Quantum chromodynamics: high energy experiments and theory

INIIÌHN.VI IONAI II'I

(I

DIKMIIOIÌ,

I

I;

SKKIIÌS OI

Knowles,

M.

MONOCIIAIMIS

ON

PHYSICH

Hclimelling: Quantum cliromodynamics: Ini/li imi yi/

cj/icriiiniils and limil i/

III II DeWill: l'In• 1/fallili appnuich lo illuminiti fichi theory I li! I /.imi .Itisi ¡11: Quantum Jiclil III tori/ nini criticai phcnomc.na. l'ornili editimi III!, li M Munì: liroiriiiini molimi: fine I uni iotls, dynamics, inni application» III II Nisliimuri: Stati-itimi plti/sits of spili ylasscs inni informatimi pivcessinq: 1111 intrvduclioii llli

N. M. « u p n i n : / ' / n o i i / o / noiicqttìlibrìum

supcivovductivity

UHI 108. 107. I0(i I0V 101

A Aliaroni: Introducilo 11 lo lite I licori/ of ferromagnetism, Seco ti ti editimi li DoIjUs: / Idi 11111 Ih ree li WÌKliians: Calorimetri/ I Kiibler: 'ihcori/ of itincirint chetimi mni/nclism Y «inainolo. Y. Kitaoka: Dynamics of heavy clcctrmis I). ISaldili. (I. l'assarino: The standard model in the makiny

101

C ( ' . I il lineo. IJ. I.avoura. J . I'. Silva: CP

102. 101. 100. (IO. !I8. 1)7 •IO 'l'i II I 'Il 00 HO. H8. 87, 8 BlßüOTHC* CIUMCIAS

CLARENDON PRESS • OXFORD

Germani/

Ills boo Has been printed digitally and produced in a standard specification in order to ensure ils continuing availability

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78 0 1 8 0 72

A

Quantum Climmoduiiawics ( C ) was formulated as a nou-ahclian field theory in 1 )7. 5 and in the meantime lias evolved to the generally aeeepted theory describing strong interactions. T h i s book is intended to give a comprehensive overview over C studies in e e annihilation, lepton nucleoli scattering and hadron liadron scattering. T h e book conies at a time when the analysis of the L P data is being finalized and more t han ten years of most fruit ful research deserve to be summarized. n e of the goals of the book is to bridge the gap between theory and experiment, combining into a single volume theoretical description and tin1 experimental measurements. xperimental results are discussed with emphasis on their relevance with respect to strong interactions physics at high energies. Although not necessary for understanding the physics, it might occasionally be helpful to have some background knowledge with respect to the analysis of experimental data, such as the least-squares method or the interpretation of experimental errors and confidence level intervals. For an introduction to this subject., which is beyond t he scope of this book, we would like to refer the reader to any of the many excellent textbooks on the subject. ealing with particle physics we will express energy in units of e or mult iples of it. such as. for example. Me or c . In addition the so-called natural units will be used. that. is. a system of units where li = c = I. T h u s energy, momentum and mass are measured in units of energy, lengths in unit s of energy . nly iu cases where we want to be explicit about momentum or mass we will use e /r or e / r " . respectively. If desired, conversion to M SA-units is achieved by multiplying with appropriate powers of It and c. ere it is helpful to remember that, lie 0.1 7 e fin. c = .5 x 10 s m / s and r ss 1.(502 As. lectrical charges are measured in units of the positron charge and may carry an index specifying t he ident ity of a part icle, for example, c,, for the charge of a quark. An overview of the mathematical notation used in this book is given 011 page x. T h e presentation of the subject should be understandable for graduate students. and to some extent even undergraduates, but. also contains material useful for a research physicist. T h e basics of pert urbative C are presented in quite some detail. Anybody who has heard an introduction to field theory should be able to follow also t lie derivat ions and learn how t he building blocks of the C Lagraugian can be probed iu high energy physics experiments. T h e book focuses on basics. evertheless, some space in the presentation of the theory of C is also dedicated to advanced topics aiming to convey at least some impression of the general picture. For an in-dept h discussion of, for example, renormalization or I lie pat h int egral formalism the reader is referred to specialized textbooks.

