive fermions, respectively.

where the derivative expansion fails, and the induced charges cannot be reliably calculated even within perturbation theory. As an example, let us consider a model of free left-handed fermions t?, tI and s? , a, a "1,2, N , with the mass term ? L "k tI s? #h.c. (8) I ? The model is invariant under the global S;(N ) ;S;(N ) symmetry under which t, tI , and * 0 s transform as (N ,1), (1,NM ), and (1,N ), respectively. The `baryon numbersa of fermions t,tI and s are #1, !1 and #1, respectively. Let us introduce background "elds m? (x) and their @ interaction with fermions t and tI in the same way as in Eq. (1). To see that the derivative expansion is not reliable in this model, let us again consider the background "eld of the form (3). At e"0, fermions tI and m"const ) (k s#m t) form massive Dirac "eld, while g"const ) (m s!k t) remains massless Weyl "eld. Both types of fermions interact with the background "eld e(x). In an attempt to calculate the induced baryonic current, one faces diagrams with massless internal fermion lines like the one shown in Fig. 2. It is straightforward to see that the derivative expansion of these diagrams is singular. The fact that the derivative expansion fails in this model manifests itself in di!erent values of induced charges in various limits. Namely, at k <m one can ignore the background "elds, and 1B2"0. On the other hand, at k ;m , the mass term (8) becomes irrelevant, so 1B2"N[;]. As outlined above, this phenomenon is due to the fact that not all fermions charged under S;(N ) ;S;(N ) obtain masses upon introducing the background "elds m(x). * 0 This example shows that the validity of the derivative expansion requires that the background scalar "elds provide masses to all relevant fermions. This will be the case in all examples presented in the next section. 4. Examples of induced charge matching 4.1. QCD We again consider conventional S;(N ) QCD with N massless #avors. Let us generalize slightly the discussion of Section 2 by introducing background "elds (9) mN (x)"m ;N (x) O O

152


which give x-dependent masses to N quark #avors only, L "mN (x)tI tO#h.c. The "elds m(x) O N are N ;N matrices; hereafter the indices p,q,r;p ,q ,r run from 1 to N with N (N . (N !N ) #avors remain massless. In all the examples of this section we consider background "elds that are constant at spatial in"nity; by a global S;(N ) rotation we set ;(x)P1 as "x"PR. Besides the * baryon number symmetry ;(1) , we will be interested in a vector subgroup ;(1)D of the original S;(N ) ;S;(N ) #avor group, whose (unnormalized) generator is * 0 ¹D"diag(1,2,1,!N /(N !N ),2,!N /(N !N )) . The background "elds are charged under neither ;(1) nor ;(1)D. As all fundamental fermions that interact with the background scalar "elds acquire masses due to this interaction, the derivative expansion is justi"ed, at least order by order in perturbation theory. Hence, for slowly varying m(x) one has 1B2"N N[;] , (10) 1¹D2"N N[;] . (11) Let us see that the low-energy e!ective theory of QCD * the non-linear sigma-model * indeed reproduces Eqs. (10) and (11). In the absence of the background "elds, the non-linear sigma-model action contains only derivative terms for the S;(N ) matrix valued dynamical sigma-model "eld e\-collider experiments [8]. Basing on the assumption of the short lifetime of such leptons (that could be expected from the universality of the weak interaction) the experimental search criteria have been developed, the main of which is to search for uncorrelated isolated k!e pairs in e>e\-collisions. This method has been used in experiments at the ADONE collider to search for heavy leptons [9]. The ADONE energy turned out to be insu$cient for q-lepton discovery. However, namely the uncorrelated k!e-pair observation became the "rst signal of q-lepton discovered by Perl [10]. The example with the q-lepton discovery demonstrates that the statement of the question of exotic particle existence and their properties is extremely important for their discovery, as it mainly determines the experimental strategy. The principal approach by Okun [1] (everything is allowed that is not prohibited) in connection with questions (3), (4), (5) turned out to be absolutely correct as well. The answer on these questions has been given after the origin of the idea of the supersymmetry and construction of the GUT models. It is evident, that baryons and leptons with the integer spin are the superpartners of ordinary particles. Thus, questions (3) and (4), in fact, raised the problem of the possibility of the broken supersymmetry existence. If the scale of the supersymmetry breaking were not so large, as it results from the experimental data, then the existence of mesons with half-integer spin (which consist of quarks q and their supersymmetric partner q ) could be quite possible. However, with the modern lower limits

S.S. Gershtein et al. / Physics Reports 320 (1999) 159}173

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on supersymmetric quark masses the lifetimes of such mesons turn out to be much smaller than their hadronization times, so as the bound state with ordinary quarks does not have time to be produced. From the point of view of experimental bounds the only possibility remains to be open * the existence of longliving light gluino, m (5 GeV, although it is beyond the MSSM. E The realization of this possibility could lead to the presence of the objects discussed in [1]: mesons (hybrids) with half-integer spin, udM g , etc., and baryons with half-integer spin, uudg , etc. As for baroleptons discussed in [1], they appear in a natural way in the GUT models in the form of leptoquarks. If leptoquark masses are comparable with the GUT scale, these leptoquarks are hardly to be discovered in current experiments. However, following the logics of [1], the question of possibility for `lighta leptoquarks to exist and possible bounds on their masses can be considered. Namely this question is a theme of the paper presented.

2. Models Here we present a short review of some models, in which one could have light leptoquarks with the masses in the TeV range. Among the possibilities are: E Grand Uni"ed Theories (GUTs) [12] and their supersymmetric analogues [11], where leptons and quarks usually appear in the same multiplet. E Heterotic superstring models, with and without intermediate grand uni"cation, in particular their free fermionic realizations. E TeV-scale GUTs and TeV-scale superstrings, where low scale of grand uni"cation is achieved through the appearance of extra space}time dimensions. E Extended technicolour theories [13], where quarks and leptons individually appear in multiplets of the dynamically broken extended technicolour group. The other particles in each multiplet are new fermions, that would appear at low energies in fermion}anti-fermion bound states, some of which are leptoquarks. E Substructure or compositeness models [14], where the `preonsa in a quark and lepton could combine to form a scalar or vector leptoquark. Any discussion of the above leptoquark (LQ) models has been historically based on a set of assumptions due to BuchmuK ller, RuK ckl and Wyler (BRW) under which consistent LQ models can be constructed [15]. These authors had also classi"ed the possible leptoquark states, according to their possible spins and fermion number thus leading to the 10 states displayed in Table 1. These assumptions may be stated as follows: (a) LQ couplings must be invariant with respect to the SM gauge interactions. (b) LQ interactions must be renormalizable. (c) LQs couple to only a single generation of SM fermions. (d) LQ couplings to fermions are chiral. (e) LQ couplings separately conserve baryon and lepton numbers. (f ) LQs only couple to the SM fermions and gauge bosons.

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2.1. Non-SUSY and SUSY GUTs Historically, leptoquarks "rst appeared in Pati and Salam's SU(4) model [16], where the idea of lepton number as a fourth colour was implemented. The minimal group for models of this type is S;(4) ;S;(2) ;;(1) 0 and matter of one generation is sitting in the following representations: * 2 (4,2,0), (4,1,#), (4,1,!). The leptoquarks, arising in these models induce #avour-changing neu tral currents and lepton family and baryon number violation, but their contribution to proton decay is severely suppressed [17]. Constraints on Pati}Salam-type leptoquarks have been discussed in [16] and [18,19]. One would expect these vector leptoquarks to have full-strength couplings to a lepton and a quark of the same generation, giving, for instance, a large (&j/m , j&1) contribution to KPek (s#kPd#e). In general, in grand uni"ed models the intermediate scale of symmetry breaking is large of the order of 10}10 GeV, and so the vector leptoquarks, appearing in these models, are of no interest to us. In paper [20] the model with Pati}Salam SU(4)SU(2) SU(2) group, broken at * 0 the lowest energy scale that phenomenology [21] allows (1000 TeV) was discussed. There, in order to generate large mass for right-handed q-neutrino the `3;3 see-saw mechanisma was used [22]. For this one introduces a singlet fermion S &(1,1,1) and the simple Higgs multiplet s&(4,1,2). * The additional Yukawa term then has the following form: "nSM Tr[sRf ]#h.c. (1) 7 * 0 when combined with the standard electroweak Yukawa terms this yield a neutrino mass matrix of the form L

0 m 0

m 0

n1s2

n1s2

0

0

(2)

in the [l , (l ), S ] basis (m is the top-quark mass). This will give us one massless eigenstate which * 0 * we identify with the standard neutrino, and a massive Dirac neutrino. For n1s2<m , the massless state has approximately standard electroweak interactions. Because the light eigenstate is massless for all values of the nonzero entries of the mass matrix, 1s2 can be reduced to about 1000 TeV. The implementation of this mechanism of symmetry breaking in SUSY grand uni"ed theories with the SO(10) gauge group and symmetry breaking chain SO(10)PS;(2) ;S;(2) ;S;(4) PMSSM * 0 .1 are also possible, because, as it is shown in the analysis of [23], the reason for high intermediate scale of mass breaking is the use of (1,3,10) Higgs "eld to generate right-handed neutrino masses, which is not needed in the mechanism, described above. In papers [24,25] in the framework of left}right supersymmetric models the phenomenology of light doubly charged particles was analysed. In the Pati}Salam case they "nd that entire leptoquark (3,1,10) multiplets can remain light. The discussion of prospects for including additional light colour triplets and anti-triplets in the spectrum of SUSY grand uni"ed theories was done in [26]. In this paper the authors proposed


163

particular string-inspired Pati}Salam model, which has light colour triplet of charge ! plus antitriplet of charge with masses of order 200 GeV in its low-energy spectrum. It was shown in [27] that the minimal S;(5) model plus a light (m &100 GeV) S;(2) doublet leptoquark would be compatible with present experimental results on proton decay and sin h . 5 The authors of [28] have built a low-energy compatible S;(4)-type model for vector leptoquarks with mass &1 TeV. And "nally, we would like to mention S;(15) grand uni"ed theory as a candidate theory with light leptoquarks, where all 14 types of leptoquarks, listed in Table 1 can be present and the GUT scale may be about 10 GeV. The main motivation for the models with this gauge group is the gauge uni"cation without proton decay. In this model gauge bosons have de"nite B and ¸ numbers, because of the only de"ning representation 15 for the fermions. This situation is contrary to that with the conventional GUTs based on S;(5), SO(10) and E(6), in which proton decay occurs at leading order via gauge bosons with de"nite B!¸ but inde"nite separate B and ¸. The attractive feature of these models is that they predict a large set of weak-scale scalar leptoquarks (for more details see [29]). 2.2. Heterotic superstring models The "eld theory limit of the heterotic superstring is some GUT-like gauge "eld theory, which is broken at or below the string scale to the Standard Model gauge group and possibly some extra ;(1)s:S;(3);S;(2);;(1)L>. Thus, this low-energy limit of heterotic string theory may contain leptoquarks for the same reason as grand uni"ed theories themselves. For example, certain Calabi}Yau or orbifold compacti"cations have G"E /H as the four dimensional gauge group below the compacti"cation scale, where H is some discrete symmetry group that can be chosen such that G is the Standard Model (;;(1)L as we already noted). The low-energy "elds are then generally sit in the 27 of E . For each generation of quarks and leptons one then would expect a right-handed neutrino, an extra SO(10) singlet, the two Higgs doublets of the MSSM, and a pair of S;(2) singlets, which may have their couplings of either diquarks or leptoquarks, but not both simultaneously [58], as this would lead to the proton decay [50]. Detailed investigation of the uni"cation of gauge couplings within the framework of a wide class of realistic free-fermionic string models, including the #ipped S;(5), SO(6);SO(4), and various S;(3);S;(2);;(1) models has been done in [30]. It was shown, that if the matter spectrum below the string scale is that of the MSSM, then string uni"cation is in disagreement with experiment. The one-loop string threshold corrections in free-fermionic string models, the e!ect of non-standard hypercharge normalization, light SUSY thresholds and intermediate-scale gauge structure cannot resolve the disagreement with low-energy data, and, only the inclusion of extra colour triplets and electroweak doublets beyond the MSSM at the appropriate thresholds lead to the gauge couplings uni"cation. The constraints on leptoquarks in such superstring-derived models were calculated in [58], and are extensively reviewed in [59]. 2.3. Extra dimensions Recently, in [31] extra large space dimensions was used as a radical new proposal for avoiding the gauge hierarchy problem by lowering the Planck scale to the TeV scale. The extra dimensions

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Table 1 Quantum numbers and fermionic coupling of the leptoquark states. No distinction is made between the representation and its conjugate. Bl is the branching fraction of the LQ into the ej "nal state, Q is its electric charge and F * fermion number (F"3B#¸) Leptoquark

SU(5) Rep

Q

Coupling

Bl

S * S 0 SI 0 S *

5 5 45

j (e>u ), j (ldM ) * * j (e>u ) 0 j (e>dM ) 0 !(2j (e>dM ) * !j (e>u ), !j (ldM ) * * (2j (lu ) * j (e>u) * j (lu) * j (e>u) 0 !j (e>d) 0 j (e>d) * j (ld) *

1/2 1 1

Scalars F"!2

F"0

45

S *

45

S 0

45

SI *

10/15

! !

1 1/2 0 1 0 1 1 1 0

Vectors F"!2

F"0

< *

24

< 0

24

dM ) 0 j (e>u ) 0

1

j (e>u ) * j (lu ) * j (e>d), j (lu) * * j (e>dM ) 0 j (e>u) 0 (2j (e>u) * !j (e>d), j (lu) * * (2j (ld) *

1

0 1 1

0 1/2 1 1 1 1/2 0

required to achieve this are in the (sub-)millimeter range, and thus imply a profound change in Newton's gravitational force law at these distances. After that in [33}37], it was shown how extra large dimensions could also be used to lower the fundamental string scale to the TeV scale. The idea of taking the string scale in the TeV range was "rst considered in [38]. In [32,35]


165

a complementary proposal of lowering the fundamental GUT scale to the TeV range was presented. As it was shown in these papers extra space}time dimensions modify the running of the Standard Model gauge couplings in such a way that not only is the uni"cation preserved, but in fact it occurs more rapidly. So, with the help of extra dimensions, it becomes possible to have grand uni"cation as low as the TeV scale. In these models Standard Model gauge "elds (together with corresponding charged matter) may reside inside of p49 spatial dimensions p * branes (or a set of overlapping branes), while gravity lives in a larger (10 or 11) dimensional bulk of space}time. So, in this scenario we might have light TeV * scale vector leptoquarks. It could turn out that coupling of these leptoquarks to the Standard Models quarks and leptons are severely suppressed due to some anomaly free gauge discrete symmetries, as it is the case for B and ¸ non-preserving leptoquarks, mediating proton decay. Here its worth to mention the TeV * scale supersymmetric Standard Model (TSSM) [39]. However, there are some Pati}Salam like Type I string models [34], so that the leptoquarks, which arise here can give sizeable contributions to rare processes. 2.4. Extended technicolour Another possibility to have vector leptoquarks in the TeV range is the models with the dynamical electroweak and #avour symmetry breaking. One such class of models is extended technicolour (ETC) [40,41]. In these models it is assumed that there is a new gauge interaction, besides Standard Model interactions, called `technicoloura, with gauge group G , and gauge 2! coupling a that becomes strong in the vicinity of a few hundred GeV. The dynamical symmetry 2! breaking is than realized via the condensation of technifermions at this scale, in the same way as chiral symmetry breaking in QCD via light quark condensation. The breaking of quark, lepton and technifermion #avour symmetries is achieved by embedding technicolour, colour and part of electromagnetic ;(1) into the gauge group G , which breaks at a high scale K CK N * Corresponding author. Tel. #49-2461-616472; fax: #49-2461-613930. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 7 2 - 1

(1)

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I.P. Ivanov et al. / Physics Reports 320 (1999) 175}186

where D is the momentum transfer and k,l"$1,¸ and o,j"$ are helicities of particles in cHp PcHp scattering, Q,= and x"Q/(Q#=) are standard DIS variable. The motto of J H I M high-energy QCD * the quark helicity conservation, the common wisdom that high-energy scattering is spin-independent, some model considerations including [1] the vanishing onepomeron exchange contribution to A (D"0), all suggest that the corresponding spin \\*> asymmetry A "p /p vanishes in small-x limit of DIS. *2 2 In this communication we demonstrate that this is not the case. We "nd about x-independent spin asymmetry A and scaling and steeply rising g (x,Q) at small x, *2 g (x,Q)&(G(x,QM ))/x , *2

(2)

where G(x,Q)"xg(x,Q)&(1/x)B is the conventional unpolarized gluon structure function of the target nucleon and QM is #avour-dependent scale to be speci"ed below. The case of the helicity amplitude A (D) is quite tricky. On the one hand, \\*> QCD-motivated considerations strongly suggest a nonvanishing pomeron spin-#ip in di!ractive nucleon}nucleon scattering [2]. On the other hand, recent studies have shown that the s-channel helicity nonconserving (SCHNC) LT interference cross section p" of di!ractive *2 DIS [3] and related SHCNC spin-#ip amplitudes of di!ractive vector meson production do not vanish [4,5] at small x. As Zakharov emphasized [2] such spin-#ip does not con#ict the quark helicity conservation because in scattering of composite objects helicity of composite states is not equal to the sum of helicities of quarks, which arguably holds way beyond the perturbative QCD (pQCD) domain. The recent work on SCHNC vector meson production illustrates this point nicely [3}5]. Consequently, pomeron exchange well contributes to this helicity amplitude but the Procrustean bed of Regge factorization enforces the forward zero, A (D)JD, and vanishing p in \\*> *2 one-pomeron exchange approximation. The principal point behind our result (2) is Gribov's observation [6] that such kinematical zeros can be lifted by two-pomeron exchange (two-pomeron cut) which can contribute to helicity amplitudes vanishing in one-pomeron exchange approximation. A good example is a recent derivation [7] of a rising tensor structure function b (x,Q) for DIS o! spin-1 deuterons. In de"ance of common wisdom, it gives rise to dependence of total cross section on the deuteron tensor polarization which persists at small x. Such a rise of b (x,Q) invalidates the Close}Kumano sum rule [8]. Incidentally, it derives for the most part from di!ractive mechanism which we pursue in this paper. Another example due to Karnakov [9] is a di!erence of cc total cross sections for parallel and perpendicular linear polarizations of colliding photons * the quantity which vanishes in one-pomeron exchange approximation. The keyword behind these new e!ects is unitarity [10], two-pomeron cut is simply a "rst approximation to imposition of unitarity constraints.

2. Regge theory expectations and sum rules We recall that our expectations for small-x behaviour of di!erent structure functions, &(1/x)B, have been habitually driven by the Regge picture of soft interactions, in which the exponent (intercept) d"a!1 is controlled by quantum numbers of the relevant t-channel exchange (a good


177

summary is found in textbook [11]). For instance, the Regge theory suggests a/&1 for helicitydiagonal pomeron (vacuum) exchange dominated F (x,Q) and F (x,Q) and a & for the * 0 secondary reggeon (o,A )-exchange quantities like F (x,Q)!F (x,Q) and u-exchange F (x,Q). N L The dominant A and f reggeon exchange in the axial vector channel suggests a &0 for xg (x,Q). These Regge theory intercepts are not stable against QCD evolution, but extensive studies of small-x asymptotics of generalized two-gluon and quark}antiquark ladder diagrams have revealed only marginal modi"cations of the above hierarchy of intercepts (for the BFKL pomeron exchange see [12], for reggeon exchange and/or non-singlet structure function see [13], for di!erent spin structure functions see: g (x,Q) in [14], g (x,Q) in [15], F (x,Q) in [16]). The corollary of these A studies is that g (x,Q) and g (x,Q) of two-parton ladder approximation have the x-dependence typical of the reggeon exchange and their contributions to spin asymmetries A and A do indeed vanish in the small-x limit. We recall that works [14}16] focused on exactly forward, D"0, Compton scattering amplitudes. Burkhardt and Cottingham [1] argued that because neither pomeron nor high-lying reggeon exchanges contribute to A (D"0), then unsubtracted (superconvergent) dispersion \\*> relation holds for this Compton scattering amplitudes. Precisely superconvergence has been the principal assumption behind the much discussed BC sum rule [1]

dl Im A (Q,l,D"0)"0 (3) / for thorough reviews see [11,17}19]. The tricky point is that the BC amplitude A (Q,l,D) (which di!ers from our A (D) only insigni"cantly) receives a contribution from pomeron \\*> exchange, and the integral

dxg (x,Q)J

dl Im A (Q,l,D) / would diverge at any "nite DO0, which makes the BC sum rule quite a singular one. As we emphasized above, A (D)JD and vanishes at D"0 only because of rigours of Regge \\*> factorization, Gribov's two-pomeron exchange breaks Regge factorization and gives A (D"0)O0, see also [20]. The specter of resulting dramatic small-x rise of g (x,Q) \\*> and of divergence of the BC integral permeates the modern literature on spin structure functions (see textbook [11] and recent reviews [17}19]). The aforementioned breaking of the Close}Kumano sum rule is of the same origin; if cast in the Regge language, the scaling and rising tensor structure function found in [7] falls into the pomeron-cut category. Numerical estimates show that tensor asymmetry is quite large, the related evaluations of g (x,Q) are as yet lacking. *2 (x,Q) 3. Di4ractive DIS and unitarity-driven gU LT In this communication we "ll this gap and report the "rst ever evaluation of unitarity or di!ractive driven contribution to g (x,Q) in terms of the two other experimentally accessible spin *2 observables: the SCHNC LT interference di!ractive DIS structure function [3] and the pomeron spin-#ip amplitude in nucleon}nucleon and pion}nucleon scattering [2]. By unitarity relation, the

178


Fig. 1. The unitarity diagram with (a) &elastic' pX intermediate state and (b) &inelastic' pHX intermediate state with excitation of the proton to resonances or low-mass continuum pH.

opening of di!ractive DIS channel cHpPpX a!ects the elastic scattering amplitude ([10] and references therein). The best-known unitarity e!ect is Gribov's absorption or shadowing correction [21] to one-pomeron exchange. Besides simple shadowing, for spinning particles unitarity corrections can give rise to new spin amplitudes absent in one-pomeron exchange, which was precisely the case with tensor structure function for DIS o! spin-1 deuteron [7]. In the related evaluation of unitarity-driven p3 we start with the eikonal unitarity diagram in *2 Fig. 1. Here the eikonal refers to the &elastic' pX intermediate state, the e!ect of so-called &inelastic' intermediate states pHX when the proton excites into resonances or low-mass continuum states will be commented on below. Hereafter all unitarity corrections will be supplied by a superscript (U). As an input we need amplitudes A" of di!ractive DIS cHp PX p , where k stands for spin states of IMJH J H I M the di!ractive state X. Applying the optical theorem to this unitarity contribution to forward scattering amplitude, we "nd [10]

A" , (4) p3"Re 1/(16p(=#Q)) dD dM A" \\IM IM*> *2 IM where M is the invariant mass of the intermediate state. In order for this unitarity diagram to contribute to p , the r.h.s. of (4) must have a structure which in the convenient polarization *2 vector-spinor representation has the form p J1 f "(r[neR(!)])"in2 , *2 where r is the nucleon spin operator, n is the unit vector along the cHp collision axis, and

(5)

e(l)"!1/(2(l,i) is the photon polarization vector for helicity l. In the polarization-vector representation the factorized one-pomeron amplitude for di!ractive DIS reads [4,5]

p A"" i# d/ +¹ \* which, as we mentioned in the Introduction, vanishes in the forward case D"0. We need the ¸¹ transition in either of the di!ractive cHX vertices and spin-#ip transition in either of the pomeron}nucleon vertices in unitarity diagram of Fig. 1, the other two vertices are spin non-#ip ones. The both spin-#ip transitions are o!-forward with "nite momentum transfer D to the intermediate state and the integrand of Eq. (4) will be J(eRD)(r[nD]). Summing over the phase space of the intermediate state X includes an integration over azimuthal angle of D of the form

d

1 (eRD)(r[nD])" D(r[neR]) , 2p 2

(10)

which has precisely the desired spin structure (5). Now we notice that the LT interference term in di!erential cross section of di!ractive DIS on unpolarized nucleons cHp PX p equals J H I M dp" /dMdD"1/(16p(=#Q)) A"H A" (11) *2 \HIM IM*H IMH and di!ers from the r.h.s. of (4) only by complex conjugation of one of di!ractive amplitudes. In principle, p" can be measured experimentally. The scaling properties of dp" have been estab*2 *2 lished in [3]. The conventionally de"ned LT interference di!ractive structure function F is *2 twist-3 [3], for the purposes of the present discussion it is convenient to factor out the kinematical factor D/Q and de"ne the scaling and dimensionless LT di!ractive structure function g" (x/,b,Q) *2 such that (M#Q)p" dMdD"4pa /Q ) (De)/Q ) (1#"r "D/m) *2 CK N (12) ) g" (x/,b,Q)B exp(!B D) , *2 *2 *2 where b"Q/(Q#M) and x/"x/b are di!ractive DIS variables. In what follows we shall neglect corrections J"r ", because nucleon spin-#ip e!ects are numerically very small within the di!raction cone. Then, making use of (11), (12) and of the factorization property of one-pomeron amplitude (6), we obtain g3 (x,Q)"1/x ) r (0)sin(pa/) ) *2

db B *2 ) ) g" (x/,b,Q) . b 4m(B #B ) *2 N *2 V

(13)

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I.P. Ivanov et al. / Physics Reports 320 (1999) 175}186 (U)

4. The model evaluations of gLT

Eq. (13) is our central result and up to now we have been completely model independent. In principle, the g" can be measured experimentally, in the lack of such direct data in our numerical *2 estimates of p we resort to QCD model for di!ractive DIS developed in [3]. We refer to this *2 paper for details, here we only recall the salient results. The driving term of di!ractive DIS is excitation of qq Fock states of the photon (Fig. 2). We notice that only qq pairs with the sum of helicities zero contribute to p . Consider "rst the *2 contribution from intermediate heavy #avour excitation, in which case the mass m of a heavy D quark provides the large pQCD hard scale [22,23] QM +m/1!b . (14) D The lower blobs in the diagram of Fig. 2a are related to skewed unintegrated gluon structure functions of the proton which can be approximated by the conventional diagonal unintegrated gluon structure functions taken at x "x/. To a log QM accuracy, gross features of g" (x/,b,Q) *2 are described by [3] (15) g" +e/3B m ) b(1!b)(2!3b)a(QM )G(x/,QM ) , *2 D *2 D 1 where e is the quark charge in units of the electron charge, a is the strong coupling, and we D 1 assumed Q "h /(4p) for low tan b in the framework of the MSSM. In this case the corresponding one-loop R R RGE has exact solution > (t)"> E(t)/(1#6> F(t)) , (9) R R R where E(t) and F(t) are some known functions. It exhibits the IRQFP behaviour in the limit > "> (0)PR [15,18}20,22,30] where the solution becomes independent of the initial R R conditions: > (t)N>$."E(t)/6F(t) . (10) R R A similar conclusion is valid for the other couplings [20,22}24,29,30]. It has been pointed out that the IRQFPs exist for the trilinear SUSY breaking parameter A [22], for the squark R masses [20,23] and for the other soft supersymmetry breaking parameters in the Higgs and squark sector [29]. In the case of large tan b the system of the RGEs has no analytical solution and one can use either numerical or approximate ones. It has been shown [31] that almost all SUSY breaking parameters exhibit IRQFP behaviour. For the IRQFP solutions the dependence on initial conditions > , A and m disappears at low G energies. This allows one to reduce the number of unknown parameters and make predictions for the MSSM particle masses as functions the only free parameter, namely m , or the gaugino mass, while the other parameters are strongly restricted. The strategy is the following [32,31]. As input parameters one takes the known values of the top-quark, bottom-quark and q-lepton masses (m , m , m ), the experimental values of the gauge R @ O couplings [2] a "0.118, a "0.034, a "0.017, the sum of Higgs vev's squared v"v# v"(174.1 GeV) and the "xed-point values for the Yukawa couplings and SUSY breaking parameters. To determine tan b the relations between the running quark masses and the Higgs v.e.v.s in the MSSM are used: m "h v sin b , (11) R R m "h v cos b , (12) @ @ m "h v cos b . (13) O O The Higgs mixing parameter k is de"ned from the minimization conditions for the Higgs potential. Then, one is left with a single free parameter, namely m , which is directly related to the gluino mass M . Varying this parameter within the experimentally allowed range, one gets all the masses as functions of this parameter.

D.I. Kazakov / Physics Reports 320 (1999) 187}198

193

For low tan b the value of sin b is determined from Eq. (11), while for high tan b it is more convenient to use the relation tan b"(m /m )/(h /h ), since the ratio h /h is almost a constant in the R @ @ R R @ range of possible values of h and h . R @ For the evaluation of tan b one "rst needs to determine the running top- and bottom-quark masses. One can "nd them using the well-known relations to the pole masses (see e.g. [33,34,30]), including both QCD and SUSY corrections. For the top-quark one has m R , (14) m (m )" R R 1#(*m /m ) #(*m /m ) R R /!" R R 1317 where m "(174.1$5.4) GeV [35]. Then, the following procedure is used to evaluate the R running top mass. First, only the QCD correction is taken into account and m (m ) is found in the R R "rst approximation. This allows one to determine both the stop masses and the stop mixing angle. Next, having at hand the stop and gluino masses, one takes into account the stop/gluino corrections. For the bottom quark the situation is more complicated because the mass of the bottom quark m is essentially smaller than the scale M and so one has to take into account the running of this @ 8 mass from the scale m to the scale M . The procedure is the following [34,36,37]: one starts with @ 8 the bottom-quark pole mass, m "4.94$0.15 [38] and "nds the SM bottom-quark mass at the @ scale m using the two-loop QCD corrections @ m (m )1+"m /(1#(*m /m ) ) . (15) @ @ @ @ @ /!" Then, evolving this mass to the scale M and using a numerical solution of the two-loop SM RGEs 8 [34,37] with a (M )"0.12 one obtains m (M ) "2.91 GeV. Using this value one can calculate 8 @ 8 1+ the sbottom masses and then return back to take into account the SUSY corrections from massive SUSY particles m (M )"m (M )1+/1#(*m /m ) . (16) @ 8 @ 8 @ @ 1317 When calculating the stop and sbottom masses one needs to know the Higgs mixing parameter k. For the determination of this parameter one uses the relation between the Z-boson mass and the low-energy values of m and m which comes from the minimization of the Higgs potential: & & M m #R !(m #R )tan b 8#k" & & , (17) 2 tan b!1 where R and R are the one-loop corrections [11]. Large contributions to these functions come from stops and sbottoms. This equation allows one to obtain the absolute value of k, the sign of k remains a free parameter. Whence the quark running masses and the k parameter are found, one can determine the corresponding values of tan b with the help of Eqs. (11) and (12). This gives in low and high tan b cases, respectively, tan b"1.47$0.15$0.05 for k'0 , tan b"1.56$0.15$0.05 for k(0 , tan b"76.3$0.6$0.3

for k'0 ,

tan b"45.7$0.9$0.4

for k(0 .

194


The deviations from the central value are connected with the experimental uncertainties of the top-quark mass, a (M ) and uncertainty due to the "xed-point values of h (M ) and h (M ). 8 R 8 @ 8 Having all the relevant parameters at hand it is possible to estimate the masses of the Higgs bosons. With the "xed-point behaviour one has the only dependence left, namely on m or the gluino mass M . It is restricted only experimentally: M '144 GeV [2] for arbitrary values of the squarks masses. Let us start with low tan b case. The masses of CP-odd, charged and CP-even heavy Higgses increase almost linearly with M . The main restriction comes from the experimental limit on the lightest Higgs boson mass. It excludes the k(0 case and for k'0 requires the heavy gluino mass M 5750 GeV. Subsequently one obtains m '844 GeV,

m !'846 GeV, &

m '848 GeV for k'0 , &

i.e. these particles are too heavy to be detected in the nearest experiments. For high tan b already the requirement of positivity of m excludes the region with small M . In the most promising region M '1 TeV (m '300 GeV) for the both cases k'0 and k(0 the masses of CP-odd, charged and CP-even heavy Higgses are also too heavy to be detected in the near future: m '1100 GeV for k'0, m !'1105 GeV for k'0, & m '1100 GeV for k'0, &

m '570 GeV for k(0 , m !'575 GeV for k(0 . & m '570 GeV for k(0 . &

The situation is di!erent for the lightest Higgs boson h, which is much lighter. As has been already mentioned, for low tan b the negative values of k are excluded by the experimental limits on the Higgs mass. Further on we consider only the positive values of k. Fig. 3 shows the value of m for k'0 as a function of the geometrical mean of stop masses * this parameter is often identi"ed with a supersymmetry breaking scale M . One can see that the value of m quickly 1317 saturates close to &100 GeV. For M of the order of 1 TeV the value of the lightest Higgs mass 1317 is [32] m "(94.3#1.6#0.6$5$0.4) GeV for M "1 TeV , 1317

(18)

where the "rst uncertainty comes from the deviations from the IRQFPs for the mass parameters, the second one is related to that of the top-quark Yukawa coupling, the third re#ects the uncertainty of the top-quark mass of 5.4 GeV, and the last one comes from that of the strong coupling. One can see that the main source of uncertainty is the experimental error in the top-quark mass. As for the uncertainties connected with the "xed points, they give much smaller errors, of the order of 1 GeV. Note that the obtained result (18) is very close to the upper boundary, m "97 GeV, obtained in Refs. [30,13] (see the previous section). For the high tan b case the lightest Higgs is slightly heavier, but the di!erence is crucial for LEP II. The mass of the lightest Higgs boson as a function of M is shown in Fig. 3. One has the 1317


195

Fig. 3. (A) The dependence of the mass of the lightest Higgs boson h on M "(m m ) (shaded area) for k'0, low 1317 R R tan b. The dashed line corresponds to the minimum value of m "90 GeV allowed by experiment. (B), (C) The mass of the lightest Higgs boson h as function of M for di!erent signs of k, large tan b. The curves (a, b) correspond to the upper 1317 limit of the Yukawa couplings and to m/m "0 (a) or to m/m "2 (b). The curves (c, d) correspond to the lower limit of the Yukawa couplings and to m/m "0 (c) or to m/m "2 (d). Possible values of the mass of the lightest Higgs boson are inside the areas marked by these lines.

following values of m at a typical scale M "1 TeV (M +1.3 TeV) [31]: 1317 m "128.2!0.4!7.1$5 GeV for k'0 , m "120.6!0.1!3.8$5 GeV for k(0 . The "rst uncertainty is connected with the deviations from the IRQFPs for mass parameters, the second one with the Yukawa coupling IRQFPs, and the third one is due to the experimental

196


uncertainty in the top-quark mass. One can immediately see that the deviations from the IRQFPs for mass parameters are negligible and only in#uence the steep fall of the function on the left, which is related to the restriction on the CP-odd Higgs boson mass m . In contrast with the low tan b case, where the dependence on the deviations from Yukawa "xed points was about 1 GeV, in the present case it is much stronger. The experimental uncertainty in the strong coupling constant a is Q not included because it is negligible compared to those of the top-quark mass and the Yukawa couplings and is not essential here contrary to the low tan b case. One can see that for large tan b the masses of the lightest Higgs boson are typically around 120 GeV that is too heavy for observation at LEP II.

4. Summary and conclusion Thus, one can see that in the IRQFP approach all the Higgs bosons except for the lightest one are found to be too heavy to be accessible in the nearest experiments. This conclusion essentially coincides with the results of more sophisticated analyses. The lightest neutral Higgs boson, is on the contrary always light. In the case of low tan b its mass is small enough to be detected or excluded in the next two years when the c.m.energy of LEP II reaches 200 GeV. On the other hand, for the high tan b scenario the values of the lightest Higgs boson mass are typically around 120 GeV, which is too heavy for the observation at LEP II leaving hopes for the Tevatron and LHC. However, these SUSY limits on the Higgs mass may not be so restricting if non-minimal SUSY models are considered. In a SUSY model extended by a singlet, the so-called Next-to-Minimal model, Eq. (8) is modi"ed and at the tree level the bound looks like [39] mKM cos 2b#jv sin 2b , 8

(19)

Fig. 4. Dependence of the upper bound on the lightest Higgs boson mass on tan b in MSSM (lower curve), NMSSM (middle curve) and extended SSM (upper curve) [39].


197

where j is an additional singlet Yukawa coupling. This coupling being unknown brings us back to the SM situation, though its in#uence is reduced by sin 2b. As a result, for low tan b the upper bound on the Higgs mass is slightly modi"ed (see Fig. 4). Even more dramatic changes are possible in models containing non-standard "elds at intermediate scales. These "elds appear in scenarios with gauge mediated supersymmetry breaking. In this case, the upper bound on the Higgs mass may anyway increase up to 155 GeV [39] (the upper curve in Fig. 4), though it is not necessarily saturated. One should notice, however, that these more sophisticated models do not change the generic feature of SUSY theories, the presence of the light Higgs boson.

Acknowledgements The author is grateful to A.V. Gladyshev and M. Jurc\ is\ in for useful discussions and help in preparing the manuscript. Financial support from RFBR grant C 98-02-17453 is acknowledged.

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Ultra-high-energy cosmic rays and in#ation relics Vadim A. Kuzmin *, Igor I. Tkachev Institute for Nuclear Research, Russian Academy of Sciences, 60th October Anniversary Prosp. 7a, Moscow 117312, Russia TH Division, CERN, CH-1211 Geneva 23, Switzerland Dedicated to Lev Okun, for his continuous inspiration

Abstract There are two processes of matter creation after in#ation that may be relevant to the resolution of the puzzle of cosmic rays observed with energies beyond GZK cut-o!: 1) gravitational creation of superheavy (quasi)stable particles, and 2) non-thermal phase transitions leading to the formation of topological defects. We review both possibilities. 1999 Elsevier Science B.V. All rights reserved. PACS: 96.40.!z; 95.35.#d; 98.80.Cq Keywords: Cosmic rays; Dark matter; Early universe

1. Introduction Cosmological and astrophysical considerations are able to provide the strongest restrictions on parameters of particle physics models and even rule out some classes of models entirely. This is especially valuable when the model is unrestricted by laboratory experiments (which is often the case). Among famous results which made a strong impact on model building is the cosmological domain wall problem which appears in models with spontaneous breaking of discrete symmetries [1] and the problem of magnetic monopoles in Grand Uni"ed Theories (GUT) [2]. In return, studies of cosmological phase transitions [3] and of the dynamics of bubbles of a metastable vacuum [4] lead to the change of basic concepts of the cosmology of the early Universe, and in#ationary cosmology [5}8] was born (for reviews see [9,10]). In#ation gives a possible solution to horizon, #atness and homogeneity problems of `classicala cosmology [6]. In#ation was

* Corresponding author. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 6 4 - 2

200

V.A. Kuzmin, I.I. Tkachev / Physics Reports 320 (1999) 199}221

designed to solve the problem of unwanted relics, like magnetic monopoles. It was promptly realized [11] that in#ation can generate small amplitude large-scale density #uctuations which are the necessary seeds for the galaxy and the large-scale structure formation in the Universe. This elevates in#ation from the rank of a `broad brush problem solvera to the rank of a testable hypotheses. And testable in "ne details, as rapidly accumulating data on cosmic microwave background #uctuations (CMB) (starting from COBE detection [12] through numerous balloon and ground-based CMBR experiments and with culmination at MAP [13] and PLANK [14] anticipated detailed maps of anisotropy of the microwave sky) and huge galaxy catalogs like the already collecting data SLOAN digital sky surview [15] will provide a wealth of cosmological information. In#ation is generally assumed to be driven by the special scalar "eld known as the inyaton. During in#ation, the in#aton "eld slowly rolls down towards the minimum of its potential. In#ation ends when the potential energy associated with the in#aton "eld becomes smaller than the kinetic energy, which happens when magnitude of the in#aton "eld decreases below the Plank scale, :M and `colda coherent oscillations of the in#aton "eld commence. These oscillations . contained all the energy of the Universe at that time. All matter in the Universe was created by reheating, which is nothing but decay of the zero momentum mode of in#ation oscillations. The process is obviously of such vital importance that here too one may hope to "nd some observable consequences, speci"c to the process itself and for particular models of particle physics, despite the fact that scales relevant for the reheating are very small. And, indeed, we now believe that there may be some clues left. Among those are: topological defects production in non-thermal phase transitions [16], GUT scale baryogenesis [17], generation of primordial background of stochastic gravitational waves at high frequencies [18], just to mention a few. However, the most interesting could be a possible relation to a mounting puzzle of the ultra-high-energy cosmic rays (UHECR) [19]. When a proton (or neutron) propagates in CMB, it gradually looses energy colliding with photons and creating pions. There is a threshold energy for the process, so it is e!ective for very energetic nucleons only, which leads to the Greisen}Zatsepin}Kuzmin (GZK) cuto! [20] of the high-energy tail of the spectrum of cosmic rays. All this means that the detection of, say, 3;10 eV proton would require its source to be within &50 Mpc. However, several well established events above the cut-o! were observed by Yakutsk [21], Haverah Park [22], Fly's Eye [23] and AGASA [24] collaborations (for the recent reviews see Refs. [25,26]). Results from the AGASA experiment [27] are shown in Fig. 1. The dashed curve represents the expected spectrum if conventional extragalactic sources of UHECR were distributed uniformly in the Universe. This curve exhibits the theoretical GZK cut-o!, but one observes events which are way above it. (Numbers attached to the data points show the number of events observed in each energy bin.) Note that no candidate astrophysical source, like powerful active galaxy nuclei, were found in the directions of all six events with E'10 eV [27] (at these energies cosmic rays experience little de#ection by galactic magnetic "elds). Is some unexpected astrophysics at work here or is this at last an indication of the long awaited new physics? There are two logical possibilities to produce UHE cosmic rays: either charged particles have to be accelerated to energies E'10 eV, or UHECR originate in decays of heavy X-particles,


201

Fig. 1. AGASA data set [27], February 1990}October 1997.

m '10 GeV. The maximum energy which can be achieved in an accelerating site of the size 6 R which has the magnetic "eld strength B is [28] E(10Z(B/kG) (R/Mpc) eV .

(1)

A magnetic "eld is required either to keep the particle con"ned within the accelerating region or to produce an accelerating electric "eld. For protons (Z"1) a few sources satisfy this condition: pulsars, active galactic nuclei (AGNs) and radio-galaxies. However, energy losses (pair production and meson photoproduction) restrict the maximum energy to E(10 eV in pulsars and AGNs [25,29], while radio-galaxies that lie along the arrival directions of UHECR are situated at large cosmological distances, 9100 Mpc [30], i.e. beyond the GZK radius. Similar conclusions seem to be true with respect to cosmological gamma ray bursts as a possible source of UHECRs [31]. New astrophysics which may work is a possibility to generate UHECR within GZK sphere in remnants of dead quasars [32] (these are dormant galaxies which harbour a supermassive spinning black hole). New physics suggested as an explanation of UHE cosmic rays, ranges all the way to the violation of the Lorentz invariance [33]. Among less radical extensions of the standard model are: E The existence of a particle which is immune to CMB in comparison with nucleons. In this scenario the primary particle is produced in remote astrophysical accelerators (e.g. radiogalaxies) and is able to travel larger cosmological distances while having energies above the GZK cut-o!. There are variations on this scheme. Supersymmetric partner of gluon, the gluino, can form bound states with quarks and gluons. If gluino is light and quasistable (see e.g. [34,35]), the lightest gluino containing baryons will have su$ciently large GZK threshold to be such a messenger [34] and as a hadron it will be able to produce normal air showers in the Earth's atmosphere. However, there are strong arguments due to Voloshin and Okun [36], against light quasistable gluino based on constraints on the abundance of anomalous heavy isotopes which also will be formed as bound states with gluino.

202


High-energy (anti)neutrinos produced in distant astrophysical sources will annihilate via Z resonance on the relic neutrinos and produce energetic gammas or nucleons [37]. The relic neutrino masses in the eV range are consistent with this scenario [38], as well as with the Super-Kamiokande results. The required high density of the relic neutrinos is achieved if gravitational clumping takes place [37] or if the Universe has a signi"cant lepton asymmetry in background neutrinos [38]. Even then the total luminosity of the neutrino sources in the Universe must be as high as 10\}10 of its photon luminosity, and, therefore, neutrino-only sources are called for by the upper bound from the #ux of the cosmic rays [39]. An independent constraint on the density of the relic neutrinos comes from CMBR and already the present data start to be challenging for models with large neutrino asymmetry [40]. E Another class of suggestions is related to topological defects. UHECR are produced when topological defects decompose to constituent "elds (X-particles) which in turn decay [41]. Maximum energy is not a problem here, but in models which involve string [42] or superconducting string [41] networks, the typical separation between defects is of order of the Hubble distance and thus these models are subject to a GZK cut-o!. Models in which defects can decay `locallya include networks of monopoles connected by strings (necklaces) [43], vortons (charge and current carrying loops of superconducting strings stabilized by angular momentum) [44], and monopolonium (bound monopole}antimonopole pairs) [45]. Finally, magnetic monopoles accelerated by intergalactic magnetic "elds have also been considered as primary UHECR particles [46]. E Conceptually, the simplest possibility is that UHECR are produced cosmologically locally in decays of some new particle [47,48]. GZK cut-o! is automatically avoided but the candidate X-particle must obviously obey constraints on mass, number density and lifetime.

2. UHECR from decaying particles In order to produce cosmic rays in the energy range E'10 GeV, the decaying primary particle has to be heavy, with a mass well above GZK cut-o!, m '10 GeV. The lifetime, q , cannot be 6 6 much smaller than the age of the Universe, t +10 yr. Given this shortest possible lifetime, the 3 observed #ux of UHE cosmic rays will be generated with the rather low number density of X-particles, X &10\, where X ,m n /o , n is the number density of X-particles and o is 6 6 6 6 6 the critical density. On the other hand, X-particles must not overclose the Universe, X (1. With 6 X &1, the X-particles may play the role of cold dark matter and the observed #ux of UHE cosmic 6 rays can be matched if q &10 yr. 6 Spectra of UHE cosmic rays arising in decays of relic X-particles were successfully "tted to the data for m in the range 10(m /GeV(10 [49,50]. For example, the "t of Berezinsky et al. 6 6 [49] to observed #uxes of UHECR assuming m +10 GeV is shown in Fig. 2. Beside the mass of 6 the X-particle there is another parameter which controls the #ux of the cosmic rays from decaying particles: namely, the ratio of X-particles number density and their lifetime. For the "t in Fig. 2 one used (X /X )(t /q )"5;10\. 6 !"+ 3 6 The problem of the particle physics mechanism responsible for a long but "nite lifetime of very heavy particles can be solved in several ways. For example, some otherwise conserved quantum number carried by X-particles may be broken very weakly due to instanton transitions [47],


203

Fig. 2. Predicted #uxes from decaying X-particles, as calculated in Ref. [49] and the data. Latest AGASA results, Fig. 1, are not shown.

or quantum gravity (wormhole) e!ects [48]. If instantons are responsible for X-particle decays, the lifetime is estimated as q &m\ ) exp(4p/a ), where a is the coupling constant of the relevant 6 6 6 6 gauge interaction. The lifetime will "t the allowed window if the coupling constant (at the scale m ) 6 is a +0.1 [47]. 6 A class of natural candidates for superheavy long-lived particles which arise in string and M theory was re-evaluated recently in Refs. [51] and particles with desired masses and long life-times were identi"ed. Other interesting candidates were found among adjoint messengers in gauge mediated supergravity models [52] and in models of superheavy dark matter with discrete gauge symmetries [53]. Superheavy dark matter candidates in superstrings and supergravity models were considered also in Ref. [54]. Below we address the issue of X-particle abundance.

3. Superheavy particle genesis in the early Universe Superheavy particles can be created in the early Universe by several mechanisms. Among those are: E Non-equilibrium `thermala production in scattering or decay processes in primordial plasma [47,48]. E Production during decay of in#aton oscillations (`preheatinga) [55}58]. E Direct gravitational production from vacuum #uctuations during in#ation [59,19,60]. In any case the "nal ratio of the density in X-particles to the entropy density is normalized by the reheating temperature. The reheating temperature is limited to the value below 10}10 GeV in supergravity models with decaying heavy gravitino [61]. This restricts model parameters when `thermala mechanism of heavy particle production is operative (but does not rule it out [47,48,62]).

204


The last two mechanisms are closely related to each other and both can be described on equal footing within frameworks of a single uni"ed approach: particle creation in an external time varying background. However, while the outcome of the second mechanism is highly dependent upon the strength of the coupling of the X-"eld to the in#aton, no coupling (e.g. to the in#aton or plasma) is needed in the third mechanism when the temporal change of the metric is the single cause of particle production. Even absolutely sterile particles are produced by the third mechanism which may be relevant for very long-lived superheavy particles. The resulting abundance is quite independent of the detailed nature of the particle which makes the superheavy (quasi)stable X-particle a very interesting dark matter candidate. We concentrate here on the second and third mechanisms and from the start we introduce coupling of the X-"eld to the in#aton for uniformity of discussion. The limit of zero coupling will correspond to purely gravitational production. In the case of a heavy scalar "eld-X we consider the model (2) ¸"(* )!PR(1385)K> and X(2000)>PRK>. We shall summarize these data in Section 3, after a general description of the nature and expected properties of cryptoexotic baryons, as well as some promising ways for their production and observation. We also present here the SPHINX results in favour of strong violation of the OZI rule in proton di!ractive dissociation reactions [26}28] which may be connected with direct strangeness in the nucleon quark structure.

2. Exotic baryons and possible mechanisms of their production There arise three main questions tightly connected with the exotic searches in the SPHINX experiments: 1. How to identify cryptoexotic B ""qqqss 2 baryons without open exotic values of their ( quantum numbers and how to distinguish them from several dozens of well-known NH and D isobars? 2. How to produce the exotic baryons in the most e!ective way? 3. How to reduce background processes and to make easier the exotic baryon observation? We will try to "nd some qualitative answers to these questions because of the lack of theoretical models for the description of exotic hadrons. 2.1. Properties of exotic baryons with hidden strangeness As has been stated before, cryptoexotic baryons do not have external exotic quantum numbers, and their complicated internal valence structure can be established only indirectly by examining their unusual dynamical properties that are quite di!erent from those of ordinary baryons "qqq2. In this connection, we consider the properties of multiquark baryons with hidden strangeness "qqqss 2. If such cryptoexotic baryon structure consists of two color parts spatially separated by a centrifugal barrier, its decays into the color-singlet "nal states may be suppressed because of a complicated quark rearrangement in decay processes. The properties of multiquark exotic baryons with the internal color structure "qqqqq 2 ""(qqq) ;(qq ) 2 (color octet bonds) or

(1)

(2) "qqqqq 2 ""(qqq ) ;(qq) 2 (color sextet}antisextet bonds) are discussed in [29}32]. Here, subscripts 1c, 8c, and so on specify representations of the color S;(3) group. If the mass of a nonstrange baryon with hidden strangeness is above the threshold for decay modes involving strange particles in "nal states, the main decay channels must be of the type B ""qqqss 2P>K#kn (

(3)

226

L.G. Landsberg / Physics Reports 320 (1999) 223}248

(k"0, 1,2). Another possibility is associated with the decays B P"qqqss 2P N(gN; gN)#kn (4) ( which involve the emission of particles with a signi"cant ss component in their valence-quark structure. It should also be emphasized that g and, particularly, g mesons are strongly coupled to gluon "elds and, hence, to the states with an enriched gluon component. Therefore, baryon decays of the type BPNg, Ng may be speci"c decay modes for hybrid baryons (see, for example, [2]). The nonstrange decays of baryons with hidden strangeness, B PN#kn, must be suppressed by ( the OZI rule. Thus, the e!ective phase-space factors for the decays of the massive baryons B would ( be signi"cantly reduced because of this OZI suppression (owing to a high mass threshold for the allowed decays B P>K with respect to the suppressed decay B PNn, Dn). The mechanism of ( ( quark rearrangement of color clusters in the decays of particles with complicated inner structure of type (1) or (2) can further reduce the decay width of cryptoexotic baryons and make them anomalously narrow (their widths may become as small as several tens of MeV). Here, theoretical predictions are rather uncertain. For this reason, only experiments can answer conclusively the question of whether such narrow baryon resonances with hidden strangeness really exist. Thus, it is desirable to perform systematic searches for the cryptoexotic baryons B with ( anomalous dynamical features listed below. (i) The dominant OZI-allowed decay modes of the baryons B are those with strange particles in ( the "nal states: R("qqqss 2"BR["qqqss 2P>K]/BR["qqqss 2Ppnn; Dn]91.

(5)

For ordinary "qqq2 isobars, R(D; NH)&(several %) [33]. (ii) The cryptoexotic baryons B can simultaneously possess a large mass (M'1.8}2.0 GeV) and ( a narrow decay width (C450}100 MeV). This is due to a complicated internal color structure of these baryons, which leads to a signi"cant quark rearrangement of color clusters in decay processes, and due to a limited phase space for the OZI-allowed decays B P>K. At the same time, ( typical decay widths of the well-established "qqq2 isobars with similar masses are not less than 300 MeV. 2.2. Diwractive production mechanism and search for exotics It was emphasized in a number of studies [1,2,8,10,30,32,34] that di!ractive production processes featuring Pomeron exchange o!er new possibilities in searches for exotic hadrons. Originally, interest was focused on the model of Pomeron with a small cryptoexotic (qqq q ) component [30,32]. According to modern concepts, the Pomeron is a multigluon system owing to which exotic hadrons can be produced in gluon-rich di!ractive processes (see Fig. 1). It is apparent that only the states with the same charge and #avor as those of the primary hadrons can be produced in di!ractive processes. Moreover, there are some additional restrictions on the spin and parity of the formed hadrons which are stipulated by the Gribov}Morrison selection rule. According to this rule, the change in parity *P occurring as a result of the transition from the primary hadron to the di!ractively produced hadronic system, is connected with the corresponding change in the spin *J through the relation *P"(!1) (. For example, because of this rule, in the proton di!ractive dissociation (for proton J."1/2>), only baryonic states with


227

Fig. 1. Diagrams for exotic baryon production in the di!raction processes with the Pomeron exchange. The Pomeron P is a multigluon system.

natural sets of quantum numbers 1/2>, 3/2\, 5/2>, 7/2\, etc., can be excited. The Gribov}Morrison selection rule is not a rigorous law and has an approximate character. The Pomeron exchange mechanism in di!ractive production reactions can induce the coherent processes on the target nucleus. In such processes the nucleus acts as descreet unit. Coherent processes can be easily identi"ed by the transverse momentum spectrum of the "nal-state particle system. They manifest themselves as di!ractive peaks with large values of the slope parameters determined by the size of the nucleus: dN/dP Kexp(!bP ), where bK10A GeV \. Owing to 2 2 the di!erence in the absorption of single-particle and multiparticle objects in nuclei, coherent processes could serve as an e!ective tool for separation of resonance production against nonresonant multiparticle background (see e.g. [35]). The conditions for coherent reactions are destroyed by absorption processes in nuclei. Thus, the coherent suppression of nonresonant background takes place: p (res) p (res) ' . (6) (nonres. BG) p (nonres. BG) p The separation of coherent reactions can be achieved by studying dN/dP distributions for 2 processes under investigation. As is seen from dN/dP spectra in the SPHINX experiments, there are strong narrow forward 2 cones in these distributions with the slope b950 GeV\, which correspond to a coherent di!ractive production on carbon nuclei (see, for example, Fig. 2). For identi"cation of the coherently produced events we used `softa transverse momentum cut P (0.075}0.1 GeV. With this cut noncoherent 2 background in the event sample can be as large as 30}40%. It is possible to reduce this background with more stringent P cut at the cost of some decrease of the signal statistics. 2

228


Fig. 2. dN/dP distribution for typical di!ractive production reaction (p#NP[RK>]#N). The P -regions for 2 2 coherent reaction on carbon nuclei (P 10.075}0.1 GeV) and for nonperipheral processes (P '0.3 GeV) are shown. 2 2

2.3. Processes with large transverse momenta As was discussed above, coherent di!ractive production reactions with small transverse momenta seem quite promising for the search for exotic hadrons, but, of course, they do not exhaust the existing opportunities. Certainly, these searches can be also performed for all di!ractive-type processes (e.g. without coherent cuts for small P ). And of special interest is the study of 2 nonperipheral production reactions which can be the most e!ective way to seek for certain exotic states, especially those that are formed at short ranges and are characterized by broad enough transverse momentum distributions. In this case, the most favorable conditions for identifying exotic hadrons are achieved at higher transverse momenta (P '0.3}0.5 GeV), where the back2 ground from peripheral processes is strongly suppressed. For instance, the unusually narrow meson states X(1740)Pgg [36] and X(1910)Pgg [37] were observed in studying the chargeexchange reactions n\#pP[gg]#D and n\#pP[gg]#n after the selection of events with


229

Fig. 3. Diagram for di!ractive-type reaction with multiple Pomeron rescattering (these processes might be signi"cant at P 90.3 GeV). 2

P 50.3 GeV. These anomalous states are good candidates for cryptoexotic mesons. The rescat2 tering mechanism involving multipomeron exchange in the "nal state (a gluon-rich process) may explain X(1740) and X(1910) nonperipheral production [38]. In the very high primary energy region which, for example, corresponds to the search for exotic states with heavy quarks, di!ractive production reactions with rescattering can be used, instead of the charge-exchange processes with rescattering (see the diagram in Fig. 3). The cross sections of these di!ractive processes also do not die out with energy rise. The nonperipheral P region 2 for these processes is shown in Fig. 2. 2.4. Electromagnetic mechanism The search for exotic hadrons with hidden strangeness can be also carried out in another type of hadron production processes caused by electromagnetic interactions. The example of such process is the formation reaction with s-channel resonance photoproduction of strange particles c#NP"qqqss 2P>K

(7)

(see diagram in Fig. 4a). It is possible in principle to study the s-channel resonance production by detailed energetic scanning of the cross sections and angular distributions for Eq. (7) and by performing the subsequent partial-wave analysis. As is seen from Fig. 4a and from VDM (with its signi"cant coupling of photon with ss pair through -meson), reaction (7) can provide a natural way to embed the ss quark pair into intermediate resonance state and to produce the exotic baryon with hidden strangeness. The existing data on reactions (7) are rather poor and insu$cient for such systematical studies. But one can hope that good data would be produced in the near future in the experiments on strong current electron accelerators CEBAF and ELSA (see, for example, [39]). Electromagnetic production of exotic hadrons can be searched for not only in the resonance photoproduction reactions but in the collisions of the primary hadrons with virtual photons of the Coulomb "eld of the target nuclei [40}44], e.g. in the Primako! production reactions h#ZPa#Z

(8)

230


Fig. 4. Electromagnetic mechanisms for production of exotic baryons with hidden strangeness: (a) formation reaction with s-channel resonance photoproduction; (b) Coulomb production reaction h#ZPa#Z.

(see diagram in Fig. 4b). The Coulomb production mechanism plays the leading role in the region of very small transfer momenta, where it dominates over the strong interaction process [40}42]. The total cross section of the Coulomb reaction (8) is 2J #1 ? C (aPh#c) . p[h#ZPa#Z]" Kp ! 2J #1 F

(9)

The value p is obtained in the QED calculations. In the narrow width approximation for the resonance as in Eq. (9) p has the form

M O [q!q ] ?

"F (q)" dq . p "8naZ X M!m q O ? F

(10)

Here Z is the charge of nucleus; a"1/137 is the narrow structure constant; C(aPh#c) is the radiative width of a; J , J and M , m are the spins and masses of a and h particles; F (q) is ? F ? F X the electromagnetic formfactor of nucleus; q "[[M!m]/2E ] is the minimum square

? F F momentum transfer q; E is the primary hadron energy. In the high-energy region, q is very F

small and q"P #q KP . 2

2 It must be borne in mind that in the Coulomb production reactions with primary protons one studies the same processes as in ordinary photoproduction reactions. But the Coulomb production in the experiments with unstable primary particles (pions, kaons, hyperons) opens the unique possibility to study the photoproduction reactions with these unstable `targetsa.


231

3. The experiments with the SPHINX setup The search for exotic baryons with hidden strangeness was performed in the experiments on the proton beam of the 70 GeV IHEP accelerator with the SPHINX spectrometer. The SPHINX setup includes the following basic components: 1. A wide aperture magnetic spectrometer with proportional wire chambers, drift chambers, drift tubes, scintillator hodoscopes. 2. Multichannel c-spectrometer with lead}glass Cherenkov total absorption counters. 3. A system of gas Cherenkov detectors for identi"cation of secondary charged particles (including a RICH detector with photomatrix equipped with 736 small phototubes; this is the "rst counter of this type used in the experiments). 4. Trigger electronics, data acquisition system and on-line control system. The SPHINX spectrometer works in the proton beam with energy E "70 GeV and intensity N I+(2}4);10p/cycle. The measurements were performed with a polyethylene target to optimize the acceptance, sensitivity and secondary photon losses. The "rst version of the SPHINX setup was described in [12]. The next version of this setup after partial modi"cation (with a new c-spectrometer and with better conditions for K and R identi"cation) was discussed in [21]. To separate di!erent exclusive reactions, a complete kinematical reconstruction of events was performed by taking into account information from the tracking detectors, from the magnetic spectrometer, from the c-spectrometer, and from all Cherenkov counters of the SPHINX setup. At the "nal stage of this reconstruction procedure, the reactions under study were identi"ed by examining the e!ective-mass spectra for subsystems of secondary particles. Several photon-induced di!ractive production processes were studied in the experiments of the SPHINX Collaboration [12}25,27,28]: p#NP[pK>K\]#N ,

(11)

P[p ]#N vK>K\ P[K(1520)K>]#N , vK\p

(12)

(13)

P[RH(1385)K>]#N vKn P[RH(1385)K>]#N#(neutrals) , vKn P[RK>]#N , vKc P[R> K]#N vpnvn>n\

(14)

(15) (16) (17)

232


P[pg]#N, vn>n\n

(18)

P[pg]#N, vn>n\gPn>n\2c

(19)

P[pu]#N, vn>n\n P[pn>n\]#N, P[D>>p\]#N , vpp>

(20) (21) (22)

and several other processes. Here N is nucleon or C nucleus for the coherent processes. The separation of coherent di!ractive processes is obtained by studying their dN/dP distributions, as 2 is shown in Fig. 2.