vl

CHI I \< I

In order lo achieve a decpet understand ill}!,, some ol llir concepts discussed m tin- main text are illustrated hy means of problems. T h e solutions are given in I lie appendix. I )cpcndiug on I lie background of I he reader I lie apparent difficulty ol the problems may vary considerably. As a rough guideline we introduced a star-rating system for grading the complexity. A single star signals moderate difliculty. two stars are attached to what we consider really hard problems. Alter a short historical introduction, summarizing the steps which led to the formulation of QCI). the book focuses on the tests of QCD. First the t heory and phenomenology of pcrturbative Q C D for different kinds of reactions are explained at Horn level, t hen higher order correct ions are discussed. Considerable weight is also given to non-pert urbative effects such as the hadronization process. The currently available models are discussed in some detail both concerning i lie dilferent phcnomenological ideas and t heir implementation in Monte Carlo models. T h e experimental section starts with a short overview about accelerator and detector techniques and a detailed description of the general strategies of QCI) analyses. Then structure function measurements are discussed, which were vital in establishing (¿CD as a field theory and st ill are a t lu iving field of research. T h e focus then shifts to measurements done mainly in e + e ~ annihilation, which test I he detailed st ructure of stong interact ions as t hey can be probed by perturbat ive QCD. This covers measurements of the strong coupling constant as well as tests of thi- structure of Q C D . which finally led to the unambiguous proof for the existence of the gluon self-interaction. d o i n g from hard scattering to semi-soft, processes, the book describes how interference effects in higher order amplitudes alFeet. the properties of hadronic linal states. Finally studies of the hadronization process, the conversion from coloured parlous to colour neutral hadrons. are presented, as it can be probed by means of particle multiplicities or particle particle correlat ions. Clearly, within the limited space available it is only possible to cover a subset of all topics connected to QCD. T h e emphasis is put. 011 hard-scattering processes. Ot her subjects such as quark mass determinations, polarization phenomena. photon photon scattering, Q C D at thresholds, non-relativistic Q C D . nuclear collisions, lattice gauge theories, clnral perturbation theory, quark gluon plasma, or exclusive processes, if at all. are only briefly ment ioned. Again, for an in-depth discussion the reader is referred to external references specializing on the respective subject. Even after almost 30 years of research t here are st ill many open questions and plenty of opportunity for significant, cont ributions both iu t heory and experiment. Having worked through the book, we hope that the reader will have gained an overview of how Q C D developed in the twentieth century and where we stand with respect to a quantitative understanding after the turn of the millcnium. Many of the results collected for this book are likely to be superseded by improved measurements in the future, but the basic facts of (¿CD and the methods to extinct its defining parameters will remain valid.

r n t 1 ,\< 1

vil

lu collecting the material for this book we tried to do as complete as possible a survey of the relevant publications. Still, some important paper will have been missed, and we would like to apologize to all authors whose work is not yet fnllv appreciated in this book. Updates and amendments to the book, covering new developments as well as improvements to the exercises or error corrections can be found on the book's World Wide Web site. T h e link can be located via the catalogue of the Oxford University Press home page at: http://www.oup.co.uk/ November 2002

G.D.. I.G.K.. M.S.

Preface to the 2nd Edition For the second edition we would like to express sincere thanks to many readers, who gave feedback or pointed out errors in the first, version of the book. T h e mistakes have been corrected, and the? discussion of some aspects of (¿CD lias been re-worked. We hope that t his has led to an improved presentation of t he subject, which very recently has earned the highest recognition by the assignment, of the Nobel Prize in Physics 200. - American Institute» of Physics, publishers of AIP Conf. five. ~>.U Particles (mil Fit Ids: Seventh Mexican Workshop 1999. ©2000 American histitiili of Physics . for Figure 7.1. - American Physical Society, publishers of Pliys. lice. Lett. 78, 79, © 1997 hi/ the American Physical Society. Phys. Rev. Lett. SO. SI and Plry.s. Iter. D'nS'. © 190S by the American Physical Society, for Figures 7.13. 7.1 I. 7.1!)(a). 7.19(b). 7.20. 7.21. 7.25(b). 10.9(a) and 10.9(b). - Annual Reviews, publishers of Annual Review of Nuclear and Purtich ence. Volume /,!l ©I999 by Annual Reviews www.unnualiTviews.oty, Figures 6.9, (i.10. (i.ll and (i.l2.

Scifor

- Elsevier Science Ltd.. publishers of Nuel. Phys. B (Proe. Suppl.) (¡5 ©1998. Nut'.l. Phys. B/,70 ©1996 and Br>J,r, ©1999, Phys. Lett. B.','>(>. B4-2. lor Figures 5.1, 5.2, 5.3, 5.1. 5.5 and 5.6. Springer Verlag. publishers of Z. Phys. C57 ©199:}. C(>2 ©199/,. 67.7. C75 and C70 ©1997 and Eur. Phys. .1. Cl ©1998. C7. ClJ ©1999. CI2. Ct:l. ClJ, and Cl7 ©2000 and Cl9 ©2001. for Figures 6.3. 6.5. 7.3. 7.4. 7.6. 7.7, 7.10. 7.11. 7.12(a). 7.22. 7.23. 7.24. 7.25(a), 10.8, 11.6. 11.7. 12.1. 12.2. 12.4. 12.5. 12.6 and F.5. The Royal Swedish Academy of Sciences, publishers of Physica Scripta. Volume 51. for Figures 2.9. 9.2. I I.l. 11.2 and 11.3(left). World Scientific Publishing, publishers of the Procccdinys of tin 9th international Workshop DfS 2001. Boloyna. Italy, for Figure 7.8.

NOTATION Lorent.z four-vector indices {0.1.2.3} Euclidean three-vector indices {1,2,3}.«/' colour indices of the fundamental representation h. l>. hwp" scalar product, of two three-vectors V'l P\N

n growt h of t he interact ion st length wit h a divergence in t he ln(Q~'/A")-tenn. which implies t hat around and below a certain cut,-oil" energy A t his simple expression no longer describes t he physics of st rong interact ions. On t he other hand the coupling strength decreases with increasing Q 1 and eventually approaches the same value as that of the unified electroweak interaction. T h e exact point of this grand unification depends on the details of the theory. A hint that, it might exist can already be inferred from t he rough sketch given by Table 1.1. T h e fundamental fields known today are leptons and (¡narks, which are both spin-1 / 2 fermions. and spin-1 gauge boson fields such as the f/lnoil (g). the photon (",) and the \ \ : ± and the Z bosons, which mediate strong, electromagnetic and weak interactions, respectively. Isolated free quarks have never been observed experimentally. Bound states of three quarks form the so-called bart/ons. such as t he proton or the neutron, combinations of a quark and an antiquark yield a meson, such as. for example, the pion or tin» kaon. Mesons and baryons are collectively referred to as liadrons. heavy particles which are subject to strong interactions. In comparison to quarks, leptons. with the elect ron and its neut rino being the most prominent representatives, are rather light particles which do exist as free fields and are oblivious to strong interactions. (¿narks carry colour charge, electric charge and weak isospin and thus couple to gluoiis. photons and \ V ± and Z bosons. All leptons carry weak isospin and t hus are subject to the weak interaction, but only t he charged leptons have electric charge and thus also interact electromagnet ically. In the context of the Standard Model, all massive particles acquire their mass by coupling to the scalar Higgs field II. Finally, all energy couples to the spin-2 gravit.on field (G). which in a quant um theory of gravity is responsible for t he gravitational interaction. Table 1.2 .summarizes basic properties of the fields of t he Standard Model. T h e masses quoted in the table are taken from the 'Review of Particle Properties' ( P D G . 200(1). T h e considerable ranges given for the quark masses reflect t he difficult ies in dealing with masses of strongly interacting particles which are not observable as free fields. Tliey have to be understood as mass parameters of the theory rather than mass contributions to the bound states corresponding to observable liadrons. Table 1.2 shows that we currently have to deal with IS fundamental fields, and the natural question arises whether there might be a more fundamental level at which the picture becomes much simpler. Unification of the different interact ions including gravity addresses this quest ion, for example, in the context of the so-called sir hit/ theories. In such theories also the apparent, symmetry bet ween leptons and quarks and the repetition of weak-isospiu doublets iu both sectors may find a natural explanation. Promising alternative models to extend the minimal Standard Model exist, based on very good theoretical arguments. However, all experimental findings are perfectly described by t he Standard Model so far. T h e main focus of this book is Q C D in hard interactions. Structurally. (¿CD is a very straightforward theory, being a Yang Mills gauge theory based on an

I IN I l1 M M

I II t[N

T a b l e 1.2 ¡'a ids of the iininiinitl Slandunl Moth l. The Jirst uroui> contains III< i/uarl.'s. tin second the Irjiluns and tile last one the bosoinc Jit Ids. All chaiycs arc i/inrii in anils of tin positron cliuiyc. Field

Mass/McV/r"

Spin

Charge/e

ll a l~ ·s i s. and hen 1turn to ilt actual measurements off :;l.rll structure functions, the' st strong coupling /l lId tlliell ,111'11 Lo l.ll llJ 11I(,f1S lIl'('II IC II Ls o d m c fUll d iolls. cit rollg tO llpli ll:.( constant. he stru structure of Q (¿CD st udies of of l.hp the had hadroni/.ation process. (" lIls\ll IlI.. tests LeSt.s of tIIll' ct ure of '0 aand nd s\.lIdies l'OlIi zat.io ll pro(:css. Since it is pr'l('t practically impossible in O one book, we Sill(,(' ir'u lly ill l()(Issi hlp to review I'(' \'i('w all relevant n' I(,\o':1 l1 t results r('snlt,s ill il!' hn k. \\'(' focus studies performed high energy (:ollidl'l's. colliders. ro ' li S on ( II Q C'ID stlldies pf'\' for nwd at at. hi gh ('IIl'rg,v

2 T H E DEVELOPMENT OF QCD 2.1

Experimental evidence

I'lie liistorv f high energy physios in t.lie second half of I he twent i< * I h century was driven l>.v a sequence of increasingly more powerful particle accelerators, which allowed matter to !>' of the hadronic system with an invariant mass IF.