4. Previous data on the coherent di4ractive reactions pⴙCP[R 0Kⴙ ]ⴙC and pⴙCP[R*(1385)0Kⴙ ]ⴙC One of the major results obtained with the SPHINX setup was the study of RK> system produced in di!ractive process (16). These data were obtained in two di!erent runs on the SPHINX facility: (a) the "rst run with the old version of this setup (òld runa, [12,17}20]); (b) the second run with partially upgraded SPHINX apparatus (`new runa [21}23]). As a result of this upgrading the detection e$ciency and purity of K and R events were signi"cantly increased. The main results of these measurements can be summarized as follows: (1) Old [16}20] and new [21}23] data from coherent di!ractive reaction (16) were obtained under di!erent experimental conditions, with a signi"cantly modi"ed apparatus, with di!erent background and systematics. Nevertheless, the RK> invariant mass spectra from both runs are in a good agreement which makes them more reliable. (2) The combined mass spectrum M(RK>) for coherent reaction (16) from the old and new data (with P (0.1 GeV) is presented in Fig. 5. This spectrum is dominated by the X(2000) peak with 2 parameters in Table 1. (3) There is also some near threshold structure in this M(RK>) spectrum in the region of &1800 MeV (see Fig. 5 and Table 1). Such a shape of the RK> mass spectrum (with an additional structure near the threshold) proves that the X(2000) peak cannot be explained by a non-resonant Deck-type di!ractive singularity. Therefore, most likely this peak has a resonant nature. A strong in#uence of P cut for the production of this X(1810) state was established: this 2 structure is produced only at very small P (:0.01}0.02 GeV) } see below. 2


233

Fig. 5. Combined mass spectrum M(RK>) for coherent di!ractive reaction (16) in old and new runs on the SPHINX setup (P (0.1 GeV). The parameters of X(2000) peak in this spectrum are: M"1997$7 MeV; C"91$17 MeV. 2

We have also some data in studying the RH(1385)K> system in reaction (14), which were obtained only in the old run (data on this reaction from the new run are now in process of analysis). Coherent events of (14) were singled out in the analysis of dN/dP distribution as a strong forward 2 peak with the slope b950 GeV\. In order to reduce the noncoherent background and to obtain the R(1385)K> mass spectrum for the `purea coherent production reaction on carbon nuclei a `tighta requirement P (0.02 GeV was imposed and the mass spectra of R(1385)K> for the 2 coherent events of (14) were obtained (see, for example, Fig. 6). In these spectra some very narrow structure X(2050) was observed. The "ts of the spectra with Breit}Wigner peaks and polynomial smooth background were carried out, and the average values for the main parameters of X(2050) structure are presented in Table 1. Certainly, one needs further con"rmation of the existence of X(2050) in the new data with increased statistics. Up to now it is impossible also to exclude completely the feasibility for X(2000) and X(2050) to be in fact two di!erent decay modes of the same state. In studying coherent reactions (21) p#CP[pp>p\]#C and (22) p#CP[D(1232)>>p\] #C under the same kinematical conditions as of processes (14) and (16) a search for other decays channels of the X(2000) and X(2050) baryons was performed [18,19]. No peaks in 2 GeV mass range were observed in the mass spectra of pp>p\ and D(1232)>>p\ systems produced in reactions (21) and (22), respectively. Lower limits on the corresponding decay branching ratios R (see (5))

234


Table 1 The main results of the previous SPHINX data for coherent di!ractive reactions p#CP[RK>]#C and p#CP[RH(1385)K>]#C 1. Coherent di!ractive production reaction p#CP[RK>]#C with coherent cut P (0.1 GeV was studied in the 2 old and new runs. The combined mass spectrum M(RK>) is presented in Fig. 5. This spectrum is dominated by X(2000) state with parameters M"1997$7 MeV C"91$17 MeV statistical significance of the peak is 7 SD 2. There are also some near threshold structure X(1810), which is produced only in the region of very small P (:0.01}0.02 GeV). The parameters of this peak are 2 M"1812$7 MeV C"56$16 MeV 3. Coherent di!ractive production reaction p#CP[RH(1385)K>]#C with tight coherent cut P (0.02 GeV was 2 studied in the old run ([12,17}20]). The mass spectra of M[RH(1385)K>] are in Fig. 6. The peak was observed in these spectra with average value of parameters M"2052$6 MeV C"35> MeV \ (with the account of the apparatus mass resolution); statistical CL of the peak55 SD 4. The data of (14) and (16) were analyzed together with the data from (21) and (22) to obtain the branching ratios of di!erent decay channels (with strange particles and without strange particle in "nal state). The lower limits of the ratios were obtained from this comparative analysis (with 95% CL): BR+X(2050)>P[RH(1385)K]>, '1.7 R " BR+X(2050)>P[D(1232)n]>, BR+X(2050)>P[RH(1385)K]>, R " '2.6 BR+X(2050)>Ppn>n\, BR+X(2050)>PRH(1385)K>, R " '0.86 BR+X(2050)>Ppn>n\, BR+X(2000)>P[RK]>, R " '0.83 BR+X(2000)>P[D(1232)n]>, BR+X(2000)>P[RK]>, R " '7.8 BR+X(2000)>Ppn>n\, BR+X(2000)>PRK>, '2.6. R " BR+X(2000)>Ppn>n\,


235

Fig. 6. Invariant mass spectra M[RH(1385)K>] in the coherent reaction (14) at tight transverse-momentum cut P (0.02 GeV for various bin widths: (a) *M"10 MeV; (b) *M"30 MeV. The spectra are "tted to the sum of 2 a smooth polinomial background and X(2050) Breit}Wigner peak. The parameters of X(2050) peak are: (a) M"2053$4 MeV, C"40$15 MeV; (b) M"2053$5 MeV, C"35$16 MeV.

were obtained from this comparative analysis: R[X(2000); X(2050)]91}10 (95% CL)

(23)

(more details are in Table 1). The isotopic relations between the decay amplitudes of I" particles were assumed in these calculations (the X(2000) and X(2050) states belong to isodoublets since they are produced in a di!ractive dissociation of proton). In accordance with these relations BR[X> PRK>]"BR[X> P(RK)>] , ' '

(24)

BR[X> PD>>p\]"BR[X> P(*p)>] . ' '

(25)

The ratios R }R of the widths of the X(2000) and X(2050) decays into strange and nonstrange particles are much larger than those for ordinary (qqq)-isobars [18,33]. A narrow width of the X(2000) and X(2050) baryon states as well as anomalously large branching ratios for their decay channels with strange particle emission (large values of R) are the reasons to consider these states as serious candidates for cryptoexotic baryons with a hidden strangeness "uudss 2.

236


5. New analysis of the data for reaction pⴙNP[R 0Kⴙ ]ⴙN In what follows we present the results of a new analysis of the data obtained in the run with partially upgraded SPHINX spectrometer where conditions for K and R separation were greatly improved as compared with an old version of this setup. The key element of the new analysis lays in a detailed study of the RPK#c decay separation, which makes it possible to reach the reliable identi"cation of this decay and reaction (16) with the increased e$ciency in comparison with the previous analysis of Bezzubor et al. [21]. In this new analysis the data for (16) were studied with di!erent criteria for RPK#c separation (with larger e$ciency and larger background or with the reduced background at the price of lower e$ciency). We will designate these di!erent criteria for photon separation as soft, intermediate and strong photon cuts (the details will be presented in [45]). Reaction (16) was studied in di!erent kinematical regions. It was found that improved background conditions were important for the investigation of the region of small mass M(RK>) and very small transverse momenta. The e!ective mass spectra M(RK>) in p#NP[RK>]#N for all P are presented in Fig. 7. 2 The peak of X(2000) baryon state with M"1986 MeV and C"98$20 MeV is seen very clearly in these spectra with a very good statistical signi"cance. Thus, the reaction p#NP X(2000)#N ,

(26)

vRK> is well separated in the SPHINX data. The cross section for X(2000) production in (26) is p[p#NPX(2000)#N] ) BR[X(20000)PRK>]"95$20 nb/nucleon

(27)

(with respect to one nucleon under the assumption of pJA, e.g. for the e!ective number of nucleons in carbon nucleus equal to 5.24). The parameters of X(2000) are not sensitive to the di!erent photon cuts, as is seen from Table 2. The dN/dP distribution for reaction (26) is shown in 2 Fig. 8. From this distribution the coherent di!ractive production reaction on carbon nuclei is identi"ed as a di!raction peak with the slope bK63$10 GeV\. The cross section for coherent reaction is determined as p[p#CPX(2000)>#C]

! "260$60 nb/C nucleus .

) BR[X(2000)>PRK>] (28)

We must bear in mind that it is more convenient to use other relations for cross sections: p[p#NPX(2000)>#N]BR[X(2000)>P(RK)>]"285$60 nb/nucleon ,

(29)

p[p#CPX(2000)>#C]BR[X(2000)>P(RK)>]"780$180 nb/nucleus ,

(30)

which were obtained from Eqs. (27) and (28) using branching ratio (24). The errors in the cross sections of Eqs. (27)}(30) are statistical only. Additional systematic errors are K$20% due to uncertainties in the cuts, in Monte Carlo e$ciency calculations and in the absolute normalization.


237

Fig. 7. Invariant mass spectra M(RK>) in the di!ractive reaction p#NP[RK>]#N for all P (with soft photon 2 cut): (a) measured mass spectrum; (b) the same mass spectrum weighted with the e$ciency of the setup. Parameters of X(2000) peak are in Table 2.

In the mass spectra M(RK>) in Fig. 7 there is only a slight indication for X(1810) structure which was observed earlier in the study of coherent reaction (16) } see Fig. 5 and [21]. This di!erence is caused by a large background in this region for the events in Fig. 7 (all P , soft photon cut). To clarify 2 the situation in this new analysis we investigated also the M(RK>) mass spectra for coherent reaction (16), e.g. for P (0.1 GeV. In these mass spectra not only the X(2000) peak is observed, but 2 the X(1810) structure as well. These spectra (see [45]) are compatible with the data in Fig. 5. The yield of the X(1810) as function of P is shown in Fig. 9. From this "gure it is clear that 2 X(1810) is produced only in a very small P region (P (0.01}0.02 GeV). For P (0.01 GeV the 2 2 2 M(RK>) mass spectrum demonstrates a very sharp X(1810) signal (see Fig. 10) with the parameters of the peak

X(1810)PRK>

M"1807$7 MeV ,

C"62$19 MeV

(31)

which is in a good agreement with the previous data of Table 1. The cross section for coherent X(1810) production is BR[X(1810)>PRK>] p[p#CPX(1810)>#C]" 2 . %4 "215$45 nb/C nucleus .

(32)

238


Table 2 Data on M(RK>) in reaction p#NP[RK>]#N, RPKc with di!erent photon cuts (for all P ) 2 Photon cut

Soft

N events in X(2000) peak Correction factor for photon e$ciency Parameters of X(2000) M (MeV)

Weighted spectrum Measured spectrum

C (MeV)

Weighted spectrum Measured spectrum

p[p#NPX(2000)#N] BR[X(2000)PRK>] (nb/nucleon)

Average values

430$89 1.0

1986$6 1988$5

Intermediate 301$71 1.4

1991$8 1994$7

Strong 190$47 2.25

1988$6 1990$6

98$20 84$20

96$26 94$21

68$21 68$20

100$19

93$25

91$21

1M2 MeV 1C2 MeV 1p[p#NPX(2000)#N]2 BR[X(2000)PRK>] nb/nucleon

1989$6 91$20 95$20 (statist.) $20 (system)

1p[p#CPX(2000)#C]2 BR[X(2000)PRK>] nb/C nucleus

285$60 (statist.) $60 (system)

The additional systematic error for this value is $30%. It increased as compared with the same errors in Eqs. (27)}(30) due to the uncertainty in the evaluation of P smearing in the region of 2 P (0.01 GeV, which is more sensitive to P resolution. 2 2 It is possible also to demonstrate the coherent di!ractive X(2000) production in the clearest way by using the `restricted coherent regiona 0.02(P (0.1 GeV (see Fig. 11) where there is no 2 in#uence of X(1810) structure. To explain the unusual properties of X(1810) state in a very small P region the hypothesis of the 2 electromagnetic production of this state in the Coulomb "eld of carbon nucleus was proposed earlier [46]. It is possible to estimate the cross section for the Coulomb X(1810) production from (9) and (10): p[p#CPX(1810)>#C]" 2 . %4_ ! "(2J #1)+C[X(1810)>Pp#c][MeV],2.8;10\ cm/C nucleus V 55.6;10\ cm+C[X(1810)>Pp#c][MeV], (J 5 is the spin of X(1810)). V

(33)


239

Fig. 8. dN/dP distribution for the di!ractive production reaction p#NPX(2000)#N. The distribution is "tted 2 in the form dN/dP "a exp(!b P )#a exp(!b P ) with the slope parameters b "63$10 GeV\; 2 2 2 b "5.8$0.6 GeV\.

Fig. 9. The P dependence for the X(1810) structure production in the coherent reaction p#CPX(1810)#C. 2

240


Fig. 10. Invariant mass spectra M(RK>) in the coherent di!ractive production reaction p#CP[RK>]#C in the region of very small P (0.01 GeV (with strong photon cut) weighted with the e$ciency of the setup. 2

Let us compare this Coulomb hypothesis prediction with the experimental value p[p#CPX(1810)>#C]" 2 9645 nb/C nucleus . (34) . %4 To obtain (34) we assumed in (16) that X(1810) is isodoublet, and then we use from (24) the branching BR(X>PRK>): (here K means that BR[X>P(RK)>]K1, i.e. this decay is dominating). If the value of radiative width C[X(1810)Pp#c] is around 0.1}0.3 MeV and the branching BR[X(1810)>P(RK)>] is signi"cant, then the experimental data for cross section of the coherent X(1810) production (34) can be in agreement with the Coulomb mechanism prediction (33). It seems that such value of radiative width is quite reasonable. For example, the radiative width for D(1232) isobar is C[D(1232)>Pp#c]K0.7 MeV. The value of radiative width depends on the amplitude of this process A and on kinematical factor: C""A" ) (P )J> (P is the momentum of A A photon in the rest frame of the decay baryon and l is orbital momentum). For X(1810)Pp#c decay the kinematical factor may be by an order of magnitude larger than for D(1232)>Pp#c because of the large mass of X(1810) baryon. Certainly, the predictions for amplitude A are quite speculative. But if, for example, (X1810) is the state with hidden strangeness "qqqss 2, then the amplitude A might be not very small due to a possible VDM decay mechanism


241

Fig. 11. Weighted invariant mass spectrum M(RK>) for the reaction p#CP[RK>]#C in the `restricted coherent regiona 0.02(P (0.1 GeV (with intermediate photon cut). 2

(qqqss )P(qqq)# P(qqq)#c. Thus, it seems that the experimental data for the coherent production of X(1810) (34) is not in contradiction with the Coulomb production hypothesis. It is possible that the candidate state X(2050)PRH(1385)K> which was observed in coherent reaction (14) in the region of very small transverse momenta (P (0.02 GeV) is also produced not 2 by di!ractive, but by the electromagnetic Coulomb production mechanism [46]. The feasibility to separate the Coulomb production processes in the coherent proton reactions at E "70 GeV on the carbon target in the measurements with the SPHINX setup was demonstrated N recently by observation of the Coulomb production of D(1232)> isobar with I" in the reaction p#CPD(1232)>#C (35) (see [46]).

6. Reliability of X(2000) baryon state The data on X(2000) baryon state with unusual dynamical properties (large decay branching with strange particle emission, limited decay width) were obtained with a good statistical signi"cance in di!erent SPHINX runs with widely di!erent experimental conditions and for several

242


kinematical regions of reaction (16). The average values of the mass and width of X(2000) state are

X(2000)PRK>

M"1989$6 MeV ,

C"91$20 MeV .

(36)

Due to its anomalous properties X(2000) state can be considered as a serious candidate for pentaquark exotic baryon with hidden strangeness: "X(2000)2""uudss 2. Recently some new additional data have been obtained which are in favor of the reality of X(2000) state. (a) In the experiment of the SPHINX Collaboration reaction (17) was studied. The data for the e!ective mass spectrum M(R>K) in this reaction are presented in Fig. 12. In spite of limited statistics the X(2000) peak and the indication for X(1810) structure are seen in this mass spectrum and are quite compatible with the data for reaction (16). (b) In the experiment on the SELEX(E781) spectrometer with the R\ hyperon beam of the Fermilab Tevatron the di!ractive production reaction R\#NP[R\K>K\]#N

Fig. 12. The e!ective mass spectrum M(R>K) in reaction (17) for P (0.1 GeV. 2

(37)


243

Fig. 13. Study of M(R\K>) in the reaction R\#NP[R\K>K\]#N in the SELEX experiment. Here the spectrum M(R\K\) with open exotic quantum number used for subtraction of nonresonance background in M(R\K>) after some normalization. One presents in this "gure M(R\K>)!0.95M(R\K\) (here 0.95 } normalization factor): (a) all the events; (b) after subtraction of the events in band to suppress the in#uence of the reaction R\#NP[R\ ]#N.

was studied at the beam momentum PR\K600 GeV. In the invariant mass spectrum M(R\K>) for this reaction a peak with parameters M"1962$12 MeV and C"96$32 MeV was observed (Fig. 13). The parameters of this structure are very near to the parameters of X(2000)PRK> state which was observed in the experiments on the SPHINX spectrometer. It seems that the real existence of X(2000) baryon is supported by the data of another experiment and in another process. Preliminary results of studying reactions (17) and (37) were discussed in the talks at the last conferences [24,25,47] and are now under detailed study.

7. Nonperipheral processes As was discussed above (Section 2.3) the search for new baryons in proton-induced di!ractivelike reactions in the nonperipheral domain, with P '0.3}0.5 GeV, seems to be quite promising. 2 Here we present the very "rst results of these searches in the invariant mass spectra of the RK> and pg systems produced in the reactions p#NP[RK>]#N (16) and p#NP[pg]#N (18) for P '0.3 GeV (see [20,21]). Combined data on reaction (16) from the old and new runs are 2 shown in Fig. 14a. The data from the old run for reaction (18) are shown in Fig. 14b. Despite limited statistics, a structure with mass M+2350 MeV and width C&60 MeV can be clearly seen in these two mass spectra. They require a further study in future experiments with larger statistics. The same statement seems true for the intriguing data on the invariant mass spectrum M(pg) for reaction (19) in the region P '0.3 GeV (see Fig. 14c). It must be stressed that reaction (19) is the 2 only one (among more than a dozen of other di!ractive-like reactions studied in the SPHINX experiments) in which a strong coherent production on carbon nuclei was not observed (the absence of the forward peak in dN/dP distribution with the slope value b950 GeV\; the slope 2 for forward cone in Eq. (19) is b&6.5 GeV\ } see [20,22]).

244


Fig. 14. (a) Invariant mass spectrum of the RK> system produced in the reaction p#NP[RK>]#N (16) for P '0.3 GeV (combined data from the old and new runs). (b) Invariant mass spectrum of the pg system produced in the 2 reaction p#NP[pg]#N (18) for P '0.3 GeV (old data, see [20]). A narrow structure with M&2350 MeV and 2 C&60 MeV in the nonperipheral region of reactions (16) and (18) is seen in these mass spectra. (c) Invariant mass spectrum M(pg) for the reaction p#NP[pg]#N (19) in the region P '0.3 GeV (old run). This spectrum is 2 dominated by a threshold structure with M&2000 MeV and C&100 MeV.

8. OZI rule and di4ractive proton reactions The selection rule for connected and disconnected quark diagrams, which is referred to as the OZI rule, has been known for a long time [48}50]. According to this rule, processes involving the annihilation or creation of a quark}antiquark pair entering into the composition of the same hadron are forbidden or, strictly speaking, strongly suppressed. The OZI rule can be illustrated by results of studying the charge exchange reactions p\#pP #n

(38)

(OZI forbidden process) and p\#pPu#n (OZI allowed process). As is well known, the quark structure of vector and u mesons has the form 1 " 2"!cos a ) "ss 2!sin a ) " (uu #ddM ) , (2 1 (uu #ddM ) . "u2"!sin a ) "ss 2#cos a ) " (2

(39)

(40)


245

Here a"*0 "0 !0 ; 0 is the mixing angle in the vector meson nonet and 0"arctg (1/(2)" 4 4 4 4 4 35.33 is ideal mixing angle. For ideal mixing a"0 and " 2"ss ; "u2""(1/(2)(uu #ddM ). From several data the value of "a" for vector nonet is "a"K(3}43). Thus, in accordance with the OZI rule, the ratio of the cross sections for reactions (38) and (39) is R( /u) "p( )/p(u)"tg a&(3}5);10\ -8'

(41)

in a very good agreement with several experimental data. For example, the recent result of LEPTON-F Collaboration for charge exchange reactions (38), (39) at P \"32.5 GeV [26] is: p R( /u) "(2.9$0.9);10\ . _p

(42)

Recent years have seen revived interest in the OZI rule. First, intensive searches for exotic hadrons (see the surveys in [1}7] and references therein) are greatly facilitated by choosing production processes in which the formation of conventional particles is suppressed by the OZI rule [51}53,34]. Since this rule may be signi"cantly violated for exotic hadrons because of their complex color structure, signals from exotic and cryptoexotic particles are expected to bene"t from the more favorable background conditions in OZI-suppressed production processes. While the p and o decays of isovector mesons with conventional quark structure (uu !ddM )/(2 are suppressed by the OZI rule (the corresponding probabilities are reduced by more than two orders of magnitude), the same decay channels may prove much more probable for exotic multiquark mesons with hidden strangeness [(uu !ddM )ss /(2] and hybrid states like (uu !ddM )g/(2. Furthermore, unexpected results obtained in polarization measurements for deep-inelastic lepton scattering on nucleons give rise to the well-known problem of nucleon-spin crisis [54,55]. To explain this phenomenon, it was hypothesized that nucleons involved an enhanced quark component with hidden strangeness (direct strangeness in nucleons). This may induce signi"cant violations of the OZI rule in nucleon processes [56,57]. Strong violations of the OZI rule were indeed observed in the relative and u yields from some channels of p annihilation (reactions p pP p, p pPup, p pP c, and p pPuc [57,58]). The above arguments suggest that the OZI rule should be further tested in various production and decay processes and "rst of all in the nucleon reactions. In the experiments of the SPHINX collaboration this test was performed in studying di!ractive production reactions with and u mesons (12) and (20) [27,28]. It was observed that the average value of the e!ective ratio of yields of and u mesons in these reactions is 1R( /u)2 "(4}7);10\ . _N

(43)

Thus, the strong violation of the OZI rule (by more than an order of magnitude) is observed in proton di!ractive production reactions. The intriguing large violation of the OZI rule in di!erent nucleon reactions and, particularly, in proton di!ractive production processes may suggest an enhanced component with hidden strangeness in a quark structure of nucleons (the model of direct strangeness in nucleon). The OZI rule in proton reactions and the possible existence of direct strangeness are illustrated by diagrams in Fig. 15.

246


Fig. 15. production in proton}nucleon or proton}nucleus di!ractive reactions. (a) Disconnected quark diagram (suppressed by OZI rule). (b) Connected diagram for production in the model with some small direct component of ss in proton wave function. In this model one observes the apparent OZI rule violation due to a small exotic ss component (model with direct strangeness in nucleon).

9. Conclusion The extensive research program of studying di!ractive production in E "70 GeV proton N reactions is being carried out in experiments with the SPHINX setup [2,12}28,45}47]. This program is aimed primarily at searches for cryptoexotic baryons with hidden strangeness ("uudss 2). Only a part of these experiments is discussed here. The most important results of these searches were obtained in studies of the hyperon-kaon systems produced in proton di!ractive dissociation processes and "rst of all in reaction p#NPRK>#N (16). New data for this di!ractive production reaction were obtained with the partially upgraded SPHINX detector (with new c-spectrometer and with better possibilities to detect KPpn\ and RPKc decays). New data are in a good agreement with previous SPHINX results on the invariant mass spectrum M(RK>) in this reaction. A strong X (2000) peak with M"1989$6 GeV and C"91$20 MeV together with a narrow threshold structure (with M&1810 MeV and C&60 MeV) are clearly seen in the (RK>) invariant mass spectra. The latter structure is produced at very small transverse momenta, P 10.01!0.02 2 GeV. The unusual properties of the X (2000) baryon state (narrow decay width, anomalously large branching ratio for the decays with strange particle emission) make this state a serious candidate for a cryptoexotic pentaquark baryon with hidden strangeness "qqqss 2. Preliminary data for "RK2 states in other reactions (17) and (37) con"rm the real existence of X (2000) baryon. The OZI selection rule was investigated by comparing the cross sections for pion-induced charge-exchange reactions n\#pP #n and n\#pPu#n at P \"32.5 GeV, as well L as the cross sections for the proton-induced di!ractive reactions p#NP[p ]#N and p#NP[pu]#N at E "70 GeV, in experiments with the LEPTON-F and SPHINX spectromN eters. It was shown that in pion reactions the ratio R( /u)K(3$1);10\ is in good agreement with the naive-quark model and with the OZI rule prediction (R( /u) "tgDh K4;10\). -8' 4


247

At the same time, this ratio in proton reactions is 1R( /u)2K(4}7);10\. That is, a strong violation of the OZI rule is observed in proton}nucleon interactions. The large violation of OZI rule in proton di!ractive reactions may be in favor of some enhanced hidden strange component in nucleons (the model with direct strangeness). Several other interested phenomena were observed in the nonperipheral domain (with P 90.3 GeV), in the study of di!erent reactions (for example, reactions (14), (18) and (19)). But 2 they need experimental veri"cation with much better statistics. Now the "rst stage of the experimental program on the SPHINX setup has been completed. In the last years the SPHINX spectrometer was totally upgrated. The luminosity and the data taking rate were greatly increased. In the recent runs with this upgrated setup we obtained large statistics which is now under data analysis. In the near future, we expect to increase statistics by an order of magnitude for the processes discussed above and for some other proton reactions. Acknowledgements It is a great pleasure and a great honor for me to participate in this volume dedicated to the 70th anniversary of the birth of the outstanding scientist Lev Okun. I am very lucky to share with him many years of close contacts and a real friendship. I am deeply obliged to Lev for many illuminating discussions which helped me tremendously in many ways and, in particular, in the studies which are covered in this paper. Even my original interest in exotic hadrons was initiated many years ago by his brilliant lectures at physics schools in IHEP, ITEP and MEPI (see, for example, Ref. [59]). I wish Lev many further productive and happy years in his life and work. This work is partially supported by Russian Foundation for Basic Research (grant 99-02-18251). References [1] L.G. Landsberg, Surv. High Energy Phys. 6 (1992) 257. [2] L.G. Landsberg, Yad. Fiz. 57 (1994) 47 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 42]; USP. Fiz. Nauk. 164 (1994) 1129 [Physics-Uspekhi (Engl. Transl.) 37 (1994) 1043]. [3] K. Peters, Nucl. Phys. A 558 (1993) 92. [4] C.B. Dover, Nucl. Phys. A 558 (1993) 721. [5] C. Amsler, Rapporter talk, Proc. Conf. on High Energy Phys. (ICHER), Glasgow, Scotland, July 1994. [6] F.E. Close, preprint RAL-87-072, Chilton, 1987. [7] P. Blum, Int. J. Mod. Phys. 11 (1996) 3003. [8] T. Hirose et al., Nuov. Cim. 50 (1979) 120; C. Fucunage et al., Nuov. Cim. 58 (1980) 199. [9] J. Amizzadeh et al., Phys. Lett. B 89 (1979) 120. [10] A.N. Aleev et al., Z. Phys. C 25 (1984) 205. [11] V.M. Karnaukhov et al., Phys. Lett. B 281 (1992) 148. [12] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fiz. 57 (1994) 241 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 227]; M.Ya. Balatz et al. (SPHINX Collab.), Z. Phys. C 61 (1994) 220. [13] M.Ya. Balatz et al. (SPHINX Collab.), Z. Phys. C 61 (1994) 399. [14] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fiz. 57 (1994) 253 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 238]. [15] L.G. Landsberg et al. (SPHINX Collab.), Nuov. Cim. A 107 (1994) 2441. [16] V.F. Kurshetsov, L.G. Landsberg, Yad. Fiz. 57 (1994) 2030 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 1954]. [17] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fiz. 57 (1994) 1449 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 1376]. [18] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fizz. 58 (1995) 1426 [Phys. At. Nucl. (Engl. Transl.) 58 (1995) 1342.

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Hamiltonian for Reggeon interactions in QCD L.N. Lipatov* Petersburg Nuclear Physics Institute, Gatchina, 188350, St. Petersburg, Russia

Abstract It is shown, that the interaction of the reggeized gluons in the leading logarithmic approximation of the multicolour QCD has a number of remarkable properties including the duality symmetry. The duality relation for the Odderon wave function takes a form of the one-dimensional SchroK dinger equation. It gives a possibility to express the Odderon Hamiltonian as a function of its integrals of motion. 1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.!t; 14.70.Dj

1. Introduction In QCD the scattering amplitudes at high energies (s are obtained by calculating and summing all large contributions (g ln(s))L, where g is the coupling constant. In the leading logarithmic approximation (LLA) the gluon is reggeized and the BFKL Pomeron is a compound state of two reggeized gluons [1}4]. Next-to-leading corrections to the BFKL equation were recently calculated [5], which gives a possibility to "nd its region of applicability. In the framework of the optimal renormalization approach one can verify that the MoK bius invariance is valid approximately even after taking into account next-to-leading terms [6]. The power asymptotics sH of scattering amplitudes is governed by the j-plane singularities of the t-channel partial waves f (t). The position of these singularities u "j !1 for the Feynman H diagrams with n reggeized gluons in the t-channel is proportional to the ground state energy E KK u "!(gN /8p) E (1) KK A KK

* Tel.: #7-812-2949196; fax: #7-812-4131963. E-mail address: [email protected] (L.N. Lipatov) Supported by the CRDF and INTAS-RFBR grants: RP1-253, 95-0311. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 6 2 - 9

250

L.N. Lipatov / Physics Reports 320 (1999) 249}260

of a SchroK dinger-like equation [7}12]: (2) E f "Hf . KK KK KK Its eigenfunction f (q , q ,2, q ; q ) describing the composite state W (q ) of the reggeized L KK KK gluons depends on their impact parameters q , q ,2, q . In LLA it belongs to the basic series of L unitary representations of the MoK bius group [9}12] o P(a o #b)/(c o #d) , (3) I I I where o "x #iy , oH"x !iy complex coordinates of gluons and a, b, c, d are arbitrary comI I I I I I plex parameters. For this series the conformal weights m"1/2#il#n/2, m "1/2#il!n/2

(4)

are expressed in terms of the anomalous dimension c"1#2il and the integer conformal spin n of a local operator O (q ). They are related to the eigenvalues KK Mf "m(m!1) f , MHf "m (m !1) f (5) KK KK KK KK of the Casimir operators M and MH:

L M" M? " 2 M? M?"! o R R , MH"(M)H . (6) I P Q PQ P Q I PQ PQ Here M? are the MoK bius group generators I M"o R , M\"R , M>"!oR (7) I I I I I I I I and R "R/(Ro ). I I In the particular case of the Odderon, being a compound state of three reggeized gluons with the charge parity C"!1 and the signature P "!1, the eigenvalue u is related to the highH KK energy behaviour of the di!erence of the total cross-sections p and p for interactions of particles NN NN p and antiparticles p with a target: KK . p !p &sS (8) NN NN The Hamiltonian H in the multicolour QCD has the property of the holomorphic separability [9}12]:

H"(h#hH), [h, hH]"0 , where the holomorphic and anti-holomorphic Hamiltonians L L h" h , hH" hH II> II> I I are expressed in terms of the BFKL operator [9}12]:

(9)

(10)

"log(p )#log(p )#(1/p )log(o )p #(1/p ) log(o )p #2 c . (11) II> I I> I II> I I> II> I> Here o "o !o , p "iR/(Ro ), pH"iR/(RoH) , and c"!t(1) is the Euler constant. II> I I> I I I I Owing to the holomorphic separability of H, the wave function f (q , q ,2, q ; q ) has the L KK property of the holomorphic factorization [9}12]: h

f (q , q ,2, q ; q )" c f P(o , o ,2, o ; o ) f J (oH, oH,2, oH; oH) , L KK L PJ K L K PJ

(12)


251

where r and l enumerate degenerate solutions of the SchroK dinger equations in the holomorphic and anti-holomorphic sub-spaces: (13) e f "h f , e f "hH f , E "(e #e ) . K KK K K K K K K K Similarly in the case of two-dimensional conformal "eld theories, the coe$cients c are "xed by the PJ single-valuedness condition for the function f (q , q ,2, q ; q ) in the two-dimensional q-space. L KK For the holomorphic hamiltonian h there is a family +q , of mutually commuting integrals of P motion [9}12]: [q , q ]"0, [q , h]"0 . (14) P Q P The generating function for these integrals of motion coincides with the transfer matrix ¹ for the XXX model [9}12]: L ¹(u)"tr (¸ (u)¸ (u)2¸ (u))" uL\P q , L P P where the ¸-operators are

u#o p p I I I . ¸ (u)" I !o p u!o p I I I I The transfer matrix is the trace of the monodromy matrix t(u) [13,14] ¹(u)"tr (t(u)), t(u)"¸ (u)¸ (u)2¸ (u) . L It can be veri"ed that t(u) satis"es the Yang}Baxter equation [9}14] tPQ(u) tQ(v) lPP(v!u)"lQQ(v!u) tQ(v) tQ(u) , P PP QQ P P where l(w) is the ¸-operator for the well-known Heisenberg spin model

(15)

(16)

(17)

(18)

(19) lQQ(w)"w dQ dQ#i dQ dQ . QQ Q Q Q Q To "nd a representation of the Yang}Baxter commutation relations, the algebraic Bethe anzatz can be used [13,14]. It is reduced to the solution of the Baxter equation [13}16]. Up to now the Baxter equation was solved only for the case of the BFKL Pomeron (n"2). This is the reason why we use below another approach, based on the diagonalization of the transfer matrix.