FtC. 2 . 1 . Kinematics of deep inelastic electron nucleoli scattering

Starting from this comparatively simple diagram we will now show explicitly what kind of phenomenology one expects if the proton is a bound state of point like charged ob jects. As a first step it is convenient, to int roduce two now quantities, the energy transfer v from the electron to the hadronic system in the had roll's rest frame ,,=

E-E'=

JWi,

'J-f

(2.2)

and the squared momentum transfer Q~ carried bv the virtual photon. '-

Ml

Mil. I M'.\ I I I MAI I MM I III' I ¿1 II

F.N l'I UIMUN'f \ l

Willi I lie quantities delincd in Fig. 2.1 o n e finds 2

Cf -- 2 A l h ( E + Mu - E') - A/,; - II" = 2A/,,i/ + M£ - W* < 2 M u « .

(2.4)

whore tlir equality for the hist term is obtained in the limiting case U'J. that is. for the case of clastic scattering. T h e deviation from clastic scattering thus can be described by the Bjorkeu-variable x n Q~ •>n = 7 r l T ~

with

» 2.5

I n proceed o n e needs to know the cross section for elastic scattering of an electron with a s p i n - 1 / 2 fertnion of mass M\, and charge See Ex. (2-1) for the elementary but rather lengthy calculation. O n e o b t a i n s da

I-";,,/',-;

E' f

dep = —QT-"~E V

2

Q2

ff

O S

2

+

2^

. S 1

"

2

o \ (

2/-

-(,)

l-;VII»KN( I-

11

does not happen with the nucleoli as a whole, but with exactly one of its constituents. Physically this picture makes sense when the energy of the projectile is sufficiently large to resolve the inner structure of the target. To describe this situation one has to partition the total four-momentum of the nucleoli between its constituents. Each constituent / thus carries the fraction .c, with a probability density f,(x,). the so-called par ton density function p.d.f.. meaning that the probability for .r, to fall into the infinitesimal range [r. .r-f-d.'] is given by /,(;e)d.r. From these a s s u m p t i o n s the structure functions 1F| and 11% can be calculated as superpositions of the elastic structure functions eqn (2.!)) with weights / , ( . r ) . To do this we n o t e that t he mass-shell constraint for the scattered constituent can be written (.t',/>)2 = (•'")/'+'/) 2 equivalent t o Q2 = 2x,M\,i/. We emphasise at this point that any constituent mass or transverse m o m e n t u m is negligible in a full calculation. T h i s suggest using the trick of replacing Mi, by x,M\, in eqn (2.!)) lo obtain the required structure functions, after s u m m i n g over all constituents and integrating out. the modilied ¿-functions. This yields

I'Viim this the double differential cross section with respect t o Q 2 and u can be derived. Start ing wit h the t rivial case of elastic scattering, where the mass-shell constraint on the scattered particle imposes the relation Q~ - 2M\,i'. one g e t s 2

d { Q 2 . t ' ) . can he written as structure functions W\{Q2.i') ^
I III: I »KVKI.OI'MKN I !• (J( I)

12

ns fiiiu t ion of x while F> describes the inoiiienl 11111 density, botli weighted with I he coupling strength to the photon probe. T h e fact that the observed cross sections depend only on a single diuiensionless variable x is also referred to as sealing behaviour. As shown above, it is a direct consequence of having point like diinensionsless scattering centres. Extended o b j e c t s would introduce a new energy scale into t in1 problem. 2.1.2..'5 Experimental findings T h e experimental observat ion of scaling was t lie first clear evidence for a partonic sub-structure in the nucleón, giving s u p p o r t to the concept of quarks as the building blocks of hadronic matter. S o m e early results (M1T-SI,AC C'ollab.. 1970) are shown in Fig. 2.2 and Fig. 2.3. T h a t the structure functions for deep inelastic electron nucleón scattering are mainly a function of j n and essentially independent of Q2 is illustrated by Fig. 2.2. An alternative way of showing scaling is to plot F> at. for example. Xn 0.25, as a function of Q~. T h i s is done in Fig. 2.3, where one sees that F> is indeed independent of Q2. 0.5 r

FIG. 2 . 2 . Scaling behaviour of //lF¿(u;) = F>(u>). uj = ranges. Figure from Ml I'-SLAC" Collab.(1970).