2. Duality property of Reggeon interactions The di!erential operators q and the Hamiltonian h are invariant under the cyclic permutation of P gluon indices iPi#1 (i"1, 2,2, n), corresponding to the Bose symmetry of the Reggeon wave function at N PR. It is remarkable that above operators are invariant also under the more A general canonical transformation [17]: o Pp Po , G\G G GG> combined with reversing the order of the operator multiplication.

(20)

252


This duality symmetry is realized as an unitary transformation only for the vanishing total momentum: L p" p "0 . (21) P P The wave function t of the composite state with p"0 can be written in terms of the KK eigenfunction f of a commuting set of the operators q and qH for k"1, 2,2, n as follows: I I KK do f (q , q , , q ; q ) . t (q , q ,2, q )" (22) KK L 2 p KK 2 L

Taking into account the hermicity properties of the total Hamiltonian [9}12] L L L L H>" "o "\ H "o "" "p " H "p "\ , (23) II> II> I I I I I I the solution t> of the complex-conjugated SchroK dinger equation for p"0 can be expressed in KK terms of t as follows: KK L t> (q , q ,2, q )" "o "\(t (q , q ,2, q ))H . (24) KK L II> KK L I Because t is also an eigenfunction of the integrals of motion A"q and AH with their KK L eigenvalues j and jH"j [9}12]: K K K (25) A t "j t , AH t "j t , A"o 2o p 2p , K KK KK K KK L L KK one can verify that the duality symmetry takes the form of the following integral equation for t [17]: KK L\ do L e qII> qI H I\I t (q ,2, q )""j " 2L t (q , , q ) . (26) KK L K 2p "o " KK 2 L I I II> In the case of the Odderon the conformal invariance "xes the solution of the SchroK dinger equation [9}12]

f (q , q , q ; q )"(o o o /o o o )K(oH oH oH /oH oH oH)K f (x) (27) KK KK up to an arbitrary function f (x) of one complex variable x being the anharmonic ratio of four KK coordinates x"o o /o o . (28) Note that, owing to the Bose symmetry of the Odderon wave function, f (x) has certain KK modular properties [17]. The Odderon wave function t (q ) at q"0 can be written as KK GH (29) t (q )"(o /o o )K\(oH /oH oH )K \s (z), z"o /o , KK KK GH


253

where s

dx f (x) (x!z) K (xH!zH) K KK . (z)" KK 2p"x!z" x(1!x) xH(1!xH)

(30)

In fact this function is proportional to f (z): \K\K s (z)&(x(1!x))K\(xH(1!xH))K \ f (z) , (31) KK \K\K which is a realization of the linear dependence between two representations (m, m ) and (1!m, 1!m ). The corresponding reality property for the MoK bius group representations can be presented in the form of the integral relation s

dx (z)" (x!z)K\ (xH!zH)K \ s (x) KK \K\K 2p

. for an appropriate choice of phases of the functions s and s \K\K KK The duality equation for s (z) can be presented in the pseudo-di!erential form [17]: KK z(1!z) (iR)\K zH(1!zH) (iRH)\K u (z)""j " (u (z))H , \K\K KK \K\K where u (z)" (z(1!z))\K(zH(1!zH))\K s (z) . \K\K KK The normalization condition for the wave function

dx ""u """ "u (x)" KK "x(1!x)" KK

(32)

(33)

(34)

(35)

is compatible with the duality symmetry. For the holomorphic factors uK(x) the duality equations have the form [17]: a uK(x)"jKu\K(x), a u\K(x)"j\KuK(x) , K \K where

(36)

a "x(1!x) p>K . (37) K If we consider p as a coordinate and x!1/2 as a momentum, the duality equation for the most important case m"1/2 can be reduced to the SchroK dinger equation with the potential K, cK(j)"1 . (41) D I>K K I Due to this di!erential equation the coe$cients a, c and d satisfy certain recurrence relations. From the single-valuedness condition near x"0, we obtain for the total wave function the following representation: u (x, xH)"uK(x, j) uK (xH, jH)#c uK(x, j) uK (xH, jH) D D P P KK #c (uK(x, j) uK(xH, jH)#uK(x, j) uK (xH, jH))#(jP!j) . (42) Q P P Q The complex coe$cients c , c and the eigenvalues j are "xed from the conditions of the single-valuedness of f (q , q , q ; q ) at q "q (i"1, 2) and the Bose symmetry [18}22]. G KK From the duality equation we have [17] "c """j" . Another relation

(43)

Im c /c "Im (m\#m \) . (44) can be derived [17] if we shall take into account, that the complex conjugated representations u and u of the MoK bius group are related by the above discussed linear transformation. KK \K\K One can verify from the numerical results of Refs. [18}22] that both relations for c and c are ful"lled.

4. Conformal weight representation If we introduce the time-dependent pair hamiltonian h h

(t)"exp(i ¹(u) t)h exp(!i ¹(u) t) , II> II> in the total hamiltonian h we can substitute

(t) in the form II> (45)

h

Ph (t) (46) II> II> due to the commutativity of h and ¹(u). Due to its rapid oscillations at tPR each pair Hamiltonian is diagonalized in the representation, where the transfer matrix ¹(u) is diagonal: h

(R)"f (q( , q( , , q( ) . II> II> 2 L

(47)


255

In the case of the Odderon h (R) is a function of the total conformal momentum q and of II> the integral of motion q "A which can be written as follows: (48) A" i [M , M ]"i [M , M ]"i [M , M ] . In a general case of n reggeized gluons, one can use the Clebsch}Gordan approach, based on the construction of common eigenfunctions of the total momentum q with the eigen value m(m!1) and a set +MK , of the commuting sub-momenta, to "nd all operators M in the corresponding I II> representation. However to diagonalize h we should perform a unitary transformation to the representation, where ¹(u) is diagonal. Let us consider, for example, the interaction between particles 1 and 2. The transfer matrix, which should be diagonalized, can be written as follows: (49) ¹(u)"(u!L)d 2 (u)#(i u L![L, N ])d 2 (u) , L L where the di!erential operators d 2 (u) and d 2 (u) are independent of q and q . They are related L L to the monodromy matrix t 2 (u) for particles 3, 4,2, n as follows: L (50) d 2 (u)"tr t 2 (u), d 2 (u)"tr(rt 2 (u)), t 2 (u)"¸ (u)2¸ (u) L L L L L L and the matrix t 2 (u) satis"es the Yang}Baxter equations. L The operators L and N are constructed in terms of the MoK bius group generators of particles 1 and 2: L"M #M , N"M !M , MX"o R , M>"!oR , M\"R . I I I I I I I I They have the commutation relations, corresponding to the Lorentz algebra: [¸X, ¸!]"$¸!, [¸>, ¸\]"2¸!, [¸X, N!]"$N! , [¸>, N\]"2NX, [NX, N!]"$¸!, [N>, N\]"2¸X

(51)

(52)

and can be constructed in a closed form in the in"nite dimensional representation "o , M2 for the composite states with the coordinate o and the pair conformal weight M. Because of its MoK bius invariance, the transfer matrix ¹(u) after acting on f written as a K superposition of "o , M2 gives again a superposition of the states "o , M2, but with the coe$cients Y fI which are linear combinations of the initial coe$cients f and f . Therefore, for the eigen K+ K+ K+! function of ¹(u) the coe$cients satisfy some recurrence relations, and the problem of its diagonalization is reduced to the solution of these recurrence relations. In the case of the Odderon (n"3) we can introduce the special functions describing the states of three gluons with the total conformal weight m and the pair conformal weight M for gluons 1 and 2: FK (x)"x+(1!x)K F(m#M, M; 2M; x) , + where F(a, b; 2c; x) is the hypergeometric function, and UK(x)" lim (d/dM)FK (x) . P + +P The matrix element of the pair hamiltonian h is diagonal in the M-representation, but we should pass to the j-representation. The three independent eigen functions of the integral of motion A can

256


be written as follows: uK(x, j)" DK(j) FK(x) . P I I I uK(x, j)" (aK(j) FK(x)#DK(j)UK (x)) , Q I I I I I uK(x, j)" cK(j)FK (x) , D I I>K I where the coe$cients satisfy the reccurrence relations [17]:

d ijaK(j)" aK (j)#bK (j) k(k#1)(k!m#1) I I> I> dk

1 d (k#m!1)(k#m!2) ! aK (j)#bK (j) (k!1)(k!2)(k!m!1) , I\ I\ 4 dk (2k!3)(2k!1) 1 (k#m!1)(k#m!2) ijDK(j)"! (k!1)(k!2)(k!m!1) DK (j) I I\ 4 (2k!3)(2k!1) #k(k#1)(k!m#1)DK (j), DK(j)"1 , I> 1 (k#2m!1)(k#2m!2) ijcK(j)"! (k#m!1)(k#m!2)(k!1) cK (j) I 4 (2k#2m!3)(2k#2m!1) I\ #(k#m)(k#m#1)(k#1)cK (j), cK(j)"1 . I> In the next section the relation between the Odderon Hamiltonian and its integral of motion A is discussed from another point of view.

5. Odderon Hamiltonian and integrals of motion One can present the holomorphic Hamiltonian for n reggeized gluons in the form explicitly invariant under the MoK bius transformations

L o o o o I> II> R #log I\ II\ R !2t(1) , h" log (53) I I o o o o I> I>I> I\ I\I\ I where o is the coordinate of the composite state. Let us consider in more detail the Odderon case. Using for its wave function the conformal anzatz f (o , o , o ; o )"(o /o o )Ku (x), x"o o /o o , K K

(54)


257

one can obtain the following Hamiltonian for the function u (x) in the space of the anharmonic K ratio x [9}12]

h"6c#log(xR)#log((1!x)R)#log

#log

x ((1!x)R#m) 1!x

1 (1!x) 1 ((1!x)R#m) #log (xR!m) #log (xR!m) . 1!x x x

(55)

It is convenient to introduce the logarithmic derivative P,xR as a new momentum. In this case one can transform this Hamiltonian to the normal form [17]: h "!log(x)#t(1!P)#t(!P)#t(m!P)!3t(1)# xIf (P) , I 2 I where

(56)

1 2 1 1 I c (k) f (P)"! # # # R . I P#t k 2 P#k!m P#k R

(57)

(!1)I\RC(m#t)((t!k)(m#t)#mk/2) . c (k)" R kC(m!k#t#1)C(t#1)C(k!t#1)

(58)

Here

Because h and B"iA commute each with another, h/2 is a function of B. In particular for large B this function should have the form: h c "log(B)#3c# P . (59) 2 BP P The "rst two terms of this asymptotic expansion were calculated in Refs. [9}12]. The series is constructed in inverse powers of B, because h should be invariant under all modular transformations, including the inversion xP1/x under which B changes its sign. The same functional relation should be valid for the eigenvalues e/2 and k"ij of these operators. For large k it is convenient to consider the corresponding eigenvalue equations in the P representation, where x is the shift operator x"exp(!d/dP) ,

(60)

after extracting from eigenfunctions of B and h the common factor u (P)"C(!P)C(1!P)C(m!P) exp(ipP)U (P) . K K The function U (P) can be expanded in series over 1/k K U (P)" k\LUL (P), U (P)"1 , K K K L

(61)

(62)

258


where the coe$cients UL (P) turn out to be the polynomials of order 4n satisfying the recurrence K relation: . UL (P)" (k!1)(k!1!m)((k!m)UL\(k!1)#(k!2)UL\ (k!1!m)) K K \K I 1 K (63) ! (k!1)(k!1!m)((k!m)UL\(k!1)#(k!2)UL\ (k!1!m)) , K \K 2 I valid due to the duality equation written below for a de"nite choice of the phase of U (P) K x\K(1!xP(P!m)(P!m#1))U (P)"kKU (P) (64) K \K with the use of the substitution xkPx. Note that the summation constants UL (0) in the above recurrence relation have the antiK symmetry property UL (0)"!UL (0) , K \K which guarantees the ful"lment of the relation

(65)

UL (m)"UL (0) K \K being a consequence of the duality relation. The energy can be expressed in terms of U (P) as follows: K R e "log(k)#3c# log U (P) K RP 2

(66)

I #(U (P))\ k\If (P!k)U (P!k) (P!r)(P!r#1)(P!r!m#1) (67) K I K I P and it should not depend on P due to the commutativity of h and B. By solving the recurrence relations for UL (P) and putting the result in the above expression, we K obtain the following asymptotic expansion for e/2 [17]:

3 1 13 1 1 1 e "log(k)#3c# # m! ! m! 448 120 12 k 2 2 2 1 4185 2151 1 # ! ! m! #2 k 2050048 49280 2

#

965925 1 #2 #2 . 37044224 k

(68)

This expansion can be used with a certain accuracy even for the smallest eigenvalue k"0.20526, corresponding to the ground-state energy e"0.49434 [18}22]. For the "rst excited state with the same conformal weight m"1/2, where e"5.16930 and k"2.34392 [18}22], the energy can be calculated from the above asymptotic series with precision. The analytic approach, developed in this section, should be compared with the method based on the Baxter equation [23].


259

One can derive from above formulas also the representation for the Odderon Hamiltonian in the two-dimensional space x: 2H"h#hH"12c#ln("x" "R")#ln("1!x" "R") #(x!1)K(xH!1)K (ln("R")#ln("x" "R"))(x!1)\K(xH!1)\K #(!x)K(!xH)K (ln("1!x" "R")#ln("R"))(!x)\K(!xH)\K .

(69)

The logarithms in this expression can be presented as integral operators with the use of the relation

dp h("y"!e) (p) 1 exp(ipy) 2c#ln "!2 !2p ln d(y) . 2p "y" 4 e

(70)

Let us use this representation to "nd the eigen value of the Hamiltonian for the eigenfunction of the integrals of motion B and BH with their vanishing eigenvalues k"kH"0: (71) u (x)"1#(!x)K(!xH)K #(x!1)K(xH!1)K . KK The corresponding wave function f (q , q , q ; q ) is invariant under the cyclic permutation of KK coordinates q Pq Pq Pq but it is symmetric under the permutations q q only for even value of the conformal spin n"m !m, where the norm ""u "" is divergent due to the singularities KK at x"0, 1,R. It is the reason, why we consider only the case m !m"2k#1,

k"0,$1,$2,2 .

(72)

Owing to the Bose symmetry of the wave function, this state corresponds to the f-coupling and has the positive charge-parity C. Using the above representation for H, we obtain 2Hu (x)"EN u (x) , KK KK KK where EN is the corresponding eigen value for the Pomeron Hamiltonian [1}4,9}12] KK EN "eN #eN K K KK and

(73) (74)

eN "t(1!m)#t(m)!2t(1) . (75) K The minimal value of EN is obtained at m !m"$1 and corresponds to u"0. Because KK EN has the property of the holomorphic separability, it is natural to de"ne the holomorphic KK energy for k"0 as e"eN "t(1!m)#t(m)!2t(1) . K In this case its value for m"1/2 will be negative e"!4 ln 2 ,

(76) (77)

but it does not correspond to any physical Odderon state, because its wave function u (x) is not normalized. For the case of odd n"m !m, the norm of u (x) is not in"nite: KK dx "u (x)" KK "Re(t(m)#t(1!m)#t(m )#t(1!m )!4t(1)) . (78) 3p "x(1!x)"

Note, that this norm becomes negative for m"m "1/2.

260


In the conclusion, we note that the remarkable properties of the Reggeon dynamics are presumably related with supersymmetry. In the continuum limit nPR the above duality transformation coincides with the supersymmetric translation, which is presumably connected with the observation [24], that in this limit the underlying model is a twisted N"2 supersymmetric topological "eld theory. Additional arguments supporting the supersymmetric nature of the integrability of the reggeon dynamics are given in Ref. [25]. Namely, the eigenvalues of the integral kernels in the evolution equations for quasi-partonic operators in the N"4 supersymmetric Yang}Mills theory are proportional to t( j!1), which means that these evolution equations in the multicolour limit are equivalent to the SchroK dinger equation for the integrable Heisenberg spin model similar to the one found in the Regge limit [15,16]. Note that at large N the N"4 A Yang}Mills theory is guessed to be related with the low-energy asymptotics of a superstring model [26]. Acknowledgements I thank A.P. Bukhvostov for helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

L.N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 642. V.S. Fadin, E.A. Kuraev, L.N. Lipatov, Phys. Lett. B 60 (1975) 50. Ya.Ya. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. L.N. Lipatov, Sov. Phys. JETP 63 (1986) 904. V.S. Fadin, L.N. Lipatov, Phys. Lett. B 429 (1998) 127. S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov, G.V. Pivovarov, Phys. Rev. Lett. 80 (1998) 2047; 81 (1998) 2394; Phys. Lett., in preparation. J. Bartels, Nucl. Phys. B 175 (1980) 365. J. Kwiecinski, M. Prascalowicz, Phys. Lett. B 94 (1980) 413. L.N. Lipatov, Sov. Phys. JETP 63 (1986) 904. L.N. Lipatov, Phys. Lett. B 251 (1990) 284. L.N. Lipatov, Phys. Lett. B 309 (1993) 394. L.N. Lipatov, hep-th/9311037, Padua preprint DFPD/93/TH/70, unpublished. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982. V.O. Tarasov, L.A. Takhtajan, L.D. Faddeev, Theor. Math. Phys. 57 (1983) 163. L.N. Lipatov, Sov. Phys. JETP Lett. 59 (1994) 571. L.D. Faddeev, G.P. Korchemsky, Phys. Lett. B 342 (1995) 311. L.N. Lipatov, preprint CERN-TH/98-360, Nucl. Phys. B, in preparation. L.N. Lipatov, Recent Advances in Hadronic Physics, Proceedings of the Blois Conference, World Scienti"c, Singapore, 1997. R. Janik, J. Wosiek, Phys. Rev. Lett. 82 (1999) 1092. M.A. Braun, hep-ph/9801352, St. Petersburg University preprints. M.A. Braun, P. Gauron, B. Nicolescu, preprint LPTPE/UP6/10/July 98. M. Praszalowicz, A. Rostworowski, Acta Phys. Polon. B 30 (1999) 349. R. Janik, J. Wosiek, Phys. Rev. Lett. 79 (1997) 2935. J. Ellis and N.E. Mavromatos, Eur. Phys. J. C 8 (1999) 91. L.N. Lipatov, Perspectives in Hadronics Physics, Proceedings of the ICTP Conference, World Scienti"c, Singapore, 1997. J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231.


Multiparton collisions and multiplicity distribution in high-energy pp(p p) collisions Sergei Matinyan * Department of Physics, Duke University, Durham, NC 27708-0305, USA Yerevan Physics Institute, Yerevan, Armenia

Abstract We discuss the multiplicity distribution for the highest accessible energies of pp- and p p-interactions from the point of view of multiparton collisions. The inelastic cross sections for single, p , and multiple (double and, presumably, triple), p parton collisions are calculated from the analysis of experimental data on the multiplicity distribution up to Tevatron energies. It is found that p becomes energy independent while p increases with (s for (s5200 GeV. The observed growth of 1p 2 with multiplicity is attributed to the , increasing role of multiparton collisions for the high-energy p p(pp)-inelastic interactions. p reproduces > quite well the cross-section for the mini-jet production. 1999 Elsevier Science B.V. All rights reserved. PACS: 13.85.Hd Keywords: KNO scaling; Partons; Mini-jets; Proton radius

1. Introduction In the last 20 years a tremendous amount of work was done on the study of the longitudinal parton distribution in the deep inelastic scattering processes. These studies provide us with valuable information about the structure function of the proton F(x, Q). Presently, this information needs to be extended to the distribution of the partons in the transverse plane of the collision (p -distribution). This is non-perturbative information because it , gives us the new scale * size of the hadron.

* Department of Physics, Duke University, Durham, NC 27708-0305, USA. Tel.: #1-919-660-2596; fax: #1-919-6602525 E-mail address: [email protected] (S. Matinyan) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 7 1 - X

262

S. Matinyan / Physics Reports 320 (1999) 261}264

Fig. 1.

The natural way to study the p -dependence of the partons inside a proton is to perform the , multiple parton collision in the hadron}hadron interaction at high energy. We use the multiplicity distribution data obtained from pp colliders for (s5200 GeV, including new data from experiment E735 at (s"1.8 TeV, for an estimation of the inelastic cross sections of the soft single (p ) and double (p ) parton collisions. Our basis is the observation that the so-called KNO scaling [1] is violated at the high energy (pp)- and (p p)-collisions [2] ((s5200 GeV) while it is well satis"ed in the range of ISR energies. We attribute the deviation from the KNO scaling at higher energies to another new process which is incoherently superimposed on the KNO respecting process. Thus, the single inelastic parton collisions are characterized by this scaling. However, the violation is due to the double (and, maybe, triple) parton collisions. By subtracting the KNO distribution from the experimental data (Fig. 1) on multiplicity distribution we determine the shape for the competing process as shown in Fig. 2. The main characteristics of the derived distributions is that the most probable value of the distributions occurs at twice the multiplicity corresponding to the initial low-energy (ISR) KNO


263

Fig. 2.

distributions (single collisions). The width of the distribution is close to (2 times the width of that KNO shape at (s"1.8 GeV, which is in accordance with the Dual Parton Model based on the adding of double inelastic collision of partons of the colliding hadrons to the single parton collisions. These collisions are provided by the exchange of the pairs of the gluonic strings between partons. Integrating the distributions displayed in Fig. 2 over x"N/1N 2 we obtain the inelastic cross section p for the double parton collisions as a function of (s and, thus, the inelastic cross section p "p !p (Fig. 3) for the single parton collisions. ,1" From Fig. 3 we see that p equals 17 mb at 1.8 TeV which we can compare with CDFs recent value for the e!ective double-parton collision cross section [3] (14.5$1.7> ) mb. \ Single parton collision is nearly independent of (s for (s5200 GeV and has a value of (34$2) mb. p is increased with (s.

264


Fig. 3.

We see from our analysis that the multiple inelastic parton collisions account for a large fraction of the total pp -cross section and, de"nitely, are responsible for the increase of the p p inelastic cross section at (s5200 GeV. We expected that at LHC domain the triple collisions will be seen clearly in the multiplicity distribution. We observe that the cross-sections of the so-called minijet production extracted from several experiments are very similar, by their (s dependence and by absolute value to our p with the same threshold at (s,200 GeV. This indicates that there exists a smooth transition from hard (jet) to soft physics of the hadron collisions. Using our data for p and p we can, under the simplifying assumptions about the factorization of the proton's two-body parton distribution, estimate the `hadronica radius of the proton which is equal to 0.96 fm. If we take seriously the saturation of the single parton collision cross section p with (s, we can conjecture that the same will happen (at much more higher (s) with double parton collision cross section p , so asymptotically, p approaches to the constant value. Acknowledgements It is my great pleasure to thank W.D. Walker for collaboration and fruitful discussion. I am grateful to B. MuK ller for useful discussions. This work was supported in part by a grant from Department of Energy (DE-FG02-96ER4095). References [1] Z. Koba, H.B. Nielsen, P. Olesen, Nucl. Phys. B 40 (1972) 317. [2] G.J. Alner et al., UAS Collaboration, Phys. Lett. B 138 (1984) 304. [3] F. Abe et al., CDF Collaboration, Phys. Rev. D 56 (1997) 3811.


Perturbative}nonperturbative interference in the static QCD interaction at small distances Yu.A. Simonov Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117 218 Moscow, Russia

Abstract Short distance static quark}antiquark interaction is studied systematically using the background perturbation theory with nonperturbative background described by "eld correlators. A universal linear term 6N a pr/2p is observed at small distance r due to the interference between perturbative and nonperturbative A Q contributions. Possible modi"cations of this term due to additional subleading terms are discussed and implications for systematic corrections to OPE are formulated. 1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.Bx; 14.65.!q

1. Introduction It is more than 20 years ago that the power correction has been computed in OPE [1] laying ground for numerous later applications in QCD. Since then OPE is the basic formalism for study of short-distance phenomena, such as DIS, e>e\ annihilation and, with some modi"cations, heavy quark systems. Interaction of static charges at small distances has drawn a lot of attention recently [2}4]. The theoretical reason is that the appearance of linear terms in the static potential ). The #avor dependence arises in the next order, A A A m\ relative to C , and can be interpreted as due to two mechanisms: the weak scattering (WS) / and the Pauli interference (PI). The weak scattering corresponds to a cross-channel of the decay, generically QPq q q , where either the quark q is a spectator in a baryon and can undergo a weak scattering o! the heavy quark: q QPq q , or an antiquark in meson, say q , weak-scatters (annihilates) in the process q QPq q . The Pauli interference e!ect arises when one of the "nal (anti)quarks in the decay of Q is identical to the spectator (anti)quark in the hadron, so that an interference of identical particles should be taken into account. The latter interference can be either constructive or destructive, depending on the relative spin-color arrangement of the (anti)quark produced in the decay and of the spectator one, thus the sign of the PI e!ect is found only as a result of speci"c dynamical calculation. Although the WS and PI e!ects carry the relative suppression by m\, they are found to be / greatly enhanced by a large numerical factor, typically 16p/3, re#ecting mainly the di!erence of the numerical factors in the two-body versus the three-body "nal phase space, which makes these e!ects overwhelmingly essential in the charmed hadrons, while greatly reduced in the heavier b hadrons, as is con"rmed by the experimental data. In e!ect, the contribution of the O(m\) terms / is signi"cantly smaller than that of the O(m\) terms in the charmed hadrons and is slightly smaller / in the b hadrons [6]. In particular the O(m\) terms split the decay rate of the K from that of the / @ B mesons by only about 2%, which is by far insu$cient to explain the current data [7] on the ratio of the lifetimes: q(K )/q(B)"0.79$0.05 (for a recent theoretical discussion see e.g. Ref. [8]). @ A quantitative estimate of the e!ects of the O(m\) terms runs into a problem of evaluating matrix / elements over the hadrons of four-quark operators of the type (QM C Q)(q C q) with certain spin-color matrix structures C and C . Although simple estimates within a non-relativistic picture of the light quarks in hadrons (where these operators reduce to the density of the light quarks at the location of the heavy quark) allowed to correctly predict [4,5] the hierarchy of the lifetimes of the charmed hadrons, these estimates are obviously very unreliable for a quantitative description of the e!ect. Neither can this approach explain the ratio q(K )/q(B) to be less than approximately 0.9. In view of @ this di$culty it is quite worthwhile to have a better understanding of the spectator #avordependent di!erences of the rates in a possibly more model-independent way. The purpose of this paper is to point out relations between some inclusive decay rates of the charmed and b baryons in the (K , N ) triplets, which do not require explicit knowledge of / / the matrix elements of the four-quark operators, and rely only on the general expansion in m\ for / the rates and on the #avor SU(3) symmetry. Certainly, the latter symmetry is known to be not very precise, however arguably the uncertainties due to the SU(3) violation are substantially less than

M.B. Voloshin / Physics Reports 320 (1999) 275}285

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those brought in by the current model assumptions about the hadronic matrix elements. Thus it may be expected that an experimental veri"cation of these relations can serve as a test of the whole method based on the operator product expansion for weak decay rates. To an extent, such approach was pursued in the prediction [9] of a signi"cant enhancement of the total semileptonic decay rates of the N baryons over the same rate for the K and even greater enhancement of this A A rate for X . That analysis was further extended [10,11] to include the enhancement of the A Cabibbo-suppressed semileptonic decays of K . Although those papers used model considerations A for the matrix elements of the four-quark operators, in fact one can obtain, as shown in the present paper, quantitative results for the (K , N ) triplet without resorting to models of quark dynamics in A A the baryons. Namely, it will be shown that the di!erence of the semileptonic rates within the (K , N ) A A baryon triplet, both the dominant and the Cabibbo-suppressed, as well as the di!erence of the non-leptonic Cabibbo-suppressed decay rates, can all be expressed in terms of the total lifetime di!erences within the same triplet in a model-independent fashion, modulo the assumption of the #avor SU(3) symmetry. In addition the di!erences of the lifetimes within the triplet of the b baryons (K , N ) are also expressed through the same di!erences for the charmed baryons, with @ @ a possible extra uncertainty due to the quark mass ratio m/m. @ A Using the currently available data on lifetimes of the charmed hyperons, the discussed e!ects are estimated to be quite large. In particular, the conclusion of the previous analyses is con"rmed that the semileptonic decay rates of the N baryons should exceed by a factor 2 to 3 the same rate for the A K hyperon. It is also found that the lifetime of the N\ baryon can be longer than that of K by A @ @ about 14%, which is a very large e!ect for b hadrons.

2. E4ective Lagrangian for spectator}6avor-dependent e4ects in decay rates The systematic description of the leading as well as subleading e!ects in the inclusive decay rates of heavy hadrons arises [2,4,5] through the application of the operator product expansion in powers of m\ to the &e!ective Lagrangian' ¸ related to the correlator of two weak-interaction / terms ¸ : 5

¸ "2 Im i dx e OV¹+¸ (x), ¸ (0), . 5 5

(1)

In terms of ¸ the inclusive decay rate of a heavy hadron H is given by the mean value / C "1H "¸ "H 2 . (2) & / / The leading term in ¸ describes the &parton' decay rate. For instance, the term in the non-leptonic weak Lagrangian (2G l and the Cabibbo-suppressed one: cPdl>l is given by [9}11] Gm ¸ " $ A +c[¸ (c C c#c c c c)(s C s)#¸ (c C c #c c c c )(s C s )] I I I G I I G I I I IG 12p # s[¸ (c C c#c c c c)(dM C d)#¸ (c C c #c c c c )(dM C d )] I G I I G I I I I G I I (7) ! 2i(i\!1)(c C t?c#c c c t?c)j? , , I I I where the coe$cients ¸ and ¸ are ¸ "(i!1), ¸ "!3i . (8) In the next section the general expressions in Eqs. (4), (7) and (8) are used for the analysis of the relations between the splittings of various inclusive decay rates within the triplet of charmed baryons.

3. Di4erences of inclusive decay rates for charmed baryons Estimates of the absolute e!ect of the #avor-dependent operators in the e!ective Lagrangian ¸ require evaluation of the matrix elements of the four-quark operators over charmed hadrons. In mesons the same term describes the Pauli interference of the dM quark in the decays of D>, which is considered to be the major reason for the observed suppression of the D> total decay rate.