L/.r», for various Q2

,

(

i -IV,

2

4 2

Fit . 2 . . Value of v i r a IlC Collab. 1

2

GeV c

I

r

n

.2 . Figure from

.

nce it was possible to loo into the nucleoli, one could also try to determine the properties of those parlous. hile generic scaling behaviour is a universal feature of any parton model, t he details of course depend on the properties of the particles involved. The above derivation for electron nucleoli scattering, for example, explicitly assumed that the partons are s p i n - 1 2 particles. In this case a definite relation has to hold between F\ and F>. the so-called Callun Cross relation 2xFi(x)

=

F>(x)

.

2.14

T h e fact that the lepton nucleoli data are in very good agreement with this prediction shows that the struc partons indeed are s p i n - 1 2 fermions. dditional possibilities arise when different probes are used in deep inelastic scattering processes. sing, for example, neutrinos instead of electrons, the interaction is mediated by or bosons rather than photons. ow the coupling strengt h is given by the I bird component, of the wea isospin, which unli e the electric charge gives the s a m e coupling strength to all uar s. Comparing electron nucleoli and neutrino nucleoli cross sections thus allows to probe the electric charge of t he partons. Introducing uar densities for up and down uar s and their antiparticles in the proton. =

(*)

'I = dpi*)

'' =

V( ' )

d = dpix)

2.1

and assuming t he itar -charges as predicted bv the rpiar model, the st ruct ure funct ions for electron proton and electron neutron scattering are given by iT p .r

x l «

ri

i

rf

2.1

I I

K"(x)

II'.

I I I ' \

I ,1 A I L

\ I I M N

I

U L

1

I

I I

* { j | ( r / I ,!) + g ( « + « ) J .

(2.17)

Tin- transition from eqn (2.10) to eqn (2.17) is done by a simple isospin transformation: in the case of perfect isospin symmetry n-quarks in the proton are e(|iiivalcnt to d-quarks in the neutron and trice versa. It is also worth noting, that the ansatz for the structure functions takes antiquarks into account, the so-called sea-quarks, which are expected to contribute because of vacuum lluetuations. Contributions from heavier quarks are neglected at this stage, although they. too. are present in the nucleoli. Doing a scattering experiment with a targel material having equal numbers of protons and neutrons, like, for example, "'('a, the effective nucleoli structure function seen is the arithmetic average of the proton and the neutron contributions Ff

(.,-) = j-x 18

{ti + v + d + d} .

(2.18)

In the structure function describing neutrino nucleón scattering, where, for example. an incident union neutrino interacts via a charged W boson and is transformed into a because of charge conservation only negatively charged quarks cont ribute. One finds F.p\x)

=2x{a

fV'(x)

= 2x{ .2 the probability density .2 it is growing. is shrin ing with increasing 2 . for values x u 2.2

The

C

agrangian

In a nutshell the evidence presented in the previous sections can be summari ed as follows hadrons are composed of fract ionally charged

uar s

uar s arc s p i n - 1 2 fermions t hey come iu three distinct colours there is evidence that colour exhibits an

symmetry

uar s are sub ect to a strong interact ion besides

uar s there are additional parlous in t he nucleoli

those parlous feel neither the electromagnetic nor t he wea

force

ote that the colour is distinct from and must not. be confused with the flavour discussed in ection 2.1.1. T h e pu les posed by these findings can be solved in a very elegant way by assuming that colour is a charge-li e uantum number, conceptually similar to the electric charge or the wea isospin. riginally proposed only as an index to got the boo eeping right, it is then understood as the source of a colour field. If the interaction mediated by this field is strong enough, then the large scaling violations observed iu the structure functions of the nucleoli find a natural explanation. T h e colour field apparently glues the uar s together to form the observed hadrons. motivating the name i/himis for the uanta of the colour field. If those glitons couple only to colour charge, then they are invisible in all deep inelastic scattering experiments using lepton probes and also the missing m o m e n t u m found from t he integral over F>(x) can be accounted for. Evidently there are good arguments iu favour of a field theory of strong interactions based on the colour charge of the uar s. T h e re uirement, that the theory bo renorinali ablo suggests a ang- ills gauge theory ang and ills. 1 4 . ssuming an unbro en g a u g e s y m m e t r y the general form of the agrangian is

CI

- IF;

F

1

i

- m

,, ,

2.

I |ii:\ l l,OI'MI NI

wln-re s u m s over repeated indices an- implied. T h e lield st.rength tensor F" the (ovarian! derivative D,, are given by the following expressions = O„ai

- D„AI -

{D,,),j

= ¿,jd„ + \aJTjK

(m,,),j

= niqSij .

I III'. (,»1 'l> I ,A< ¡H A N ( ¡ I A N



and

!,rhrA';,K (2.30)

where A", are t he gluon fields. ). and f ( A ) is a function such that for a given A a solution exists for only one value of the gauge parameters 0. In a non-abelian theory this may be true only if we exclude topologically non-trivial gauge field configurations, which in any case give only very small contributions to the action and do not affect perturbation theory (Gribov. 1!)7<S). The situation is illustrated in Fig. .'5.1. By inserting the identity in a suitable form. c.f. I / d . < ; | d / / d j ' | i ( / ( . r ) ) , and not showing source terms, the fundamental partition function can be symbolically written as

Z = Jvij'Vii'VA

exp

p t f ddet et | d ' , x £ c | 0 „ ( t f v < M ) ) x y[vo

- J v ^ D c - D A V ^ V v exp ^

/

J d l x j£,.liiss -

^f(A)2

¿f(A")

S

Ml 60

(/{/!"))

m

2D

(3.1-1) Source terms are not. shown. In the second line the divergent. 0 integral lias lieen discarded as the remaining terms are act ually ^-independent. Formally, l his means that we have redefined the integration measure. T h e ¿-funct ion has been implemented in the action as the quadratic term. T h e parameter £ is arbitrary, contributing only to t he overall normalization, and as such it cannot, enter into any physical quantity, like S-matrix elements, though it may appear in intermediate expressions. As is made clear below, particular choices, such as £ = I. are often preferred due to the relative simplicity of the resulting gluon propagator. The determinant of the .lacobiau matrix is incorporated into the action as an integral over the octet of ghost fields. ;/". These are unphysical, complex valued. Lorentz scalars which obey Fermi Dirac statistics, that is. t hey are represented by Grassmann variables, and transform under the adjoint representation of the gauge group. Ghost fields only appear internally in loop diagrams, their physical role is discussed in a less abstract fashion ill Section 3.3.3.1. T h e result, of these manipulations is the addition of gauge fixing and ghost terms to the Lagrangian density. T h e gauge divergence in the path integral which is associated with t he gauge degeneracy of t he gluon lieltls manifests itself pert urbatively in the lack of a gluon propagator. T h e addit ion of a gauge fixing term allows this propagator to be defined. To see how t his works consider the popular choice of covariant gauge: ,,. is used to go to momentum space. T h e gluon propagator. IK/;)"""'', is given by the inverse of the bracketed term.

NU

IÏIKOUY NR

q c p

» r y - ( i - 0 />"//' IK")

,7TT7

- , r +

(i

(3.17)

-

It is easy t o sec t hat t his inverse? would not exist in t he a b s e n c e of tin? g a u g e lixing term, that is. in the limit. £ —«> oc. S i n c e t h e n t he moment tun-vector />'' would lie an eigenvector of the inverse propagator with eigenvalue zero, this results in a matrix with at least o n e vanishing eigenvalue which cannot be inverted. T h e a term enforces causality. It can be traced to a d d i n g a term +ieA",A t t t ' to the action to ensure that the action integral is convergent. Another popular choice is the axial or physical g a u g e delined by 11 • .-1" 0 where n is a fixed Lorentz four-vector. S o m e t i m e s , the additional restriction I or i r = 0 is applied. T h e required g a u g e fixing term is ii 2 (3.18)

£ n , = - ¿ ( H • A-)(M - A " ) .

Since in this axial g a u g e the corresponding g h o s t term only c o n t a i n s t h e kin d ic piece and d o e s not. couple g h o s t s to any other lields. the g h o s t s m a y he t rivially integrated out and need not be considered further. T h e corresponding, m o m e n t u m space, gluon propagator is given by jtt'ub IK/')

i>2

Cllll

-V

+

n • ¡i

(" •

I>)

(3-If)

2

Now. in any g a u g e the gluon propagator can be d e c o m p o s e d into a weighted s u m of direct p r o d u c t s of polarization vectors ((/;)'' for the oil' mass-shell gluon: \ W "

= -T^-r- £ '' T.I..S

'f M

V

W

'