280


One approach (for a review see e.g. Ref. [8]) is to use a low normalization point k of the order of the con"nement scale and invoke a constituent quark model, simplifying it further to a non-relativistic model, with possible additional (also non-relativistic) input about the wave functions of the light quarks at the origin (see e.g. Refs. [14,15] and a rather general consideration in Refs. [16,17]). Needless to mention that such approach can be used only for qualitative, or very approximate semi-quantitative estimates, since it inevitably involves poorly controllable approximations. In order to be able to "nd arguably more reliable relations we do not attempt here an absolute evaluation of those matrix elements, but rather use the #avor SU(3) properties of the operators in ¸ to relate the measurable splittings of the semileptonic and the Cabibbo-suppressed inclusive decay rates in the charmed baryon triplet to the splitting of the total decay rates. Namely, assuming the #avor SU(3) symmetry, and the applicability of the heavy quark limit to the charmed quark, it will be shown that the discussed splittings are determined by only two independent matrix elements, which can be expressed in terms of the di!erences in the measured total decay rates: D "C(N)!C(K ) and D "C(K )!C(N>). A A A A Proceeding with derivation of the relations we "rst notice that for the triplet of the baryons (K , N ) in the heavy quark limit the spin of the charmed quark is not correlated with spinorial A A characteristics of its light &environment'. Thus the operators with the axial current of the c quark give no contribution to the matrix elements. The remaining #avor non-singlet structures in ¸ involve only operators of the types (c c c)(q C q) and (c c c )(q C q ) with q being d, s or u. Due to I I G I I I I G the SU(3) symmetry the #avor non-singlet part of the matrix elements of each of these types of operators in the baryon triplet is expressed in terms of only one parameter. Indeed, the di!erence of the matrix elements between the components of a A , \ \ I I I I I I

(9)

y"1(c c c )[(u C u )!(sN C s )]2NA KA"1(c c c )[(s C s )!(dM C d )]2KA N>A \ \ I IG G I I I IG I I G G I I I I G with the shorthand notation for the di!erences of the matrix elements: 1O2 " \ 1A"O"A2!1B"O"B2, the splitting of the inclusive decay rates within the baryon triplet are expressed in terms of x and y as follows. The di!erences of the dominant Cabibbo-unsuppressed non-leptonic decay rates are given by Gm d ,C (N)!C (K )"c $ A [(C !C )x#(C !C )y] , 1\ A 1\ A 4p (10) Gm d ,C (K )!C (N>)"c $ A [(C !C )x#(C !C )y] . 1\ A 1\ A 4p ¹he once Cabibbo-suppressed decay rates of K and N> are equal, due to the *;"0 property of A A the corresponding e!ective Lagrangian ¸ (Eq. (7)). Thus the only di!erence for these decays in


281

Gm d ,C (N)!C (K )"cs $ A [(2C !C !C )x#(2C !C !C )y] . 1 A 1 A 4p

(11)

the baryon triplet is

The dominant semileptonic decay rates are equal among the two N baryons due to the isotopic A spin property *I"0 of the corresponding interaction Lagrangian, thus there is only one nontrivial splitting for these decays: Gm d ,C (N )!C (K )"!c $ A [¸ x#¸ y] . 1\ A 1\ A 12p

(12)

Finally, the Cabibbo-suppressed semileptonic decay rates are equal for K and N, due to the A A * > \d ! d . (16) d " 3(C #2C ) c 12C C (C #2C ) > \ \ > \ > The coe$cients in this relation depend only on the physical ratio of the couplings r"(a (m )/a (m )). Moreover, this dependence is in fact rather weak: in the absence of the QCD Q A Q 5 radiative e!ects, i.e. with r"1, the coe$cients in the square brackets in Eq. (16) are +0.22 and !+!0.11, while with a realistic value r+2.5 they are, respectively, 0.23 and !0.09. Similar

282


relative insensitivity of the numerical results to the exact value of r also holds for other relations between the observable splittings. In subsequent numerical estimates the numerical values r"2.5 and s"0.05 are used. We also use the data from Ref. [7] on the lifetimes of the charmed baryons in the form: C(K )"4.85$0.28 ps\, C(N>)"2.85$0.5 ps\ and C(N)"10.2$2 ps\. (The A A A error bars on the lifetimes for the N hyperons are in fact not symmetric. However the symmetry A improves for the inverse quantities, i.e. for the total widths, and a close approximation to the errors in the widths is used here in a symmetric fashion.) Solving Eqs. (14) for the Cabibbo-dominant splitting of the semileptonic widths yields d "0.13D !0.065D +0.59$0.32 ps\ (17) and, obviously, for the Cabibbo-suppressed semileptonic decays one has d "(s/c)d +0.030$0.016 ps\. The solution of Eqs. (14) for the splitting of the Cabibbo-suppressed nonleptonic rates similarly gives d "0.082D #0.054D +0.55$0.22 ps\ .

(18)

4. Splitting of lifetimes in the triplet of b baryons Once the unknown baryonic matrix elements x and y are phenomenologically determined through the di!erences of the total decay rates in the charmed baryon triplet, one can use these parameters for evaluating the di!erences of decay rates in the triplet of the b baryons: K , N, and @ @ N\. Indeed, in the limit where both the c and the b quarks are heavy the matrix elements of the @ four-quark operators over the b hyperons should be the same as for the charmed ones, provided that the operators are normalized at a low point k which does not depend on the masses m or m . A @ For proceeding in this manner we write here the expression [5] for the corresponding e!ective lagrangian for non-leptonic b decays, neglecting small kinematical e!ects O(m/m) in the relevant A @ expressions for the two-body phase space of the pair cc or cq, Gm ¸@ "c"< " $ @ +CI (bM C b)(u C u)#CI (bM C u)(u C b) I I I I @A 4p # CI (bM C b#bM c c b)(q C q)#CI (bM C b #bM c c b )(q C q ) I G I I G I I I I G I I # i(i\!1)[2(CI !CI )(bM C t?b)j? > \ I I (19) ! (5CI #CI !6CI CI )(bM C t?b#bM c c t?b)j? ], , I > \ > \ I I where again the notation (q Cq)"(dM Cd)#(s Cs) is used, and the renormalization coe$cients are determined by a (m ) instead of a (m ): CI "CI \"(a (m )/a (m ))@, i"(a (k)/a (m )). The coe$Q @ Q A \ > Q @ Q 5 Q Q @ cients CI are related to CI , CI , and i in the same way as in Eqs. (5). \ > The expression with these small terms included can be found in Ref. [18]. Also only the CKM-dominant processes bPcu d and bPcc s are taken into account in order to keep the formulas simple. The contribution of sub-dominant processes to the total rates is below the expected uncertainty.


283

The dominant semileptonic decay bPcll does not involve light quarks, thus one expects no substantial splitting of the semileptonic decay rates within #avor SU(3) multiplets of b hadrons. The non-leptonic e!ective lagrangian in Eq. (19) is symmetric with respect to s d, i.e. it has *;"0 (this property is broken if the small kinematical e!ects of the c quark mass are kept in ¸@). Thus at this level there is no splitting between the non-leptonic decay rates of K and N. The @ @ splitting of the decay rate between either of these and N\ is given in terms of x and y by @ Gm (20) D ,C(K )!C(N\)"c"< " $ @ [(CI !CI )x#(CI !CI )y] . @ @ @ @A 4p One can notice that in the absence of any QCD radiative e!ects the latter di!erence is simply related to d and d for the charmed baryons D ""< "/c m/m(d #d ) . (21) @ @A @ A The QCD correction coe$cients make the full expression somewhat more lengthy 1 "< " m @ D " @A +[C ((3#2m)CI #4mCI CI #6(1!m)CI ) @ > \ \ > > c m 4C C (C #2C ) A > \ \ > # 2C C (CI #2CI )!C ((1!m)CI !2mCI CI #(2#3m)CI )]d > \ \ > \ \ \ > > # 4C C (CI #2CI )d , , (22) > \ \ > where m"(i/i)"(a (m )/a (m )). One can again note that the relation (22) between the Q A Q @ physically measurable quantities does not depend on the low normalization point k. Numerically, however the full expression is not far from the simple approximation in Eq. (21): with a realistic value (a (m )/a (m ))+1.25 one "nds from Eq. (22): Q A Q @ D +"< "/c m/m(0.91d #0.93d ) . @ @A @ A When expressed in terms of the di!erences in the total decay rates D and D for the charmed baryons, using Eq. (14), the splitting of the decay rates within the b baryon triplet reads numerically as D +"< "m/m(0.85D #0.91D )+0.015D #0.016D +0.11$0.03 ps\ , (23) @ @A @ A which represents the estimate from the present analysis of the expected suppression of the total decay rate of N\ with respect to that of K , or N. @ @ @ 5. Discussion The relative di!erences of the lifetimes for charmed particles are large, even within one #avor SU(3) triplet of the hyperons. Therefore the assumption that these di!erences in the triplet are described by just one term of the expansion in m\ certainly requires additional study. It should / be noted however, that this assumption is not necessarily #awed, since the discussed O(m\) terms /

284


are singled out by large numerical coe$cient, and there is no reason for recurrence of such anomaly further in the expansion. Thus the relations between the splittings of the decay rates within the triplet charmed hyperons (K , N ) as well as of the b hyperons (K , N ) may present a good testing A A @ @ point for experimental study of this issue. As a consequence of large di!erence in the lifetimes the additional e!ects, discussed here, are also estimated to be quite large. The predicted di!erence in the semileptonic decay rates between the N and K (Eq. (17)) can be compared with the current data A A on the semileptonic width of K : C (K )"0.22$0.08 ps\. This comparison con"rms the conA A clusion of a previous analysis [9], that the semileptonic decay rates of the N hyperons can be larger A than that of K by a factor of 2 or 3. Similarly, the small Cabibbo-suppressed semileptonic decay A rate of K , should be enhanced by the same factor [11] and may in fact constitute 10% to 15% of all A semileptonic decays of K . The e!ect in the Cabibbo-suppressed non-leptonic decays evaluated in A Eq. (18) can amount to more than 10% of the di!erence in the total non-leptonic decay rates of N> and K and should be quite prominent, provided that it would be possible to separate and A A measure the inclusive Cabibbo-suppressed rates experimentally. Finally, the prediction of Eq. (23) for the di!erence of the total decay rates of K and N\ can be @ @ interesting in relation to the mentioned earlier problem of the ratio q(K )/q(B). Indeed, the central @ number in Eq. (23) amounts to about 14% of the total decay rate C(K )"0.81$0.05 ps\, and @ a di!erence of such relative magnitude is undoubtedly to be considered as very large for the b hadrons. If con"rmed, this would indicate that the spectator e!ects in heavy hyperons can be substantially larger, than usually expected, and may shed some light on the problem of the K versus B lifetime. @ Acknowledgements It is a great pleasure to use this occasion to express my gratitude and appreciation to Lev Borisovich Okun, who taught me the SU(3), quarks, leptons, and many other things in physics and beyond. This work is supported in part by DOE under the grant number DE-FG02-94ER40823.

References [1] W. Bacino et al. (DELCO Coll.), Phys. Rev. Lett. 45 (1980) 329. [2] M.A. Shifman, M.B. Voloshin (1981) unpublished, presented in the review V.A. Khoze, M.A. Shifman, Sov. Phys. Usp. 26 (1983) 387. [3] N. Bilic, B. Guberina, J. Trampetic, Nucl. Phys. B 248 (1984) 261. [4] M.A. Shifman, M.B. Voloshin, Sov. J. Nucl. Phys. 41 (1985) 120. [5] M.A. Shifman, M.B. Voloshin, Sov. Phys. JETP 64 (1986) 698. [6] I.I. Bigi, N.G. Uraltsev, A.I. Vainshtein, Phys. Lett. B 293 (1992) 430, erratum } ibid. B 297 (1993) 477. [7] Particle Data Group, Eur. Phys. J. C3 (1998) 1. [8] I. Bigi, M. Shifman, N. Uraltsev, Ann. Rev. Nucl. Part. Sci. 47 (1997) 591. [9] M.B. Voloshin, Phys. Lett. B 385 (1996) 369. [10] H.-Y. Cheng, Phys. Rev. D 56 (1997) 2783. [11] B. Guberina, B. MelicH , Eur. Phys. J. C 2 (1998) 697. [12] L.B. Okun, Leptons and Quarks, North-Holland, Amsterdam, 1982, 1984.


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[13] M.A. Shifman, M.B. Voloshin, Sov. J. Nucl. Phys. 45 (1987) 292. [14] B. Guberina, R. RuK ckl, J. TrampeticH , Z. Phys. C 33 (1986) 297. [15] B. Blok, M. Shifman, Lifetimes of Charmed Hadrons Revised } Facts and Fancy, in: J. Kirkby, R. Kirkby (Eds.), Proceedings of the Workshop on the Tau-Charm Factory, Marbella, Spain, 1993, Editions Frontiers, Gif-surYvette, 1994, p. 247. [16] N.G. Uraltsev, Phys. Lett. B 376 (1996) 303. [17] D. Pirjol, N. Uraltsev, Phys. Rev. D 59 (1999) 034012. [18] M. Neubert, C.T. Sachrajda, Nucl. Phys. B 483 (1997) 339.


Gluon condensate from superconvergent QCD sum rule F.J. YnduraH in Departamento de Fn& sica Teo& rica, C-XI, Universidad Auto& noma de Madrid, Canto Blanco, 28049-Madrid, Spain

Abstract Sum rules for the nonperturbative piece of correlators (speci"cally, the vector current correlator) are discussed. The sum rule subtracting the perturbative part is of the superconvergent type. Thus it is dominated by the bound states and the low-energy production cross section. It leads to a determination of the gluon condensate 1a G2. We "nd 1a G2K0.048$0.030 GeV. 1999 Elsevier Science B.V. All rights Q Q reserved. PACS: 12.38.!t; 12.38.Aw Keywords: Gluon condensate

1. Sum rule The potential, or more generally the spectrum of a system of heavy quarks cannot be directly discussed in terms of the operator product expansion (OPE). However, one can use dispersion relations to deduce a number of sum rules relating bound-state properties to quantities obtainable via the OPE (ÌTEP-typea sum rules). One can then use the estimates of nonperturbative contributions to bound states energies and wave functions to actually go beyond the traditional analysis. Although the sum rules, being global relations, cannot discriminate details one can check consistency and even obtain reasonable estimates on nonperturbative quantities, speci"cally on the gluon condensate. This is the last aim of the present note, where we will use a method generalizing that proposed by Novikov [1]. To do so we consider the correlator for the vector current of heavy quarks:

P "(pg !p p )P(p)"i dx e NV1TJ (x)J (0)2 , I J IJ IJ I J E-mail address: [email protected] (F.J. YnduraH in) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 7 9 - 4

(1)

F.J. Yndura& in / Physics Reports 320 (1999) 287}293

288

where J "tM c t and sum over omitted colour indices is understood. This will give information on I I triplet, l"0 states; information on states with other quantum numbers would be obtained with other correlators. The function P(t) satis"es a dispersion relation

o(s) 1 , P(t)" ds s!t p

(2)

where o(s),Im P(s). Actually, this equation should have been written with one subtraction. We will not bother to do so as its contribution drops out for the quantities of interest for us here. Let us denote by P , o to the corresponding quantities calculated in perturbation theory, albeit to all orders, but nonperturbative e!ects are neglected in P , o . (In actual calculations we cannot of course include all orders. We will sum the one-gluon exchange to all orders, which can be done explicitly in the nonrelativistic regime, and add one-loop radiative corrections to this.) In particular, for example, the gluon condensate contribution is not included in the `p.t.a pieces. At large t, both spacelike and timelike, the OPE is applicable to P(t), and we have the well-known results [2]: P(t)KP

(t)#1a G2/12pt Q

(3)

and o(s)Ko

N C 1a G2 (1#v)(1!v) (s)! A $ Q s v 128

(4)

with v"(1!4m/s) the velocity of the quarks. Moreover,

N !t 3C !t P (t) K ! A log # $ log log #2 , 12p l l b R N 3C a A 1# $ Q#2 , N "3, C " . Im P (s) K A $ 12p 4p R If we then de"ne P , o as the results of subtracting the perturbative parts, ,. ,. P ,P!P , ,.

o ,o!o , ,.

it follows from the OPE, Eq. (3), that P decreases at in"nity like t\ and hence satis"es ,. a superconvergent dispersion relation. We thus have a "rst sum rule

ds o (s)"0 . ,.

(5)

In fact it would appear that one still has another sum rule because of the following argument. At large t, P (t) behaves like (cf. Eq. (3)) ,. P (t)K1a G2/12pt , ,. Q


289

while the contribution from the bound states to the dispersion relation (see below) P (t)&1a G2/ta ,._ Q Q dominates over this. Therefore, we have the extra relation

ds so (s)"0 . ,.

(6)

(7)

It turns out that (7) is actually equivalent to (5), up to radiative corrections. This is because the region where any of the integrals in (5), (7) are appreciably di!erent from zero is for sK4m(1#O(a)), so (7) di!ers from (5) by terms of order a, smaller than the radiative Q Q corrections which neither (5) nor (7) take into account. Let us return to sum rule (5). The function o(s) consists of a continuum part, for s above threshold for open bottom production, and a sum of bound states. Both can be calculated theoretically provided that s is larger than a certain critical s(v ), and n smaller or equal than a critical n . s(v ) and n are de"ned as the points where the perturbation theoretic contribution to o and the nonperturbative one are of equal magnitude, and form the limits of the regions where a full theoretical evaluation is possible. To be precise, for the continuum we use (4) so that above the critical s(v ) N C 1a G2 (1#v)(1!v) , s's(v ) (8a) o(s)" A $ Q ,. s v 128 and v is such that o(s(v ))Ko(s(v )); numerically, and for bM b, v K0.2. For the bound states ,. o is proportional to the square of the wave function at the origin: N o(s)" A "R (0)"d(s!M ) . L M L L We may get o (s) and o (s) by splitting the residue "R (0)" into a Coulombic piece; ,. L mCa $ Q (1!d a ) , "R! (0)"" L Q L 2n where the one-loop corrections d a may be found in Refs. [3}6], and the (leading) nonperturL Q bative correction are given by the Leutwyler}Voloshin analysis (Refs. [3}9]). So we have "R (0)"K"R! (0)"#"R! (0)"d,. L L L L the numbers d,. have been calculated by Leutwyler and Voloshin. For n"1, L 38.31a G2 Q . d,." mCa $ Q This is all we really need since, for bottomium, n "1. Thus we have 3N Cpm1a G2 L g Q L d(s!M), n4n o (s)" A $ L ,. M 8am Q L L the g known in terms of the d,.. L L

(8b)


290

Sum rule (5) can then be written schematically as

L o # Residue of o " ,. ,. QT L

QT

Residue of o o # . ,. ,. LL>

The left-hand side is given in terms of 1a G2 by Eq. (8); the right-hand side can be connected with Q experiment with the following argument. The sum over higher bound states, ` Residue of o a ,. LL> may be identi"ed as the di!erence between the sum over the experimental residues of the poles of the bound states, and what we would get by a Coulombic formula, for all n5n #1. Certainly, this Coulombic formula will not be valid for large n because here the radiative corrections will become large; but, because the residues decrease like 1/n the contribution of these states will be negligible. We write this decomposition as (s) . (bound states with n'n )"o (s)!o ! LL LL As for the continuum piece below s(v ) we may likewise interpret it as the di!erence between experiment and a perturbative evaluation, which we write as o(s)"o(s)!o(s), s(s(v ) ,. and, because we are close to threshold, we have NC a 1 o(s)" A $ Q (1#d a ) Q 8 1!e\p!$?QT NC a K A $ Q (1#d a ) Q 8 and the value of the one-loop radiative correction d a may be found in Ref. [10]. Q Taking everything into account sum rule (5) becomes, 1 "R(0)"#f (v ) mM L L LL > 1 p1a G2 L 2 1a G2 Q "2C a ! j n # 8ea# Q . (9) $ Q L Q 48eam n ma 3 Q L Q LL> We have de"ned v ,ea and the expression is valid up to corrections of relative order a . The Q Q function f (v ) is the contribution of the background which, when added to the resonances above threshold (included in the sum in the l.h.s. of (9)), give the experimental value of QT o . The ,. function f would be obtained by integrating the cross sections for production of B#G and BBM , where by G we mean a `glueballa decaying into 2p, and B is any of the states B, B!, BH. Because


291

we may assume that the structure is provided by the resonances, we can take f given by phase space only. So we have f (v )"f v#f v , where the "rst term refers to the channel B#G , and the second to BBM . We have in this expression neglected m . % 2. Numerology In principle the procedure would appear straightforward. One would "t the resonance and bound state residues and f to the data, and then, after substituting into (9), obtain a determination of 1a G2. In practice, however, things do not work out so nicely. The quality of the experimental Q data does not allow any precise determination of the constants f ; any values in the range f &0.03,0.1 would do the job. Secondly, the e!ective dependence of 1a G2 in Eq. (9) on Q experiment is proportional to a\: so the result will depend very strongly on the value of a we Q Q choose. This is particularly true because radiative corrections to the nonperturbative contribution to the bound states have not been calculated, so there is not even a `naturala renormalization point. These two di$culties may be partially overcome with the following tricks. First of all, since we are assuming that the n"1 bound state is described with the bound state analysis as discussed in Refs. [3}6], we may "x the value of a that produces such agreement. This means that we will take Q 0.354a 40.4. Secondly, we may alter the treatment of the continuum in the following manner. Q We split not from v , but from v , arbitrary provided only that v 5v . Thus, for s4s(v ), we use o(s)"o(s)!o(s), and for s5s(v ) we take the theoretical expression ,. N C 1a G2 (1#v)(1!v) . o(s)" A $ Q ,. s v 128 The sum rule is thus written as 1 "R(0)"#f (v ) mM L L L 4.91a G2 2 1a G2 Q "2C [f(3)!1]a! # 8ea# Q , e a "v . $ Q Q Q ma 3 48eam Q Q Then we may pro"t from the fact that the sum rule should be valid for all values of v 5v to "x f requiring this independence, at least in the mean. That is to say, that when we increase v past a particle threshold from B(2) to B(6) the variation of the corresponding determinations of 1a G2 Q around their average be minimum. The calculation may be further simpli"ed replacing

f (v ) v . The results of the analysis are summarized in the following tables, where the column `Resa indicates at which resonance the cut in v occurs. We have taken two rather extreme values of f .


292

Res.

v

1a G2 Q

B(2) B(3) B(4) B(5) B(6)

0.21 0.34 0.40 0.43 0.46

0.014 0.034 0.048 0.039 0.046

Res.

v

1a G2 Q

B(2) B(3) B(4) B(5) B(6)

0.21 0.34 0.40 0.43 0.46

0.037 0.057 0.067 0.048 0.052

For a "0.35, f "0.04 Q

For a "0.40, f "0.09 Q This derivation shows very clearly the kind of errors one encounters. To the variations that may be called `statisticala, apparent in the di!erent values found in the tables above 0.01441a G240.067 Q we have to add `systematica ones, e.g., the in#uence of the not calculated radiative corrections, easily of some 30%: not to mention our including the Coulombic wave functions at the origin for large values of n, or the lack of de"nition of the expression `perturbation theory to all ordersa because of renormalon ambiguities. Given all these uncertainties, which do even make it dubious that one can really de"ne with precision the condensate in terms of experimental observables, it is not surprising that one cannot pin down the gluon condensate with more accuracy than an estimate, taking into account above "gures, of 1a G2K0.048$0.03 Gev . Q To get this average we have taken into account all determinations in the tables above, excluding the lowest (B(2)) and highest, B(6). This is slightly larger than old averages, and slightly lower than more recent ones [11,12] which tended to give, respectively, 1a G2K0.042, 1a G2K0.065 Gev. Q Q Acknowledgements Discussions with R. Akhoury and V. Zakharov on some aspects of the sum rule are gratefully acknowledged.


References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

See, e.g., V.A. Novikov et al., Phys. Rep. C 41 (1978) 1. M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 (1978) 385. S. Titard, F.J. YnduraH in, Phys. Rev. D 49 (1994) 6007. S. Titard, F.J. YnduraH in, Phys. Rev. D 51 (1995) 6348. A. Pineda, F.J. YnduraH in, Phys. Rev. D 58 (1998) 094022. A. Pineda, F.J. YnduraH in, CERN-TH/98-402 (hep-ph/9812371). H. Leutwyler, Phys. Lett. B 98 (1981) 447. M.B. Voloshin, Nucl. Phys. B 154 (1979) 155. M.B. Voloshin, Sov. J. Nucl. Phys. 36 (1982) 143. K. Adel, F.J. YnduraH in, Phys. Rev. D 52 (1995) 6577. Cf. the reviews of S. Narison, QCD Spectral Sum Rules, World Scienti"c, Singapore, 1989. S. Narison, Nucl. Phys. Suppl. 54A (1997) 238.

293


Neutrino masses and mixings: a theoretical perspective Guido Altarelli *, Ferruccio Feruglio Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland Universita% di Roma Tre, Rome, Italy Universita% di Padova and I.N.F.N., Sezione di Padova, Padua, Italy

Abstract We brie#y review the recent activity on neutrino masses and mixings which was prompted by the con"rmation of neutrino oscillations by the Superkamiokande experiment. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.30.Hv; 12.10.!g; 12.15.Ft; 14.60.Pq Keywords: Solar and atmospheric neutrinos; Beyond the Standard Model; Neutrino physics; Grand Uni"ed Theories

1. Introduction It is for us a great pleasure to contribute to the celebration of Lev Okun anniversary with this article. Considering the continuous interest of Lev on neutrinos we thought that this subject is particularly appropriate to the occasion. Recent data from Superkamiokande [1] have provided a more solid experimental basis for neutrino oscillations as an explanation of the atmospheric neutrino anomaly. In addition the solar neutrino de"cit, observed by several experiments [2], is also probably an indication of a di!erent sort of neutrino oscillations. Results from the laboratory experiment by the LSND collaboration [3] can also be considered as a possible indication of yet another type of neutrino oscillation. Neutrino oscillations imply neutrino masses. The extreme smallness of neutrino masses in comparison with quark and charged lepton masses indicates a di!erent nature of neutrino masses, * Corresponding author. Theoretical Physics Division, CERN, CH-1211 Geneva 23, Italy. E-mail addresses: [email protected] (G. Altarelli), [email protected] (F. Feruglio) Alternative explanations such as neutrino decay and violations of the equivalence principle appear to be disfavoured by the present data [4]. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 6 7 - 8

296

G. Altarelli, F. Feruglio / Physics Reports 320 (1999) 295}318

linked to lepton number violation and the Majorana nature of neutrinos. Thus neutrino masses provide a window on the very large energy scale where lepton number is violated and on grand uni"ed theories (GUTs) [5]. The new experimental evidence on neutrino masses could also give an important feedback on the problem of quark and charged lepton masses, as all these masses are possibly related in GUTs. In particular the observation of a nearly maximal mixing angle for l Pl is particularly interesting. Perhaps also solar neutrinos may occur with I O large mixing angle. At present solar neutrino mixings can be either large or very small, depending on which particular solution will eventually be established by the data. Large mixings are very interesting because a "rst guess was in favour of small mixings in the neutrino sector in analogy to what is observed for quarks. If con"rmed, single or double maximal mixings can provide an important hint on the mechanisms that generate neutrino masses. The purpose of this article is to provide a concise review of the implication of neutrino masses and mixings on our picture of particle physics. We will not review in detail the status of the data but rather concentrate on their conceptual impact. The experimental status of neutrino oscillations is still very preliminary. While the evidence for the existence of neutrino oscillations from solar and atmospheric neutrino data is rather convincing by now, the values of the mass squared di!erences *m and mixing angles are not "rmly established. For the observed l suppression of solar neutrinos, for example, three possible C solutions are still possible [6]. Two are based on the MSW mechanism [7], one with small (MSW-SA: sin 2h &5.5;10\) and one with large mixing angle (MSW-LA: sin 2h 90.2). The third solution is in terms of vacuum oscillations (VO) with large mixing angle (VO: sin 2h &0.75). However, it is important to keep in mind that the *m values of the above solutions are determined by the experimental result that the suppression is energy dependent. This is obtained by comparing experiments with di!erent thresholds. The Cl experiment shows a suppression larger than by a factor of 2, which is what is shown by Ga and water experiments [2]. If the Cl indication is disregarded, then new energy-independent solutions would emerge, with large *m and maximal mixing. For example, good "t of all data, leaving those on Cl aside, can be obtained with *m as large as *m:10\ eV [8,9]. For atmospheric neutrinos the preferred value of *m, in the range 10\}10\ eV, is still a!ected by experimental uncertainties and could sizeably drift in one sense or the other, but the fact that the mixing angle is large appears established (sin 2h 90.9 at 90% C.L.) [10}12]. Another issue which is still open is the claim by the LSND collaboration of an additional signal of neutrino oscillations in a terrestrial experiment [3]. This claim was not so-far supported by a second recent experiment, Karmen [13], but the issue is far from being closed. Given the present experimental uncertainties the theorist has to make some assumptions on how the data will "nally look like in the future. Here we tentatively assume that the LSND evidence will disappear. If so then we only have two oscillations frequencies, which can be given in terms of the three known species of light neutrinos without additional sterile kinds (i.e. without weak interactions, so that they are not excluded by LEP). We then take for granted that the frequency of atmospheric neutrino oscillations will remain well separated from the solar neutrino frequency, even for the MSW solutions. The present best values are [6,10}12,14] (*m) &3.5;10\ eV and (*m) &5;10\ eV or (*m) &10\ eV. We also assume that the electron neu+15U1 4trino does not participate in the atmospheric oscillations, which (in the absence of sterile neutrinos) are interpreted as nearly maximal l Pl oscillations as indicated by the Superkamiokande [1] I O


297

and Chooz [15] data. However, the data do not exclude a nonvanishing ; element. In C the Superkamiokande allowed region the bound by Chooz [15] amounts to "; ":0.2 C [10,11,14].

2. Direct limits on neutrino masses Neutrino oscillations are due to a misalignment between the #avour basis, l,(l ,l ,l ), where C I O l is the partner of the mass and #avour eigenstate e\ in a left-handed weak isospin SU(2) doublet C (similarly for l and l ) and the neutrino mass eigenstates l,(l , l , l ) I O "l2";"l2 , (1) where ; is the 3;3 mixing matrix [54]. Thus, in the presence of mixing, neutrinos cannot be all massless and actually the presence of two di!erent oscillation frequencies implies at least two di!erent nonzero masses. Neutrino oscillations are practically only sensitive to di!erences *m so that the absolute scale of squared masses is not "xed by the observed frequencies. But the existing direct bounds on neutrino masses, together with the observed frequencies, imply that all neutrino masses are by far smaller than any quark or lepton masses. In fact the following direct bounds hold: m C:&5 eV, m I:170 KeV and m O:18 MeV [16]. Since the observed *m indicate mass splitJ J J tings much smaller than that, the limit on m C is actually a limit on all neutrino masses. Moreover J from cosmology we know [17] that the sum of masses of (practically) stable neutrinos cannot exceed a few eV, say m G:6 eV, corresponding to a fraction of the critical density for neutrino hot J dark matter X h:0.06 (the present value of the reduced Hubble constant h being around 0.7). In J conclusion, the heaviest light neutrinos that are allowed are three nearly degenerate neutrinos of mass around or somewhat below 2 eV. In this case neutrinos would be of cosmological relevance as hot dark matter and contribute a relevant fraction of the critical density. But at present there is no compelling experimental evidence for the necessity of hot dark matter [17,18]. As a consequence, neutrino masses can possibly be much smaller than that. In fact, for widely split neutrino masses the heaviest neutrino would have a mass around &0.06 eV as implied by the atmospheric neutrino frequency. An additional important constraint on neutrino masses, which will be relevant in the following, is obtained from the nonobservation of neutrino-less double beta decay. This is an upper limit on the l Majorana mass, or equivalently, on mCC" ; m , which is at present quoted to be G CG G C J mCC:0.2 eV [19]. J 3. Neutrino masses and lepton number violation Neutrino oscillations imply neutrino masses which in turn demand either the existence of right-handed neutrinos (Dirac masses) or lepton number (¸) violation (Majorana masses) or both. Given that neutrino masses are extremely small, it is really di$cult from the theory point of view to avoid the conclusion that ¸ must be violated. In fact, it is only in terms of lepton number violation that the smallness of neutrino masses can be explained as inversely proportional to the very large scale where L is violated, of order M or even M . %32 .

298


Recall that an ordinary Dirac mass term is of the form m "l m l plus its hermitian conjugate 0* 0 " * m , where l " (1$c )l and l is the neutrino "eld. The "eld l annihilates a left-handed * *0 0* neutrino and creates a right-handed antineutrino. Correspondingly, the "eld l creates a left* handed neutrino and annihilates a right-handed antineutrino. Now left-handed neutrinos l and * right-handed antineutrinos (l) are indeed the only observed neutrino states. Thus, in principle, one 0 could assume that right-handed neutrinos and left-handed antineutrinos do not exist at all. The "eld l and its conjugate l would then su$ce and there would be no l and l "elds in the theory. 0 0 * * In the SM Lagrangian density only the term i l D/ l appears, where D is the SU(2);(1) gauge I * * covariant derivative. Clearly if no l is allowed there is no possible Dirac mass for a neutrino. 0 But if ¸ is violated, there is no conserved quantum number that really makes neutrinos and antineutrinos di!erent and a new type of mass term is possible. For a massive neutrino, the positive helicity state that Lorentz invariance demands to be associated with the state l of given * momentum can be (l) (for a charged particle with mass, say an e\, one can go to the rest frame, 0 * rotate the spin by 1803 and boost again to the original momentum to obtain e\). If neutrinos and 0 antineutrinos are not really distinct particles, both Lorentz and TCP invariance are satis"ed by just allowing l and (l) (TCP changes one into the other at "xed momentum). For a massive charged * 0 particle of spin one needs four states, while only two are enough for an intrinsically neutral particle. If ¸ is violated we can have a Majorana mass term m "(l) ml "l2Cml where * * ** 0 * (l) "Cl2 and C is the 4;4 matrix in Dirac space that implements charge conjugation (the "eld 0 * (l) annihilates a (l) exactly as the "eld l2 does, the transposition only indicating that we want it 0 0 * as a column vector). Clearly, the Majorana mass term m violates ¸ by two units. Also, since in the ** SM l is a weak isospin doublet, m transforms as a component of an isospin triplet. In the * ** following, as we are only interested in #avour indices and not in Dirac indices, we will simply denote m by m "l2ml , omitting the Dirac matrix C. Note that if ¸ is violated and l also ** ** * * 0 exists, then a second type of Majorana mass is also possible which is m "l2ml , where we again 00 0 0 omitted C. Clearly also m violates ¸ by two units, but, since l is a gauge singlet, m is invariant 00 0 00 under the SM gauge group. In conclusion, if l does not exist, we can only have a Majorana mass 0 m if ¸ is violated. If l exists and ¸ is violated, we can have both Dirac m and Majorana masses ** 0 *0 m and m . ** 00 Imagine that one wanted to give masses to neutrinos and, at the same time, avoid the conclusion that lepton number is violated. Then he/she must assume that l exists and that neutrinos acquire 0 Dirac masses through the usual Higgs mechanism as quark and leptons do. Technically this is possible. But there are two arguments against this possibility. The "rst argument is that neutrino masses are extremely small so that the corresponding Yukawa couplings would be enormously smaller than those of any other fermion. Note that within each generation the spread of masses is by no more than a factor 10\. But the spread between the t quark and the heaviest neutrino would exceed a factor of 10. A second argument arises from the fact that once we introduce l in 0 the theory, then the ¸ violating term m "l2ml is allowed in the lagrangian density by the gauge 00 0 0 symmetry. In the minimal SM, i.e. without l , we understand ¸ and B conservation as accidental 0 global symmetries that hold because there is no operator term of dimension 44 that violates B and ¸ but respects the gauge symmetry. For example, the transition u#uPe>#dM is allowed by colour, weak isospin and hypercharge gauge symmetries, but corresponds to a four-fermion operator of dimension 6: O "(j/M)(e2d)(u2u). This term is suppressed by the dimensional factor 1/M. In the assumption that the SM extended by supersymmetry is an e!ective low-energy theory


299

which is valid up to the GUT scale, as suggested by the compatibility of observed low-energy gauge couplings with the notion of uni"cation at M &10 GeV, the large mass M is identi"ed with %32 M . In fact the factor 1/M can be obtained from the propagator of a superheavy intermediate %32 %32 gauge boson with the right quantum numbers (e.g. those of an SU(5) generator). In supersymmetric models with R invariance the status of B and L conservation as an accidental symmetry is maintained. In the presence of l , the dimension 3 operator corresponding to m is gauge 0 00 symmetric but violates ¸. By dimensions we expect a mass factor in front of this operator in the lagrangian density, and in the absence of a protective symmetry, we expect it of the order of the cut-o!, i.e. of order M or larger. Thus, ¸ number violation is naturally induced by the presence %32 of l , unless we enforce it by hand. 0 4. L violation explains the smallness of neutrino masses: the see-saw mechanism Once we accept ¸ violation we gain an elegant explanation for the smallness of neutrino masses as they turn out to be inversely proportional to the large scale where lepton number is violated. If ¸ is not conserved, even in the absence of l , Majorana masses can be generated for neutrinos by 0 dimension "ve operators of the form O "¸2j ¸

/M (2) G GH H with being the ordinary Higgs doublet, j a matrix in #avour space and M a large scale of mass, of order M or M . Neutrino masses generated by O are of the order m &v/M for j &O(1), %32 . J GH where v&O(100 GeV) is the vacuum expectation value of the ordinary Higgs. We consider that the existence of l is quite plausible because all GUT groups larger than 0 SU(5) require them. In particular the fact that l completes the representation 16 of 0 SO(10): 16"5 #10#1, so that all fermions of each family are contained in a single representation of the unifying group, is too impressive not to be signi"cant. At least as a classi"cation group SO(10) must be of some relevance. Thus in the following we assume that there are both l and 0 lepton number violation. With these assumptions the see-saw mechanism [20] is possible. In its simplest form it arises as follows. Consider the mass terms in the lagrangian corresponding to Dirac and RR Majorana masses (for the time being we consider LL Majorana mass terms as comparatively negligible): (3) L"!RM m ¸#RM MRM 2#h.c . " For notational simplicity we denoted l and l by ¸ and R, respectively (the prime de"ning the * 0 l #avour basis, see Eq. (1)). The 3;3 matrices m and M are the Dirac and Majorana mass matrices " in #avour space (M is symmetric, M"M2, while m is, in general, nonhermitian and nonsymmet" ric). We expect the eigenvalues of M to be of order M or more because RR Majorana masses are %32 SU(3);SU(2);;(1) invariant, hence unprotected and naturally of the order of the cuto! of the low-energy theory. Since all l are very heavy we can integrate them away. For this purpose we 0 write down the equations of motion for RM in the static limit, i.e. neglecting their kinetic terms !RL/RRM "m ¸!MRM 2"0 . "

(4)

300


From this, by solving for RM 2 and transposing, we obtain: RM "¸2m2 M\ . " We now replace in the Lagrangian, Eq. (3), this expression for RM and we get

(5)

L"!¸2m ¸ , J where the resulting neutrino mass matrix is

(6)

m "m2 M\m . (7) J " " This is the well-known see-saw mechanism [20] result: the light neutrino masses are quadratic in the Dirac masses and inversely proportional to the large Majorana mass. If some l are massless or 0 light they would not be integrated away but simply added to the light neutrinos. Notice that the above results hold true for any number n of heavy neutral fermions R coupled to the three known neutrinos. In this more general case M is an n;n symmetric matrix and the coupling between heavy and light "elds is described by the rectangular n;3 matrix m . " Here we assumed that the additional non renormalizable terms from O are comparatively negligible, otherwise they should simply be added. After elimination of the heavy right-handed "elds, at the level of the e!ective low-energy theory, the two types of terms are equivalent. In particular, they have identical transformation properties under a chiral change of basis in #avour space. The di!erence is, however, that in the see-saw mechanism, the Dirac matrix m is presum" ably related to ordinary fermion masses because they are both generated by the Higgs mechanism and both must obey GUT-induced constraints. Thus if we assume the see-saw mechanism more constraints are implied. In particular we are led to the natural hypothesis that m has a largely " dominant third family eigenvalue in analogy to m , m and m which are by far the largest masses R @ O among u quarks, d quarks and charged leptons. Once we accept that m is hierarchical it is very " di$cult to imagine that the e!ective light neutrino matrix, generated by the see-saw mechanism, could have eigenvalues very close in absolute value. 5. The neutrino mixing matrix Given the de"nition of the mixing matrix ; in Eq. (1) and the transformation properties of the e!ective light neutrino mass matrix m : J ¸2m ¸"¸2;2m ;¸ , J J , (8) ;2m ;"Diag[e (m , e (m , m ],m J we obtain the general form of m : J m ";m ;2 . (9) J The matrix ; can be parameterized in terms of three mixing angles and one phase, exactly as for the quark mixing matrix < . In addition we have the two phases and that are present !)+ because of the Majorana nature of neutrinos. Thus, in general, nine parameters are added to the SM when nonvanishing neutrino masses are included: three eigenvalues, three mixing angles and three CP violating phases [55]. Maximal atmospheric neutrino mixing and the requirement that the electron neutrino does not participate in the atmospheric oscillations, as indicated by the Superkamiokande [1] and Chooz


301

[15] data, lead directly to the following structure of the ; ( f"e, k, q, i"1, 2, 3) mixing matrix, DG apart from sign convention rede"nitions

c

0

!s

; " s/(2 c/(2 !1/(2 . DG s/(2 c/(2 #1/(2

(10)

This result is obtained by a simple generalization of the analysis of Refs. [21,22] to the case of arbitrary solar mixing angle (s,sin h , c,cos h ): c"s"1/(2 for maximal solar mixing (e.g. for vacuum oscillations sin 2h &0.75) , while sin 2h &4s&5.5;10\ for the small-angle MSW [7] solution. The vanishing of ; guarantees that l does not participate in the atmospheric C C oscillations and the relation "; """; ""1/(2 implies maximal mixing for atmospheric neuI O trinos. Also, in the limit ; "0, all CP violating e!ects vanish and we can neglect the additional C phase parameter generally present in ; : the matrix ; is real and orthogonal and is equal to the DG product of a rotation by p/4 in the 23 plane times a rotation in the 12 plane:

1

0

0

; " 0 1/(2 !1/(2 DG 0 1/(2 1/(2

c

!s

s

c

0

0 1

0

0 .

(11)

Note that we are assuming only two frequencies, given by * Jm!m, * Jm!m . (12) The numbering 1, 2, 3 corresponds to our de"nition of the frequencies and, in principle, may not coincide with the family index although this will be the case in the models that we favour. The e!ective light neutrino mass matrix is given by Eq. (9). We disregard the phases but in the following m can be of either sign. For generic s, using Eq. (9), one "nds

2e

m" d J d

d

d

(13) m /2#e !m /2#e !m /2#e m /2#e with e"(m c#m s)/2, d"(m !m )cs/(2, e "(m s#m c)/2. (14) We see that the existence of one maximal mixing and ; "0 are the most important input that lead C to the matrix form in Eqs. [13,14]. The value of the solar neutrino mixing angle can be left free. While the simple parametrization of the matrix ; in Eq. (10) is quite useful to guide the search for a realistic pattern of neutrino mass matrices, it should not be taken too literally. In particular the data do not exclude a nonvanishing ; element. As already mentioned, the bound by Chooz [15] amounts to C "; ":0.2. Thus neglecting "; " with respect to s in Eq. (10) is not completely justi"ed. Also note that C C in presence of a large hierarchy "m "e\ has provided new limits on the cosmic density 24 of infrared photons and thus to neutrino radiative decays [45]. The envelope of these limits is well approximated by the dashed line in Fig. 2, corresponding to the bottom line in Eq. (6). More restrictive limits obtain for certain neutrino masses above 3 eV from the absence of emission features from several galaxy clusters [55}57] and from the observation of singly ionized helium in the di!use intergalactic medium [58]. For low-mass neutrinos, the m phase-space factor in Eq. (5) is so punishing that the globularJ cluster limit is the most restrictive one for m below a few eV, i.e. in the mass range which today J

324

G.G. Rawelt / Physics Reports 320 (1999) 319}327

Fig. 2. Astrophysical limits on neutrino transition moments. The light-shaded background-radiation limits are from Ressell and Turner [44], the dark-shaded ones from Biller et al. [45] and Ra!elt [46], the dashed line is the approximation formula in Eq. (6), bottom line.

appears favored from neutrino oscillation experiments. Turning this around, the globular-cluster limit implies that radiative decays of low-mass neutrinos do not seem to have observable consequences. Another form of `radiative decaya is the Cherenkov e!ect lPl#c involving the same initial- and "nal-state neutrino. This process is kinematically allowed for photons with u!k(0, which obtains in certain media (for example air or water) or in external magnetic "elds. The neutrino may have an anomalous dipole moment, but there is also a standard-model photon coupling induced by the medium or the external "eld. Thus far it does not look as if the neutrino Cherenkov e!ect had any strong astrophysical or laboratory signi"cance * for a review of the literature see [59].

6. Laboratory limits Laboratory limits on neutrino dipole moments arise from measurements of the l}e-scattering cross section. The current limits are

1.8;10\k

k ( 7.4;10\k J 5.4;10\k

for l [60] , C for l [61] , I for l [62] , O

(7)

see also the Review of Particle Properties [63]. These limits apply also to electric dipole moments and to electric and magnetic transition moments. For example, the limit on k C applies to all J transition moments which connect l to another #avor. It should be noted, however, that the C scattering amplitudes from electric and magnetic dipole moments can interfere destructively, providing a loop-hole from these limits [64].


325

An improvement of the k C limit to something like 3;10\k is to be expected from the MUNU J experiment which has been installed at the Bugey nuclear reactor [65,66]. Other projects aiming at a similar sensitivity are in a much earlier stage of development [67}69]. The only electromagnetic from factor for which laboratory measurements provide more restrictive limits than astrophysical arguments is the l electric charge. If electric charge conservation is C assumed to hold in b processes such as neutron decay, one "nds (8) e C:3;10\e . J This limit is based on a bound for the neutron charge of e "(!0.4$1.1);10\e [70] and on L the neutrality of matter which was found to be e #e "(0.8$0.8);10\e [71]. N C 7. Conclusions The recent evidence for neutrino oscillations from the solar and the atmospheric neutrino anomaly and from the LSND experiment indicate that the neutrino mass di!erences are very small, at most in the eV range. Moreover, the absolute neutrino mass scale cannot exceed a few eV as indicated by the tritium decay limits on the l mass and by cosmological arguments. Therefore, C speculations about neutrino masses far in excess of a few eV are becoming more and more unattractive. If neutrino masses are indeed that small, it is no longer possible to invoke threshold e!ects to avoid the stellar plasmon-decay limits on neutrino dipole moments and electric charges. Moreover, Fig. 2 illustrates that for neutrino masses below about 2 eV the stellar limits on transition moments are more restrictive than those from searches for radiative decays. Turning this around, if neutrino masses are indeed below a few eV one cannot expect neutrino radiative decays to have any observable consequences. The current round of experiments to improve the laboratory limits on k C will not be able to J come even close to the globular-cluster limit so that a positive discovery would indicate extremely serious problems with our understanding of low-mass stars. Barring this unlikely possibility, one cannot hope to discover neutrino dipole moments anytime soon in a laboratory experiment. On the other hand, unless a completely new argument is put forth, the stellar-evolution limits have probably gone about as far as they can, although one could still achieve a signi"cant reduction of their uncertainties. The possibility that neutrino dipole or transition moments in the general 10\k range play an important role in astrophysical environments with large magnetic "elds cannot be ruled out in the foreseeable future. Scenarios with magnetic spin-#avor oscillations in the Sun, supernovae, active galactic nuclei, or the early universe are in no danger of being ruled out anytime soon!

Acknowledgements Partial support by the Deutsche Forschungsgemeinschaft under grant No. SFB-375 is acknowledged.

326


Appendix A: Helioseismological limit The interior of the Sun is a nonrelativistic plasma where the neutrino energy-loss rate per unit volume from transverse-plasmon decay is approximately given by [5]

a u/4p Charge , J . Dipole Moment , (A.1) Q"(8f /3p)¹; (k/2)(u/4p) J . (C G/a)(u/4p) Standard Model . 4 $ . Here, f +1.202 refers to the Riemann Zeta function, a "e /4p is the neutrino "ne-structure J J constant, C the vector}current coupling constant between neutrinos and electrons, G the Fermi 4 $ constant, and u "4pan /m is the plasma frequency with n the electron density. Natural units . C C C with "c"k "1 are used. Longitudinal-plasmon decay is not important for these conditions. Integrating these energy-loss rates over a standard solar model yields ¸ "(e /e)3.2;10 J J ¸ and ¸ "(k /k )6.0;10 ¸ (solar luminosity ¸ ), respectively. Helioseismology requires > J J > > that a new energy-loss channel of the Sun does not exceed about 10% ¸ [30], leading to > e :6;10\e and k :4;10\k . J J The globular-cluster limit on e is not much more restrictive than this result, while one gains a lot J for k . The reason is that the energy-loss rate per unit mass for the e case does not depend on the J J matter density, while for the k case it depends linearly on o. The cores of low-mass red giants before J helium ignition are about 10 times denser than the Sun, explaining the improvement of the k limit. J References [1] W. Pauli, Public letter to the group of the Radioactives at the district society meeting in TuK bingen, in: K. Winter (Ed.), Neutrino Physics, Cambridge University Press, Cambridge, 1930. [2] K. Winter (Ed.), Neutrino Physics, Cambridge University Press, Cambridge, 1991. [3] R.N. Mohapatra, P. Pal, Massive Neutrinos in Physics and Astrophysics, World Scienti"c, Singapore, 1991. [4] J.N. Bahcall, Neutrino Astrophysics, Cambridge University Press, Cambridge, 1989. [5] G.G. Ra!elt, Stars as Laboratories for Fundamental Physics, University of Chicago Press, Chicago, 1996. [6] M.B. Voloshin, M.I. Vysotskimy , Yad. Fiz. 44 (1986) 845 [Sov. J. Nucl. Phys. 44 (1986) 544]. [7] L.B. Okun, Yad. Fiz. 44 (1986) 847 [Sov. J. Nucl. Phys. 44 (1986) 546]. [8] M.B. Voloshin, M.I. Vysotskimy , L.B. Okun, Yad. Fiz. 44 (1986) 677 [Sov. J. Nucl. Phys. 44 (1986) 440]. [9] M.B. Voloshin, M.I. Vysotskimy , L.B. Okun, Zh. Eksp. Teor. Fiz. 91 (1986) 754; (E) ibid. 92 (1987) 368 [Sov. Phys. JETP 64 (1986) 446; (E) ibid. 65 (1987) 209]. [10] L.B. Okun, Yad. Fiz. 48 (1988) 1519 [Sov. J. Nucl. Phys. 48 (1988) 967]. [11] S.I. Blinnikov, L.B. Okun, Pis'ma Astron. Zh. 14 (1988) 867 [Sov. Astron. Lett. 14 (1988) 368]. [12] G.G. Ra!elt, Astrophys. J. 365 (1990) 559; Phys. Rev. Lett. 64 (1990) 2856. [13] G.G. Ra!elt, A. Weiss, Astron. Astrophys. 264 (1992) 536. [14] M. Haft, G.G. Ra!elt, A. Weiss, Astrophys. J. 425 (1994) 222; (E) ibid. 438 (1995) 1017. [15] G. Walther, Phys. Rev. Lett. 79 (1997) 4522. [16] P.A. Sturrock, G. Walther, M.S. Wheatland, Astrophys. J. 491 (1997) 409; ibid. 507 (1998) 978. [17] J. Pulido, Phys. Rep. 211 (1992) 211; Phys. Rev. D 48 (1993) 1492; Phys. Lett. B 323 (1994) 36; Z. Phys. C 70 (1996) 333. [18] E.Kh. Akhmedov, Phys. Lett. B 348 (1995) 124; hep-ph/9705451, 1997. [19] M.M. Guzzo, H. Nunokawa, Astropart. Phys. 12 (1999) 87. [20] H. Athar, J.T. Peltoniemi, A.Yu. Smirnov, Phys. Rev. D 51 (1995) 6647. [21] T. Totani, K. Sato, Phys. Rev. D 54 (1996) 5975.

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Matter}antimatter asymmetry and neutrino properties Wilfried BuchmuK ller *, Michael PluK macher Deutsches Elektronen-Synchrotron DESY, Gruppe Theorie, Notkestrasse 85, D-22603 Hamburg, Germany Department of Physics and Astronomy, University of Pennsylvania, Philadelphia PA 19104, USA

Abstract The cosmological baryon asymmetry can be explained as remnant of heavy Majorana neutrino decays in the early universe. We study this mechanism for two models of neutrino masses with a large l !l mixing I O angle which are based on the symmetries S;(5);;(1) and S;(3) ;S;(3) ;S;(3) ;;(1) , respectively. In $ A * 0 $ both cases B!¸ is broken at the uni"cation scale K . The models make di!erent predictions for the %32 baryogenesis temperature and the gravitino abundance. 1999 Elsevier Science B.V. All rights reserved. PACS: 14.20.!c; 13.35.Hb

1. Baryogenesis and lepton number violation The cosmological matter}antimatter asymmetry, the ratio of the baryon density to the entropy density of the universe, > "(n !n M )/s"(0.6!1);10\ ,

(1)

can in principle be understood in theories where baryon number, C and CP are not conserved [1]. The presently observed value of the baryon asymmetry is then explained as a consequence of the spectrum and interactions of elementary particles, together with the cosmological evolution. A crucial ingredient of baryogenesis is the connection between baryon number (B) and lepton number (¸) in the high-temperature, symmetric phase of the standard model. Due to the chiral nature of the weak interactions B and ¸ are not conserved [2]. At zero temperature this has no observable e!ect due to the smallness of the weak coupling. However, as the temperature approaches the critical temperature ¹ of the electroweak phase transition, B and ¸ violating #5 * Corresponding author. Tel.: 040-89-98-0; fax: 040-89-98-32-82. E-mail address: [email protected] (W. BuchmuK ller) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 5 7 - 5

W. Buchmu( ller, M. Plu( macher / Physics Reports 320 (1999) 329}339

330

processes come into thermal equilibrium [3]. These &sphaleron processes' violate baryon and lepton number by three units, *B"*¸"3 .

(2)

It is generally believed that B and ¸ changing processes are in thermal equilibrium for temperatures in the range ¹#5&100 GeV(¹(¹1.&&10 GeV .

(3)

The non-conservation of baryon and lepton number has a profound e!ect on the generation of the cosmological baryon asymmetry. Eq. (2) suggests that any B#¸ asymmetry generated before the electroweak phase transition, i.e., at temperatures ¹'¹#5, will be washed out. However, since only left-handed "elds couple to sphalerons, a non-zero value of B#¸ can persist in the high-temperature, symmetric phase if there exists a non-vanishing B!¸ asymmetry. An analysis of the chemical potentials of all particle species in the high-temperature phase yields the following relation between the baryon asymmetry > and the corresponding ¸ and B!¸ asymmetries >* and > \*, respectively [4], a > , > "a> \*" a!1 *

(4)

where a is a number O(1). In the standard model with three generations and two Higgs doublets one has a" . We conclude that B!¸ violation is needed if the baryon asymmetry is generated before the electroweak transition, i.e. at temperatures ¹'¹ &100 GeV. In the standard model, as well as #5 its supersymmetric version and its uni"ed extensions based on the gauge group SU(5), B!¸ is a conserved quantity. Hence, no baryon asymmetry can be generated dynamically in these models. The remnant of lepton number violation at low energies is an e!ective *¸"2 interaction between lepton and Higgs "elds, L

" f l2 H C l H #h.c. * GH *G *H

(5)

Such an interaction arises in particular from the exchange of heavy Majorana neutrinos. In the Higgs phase of the standard model, where the Higgs "eld acquires a vacuum expectation value 1H 2"v , it gives rise to Majorana masses of the light neutrinos l , l and l . C I O At "nite temperature the *¸"2 processes described by (5) take place with the rate [5] C (¹)"(1/p)(¹/v) mG . * J GCIO

(6)

In thermal equilibrium this yields an additional relation between the chemical potentials which implies > ">

\*

"> "0 . *

(7)


331

To avoid this conclusion, the *¸"2 interaction (5) must not reach thermal equilibrium. For baryogenesis at a temperature ¹ (¹ &10 GeV, one has to require C (H " , where 1.& * 2 H is the Hubble parameter. This yields a stringent upper bound on Majorana neutrino masses, (8) mG((0.2 eV (¹ /¹ )) . 1.& J GCIO For ¹ &¹ , this bound would be comparable to the upper bound on the electron neutrino mass 1.& obtained from neutrinoless double beta decay. However, Eq. (8) also applies to the q-neutrino mass. Note, that the bound can be evaded if appropriate asymmetries are present for particles which reach thermal equilibrium only at temperatures below ¹ [6]. The connection between lepton number and the baryon asymmetry is lost if baryogenesis takes place at or below the Fermi scale [7}9]. However, detailed studies of the thermodynamics of the electroweak transition have shown that, at least in the standard model, the deviation from thermal equilibrium is not su$cient for baryogenesis [10}12]. In the minimal supersymmetric extension of the standard model (MSSM) such a scenario appears still possible for a limited range of parameters [7}9].

2. Decays of heavy Majorana neutrinos Baryogenesis above the Fermi scale requires B!¸ violation, and therefore ¸ violation. Lepton number violation is most simply realized by adding right-handed Majorana neutrinos to the standard model. Heavy right-handed Majorana neutrinos, whose existence is predicted by all extensions of the standard model containing B!¸ as a local symmetry, can also explain the smallness of the light neutrino masses via the see-saw mechanism [13,14]. The most general Lagrangian for couplings and masses of charged leptons and neutrinos reads (9) L "!h e l H !h l l H !h lA l R#h.c. JGH 0G *H PGH 0G 0H 7 CGH 0G *H The vacuum expectation values of the Higgs "eld 1H 2"v and 1H 2"v "tan bv generate Dirac masses m and m for charged leptons and neutrinos, m "h v and m "h v , respectively, C " C C " J which are assumed to be much smaller than the Majorana masses M"h 1R2. This yields light and P heavy neutrino mass eigenstates lKKRl #lA K, NKl #lA , * * 0 0 with masses 1 m2KH, m KM . m K!KRm , J "M "

(10)

(11)

Here K is a unitary matrix which relates weak and mass eigenstates. The right-handed neutrinos, whose exchange may erase any lepton asymmetry, can also generate a lepton asymmetry by means of out-of-equilibrium decays. This lepton asymmetry is then partially transformed into a baryon asymmetry by sphaleron processes [15]. The decay width of the heavy


332

neutrino N reads at tree level, G C "C(N PH #l)#C(N PHA #lA)"1/8p(h hR) M . (12) "G G G J J GG G From the decay width one obtains an upper bound on the light neutrino masses via the yields the constraint out-of-equilibrium condition [16]. Requiring C (H" " 2+ m "(h hR) v/M (10\ eV . (13) J J More direct bounds on the light neutrino masses depend on the structure of the Dirac neutrino mass matrix. Interference between the tree-level amplitude and the one-loop self-energy and vertex corrections yields CP asymmetries in the heavy Majorana neutrino decays. In a basis, where the right-handed neutrino mass matrix M"h 1R2 is diagonal, one obtains [17}20]. P C(N PlH )!C(N PlAHA ) e " C(N PlH )#C(N PlAHA ) 1 3 M K! Im[(h hR) ] . (14) J J G M 16p (h hR) J J G G Here we have assumed M (M , M , which is satis"ed in the applications considered in the following sections. In the early universe at temperatures ¹&M the CP asymmetry (14) leads to a lepton asymmetry [21}23], e n !n M *"i . (15) > " * * g s H Here the factor i(1 represents the e!ect of washout processes. In order to determine i one has to solve the full Boltzmann equations [24}26]. In the examples discussed below one has iK10\210\.

3. Neutrino masses and mixings The CP asymmetry (14) is given in terms of the Dirac and the Majorana neutrino mass matrices. Depending on the neutrino mass hierarchy and the size of the mixing angles the CP asymmetry can vary over many orders of magnitude. It is therefore interesting to see whether a pattern of neutrino masses motivated by other considerations is consistent with leptogenesis. An attractive framework to explain the observed mass hierarchies of quarks and charged leptons is the Froggatt}Nielsen mechanism [28] based on a spontaneously broken ;(1) generation $ symmetry. The Yukawa couplings arise from non-renormalizable interactions after a gauge singlet "eld U acquires a vacuum expectation value,

h "g GH GH

1U2 /G>/H . K

(16)


333

Table 1 Chiral charges of charged and neutral leptons with S;(5);;(1) symmetry [33] $ t G

eA 0

eA 0

eA 0

l *

l *

l *

lA 0

lA 0

lA 0

Q G

0

1

2

0

0

1

0

1

2

Here g are couplings O(1) and Q are the ;(1) charges of the various fermions with QU"!1. The GH G interaction scale K is expected to be very large, K'K . In the following we shall discuss two %32 di!erent realizations of this idea which are motivated by the recently reported atmospheric neutrino anomaly [27]. Both scenarios have a large l !l mixing angle. They di!er, however, by I O the symmetry structure and by the size of the parameter e which characterizes the #avour mixing. 3.1. S;(5);;(1) $ This symmetry has been considered by a number of authors [32]. Particularly interesting is the case with a nonparallel family structure where the chiral ;(1) charges are di!erent for the 5H-plets $ and the 10-plets of the same family [29}31]. An example of possible charges Q is given in Table 1. G The assignment of the same charge to the lepton doublets of the second and third generation leads to a neutrino mass matrix of the form [29,30],

e e

m GH& e J e

e

1 1 1 1

v . 1R2

(17)

This structure immediately yields a large l !l mixing angle. The phenomenology of neutrino I O oscillations depends on the unspeci"ed coe$cients O(1) [34,35]. The parameter e which gives the #avour mixing is chosen to be 1 1U2/K"e& . 17

(18)

The three Yukawa matrices for the leptons have the structure,

e e e

h & e e C e 1

e e e

e , h & e e J 1 e 1

e

e , h & e P 1 e

e e e e e

1

.

(19)

Note, that h and h have the same, non-symmetric structure. One easily veri"es that the mass C J ratios for charged leptons, heavy and light Majorana neutrinos are given by m : m : m &e : e : 1, M : M : M &e : e : 1 , C I O m : m : m &e : 1 : 1 .