(:{"2,))

where, in general, the sum includes contributions from two transverse ('/'), o n e longitudinal ( L ) and o n e scalar c o m p o n e n t (S). Significantly, for an axial g a u g e in the on mass-shell limit, / r = t), only the physical, transverse polarizations propagate (C/. = C's -nuclear scale. At these scales had mi is appear I • t lie c< imposed of the (ant¡) D K K r m i ' T I O N 01'' I I A S I C U F A C T I O N S

3tt

many g a u g e couplings, g , ^/-ITTO,,, they contain. The simplest set of (tree) diagrams contribute to the cross section, which is proportional to the amplitude ,i|uared. at 0 ( o " ) where the power n is characteristic of the process. In gluon gluon scattering, for example, o n e has n = 2. whilst for three-jet production in 1. T h i s is the lauding oiiler (LO) approximation. T h e e 1 e annihilation it is n next simplest set of (one-loop) diagrams cont ribute at. O ( o " + I ) : this is the nextto-leading order ( N L O ) approximation, etc. Given a sufficiently small coupling, this perturbation series should converge t o the correct answer as more terms are added. In practice, the series is expected to be only asymptotically convergent, so that beyond a certain order the numerical evaluation of the series begins to diverge from the t rue answer. A complication arises in this approach because tree-level diagrams diverge whenever external partons become soft, or colli near and related divergences arise in virtual (loop) diagrams. T h i s is in addition to the ultraviolet divergences treated by renorinalizat.ion. Fortunately, in sufficiently inclusive measurements, such as the" total hadronic cross section, it. is guaranteed that, the two sets of divergences cancel. Unfort unately, in more exclusive quantities, which involve restricted regions of the external partons' available phase space, the cancellation is less c o m p l e t e and large logarithmic terms remain, generically of the form Since o S ( Q ' ) L is of order unity for Q2 » Q2. see eqn (3.22), L = \n{Q2/Ql). this can spoil the convergence of Unite order perturbation theory. In the second approach, the original perturbation series is rearranged in terms of powers of o s L . .la = £

rt„(asL)" n

+ as(Q2)

£ h„(a„¿)" n

+ •••

(3.23)

T h e first, infinite set of terms represent, the lending logarithm approximation (LI.A), then c o m e s the genuinely «„-suppressed next.-l.o-LLA ( N L L A ) and s o o n . Since t he enhanced regions of phase space involve near collinear or soft gluon emission, it is favourable for the primary partons t o dress themselves with a shower of near collinear or soft partons. T h e s e are the parton precursors of hadronic jets. An important feature of such multipartou matrix elements is that in the enhanced regions of phase space they factorize into products of relatively simple expressions allowing significant, simplifications in the treatment, of leading logarithms. In s o m e cases, it is actually possible to sum analytically the LLAand NLLA-series t o all orders in o s . T h e emerging picture of an event follows a sequence of decreasing scales. A genuinely hard subproeess produces a number of primary partons which then undergo semi-hard gluon radiation resulting in showers of soft partons which ult imately hadronize. T h e main features of an event are determined during its perturbative stages, thereby allowing tests of ( p ) Q C D . In the following subsections we describe the basic phenomenology of the three main types of particle collision and how QC'D applies to l lietn.

:n

nil

3.2.1

Electron

position

IIIKOHY l (VX'I)

itnn Unlnt ion

Electron p o s i t i o n annihilation to hadrons provides the simplest colliding beam processes that can be described using p Q C D . T h e simplicity follows from both I lie well-defined energies of t lie initial s t a t e part icles and the fact t hat t he Icptons interact via a weakly coupled, colour singlet, virtual photon. T h i s allows a clean separation of the initial and final s t a t e particles. T h e combined m o m e n t u m of t he incoming Icptons provides a large scale justifying the use of pQC'D. In t he parton model t he basic interaction is an electroweak process. e + e ~ — 7 * / Z —• (|(j: this has essentially the same cross section as the well established process e'e —> / / + / / ~ . It. is usually adequate t o consider single photon exchange due l o the small value of the electromagnetic coupling o ( , m = e~/(47r) s; 1/137. T h e structure of the hadromc final s t a t e d e p e n d s only on the centre-of-niomentinn (C.o.M.) energy, ^/s. of the collision and if polarized the polarizations of the incoming Icptons. T h e C.o.M. s y s t e m is often also referred to as "ccntro-of-mass" s y s t e m , since in the s y s t e m where the momenta balance, the centre-of-mass of the interacting particles is at rest. Dealing with relativist,ie particles, however, the name V-ontre-of-niomentuni' is more to the point. At low C.o.M. energies. 0 < y/s < 5 G e V . t h e most interesting quantity is the total hadromc cross section. This shows a lot of structure characterized by "steps' at quark thresholds together with strong resonances, associated with qq bound s t a t e s that possess the s a m e q u a n t u m numbers as the exchanged photon. vector meson: /i. uj. In essence the off mass-shell photon behaves as a ,I I>( ' = 1 (,'). .1 /(,'. T(L.b'). etc. T h e hadronic final s t a t e is characterised by low mult jplieitics an/* = 3()GeV. by the emergence of three-jet features in a fraction. (9(1(1%). of t h e events. By identifying these jets with primary partons it is possible to test the nature of (¿CD's basic constituents and their couplings. For e x a m p l e , three-jet events are believed to be a manifestation of vector gluon emission in t he process e ' e~ —» qqg. An e x a m p l e of a three-jet event is shown in Fig. C.l. T h e rate of this three-jet, production gives a measure of t h e strong coupling. o s . whilst the angular distribution of the j e t s reflects the spin-1 nat ure of the gluon. At even higher energies, small fractions of well separated four, live and more jet events appear, allowing tests of the triple and quartic gluon couplings. Note that these j e t s are required to be well separated to avoid the collincar and soft enhancements that would invalidate lixed order perturbation theory, thereby complicating any comparisons to theory. A more precise definition of a jet is given in Section (i.2. On dimensional grounds the total cross section must take the form