(20) (21)


334

Table 2 Chiral charges of charged and neutral leptons with S;(3) ;S;(3) ;S;(3) ;;(1) symmetry [32] A * 0 $ t G

eA 0

eA 0

eA 0

l *

l *

l *

lA 0

lA 0

lA 0

Q G

0

0

0

The masses of the two eigenstates l and l depend on unspeci"ed factors of order one, and may I O easily di!er by an order of magnitude [35]. They can therefore be consistent with the mass di!erences *mC IK4;10\!1 10\ eV [37] inferred from the MSW solution of the solar JJ neutrino problem [38,39] and *mI OK(5;10\!6;10\) eV associated with the atmospheric JJ neutrino de"cit [27]. For numerical estimates we shall use the average of the neutrino masses of the second and third family, m "(m Im O)&10\ eV. Note, that for a di!erent choice of ;(1) J J J charges the coe$cients in Eq. (17) automatically yield the hierarchy m /m &e [40]. The choice of the charges in Table 1 corresponds to large Yukawa couplings of the third generation. For the mass of the heaviest Majorana neutrino one "nds M &v/m &10 GeV . J

(22)

This implies that B!¸ is broken at the uni"cation scale K . %32 3.2. S;(3) ;S;(3) ;S;(3) ;;(1) A * 0 $ This symmetry arises in uni"ed theories based on the gauge group E . The leptons eA , l and 0 * lA are contained in a single (1, 3, 3 ) representation. Hence, all leptons of the same generation have 0 the same ;(1) charge and all leptonic Yukawa matrices are symmetric. Masses and mixings of $ quarks and charged leptons can be successfully described by using the charges given in Table 2 [32]. Clearly, the three Yukawa matrices have the same structure,

e

e

e

e

e

e

e e , h & e e e . h , h & e J C P e e 1 e e 1

(23)

Note, that the expansion parameter in h is di!erent from the one in h and h . From the quark J C P masses, which also contain e and e , one infers e Ke [32]. From Eq. (23) one obtains for the masses of charged leptons, light and heavy Majorana neutrinos, m : m : m &M : M : M &e : e : 1 , C I O

(24)

m : m : m &e : e : 1 .

(25)

Note, that with respect to Ref. [32], e and e have been interchanged.


335

Like in the example with S;(5);;(1) symmetry, the mass of the heaviest Majorana neutrino, $ M &v/m &10 GeV , (26) implies that B!¸ is broken at the uni"cation scale K . %32 The l !l mixing angle is mostly given by the mixing of the charged leptons of the second and I O third generation [32], sin H &(e#e . IO This requires large #avour mixing,

(27)

(1U2/K)"(e& . (28) In view of the unknown coe$cients O(1) the corresponding mixing angle sin H &0.7 is consistent IO with the interpretation of the atmospheric neutrino anomaly as l !l oscillation. I O It is very instructive to compare the two scenarios of lepton masses and mixings described above. In the "rst case, the large l !l mixing angle follows from a nonparallel #avour symmetry. The I O parameter e, which characterizes the #avour mixing, is small. In the second case, the large l !l I O mixing angle is a consequence of the large #avour mixing e. The ;(1) charges of all leptons are the $ same, i.e., one has a parallel family structure. Also the mass hierarchies, given in terms of e, are rather di!erent. This illustrates that the separation into a #avour mixing parameter e and coe$cients O(1) is far from unique. It is therefore important to study other observables which depend on the lepton mass matrices. A particular example is the baryon asymmetry.

4. Matter antimatter asymmetry We can now evaluate the baryon asymmetry for the two patterns of neutrino mass matrices discussed in the previous section. A rough estimate of the baryon asymmetry can be obtained from the CP asymmetry e of the heavy Majorana neutrino N . A quantitative determination requires a numerical study of the full Boltzmann equations [25,26]. 4.1. S;(5);;(1) $ In this case one obtains from Eqs. (14) and (19), e &(3/16p)e . From Eq. (15), e&1/300 (18) and g &100 one then obtains the baryon asymmetry, H > &i10\ .

(29)

(30)

For i&0.120.01 this is indeed the correct order of magnitude. The baryogenesis temperature is given by the mass of the lightest of the heavy Majorana neutrinos, ¹ &M &eM &10 GeV . (31) This set of parameters, where the CP asymmetry is given in terms of the mass hierarchy of the heavy neutrinos, has been studied in detail [36]. The generated baryon asymmetry does not depend


336

Fig. 1. Time evolution of the neutrino number density and the lepton asymmetry in the case of the S;(5);;(1) $ symmetry. The solid line shows the solution of the Boltzmann equation for the right-handed neutrinos, while the corresponding equilibrium distribution is represented by the dashed line. The absolute value of the lepton asymmetry > is given by the dotted line and the hatched area shows the lepton asymmetry corresponding to the observed baryon * asymmetry.

on the #avour mixing of the light neutrinos. The l !l mixing angle is large in the scenario I O described in the previous section whereas it was assumed to be small in [36]. The solution of the full Boltzmann equations is shown in Fig. 1 for the non-supersymmetric case [36]. The initial condition at a temperature ¹&10M is chosen to be a state without heavy neutrinos. The Yukawa interactions are su$cient to bring the heavy neutrinos into thermal equilibrium. At temperatures ¹&M this is followed by the usual out-of-equilibrium decays which lead to a non-vanishing baryon asymmetry. The "nal asymmetry agrees with the estimate (30) for i&0.1. The change of sign in the lepton asymmetry is due to the fact that inverse decay processes, which take part in producing the neutrinos, are CP violating, i.e. they generate a lepton asymmetry at high temperatures. Due to the interplay of inverse decay processes and lepton number violating scattering processes this asymmetry has a di!erent sign than the one produced by neutrino decays at lower temperatures. 4.2. S;(3) ;S;(3) ;S;(3) ;;(1) A * 0 $ In this case the neutrino Yukawa couplings (23) yield the CP asymmetry e &(3/16p)e ,

(32)


337

Fig. 2. Solution of the Boltzmann equations in the case of the S;(3) ;S;(3) ;S;(3) ;;(1) symmetry. A * 0 $

which correspond to the baryon asymmetry (cf. (15)) > &i 10\ .

(33)

Due to the large value of e the CP asymmetry is two orders of magnitude larger than in the case with S;(5);;(1)$ symmetry. However, washout processes are now also stronger. The solution of the Boltzmann equations is shown in Fig. 2. The "nal asymmetry is again > &10\ which now corresponds to i&10\. The baryogenesis temperature is considerably larger than in the "rst case, ¹ &M&eM&10 GeV .

(34)

The baryon asymmetry is largely determined by the parameter m de"ned in Eq. (13) [25]. In the "rst example, one has m &m J. In the second case one "nds m &m. Since m J and m are rather similar it is not too surprising that the generated baryon asymmetry is about the same in both cases.

5. Conclusions Detailed studies of the thermodynamics of the electroweak interactions at high temperatures have shown that in the standard model and most of its extensions the electroweak transition is too weak to a!ect the cosmological baryon asymmetry. Hence, one has to search for baryogenesis mechanisms above the Fermi scale.


338

Due to sphaleron processes baryon number and lepton number are related in the high-temperature, symmetric phase of the standard model. As a consequence, the cosmological baryon asymmetry is related to neutrino properties. Baryogenesis requires lepton number violation, which occurs in extensions of the standard model with right-handed neutrinos and Majorana neutrino masses. Although lepton number violation is needed in order to obtain a baryon asymmetry, it must not be too strong since otherwise any baryon and lepton asymmetry would be washed out. This leads to stringent upper bounds on neutrino masses which depend on the particle content of the theory. The solar and atmospheric neutrino de"cits can be interpreted as a result of neutrino oscillations. For hierarchical neutrinos the corresponding neutrino masses are very small. Assuming the see-saw mechanism, this suggests the existence of very heavy right-handed neutrinos and a large scale of B!¸ breaking. It is remarkable that these hints on the nature of lepton number violation "t very well together with the idea of leptogenesis. For hierarchical neutrino masses, with B!¸ broken at the uni"cation scale K%32&10 GeV, the observed baryon asymmetry > &10\ is naturally explained by the decay of heavy Majorana neutrinos. Although the observed baryon asymmetry imposes important constraints on neutrino properties, other observables are needed to discriminate between di!erent models. The two examples considered in this paper predict di!erent baryogenesis temperatures. Correspondingly, in supersymmetric models the predictions for the gravitino abundance are di!erent [40}44]. In the case with S;(5);;(1)$ symmetry, stable gravitinos can be the dominant component of cold dark matter [44]. The models make also di!erent predictions for the rate of lepton #avour changing radiative corrections. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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A.D. Dolgov, Ya.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1. E.W. Kolb, S. Wolfram, Nucl. Phys. B 172 (1980) 224. E.W. Kolb, S. Wolfram, Nucl. Phys. B 195 (1982) 542(E). M.A. Luty, Phys. Rev. D 45 (1992) 455. M. PluK macher, Z. Phys. C 74 (1997) 549. M. PluK macher, Nucl. Phys. B 530 (1998) 207. Y. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 81 (1998) 1562. C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B 147 (1979) 277. J. Sato, T. Yanagida, Talk at Neutrino'98, hep-ph/9809307. P. Ramond, Talk at Neutrino'98, hep-ph/9809401. J. Bijnens, C. Wetterich, Nucl. Phys. B 292 (1987) 443. S. Lola, G.G. Ross, hep-ph/9902283. W. BuchmuK ller, T. Yanagida, Phys. Lett. B 445 (1999) 399. F. Vissani, JHEP 11 (1998) 025. N. Irges, S. Lavignac, P. Ramond, Phys. Rev. D 58 (1998) 035003. W. BuchmuK ller, M. PluK macher, Phys. Lett. B 389 (1996) 73. N. Hata, P. Langacker, Phys. Rev. D 56 (1997) 6107. S.P. Mikheyev, A.Y. Smirnov, Nuovo Cim. 9C (1986) 17. L. Wolfenstein, Phys. Rev. D 17 (1978) 2369. G. Altarelli, F. Feruglio, JHEP 11 (1998) 021; hep-ph/9812475. M.Yu. Khlopov, A.D. Linde, Phys. Lett. B 138 (1984) 265. J. Ellis, J.E. Kim, D.V. Nanopoulos, Phys. Lett. B 145 (1984) 181. T. Moroi, H. Murayama, M. Yamaguchi, Phys. Lett. B 303 (1993) 289. M. Bolz, W. BuchmuK ller, M. PluK macher, Phys. Lett. B 443 (1998) 209.

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Comments on CP, T and CPT violation in neutral kaon decays John Ellis *, N.E. Mavromatos CERN, Theory Division, CH-1211 Geneva 23, Switzerland University of Oxford, Department of Physics, Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK

Abstract We comment on CP, T and CPT violation in the light of interesting new data from the CPLEAR and KTeV Collaborations on neutral kaon decay asymmetries. Other recent data from the CPLEAR experiment, constraining possible violations of CPT and the *S"*Q rule, exclude the possibility that the semileptonicdecay asymmetry A measured by CPLEAR could be solely due to CPT violation, con"rming that their data 2 constitute direct evidence for T violation. The CP-violating asymmetry in K Pe\e>n\n> recently mea* sured by the KTeV Collaboration does not by itself provide direct evidence for T violation, but we use it to place new bounds on CPT violation. 1999 Elsevier Science B.V. All rights reserved. PACS: 13.20.Eb; 11.30.Er

1. Introduction Ever since the discovery of CP violation in KP2p decay by Christenson et al. [1], its * understanding has been a high experimental and theoretical priority. Until recently, mixing in the K!KM mass matrix was the only known source of CP violation, since it was su$cient by itself to explain the observations of CP violation in other K decays, no CP violation was seen in 1* experiments on K!, charm or B-meson decays, and searches for electric dipole moments only gave upper limits [2]. There has, in parallel, been active discussion whether the observed CP violation should be associated with the violation of T or CPT [3]. Stringent upper limits on CPT violation

* Corresponding author. E-mail address: [email protected] (J. Ellis) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 5 8 - 7

342

J. Ellis, N.E. Mavromatos / Physics Reports 320 (1999) 341}354

[4] in the K!KM system have been given [5], in accord with the common theoretical prejudice based on a fundamental theorem in quantum "eld theory [6]. This suggests strongly that T must be violated, but, at least until recently, there was no direct observation of T violation. An indirect demonstration of T violation in neutral kaons, based on a phenomenological analysis of CPviolating amplitudes, was made in 1970 using data on the decay of long- and short-lived kaons into two neutral pions [7]. However, that analysis assumed unitarity, namely that kaons disappeared only into the observed states. The accumulation of experimental observations of CP and T violation has accelerated abruptly in the past few months. There have been two results on K, KM decays for which interpretations as direct observations of T violation have been proposed. One is an asymmetry in pp annihilation, pp PK\n>K or K>n\KM [8], and the other is a T-odd angular asymmetry in KPn>n\e>e\ * decay [9]. More recently, a tantalizing hint has been presented that CP may be violated at a high level in BPJ/tK decays [10]. Most recent of all, a previous measurement of direct CP violation 1 in the amplitudes for K P2p decays [11] has now been con"rmed by the KTeV Collaboration 1* [9], providing an improved determination of a second independent CP-violating experimental number, namely e/e, to test theories and to discriminate between them. One casualty of this measurement of e/e has been the superweak theory [12], according to which all CP violation should be ascribed to mass mixing in the K!KM system. Still surviving is the Kobayashi}Maskawa model of weak charged-current mixing within the Standard Model with six quarks [13]. Indeed, the new KTeV result arrives 23 years after e/e was "rst calculated within the Kobayashi}Maskawa model [14], and it was pointed out that this would be a (di$cult) way to discriminate between this and the superweak theory, providing (at least part of ) the motivation for this experiment. Coincidentally, the value estimated there agrees perfectly with the current world average for e/e, although many new diagrams and numerical improvements have intervened [15]. The latest theoretical wisdom about the possible value of e/e within the Standard Model is consistent with the value measured, at least if the strange-quark mass is su$ciently small [16]. Thus e/e does not cry out for any extension of the Standard Model, such as supersymmetry [17], though this cannot be excluded. It is not the purpose of this article to review in any detail the potential signi"cance of the e/e measurements, or of the hint of a CP-violating asymmetry in BPJ/tK . Rather, we wish to 1 comment on the suggested interpretations of the asymmetry in pp PK\n>K and K>n\KM [8], and of the T-odd angular asymmetry in KPn>n\e>e\ as possible direct evidence for T violation * [9]. We argue that the former can indeed be interpreted in this way, when combined with other CPLEAR data constraining the possible violation of the *S"*Q rule and CPT violation in semileptonic K decays [18,19]. We use the KPn>n\e>e\ decay asymmetry as a novel test of * CPT invariance in decay amplitudes, though one that may not yet be comparable in power with other tests of CPT. The layout of this article is as follows: in Section 2 we "rst introduce the semileptonic-decay asymmetry recently measured by CPLEAR, then in Section 3 we introduce a density-matrix description that includes a treatment of unstable particles as well as allowing for the possibility of stochastic CPT violation [20}22]. In Section 4 we apply this framework to show that the CPLEAR asymmetry cannot be due to CPT violation, and is indeed a direct observation of T violation. We also comment whether other examples of CP violation can be mimicked by CPT violation [22]. Then, in Section 5 we analyze the decay asymmetry observed by the KTeV collaboration, arguing


343

that it does not have an unambiguous interpretation as a direct observation of T violation. It could not be due to CPT violation in the mass-mixing matrix, but could in principle be due to &direct' CPT violation in a decay amplitude.

2. The CPLEAR asymmetry in pp PK\p>K and K>p\KM We "rst recall brie#y the key features of the asymmetry A observed by CPLEAR, motivating its 2 interpretation as direct evidence for T violation. The essential idea is to look for a violation of reciprocity in the rates for KPKM and the time-reversed reaction KM PK, denoted by P M and )) P M , respectively, as expressed in the asymmetry )) A ,!(P M !P M )/(P M #P M ) . )) )) )) 2 ))

(1)

CPLEAR has the unique capability to tag the initial K or KM by observing an accompanying K!n8 pair in a pp annihilation event. However, it is also necessary to tag the K or KM at some later time, which CPLEAR accomplishes using semileptonic decays, and constructing the observable asymmetry [8] R[n>K\n>e\l]!R[n\K>n\e>l] A ,! R[n>K\n>e\l]#R[n\K>n\e>l]

(2)

where rates are denoted by R. If one assumes the *S"*Q rule, whose validity has been con"rmed independently by CPLEAR (see below), then Eq. (2) may be re-expressed as P M (q)BR[KPn\e>l]!P M (q)BR[KM Pn>e\l] )) A " )) P M (q)BR[KPn\e>l]#P M (q)BR[KM Pn>e\l] )) ))

(3)

where decay branching ratios are denoted by BR. In our discussion below, we consider both the cases where the *S"*Q rule is assumed and where it is relaxed. If one assumes CPT invariance in the semileptonic-decay amplitudes, as was done in the CPLEAR analysis [8], then A "A and the asymmetry observed by CPLEAR can be interpreted 2 as T violation. Some doubts about this interpretation have been expressed [24], apparently based on concerns about the inapplicability of the reciprocity arguments of [25] to unstable particles. We do not believe this to be a problem, since the analysis of [25] can be extended consistently to include unstable particles [22,23,26]. However, it has also been proposed [26] that one might be able to maintain T invariance, P M "P M , interpreting the asymmetry observed by CPLEAR instead as CPT violation in the )) )) semileptonic-decay amplitudes [26]. This interpretation of the CPLEAR result would be more exciting than the conventional one in terms of T violation. It was suggested in [26] that this hypothesis of CPT violation could be tested in the semileptonic decays K Pnll. However, the 1

A recent theoretical discussion using the *S"*Q rule and CPT invariance is given in [23].

344


hypothesis of [26] can, in fact, already be excluded by other published CPLEAR data, as we see below.

3. Density-matrix formalism Before discussing this in more detail, we review the density-matrix formalism [22], which is a convenient formalism for treating unstable particles, and enables us to present a uni"ed phenomenological analysis including also the possibility of stochastic CPT violation associated with a hypothetical open quantum-mechanical formalism associated with some approaches to quantum gravity [27,20,28,21]. In fact, as we recall below, this formalism has already been used in the Appendix of [22] to discard the possibility that CP violation in the neutral-kaon system could be &mimicked' by the CPT-violating mass-matrix parameter d within conventional quantum mechanics. As we discuss later, this was possible only if Re(d)&(1.75$0.7);10\ .

(4)

This analysis is also reviewed brie#y below, taking into account recent data of the CPLEAR collaboration [18] on Re(d), which were not available at the time of writing of [22], and exclude possibility (4). When one considers an unstable-particle system in isolation, without including its decay channels, its time-evolution is non-unitary, so one uses a non-Hermitean e!ective Hamiltonian: HOHR. The temporal evolution of the density matrix, o, is given within the conventional quantum-mechanical framework by R o"!i(Ho!oHR) . (5) In the case of the neutral-kaon system, the phenomenological Hamiltonian contains the following Hermitean (mass) and anti-Hermitean (decay) components: (M#dM)!i(C#dC) MH !iCH , (6) (M!dM)!i(C!dC) M !iC in the (K, KM ) basis. The dM and dC terms violate CPT. Following [20], we de"ne components of o and H by

H"

o,o p , H,h p , a"0, 1, 2, 3 , (7) ? ? ? ? in a Pauli p-matrix representation: since the density matrix must be Hermitean, the o are real, but ? the h are complex in general. @ We may represent conventional quantum-mechanical evolution by R o "H o , in the (K, KM ) ? ?@ @ basis and allowing for the possibility of CPT violation, where

Im h

Im h

Im h Im h H , ?@ Im h Re h Im h !Re h

Im h Im h !Re h Re h . Im h !Re h Re h Im h

(8)


345

It is convenient for the rest of our discussion to transform to the K "(1/(2) (KGKM ) basis, corresponding to p p , p !p , in which H becomes ?@ !C !dC !Im C !Re C !C !2Re M !2Im M !dC . (9) H " ?@ !Im C 2Re M !C !dM !Re C !2Im M dM !C The corresponding equations of motion for the components of o in the K basis are given in [22]. The CP-violating mass-mixing parameter e and the CPT-violating mass-mixing parameter d are given by

e"(Im M )/(1/2"*C"#i*m)""e"e\ (,

d"! (!1/2dC#idM)/(1/2"*C"#i*m) . (10)

One can readily verify [22] that o decays at large t to o&e\C*R

1

eH#dH

e#d

"e#d"

,

(11)

which has a vanishing determinant, thus corresponding to a pure long-lived mass eigenstate K , * whose state vector is "K 2J(1#e!d)"K2!(1!e#d)"KM 2 . * Conversely, in the short-time limit a K state is represented by 1 "e!d" e!d o&e\C1R , eH!dH 1

(12)

(13)

which also has zero determinant and hence represents a pure state "K 2J(1#e#d)"K2#(1!e!d)"KM 2 . (14) 1 Note that the relative signs of the d terms have reversed between (11) and (13): this is the signature of mass-matrix CPT violation in the conventional quantum-mechanical formalism, as seen in the state vectors (12) and (14). The di!erential equations for the components of o may be solved in perturbation theory in "e" and the new parameters dMY ,dM/"*C",

dCY ,dC/"*C" .

(15)

We follow here the conventions of [22], which are related to the notation used elswhere [29] for CP- and CPT-violating parameters by e"!eH , d"!DH, with H denoting complex conjugation. Thus the superweak angle +

de"ned in [29] is related to the angle in (10) by " !n, so that tan "tan "2*m/"*C".

346


To "rst order, one "nds [22] o"!2"X""o (0)"[e\C*R cos( ! ! )!e\CR cos(*mt# ! ! )] , 6Y 6Y o"!2"X""o (0)"[e\C1R cos( # # )!e\CR cos(*mt! ! ! )] , 6 6 o"o (0)"X"e\ (>(6[e\C*R!e\C> KR ]#o (0)"X"e (\(6Y[e\C1R!e\C> KR ] , where the two complex constants X and X are de"ned by: Y , Y /("e"# cos dCY ) , X""e"# cos dCY #i cos dM tan "cos dM 6 Y , Y /("e"! cos dCY ) . tan "cos dM X""e"! cos dCY #i cos dM 6Y The special case that occurs when dM"0 and "e""0, namely

(16)

(17)

dC'0 : "0, "p 6 6Y (18) dC(0 : "p, "0 . 6 6Y will be of particular interest for our purposes. With the results for o through "rst order, and inserting the appropriate initial conditions [22], we can immediately write down expressions for various observables [22] of relevance to CPLEAR. The values of observables O are given in this density-matrix formalism by expressions of the G form [20] 1O 2,Tr[O o] , (19) G G where the observables O are represented by 2;2 Hermitean matrices. Those associated with the G decays of neutral kaons to 2p, 3p and pll "nal states are of particular interest to us. If one assumes the *S"*Q rule, their expressions in the K basis are 0 0 1 0 , O J , O J p p 0 1 0 0

1 1 , O \> J p J J 1 1

1 !1 O >\J . p J J !1 1

(20)

which constitute a complete Hermitean set. We consider later the possible relaxation of the *S"*Q rule, and also the possibility of direct CPT violation in the observables (20), which would give them di!erent normalizations. The small experimental value of e/e would be taken into account by di!erent magnitudes for O in the charged and neutral modes, but we can neglect this p re"nement for our purposes. In this formalism, pure K or KM states, such as those provided as initial conditions in the CPLEAR experiment, are described by the following density matrices:

1 1 1 , o " ) 2 1 1

!1 1 1 o M " . ) 2 !1 1

(21)

We note the similarity of the above density matrices (21) to the representations (20) of the semileptonic decay observables, which re#ects the strange-quark contents of the neutral kaons and our assumption of the validity of the *S"*Q rule: K U s Pu l>l, KM U sPul\l.


347

4. Interpretation of the CPLEAR asymmetry In the CPLEAR experiment [8], the generic quantities measured are asymmetries of decays from an initially pure K beam as compared to the corresponding decays from an initially pure KM beam R(KM PfM )!R(K Pf ) R R , (22) A(t)" R(KM PfM )#R(K Pf ) R R where R(K Pf ),Tr[O o(t)] denotes the decay rate into a "nal state f, starting from a pure R D K at t"0: o(t"0) is given by the "rst matrix in Eq. (21), and correspondingly, R(KM PfM ), Tr[O M o(t)] denotes the decay rate into the conjugate state fM , starting from a pure KM at t"0: D o(t"0) is given by the second matrix in Eq. (21). Several relevant asymmetries were de"ned in [22], including A (already introduced above), A , A and A . We discuss below their possible roles 2 !.2 p p in discriminating between CP- and CPT-violating e!ects, in particular when CPT violation is invoked so as to mimic CP violation whilst preserving T invariance [26]. In order to parametrize a possible CPT-violating di!erence in semileptonic-decay amplitudes as postulated there, we de"ne y: 1n>e\l"T"KM 2,(1#y)1n\e>l"T"K2

(23)

and we assume that y is real, which is justi"ed if the amplitude is T invariant [29]. We assume this here because the purpose of this analysis is to test the hypothesis [26] that the CPLEAR asymmetry can be reproduced by CPT violation alone, retaining T invariance in the mixing: P M "P M . Another important point [8] is the independence of the asymmetry A measured at 2 )) )) late times of any possible violation of *S"*Q rule. As seen from [8], violations of this rule may be taken into account simply by introducing the combination y ,y#2Re(x ) \ where, in the notation of [29,8], x parametrizes violations of the *S"*Q rule: \ 1n>e\l"T"K2,c#d, 1n\e>l"T"KM 2,cH!dH

(24)

(25)

and x,(cH!dH)/(a#b), x H,(c#d)/(aH!bH), and x ,(x$x )/2. Again, the hypothesis of ! T invariance implies the reality of x, x , x [29]. If one considers violations of *S"*Q rule, one ! should take appropriate account of the additional decay modes (25) in A (3). In the density-matrix formalism, yO0 corresponds to a di!erence in normalization between the semileptonic observables O introduced in (20). The analysis of [26], extended in the above pJJ straightforward way to take into account of possible violations of the *S"*Q rule, shows that, if one imposes reciprocity, then A K!y to lowest order in y .

(26)

For clarity and completeness, we note the following relation between the quantity y de"ned above and the quantity y de"ned in [8]: y"2y,!2b/a to lowest order in y and for real a, b [29]: 1n>e\l "T"KM 2"aH!bH, 1n\e>l"T"K2"a#b.

348


To make contact with the experimental measurement of the CPLEAR collaboration, one should take into account the di!erent normalizations of the K and KM #uxes at the production point. Because of this e!ect, the measured asymmetry [8] becomes A"A !y . 2 The measured [8] value of this asymmetry is

(27)

AK(6.6$1.3 );10\ . (28) 2 If this experimental result were to be interpreted as expressing CPT violation but T invariance, then y should have the value y "!(3.3$0.7);10\ .

(29)

Such a scenario is excluded by the current CPLEAR value of y [19]. The late-time asymmetry measured by CPLEAR can be expressed as [8] AK4Re(e)!2Re(y ) . 2 This enables a stringent upper limit to be placed [19]

(30)

y "Re(y)#Re(x )"(0.2$0.3 );10\ . (31) \ Therefore, the CPT-violating but T-conserving hypothesis is conclusively excluded independently of any assumption about the validity of the *S"*Q rule. As a side-remark, we comment on the e!ect of y on the CPT-violating width di!erence dC, assuming the validity of the *S"*Q rule (x "0), which is supported by [19]. Using \ C* "0.39;1/q K8;10\ GeV and the value (29) for y, and neglecting any possible other * CPT-violating di!erences in decay rates, we "nd dCK1.06;10\ GeV

(32)

which makes the following contribution to Re(d): 2Re(d)"dC"*C "/("*C "#4"*m ")"(dC cos / "*C ")K6.8;10\ ,

(33)

where we have used K43.493 mod p. This contribution is far below the present experimental sensitivity discussed below. Next, we comment on the possibility that what we usually regard as CP violation in the mass matrix is actually due to CPT violation. In such a case, one would have to set "e"P0 and make the following choices for the CPT-violating mixing parameters mimic CP violation:

dM"0,

dCY P2"e"/cos ,

(34)

On account of (18), then, the observable A would have the following time-independent "rst-order 2 expression: A "2"X" cos( ! )#2"X" cos( # )"4"e" cos , (35) 2 6Y 6 which is identical to the conventional case of CPT symmetry. However, this is not the case for all observables, for instance the A asymmetry, de"ned by setting f"p\e>l, fM "p>e\l in Eq. (22). !.2


In particular, one has the following asymptotic formula for A : !.2 Y Y A P4 sin cos dM!2 cos dC , !.2 which would yield the following asymptotic prediction under the &mimic' assumption (34)

349

(36)

A P!4"e" cos , (37) !.2 to be contrasted with the standard result that A "0 in the absence of CPT violation. !.2 For comparison with experimental data of CPLEAR, it is useful to express the conventional CPT-violating parameter d (10) in terms of dCY : Im(d)"!dCY sin cos . (38) Re(d)"dCY cos ""e" cos '0, The experimental asymmetry A (27), then, would be obtained upon the identi"cation of A 2 2 in (35) with A , A"4Re(d)!y (39) 2 Note that, in principle, such a situation is consistent with the experimental data, given that the combination A#y "4Re(d)'0. Taking into account (28), (31) and (39) we observe that the 2 mimic requirement would imply Re(d) &(1.75$0.7);10\ . (40)

However, the CPLEAR Collaboration has measured [18] Re(d) using the asymptotic value of the asymmetry A : B RM !R (1#4Ree ) RM !R (1#4Ree ) > * , \ *# \ (41) A, > B RM #R (1#4Ree ) RM #R (1#4Ree ) \ > * > \ * which asymptotes at large times to !8Re(d), independently of any assumption on the *S"*Q rule: Re(d)K(3.0$3.3 $0.6 );10\ , (42) in apparent con#ict with (40). The fact that the CP violation seen in the mass matrix cannot be mimicked by CPT violation [4] has been known for a long time. The possible magnitude of CPT violation is constrained in particular by the consistency between

and the superweak phase . However, it is possible to >\ mimic CP violation in any particular observable by a suitable choice of d. For example, as was shown in [22], the standard superweak result for A may be reproduced by setting "e"P0 and p using (34), which give "X"P"e" and "0. The standard CP-violating result for A may also 6 p obtained with the choices (34) [22], which give "X"P"e" and "p, since tan( !p)"tan . But, 6Y as already emphasized, the dynamical equations determining the density matrix prevent all observables from being mimicked in this way: this is what we found above with the A observable !.2 (37), to be contrasted with the standard result A "0. Moreover, as mentioned above, the mimic !.2 hypothesis is excluded by the recent CPLEAR result (42). It was also pointed out previously [22,30] that deviations from conventional closed-system quantum mechanics of the type discussed in [20], which lead to stochastic CPT violation, also cannot account for the CP violation observed in the neutral kaon system. We remind the reader

350


that generic possible deviations from closed-system quantum-mechanical evolution in the neutral kaon system } which might arise from quantum gravity or other stochastic forces } may be described by the three real parameters a, b, c of [20], if one assumes energy conservation and dominance by *S"0 stochastic e!ects. These parameters lead to entropy growth, corresponding to the appearance of an arrow of time and violation of CPT [31], as has sometimes been suggested in the context of a quantum theory of gravity. However, this CPT violation cannot be cast in the conventional quantum-mechanical form discussed above. The most stringent bounds on the stochastic CPT-violating parameters a, b, c have been placed by the CPLEAR collaboration [32]. They are not far from the characteristic magnitude O(M /M ), where the Planck mass ) . M K10 GeV, near the scale at which such e!ects might "rst set in [21] if they are due to . quantum-gravitational e!ects.

5. The KTeV asymmetry in K Pe\e>p\p> and its interpretation * Subsequent to the CPLEAR analysis, the KTeV Collaboration has reported [9] a novel measurement of a T-odd asymmetry in the decay of K Pe\e>p\p>. Since incoming and * outgoing states are not exchanged in the KTeV experiment, unlike the CPLEAR measurement comparing KM PK and KPKM transitions, it cannot provide direct evidence for T violation. However, it is interesting to discuss the information this measurement may provide about CP, T and CPT symmetry. This decay has previously been analyzed theoretically in [33], assuming CPT symmetry. The decay amplitude was decomposed as M(K Pp>p\e>e\)"M #M #M #M4#M * + # 1" !0

(43)

and the various parts of the amplitude (43) have the following interpretations: E M is the amplitude for the Bremsstrahlung process related to the standard CP-violating K P2p amplitude, violating CP just like the conventional e parameter. This amplitude is * proportional to a coupling constant [33] g "g e B+) , >\

(44)

where g is the conventional CP-violating parameter, whose phase

is that of K Pp>p\: >\ >\ * d (M ) is the relevant I"0p>p\ phase shift. ) E M is the magnetic-dipole contribution to the amplitude, which is CP-conserving. The + corresponding coupling constant has a non-trivial phase [33]: g "i"g "e BKpp>BP , + +

(45)

where d is the pp P-wave phase shift. The amplitude is invariant under CPT if du"0, leaving the prefactor i as a consequence of CPT invariance. The estimate "g ""0.76 is given in [33]. + E M denotes the electric-dipole contribution. It is CP-conserving, and its coupling constant # g has been computed in [33]. Its phase is related to that of g via arg(g /g )K . # + # + >\


351

E M4 is the contribution originating in the short-distance Hamiltonian describing the transition 1" sdM Pe>e\. Its coupling constant has been calculated in the Standard Model [33], with the result (46) g "i(5;10\)(2(M /f )e BKpp , ) p 1" where f is the pion decay constant. One could in principle introduce CPT violation into this p amplitude by allowing A(KM Pp>p\e>e\)OA(KPp>p\e>e\). As seen in (46), these amplitudes may be related to M M and M , respectively, which could be di!erent if CPT is violated. ) ) E M denotes the CP-conserving contribution due to a "nite charge radius of the K. Its coupling !0 g has the phase of K Pp>p\. . 1 The KTeV [9] Collaboration's measurement is of a CP-violating asymmetry A in the angle U between the vectors normal to the e\e> and p>p\ planes [33], which is related to the particle momenta by: sin U cos U"g ;g ) ((p #p )/("p #p "))(g ) g ) , (47) p > \ > \ p where the unit vectors g are de"ned as g ,k ;k /"k ;k " and g ,(p ;p )/("p ;p "), with p > \ > \ p > \ > \ k the lepton momenta and p the pion momenta. The observable is a CP asymmetry A of the ! ! process, which we shall discuss below. The U distribution dC/dU may be written in the following generic form [33]: dC "C cos U#C sin U#C cos U sin U , dU

(48)

where the last term changes sign under the CP transformation and is T-odd, i.e., it changes sign when the particle momenta are reversed. However, it clearly does not involve switching &in' and &out' states, and so is not a direct probe of T violation. A detailed functional form for C is given in [33]. Following the above discussion of the various terms in the decay amplitude (43), this term is interpreted [33] in terms of the dominant Bremsstrahlung, magnetic-dipole and electric-dipole contributions. For our purposes, it is su$cient to note that it involves the coupling constant combinations Re(g gH ) and Re(g gH ), which + 0 + # involve amplitudes with di!erent CP properties, and hence violate CP manifestly. It depends, in particular, on the phase

of the conventional CP-violating K Pp>p\ decay amplitude, via >\ * the K admixture in the K wave function, which enters in the M1 amplitude for K Pp>p\c. The * * following is the generic structure of the integrated asymmetry measured by KTeV [33]: p(dC/dU) dU!p (dC/dU) dU p KA cos H #A cos H "g /g " , A" # + p(dC/dU) dU#p (dC/dU) dU p where H , #d !dM !p/2!du mod p, >\

H , !p/2!du mod p >\

(49)

(50)

It is generally agreed that "nal-state electromagnetic interactions can be neglected for present purposes. The KTeV collaboration has recently reported [34] a null asymmetry in the angle between the p>p\ and e>e\ planes in the Dalitz decay K Pp>p\(pPe>e\c). This provides a nice check on the experimental technique, but does not test directly the * structure of the "nal-state interactions, since the p decays outside the Coulomb "elds of the p>p\ pair.

352


and dM and du are averages of the pp P-wave phase shift and du, respectively, in the region m (m . Numerical estimates of the quantities A in terms of the di!erent couplings in (43) p ) were given in [33]: A K0.15, A K0.38 , leading to the following prediction for A:

(51)

AK0.15 sin[ #d (m )!dM ] (52) >\ ) if the CPT-violating phase du"0. Using the experimental values d K403, dM K103 and

K433, (52) becomes >\ AK0.14 (53) As already mentioned, the experimental value "(13.5$(2.5) $(3.0) )% (54) agrees very well with the theoretical prediction (53) obtained assuming the CPT-violating phase du"0. We now analyze how well this measurement tests CPT, and assess how this test compares with other tests. Consider "rst the Bremsstrahlung contribution: as mentioned above, the coupling g has a phase . In principle, CPT violation in the neutral-kaon mass matrix could shift this >\ phase away from its superweak value by an amount d : A

(55) "m !m M "K2*m("g "/sin )"d " , ) >\ ) where (as always) we neglect e!ects that are O(e), and we recall that "g "K"e"/ cos d K"e". The >\ best limit on such a mass di!erence is now provided by the CPLEAR experiment [5] "m !m M "43.5;10\ GeV (95% CL) . (56) ) ) The limit (56) determines "d ":0.863, whereas a combination of previous data from the NA31, E731 and E773 Collaboration yields [22] d :(!0.75$0.79)3. Such a phase change "d " would change A by an amount "dA":10\, far smaller than the experimental error in (54), and also much smaller than the likely theoretical uncertainties. We consider next the magnetic-dipole contribution, with the possible incorporation of a CPTviolating phase du (45). To "rst order in du, the corresponding change in A is (57) dAK(0.15 sin H#0.38 sin H"g /g ") du # + with H evaluated using (50) and assuming du"0. However, this small-angle approximation is not justi"ed, so we use the full expression (49) for A, and interpret the experimental value (54) as implying that A90.096 at the one-standard-deviation level, corresponding to 0.14 cos du!0.04 sin du50.096 ,

(58)

which leads to !703:du:#403

(59)


353

for the allowed range of this CPT-violating parameter, where we have used the estimate [33] "g /g "K0.05, and not made any allowance for theoretical uncertainties. # + The range (59) is clearly much wider than the corresponding scope for a CPT-violating contribution d to the phase

of g , and the range would be larger still if we expanded the >\ >\ allowed range of du to the 95% CL limits. We also note in passing that the magnitude of the short-distance contribution (46) is so small that no interesting limit on direct CPT violation in it can be obtained. We now address the question whether all the KTeV asymmetry could be due to CPT violation. This would occur if A cos H#A cos H"g /g ""0 . (60) # + This possibility is disfavoured by the theoretical estimates of A , but cannot be logically excluded. If Eq. (60) were to hold, the KTeV asymmetry could be written in the form (61) AKA sin du[sin H!cos H tan H] , in which case the experimental value (54), at the one-standard-deviation level, would be reproduced if (62) 0.13:A sin du:0.22 . Unfortunately, the amplitude A has not yet been measured experimentally. However, if one adopts the estimate that A "0.15 as in (51), then the KTeV asymmetry could be reproduced if du9583. We conclude that, whilst a priori it may seem very unlikely that the KTeV asymmetry could be due to CPT violation, we are unable to exclude rigorously this possibility at the present time. We hope that future measurements of this and related decay modes will soon be able to settle this issue.

Acknowledgements We thank members of the CPLEAR and KTeV Collaborations for informative discussions. The work of N.E.M. is partially supported by a United Kingdom P.P.A.R.C. Advanced Fellowship.

References [1] J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turlay, Phys. Rev. Lett. 13 (1964) 138; V.L. Fitch, Rev. Mod. Phys. 53 (1981) 387. [2] Particle Data Group, Eur. Phys. J. C3 (1998) 1. [3] J.S. Bell, J. Steinberger, in: R.G. Moorhouse, A.E. Taylor, T.R. Walsh (Eds.), Proceedings of the Oxford International Conference on Elementary Particles, Rutherford Lab., 1965, p. 195. [4] M. Baldo-Ceolin, V.V. Barmin, V.G. Barylov, G.V. Davidenko, V.S. Demidov, A.G. Dolgolenko, E. Calimani, F. Mattioli, A.G. Meshkovskii, G. Miari, L.B. Okun, A. Sconza, S. Ciampolillo, I.V. Chuvilo, V.A. Shebanov, JETP Lett. 38 (1983) 557; V.V. Barmin, V.G. Barylov, I.V. Chuvilo, G.V. Davidenko, V.S. Demidov, A.G. Dolgolenko, A.G. Meshkovskii, L.B. Okun, V.A. Shebanov, M. Baldo-Ceolin, E. Calimani, S. Ciampolillo, F. Mattioli, G. Miari, A. Sconza, Nucl.

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[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

[34]

J. Ellis, N.E. Mavromatos / Physics Reports 320 (1999) 341}354 Phys. B 247 (1984), 293; B 254 (1985) 747 (Erratum); L.B. Okun, hep-ph/9612247, Proceedings of the Workshop on K Physics, Orsay, France, 30 May}4 June, 1996. CPLEAR Collaboration, Proceedings of the IHEP 98, Vancouver 1998, to appear. J.S. Bell, Proc. Roy. Soc. A 231 (1955) 479; G.L. LuK ders, Ann. Phys. 2 (1957) 1; R. Jost, Helv. Phys. Acta 30 (1957) 409. K.R. Schubert et al., Phys. Lett. B 31 (1970) 662. CPLEAR Collaboration, A. Angelopoulos et al., Phys. Lett. B444 (1998) 43. KTeV Collaboration, http://fnphyx-www.fnal.gov/experiments/ktev/ktev.html; J. Belz, for the KTeV Collaboration, hep-ex/9903025. CDF Collaboration, CDF/PUB/BOTTOM/CDF/4855, http://www-cdf.fnal.gov/physics/new/bottom/ publication.html; M. Paulini, CDF/PUB/BOTTOM/PUBLIC/4843, FERMILAB-PUB-99/014-E, Int. J. Mod. Phys. A, in press. NA31 Collaboration, G. Barr et al., Phys. Lett. B 317 (1993) 233. L. Wolfenstein, Phys. Rev. Lett. 13 (1964) 562. M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. J. Ellis, M.K. Gaillard, D.V. Nanopoulos, Nucl. Phys. B 109 (1976) 213. A.J. Buras, hep-ph/9806471, in: F. David, R. Gupta (Eds.), Probing the Standard Model of Particle Interactions, 1998, Elsevier Science, Amsterdam, to appear. Y.-Y. Keum, U. Nierste, A.I. Sanda, Phys. Lett. B 457 (1999) 157. A. Masiero, H. Murayama, Phys. Rev. Lett. 83 (1999) 907. CPLEAR Collaboration, A. Angelopoulos et al., Phys. Lett. B 444 (1998) 52. CPLEAR Collaboration, A. Apostolakis et al., Phys. Lett. B 456 (1999) 297. J. Ellis, J.S. Hagelin, D.V. Nanopoulos, M. Srednicki, Nucl. Phys. B 241 (1984) 381. J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Lectures presented at the Erice Summer School, 31st Course: From Supersymmetry to the Origin of Space-Time, Vol. 31, Ettore Majorana Centre, Erice, July 4}12, 1993; 1994, 1; hep-th/9403133. J. Ellis, J. Lopez, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 53 (1996) 3846. L. Alvarez-GaumeH , K. Kounnas, S. Lola, P. Pavlopoulos, Phys. Lett. B 458 (1999) 347. L. Wolfenstein, unpublished, as reported in [26]. P.A. Kabir, Phys. Rev. D 2 (1970) 540. P.K. Kabir, Phys. Lett. B 459 (1999) 335. S. Hawking, Comm. Math. Phys. 87 (1982) 395. M.S. Marinov, J.E.T.P. Lett. 15 (1972) 677; Yad. Fiz. (Sov. J. Nucl. Phys.) 19 (1974) 350; Nucl. Phys. B 253 (1985) 609. L. Maiani, in: L. Maiani, G. Pancheri, N. Paver (Eds.), The Second Da'ne Physics Handbook, Frascati, Italy, 1995, p. 3. P. Huet, M. Peskin, B 434 (1995) 3. R. Wald, Phys. Rev. D 21 (1980) 2742; D.N. Page, Gen. Rel. Grav. 14 (1982) 299. CPLEAR Collaboration, R. Adler et al., Phys. Lett. B 364 (1995) 239. L.M. Sehgal, M. Wanninger, Phys. Rev. D 46 (1992) 1035; 5209 (1992) (Erratum); P. Heiliger, L.M. Sehgal, Phys. Rev. D 48 (1993) 4146; J.K. Elwood, M.B. Wise, M.J. Savage, Phys. Rev. D 52 (1995) 5095; D 53 (1996) 2855 (Erratum); J.K. Elwood, M.B. Wise, M.J. Savage, J.W. Walden, Phys. Rev. D 53 (1996) 4078. T. Yamanaka, for the KTeV Collaboration, talk at the Rencontres de Moriond, Les Arcs, March 1999.


The value of e/e in a theory with extended Higgs sector E. Shabalin Institute of Theoretical and Experimental Physics, Moscow, Russia

Abstract The additional source of CP violation appearing in the electroweak theory with three Higgs doublets could contribute considerably to the ratio e/e. 1999 Elsevier Science B.V. All rights reserved. PACS: 12.60.Fr

The recent result [1] Re(e/e)"(2.80$0.41);10\ ,

(1)

con"rming the CERN group result [2] e/e"(2.3$0.7);10\

(2)

with better accuracy turned out to be larger than that expected in the Standard Model (SM) [3}6]:

6.7$0.7

[3] ,

3.1$2.5 (e/e) "10\ ) 1+ 4$5

[4] ,

17> \

[5] ,

(3)

[6] .

Though the result of Ref. [6] is compatible with the above experimental results inside the error bars and besides, a considerable alteration of the generally accepted magnitudes of the parameters "< /< ", m , K M , B could result in (e/e) +(2}4);10\ [7], it is pertinent to look for another SB A@ Q +1 1+ CP violating mechanism capable of production of such e/e as in Eqs. (1) and (2). The probable candidate is the additional source of CP breakdown appearing in the electroweak theory with extended Higgs sector as in Weinberg [8] containing three doublets of complex Higgs 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 5 3 - 8

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E. Shabalin / Physics Reports 320 (1999) 355}358

"elds U . The #avour conservation is assured by coupling of the quarks with di!erent electric G charges to di!erent U , i"1, 2. The U doublet does not couple directly to the quarks. The G coupling between quarks and physical charged Higgs "elds H> is de"ned by the expression (see e.g. Refs. [9,10])

1#c 1!c !KM X H> D, (4) ¸"2G;M M K > H> 3 G G " G G $ 2 2 G G where M , M and K are the mass matrices for the up and down quarks and the weak mixing 3 " matrix, respectively. In the following we shall assume for simplicity that the mass m is considerably larger than & m and take into account only the contribution of the lightest charged scalar H>,H>. The CP & violating e!ects will be proportional to Im(X>H). In the papers [11,12], it was found that being the only one, the Weinberg mechanism of CP violation would lead to magnitude of e/e&0.045 in contradiction to the experimental data. Therefore, such CP violating mechanism could be only the additional one to other mechanisms producing the main part of the parameter e. But as for e, the Weinberg mechanism could originate a contribution into e comparable, or even larger than that expected in SM [13]. This statement disagrees with conclusion in Ref. [14], based on the estimate for the upper bound on Im(X>H): 1 (5) Im(X>H)4([Br(bPsc)/C] F (x) & corresponding to a case where the real part of the scalar-exchange contribution cancels the SM contribution to bPsc transition. In Eq. (5), C+3;10\ [14], x"m/m and [10] A & F (x)"![x/6(1!x)][(3!5x)(1!x)#(4!6x)log x] . (6) & To date [15] Br(bPsc)"(3.11$0.80$0.72);10\

(7)

so that, Im(X>H)42, m "175 GeV, m "100 GeV , R & Im(X>H)43, m "M "175 GeV . (8) R & Let us show now that these bounds allow to get e/e of order (1). The Weinberg mechanism of CP violation produces two new CP violating operators [13] originating the CP-odd, *S"1 transition: O!.\ "(g Bm/3m )[M s (1!c )qq (1#c )d#M s (1#c )qq (1!c )d] 5 Q % Q B

(9)

and "(g BIf/3m )(G? )s (1#c )d . O!.\ Q ) IJ 5

(10)

E. Shabalin / Physics Reports 320 (1999) 355}358

357

These operators incorporate the short-distance and long-distance interactions, so that, a question on their renormalization is not simple. A possible renormalization is taken into account by the parameters m and f. In Eqs. (9) and (10), M and M are the constituent masses of s- and d-quarks, B is the imaginary Q B part of the e!ective coupling constant of s dG vertex H), the Eq. (15) does not exclude a possibility that the Weinberg mechanism of CP violation contributes considerably to the ratio e/e.

References [1] R. Kessler, Recent Results from the KTeV Experiment, Report at the Conf. `Les Rencontres de Physique de la Vallee d'Aostea, La Thuile, 28 February}6 March, 1999. [2] G.D. Barr et al., Phys. Lett. B 317 (1993) 233. [3] A. Buras, M. Jamin, M.E. Lautenbacher, Nucl. Phys. B 408 (1993) 209. [4] M. Chiuchini et al., in: L. Maiani, G. Pancheri, N. Paver (Eds.), The Second DAPhNE Hand Book, INFN-LNF, 1995, p. 27. [5] S. Bertolini, J.O. Eeg, M. Fabbrichesi, Preprint SISSA 103/95/EP. [6] S. Bertolini, M. Fabbrichesi, hep-ph/9802405, Vol. 2, 1998. [7] A.J. Buras, Proceedings of the Workshop on K Physics, Orsay, France, May 30}June 4, 1996, p. 459. [8] S. Weinberg, Phys. Rev. Lett. 37 (1976) 657. [9] G.C. Branco, A.J. Buras, J.-M. Gererd, Nucl. Phys. B 259 (1985) 306.

358 [10] [11] [12] [13] [14] [15]

E. Shabalin / Physics Reports 320 (1999) 355}358 P. Krawczyk, S. Pokorski, Nucl. Phys. B 364 (1991) 10. N.G. Deshpande, Phys. Rev. D 23 (1981) 654. A.I. Sanda, Phys. Rev. D 23 (1981) 2647. C. Jarlskog, E. Shabalin, Phys Rev. D 52 (1995) 6327. Y. Grossman, Y. Nir, Phys. Lett. B 313 (1993) 126. The ALEPH Collaboration, Preprint CERN-EP/98-044.


Mass and CKM matrices of quarks and leptons, the leptonic CP-phase in neutrino oscillations D.A. Ryzhikh*, K.A. Ter-Martirosyan ITEP, B.Cheremushkinskaya 25, 117259 Moscow, Russia Dedicated to our colleague } the old friend of one of us (K.T-M) and teacher of another } to Lev Okun. He always emphasizes the importance of simple and transparent physical arguments which become very impressive in his talks and discussions. He was among the "rst who discussed CP violation in neutrino oscillations.

Abstract A general approach for construction of quark and lepton mass matrices is formulated. The hierarchy of quarks and charged leptons (èlectronsa) is large, it leads using the experimental values of mixing angles to the hierarchical mass matrix slightly deviating from the ones suggested earlier by Stech and including naturally the CP-phase. The same method based on the rotation of generation numbers in the diagonal mass matrix is used in the electron}neutrino sector of theory, where neutrino mass matrix is determined by the Majorano see-saw approach. The hierarchy of neutrino masses, much smaller than for quarks, was used including all existing (even preliminary) experimental data on neutrino mixing. The leptonic mass matrix found in this way includes the unknown value of the leptonic CP-phase. It leads to large lIlO oscillations and suppresses the llO and also llI oscillations. The explicit expressions for the probabilities of neutrino oscillation were obtained in order to specify the role of leptonic CP-phase. The value of time reversal e!ect (proportional to sin d) was found to be small &1%. However, a dependence of the values of llI,llO transition probabilities, averaged over oscillations, on the leptonic CP-phase has found to be not small } of order of ten percent. 1999 Elsevier Science B.V. All rights reserved. PACS: 14.60.Pq

1. Introduction Serious e!orts have been invested recently in the natural understanding of experimental results on neutrino oscillations. They have shown that neutrinos of three generations have, perhaps, non * Corresponding author. E-mail addresses: [email protected] (D.A. Ryzhikh), [email protected] (K.A. Ter-Martirosyan) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 7 6 - 9

360

D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

vanishing small masses. The heaviest of them, the neutrino of the third generation, seems to have a mass of the order of &( ) eV and, as the Super Kamiokande data on atmospheric neutrinos show, has the maximal possible mixing with the neutrino of the second generation, and may be, also not too small mixing with that of the "rst generation. This was not expected a priori, since all similar mixing angles of quarks are small. It is the challenge of modern particle physics to include naturally these results into the framework of grand uni"cation theory together with the data on quark masses and mixing angles. For the quarks these angles are small and are known already [1]. We begin this paper by reminding the well-known picture of masses, CP-phase and mixings for the quark sector of a theory. A general method will be developed which allows one to construct consistently the 3;3 mass-matrix and the CKM mixing matrix for quarks. The same general approach will be used later for the electron}neutrino sector of a theory. Let us consider, as a useful introduction to a consistent theory of quark and lepton masses and mixings, a simple phenomenological approach suggested by Stech [2]. He has noticed the following quark and charged lepton (èlectronsa) mass hierarchies: m : m : m K1 : p : p, m : m : m K1 : p : 8p, m : m : m K1 : p : p O I R A S @ Q B

(1)

with a very small pK , pK0.058. He has also introduced the following mass matrices which reproduced approximately the masses of all the quarks as well as their mixing angles:

pg MK " ( pg> !p S p pg> ( 0

(

pg

p m, R 1

(

a p 0 B M K " a p !p 0 m /p B @ B 0 0 p 0

(2)

a p 0 MK " a p !p 0 m /p O 0 0 p 0

(3)

Here g"e B represents the CP violating phase d in the quark sector, while the values of the constants a K2, a K( correspond to the best "t of all masses of quarks and electrons (their B central values, see below). The diagonalization of the matrices (2) and (3) by means of an unitary matrix ;K : ? MK ";K M K ;K > ? ? ? ?

a"u,d,e ,

(4)

(where ;K ;K >";K >;K "1), reproduces simultaneously both the experimental (central) values ? ? ? ? of running masses of quarks and electrons and all quark mixing angles (their sines): s "sin 0 O , s "sin 0 , s "sin 0 O in the CKM matrix c c s c s e\ BY . (5) c c !s s s e BY s c " 2 0 k !s c and s"sin 0"1/(2(1!(1#t0)\), c"cos 0" Here t 0"tan 20"2o/(a!b) K one 1/(2(1#(1#t0)\), and 0 is the mixing angle. In Eqs. (2), (3) for the matrices MK and M B has a"0 and the mixing angles 0 ,0 are small since t 0B"2(2p/p/2)"8p(1 and B t 0"2(p/p"(6p is even smaller. Due to the block structure of M K and M K the unitary B matrices ;K and ;K also have the following block structure: B c s 0 c s 0 B B ;K " !s c 0 , ;K " !s c 0 , (9) B B B 0 0 1 0 0 1


363

Fig. 1. (a) The run of the upper quark masses m ,m ,m calculated in Refs. [11] in the "rst order of QCD perturbation R A S theory (the dashed lines) and in the fourth order of it (solid lines). The vertical line corresponds to the scale k"M K174.4 GeV used in the paper. The run of electron's masses m ,m ,m is disregarded (while it can be easy taken R O I into account and is not essential). (b) The same for the masses m ,m ,m of the lower quarks. @ Q B

where s "1/(2(1!(1#(8p))\)K4p(1!24p)K0.214, s "(1/(2)(1!(1#6p)\)K B (p(1!(p))K0.0705 up to the terms of the order of p. Let us note that Stech's matrices MK ,M K in Eqs. (2) and (3) can be reconstructed using their B diagonal form (i.e. the physical masses of d-quarks and electrons):

MK " B

m B

!m Q

,

m @

MK "

m

!m I

,

(10)

m O

by means of Eq. (4), which states MK ";K >M K ;K , M K ";K >M K ;K . (11) B B B B The matrices ;K "OK B (or ;K "OK ) in Eq. (9) can be considered as rotating the 12 generations of B d-quarks (or electrons).

364


A bit more complicated (but much more instructive) is the diagonalization of the matrix M K . S Similarly to M K and M K this matrix can be represented as B MK ";K >M K ;K , where ;K >"OK > OK > (d)(OK S )> . S S S S S

(12)

Here the matrices

c !s 0 c 0 s e\ B 1 0 0 S S , OK > " 0 c 0 , OK > (d)" 0 1 0 c s (OK S )>" s S S 0 0 1 !s e B 0 c 0 !s c

(13)

rotates 12, 13 and 23 generations, respectively. The 13 rotation includes naturally the complex phase d which violates the CP-parity conservation of a theory. It cannot be removed by a trivial phase transformation of the u- or t-quark "elds. However, the value of s turns out to be very small S (s &p;1) and one can really put OK S K1) . The quark CKM matrix is S OK > (d)OK > , where OK > "(OK S )>OK B KOK B . !)+ S B

(14)

Here in general

c

s 0 c 0 (OK S )>" !s 0 0 1

with s "sB c !cB s "sin(0 !0 S )Ksin 0 since s &p (see below) is neglegibly small. B B S S S m S Thus for the given upper quark masses M K " and the given quark mixing angles m S A m R (from the quark CKM mixing matrix is "xed: !)+ S

c

0 s e\ B . c c s ;K >"OK > OK > (d)" s s e B S !s c e B !s c c

(15)

The same matrix ;K > can be obtained in the form of decomposition in powers of p (up to p terms) S by a direct calculation

;K >" S

p 1! 2

p p e B 1! ! 4 (2 p p 5 !p 1! e B ! 1! p 2 4 (2

p p 1! e\ B 4

0

p

5 1! p 4 (2 p 1! 4

.

(16)


365

Multiplying ;K > in this form by ;K one obtains with the same accuracy the following CKM quark S B matrix M K ;K with ;K >"OK >OK >(d)OK J> , J J J J J where

(27)

c

s J J OK J " !s c J J 0 0

0

0 1

is de"ned similarly to OK with the substitution s "sin0 J for s in OK de"ned above (the value of J 0 J will be determined later on). The matrices:

>

OK (d)"

c

0 s e\ BY 1 0 ,

0

!s e BY 0

c

1

>

0

0

c s OK " 0 0 !s c

(28)

represent the rotation of 13 and 23 generations of M K , respectively, and d is the leptonic J CP-phase which has appeared here naturally (clearly s ,c from here and later on means the sines GH GH and cosines of the leptonic and not quark mixing angles). To calculate M K in Eq. (27) in explicit form we note that according to Eq. (8) one has J a o 0 m OK J " o b 0 (29) m OK J> m 0 0 m

with a"sm #cm Km , b"cm #sm Km , o"!c s (m !m ) and "o";m is small J J J J JJ since s in Eq. (22) is very small (actually s Ks !s K0.036 see below). Then it is easy to J J calculate the matrix

a

o

MK "OK > (d) o 0

b

0

0 OK (d) 0 m and further to "nd the following result for the neutrino mass matrix MK "OK >M K OK : J ac #m s c s s (m !a)e\ BY c c s (m !a)e\ BY #oc c !os c c s s (m !a)e BY s (m c #as ) c s (m c !b#as ) . MK " J #oc c #c b#D #D c c s (m !a)e BY c s (m c !b#as ) c (m c #as ) !os c #D #s b#D

(30)


369

The values of D "!os sin 20 J cos d, D "!os (e\ BY!2s cos d)"D> and of D "!D are small and can be neglected. Also since a;b;m , s Ks (see below) and J s ;s ;1 are very small, one can disregard in the matrix M K all the terms containing a,b and J put c Kc K1. The value of "o"Ks m is also very small "o";s m , but the terms containing it in the matrix (30) cannot be omitted as that would violate the normal complex structure of the matrix M K and of CKM matrix considered below. Therefore omitting small terms one obtains the J matrix M K in the following simple form J

s m #ac c (s s m e\ BY#oc ) c (s c m e\ BY!os ) . (31) s m #c m s c (m c !m ) MK " c (s s m e BY#oc ) J c (s c m e BY!os ) s c (m c !m ) c m #s m Here, s Kc K1/(2 and not too small values of s are determined in Eqs.(22). In conclusion of this section let us construct the leptonic CKM matrix: OK >(d)OK , !)+ J where

c

(32)

s 0 OK "OK J>OK " !s c 0 0 0 1

and s "sin(0 !0 J )"sin 0 K0.035$0.020 as it is determined by Eq. (22). This gives for the neutrino l l mixing angle: 0 J "0 !0 K0.036$0.020 (in radians, or (2.1$1.1)3). Multiplying the matrices OK > in Eq. (32) one obtains by the angle (!0 J ) and OK by the angle 0 . Therefore the product OK J>OK ,OK OK J> leads to a rotation by the angle 0 "0 !0 J , where 0 K20 (the value of s has been given above just after Eq. (9) and s in Eq. (18)) and therefore 0 J K0 , or s Ks . J Similarly to the case of the quark CKM matrix, the most natural value of the leptonic CP-phase leading to the largest possible CP violation can be d"n/2 or g"e BY"i. This CP-phase d can manifest itself in the Pontecorvo neutrino oscillation experiments. It is very di$cult to observe it now. Below we discuss shortly the possibility of these observations. The exact expressions for probabilities of neutrino oscillations are given in Appendix, since they are very cumbersome. Some of them have been obtained earlier in a number of papers [34}44].

4. The leptonic CP-phase in neutrino oscillations experiments Many papers were devoted to the studies, pioneered by Bruno Pontecorvo [34}37], of two and of three [38}44,7] neutrino oscillations. We consider them below shortly in order mainly to specify the role of the leptonic CP-phase [38,39] in these oscillations. Let us express l ,l and l "elds entering the weak interaction Lagrangian in terms of neutrino I O states l ,l ,l with de"nite masses m ,m ,m using the leptonic CKM matrix (5) (or (A.1) from the appendix) as follows: l (ct)"c l (0)e\ CR#s c l (0)e\ CR#s l (0)e\ CR\GBY , l (ct)"!(s c #s s e BY)l (0)e\ CR#c l (0)e\ CR#c s l (0)e\ CR , (35) I l (ct)"(s s !s c e BY)l (0)e\ CR!s l (0)e\ CR#c c l (0)e\ CR , O up to the terms of the second order in small quantities s ,s ;1. (see the appendix for the exact P" , I P(l l )""c (s s !c s e BY)!s c s e\ P#s c c e BY>P" , O P(l l )""(s s !s c c e\ BY)(s c #s c s e BY)#c s e P! s c c e P" I O (37)


371

where u "(e !e )¸/c"(*m /2p )¸"2.54¸(m)/(E (MeV))*m(e OK > (d)(OK S )>. MK ";K >M K ;K K (pg R S S S S S pg (p 1 The diagonalization of all these matrices M K ";K M K ;K > can be done by the same unitary ? S ? S matrices ;K , ;K , ;K , ;K with the same mixing angles (used at they construction) as were used above B S C J for the Stech matrix case. 2p

Acknowledgements Authors express their gratitude to P.A. Kovalenko for calculation of u- and d-running masses shown in Fig. 1, to N. Mikheev for stimulating discussion and also to L. Vassilevskaya and D. Kazakov for useful discussions and an essential help in edition of the paper. Especially we thank Z. Berezhiani who provided us with his and Anna Rossi paper [7] containing a number of ideas used above. They thank the RFBR and INTAS for "nancial supports by grants: RFBR 96}15}96740, 98}02}17453, INTAS 96i0155 and RFBR}INTAS: 96i0567, 95i1300. Also one of the authors (K.A.T-M) acknowledges the Soros foundation for support by professor's grant.


375

Appendix We give below the probability rates P(l l ) in the general algebraic form using the well known G H exact leptonic CKM matrix (given above for quarks in Eq. (5)):

c c s c s e\ BY . c c !s s s e BY s c

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