(3.2-1)

nii; ci

i i--sc mr i I

it .sir H

CTI

tr.

where i lie represent the relevant masses, sneli as uar or hadron masses. The function f(x,) tends to a non- ero constant as x, . ince the uar and luulron masses are mostly small, their effect becomes negligible as .s increases as prescribed by t he photon propagator. This and I he cross section falls as .sremains true until around y s -l GeV when deviations begin to be seen this is I lie tail of the resonance which becomes dominant al y s 1 GeV. part from i he large enhancement in t he total cross section, the main effect, of exchange is to modify the flavour mix of produced uar s and to introduce asymmetries into the polar angle distributions of the primary uar s, compared to pure photon exchange. bove v s 1 GeV, photon and exchange remain of comparable importance, but the total Itadronic cross sect ion continues to fall and becomes of less relative importance as other product ion channels, such as e e V , open up. n example of a less inclusive uantity in e e annihilation is the cross seel ion for the production of a specific type of hadron in the final state. uppose this hadron. h. has momentum p'1. then the differential cross section can be written in the form of a convolution

da

- -h-v(/>.*)

=

( ,*)

Dll(z)

.

2

T h e first term. d r, is the hard cross section for the product ion of a parton a such that it carries momentum p1'/z. T h e second term is a fragmentation function. D\](z)ilz, which gives the probability that the parton ii produces the hadron h carrying a fraction c of the primary part.on s momentum. This fragmentation function is the final s t a t e analogue of the previously mentioned p.d.f.s, to be discussed more fully in ection .2.2. T h e product, of t hese two terms is summed over all the possible contributing partons and integrated over the momentum fractions. T h e factori ation is between a pcrturbatively calculable, short-distance cross section and a non-perturbative fragmentation function. It is important to reali e that a does not depend on the identity of the hadron h, which would be a long-distance effect, but only on the parton n and the colliding beams. Conversely. D\) does not depend on the short-distance, hard subprocesses; in this sense it is universal and can be applied to any subprocess that, produces the outgoing parton i. t the lowest order the relevant hard subprocess is e e , so that in e n .2 the sum is over uar s with 2m,t < \/s. This gives the parton model prediction for which, as indicated, the fragmentation function depends only on t lie momentum fraet ion T h e inclusion of C corrections complicates matters, though the basic faetori ed form remains the same. In particular, renormali ation re uires the introduction of an arbitrary renormali ation scale. ;. whilst, the factori ation procedure introduces a second, arbitrary factori ation scale, .-. This acts as a cut-off on the virl.uality of intermediate particles, e uivalent to a cut-off on the inverse distance it travels. T h e exact origin of the scales , and

.Ili

I III'.

iir,i u n i ir * ¿V II

///. will licciiiiic clear in Sod inns 3. I and H.li. T h e Q C D improved parton model predict,ion is drr^

C1C,«. + .''j/',. • ) 2 = ci ./'_>.s. It is possible to calculate this structure function in Q E D perturbation theory. An approximate form is fc/c(x,

tr

) = ¡i{ 1 - x f -

1

with

/Hi'2)

=

(3.28)

In practice, one has 0 < 3 «C 1 so t hat f,./c(x. //") aciifiit accounts for I lie remaining parlous which have their origins in non-perturhative physics, perhaps associated with tin- (negative virtua l l y ) photon lluctuating into a vector meson. T h i s opens up the possibility of effective lepton liadron. known as singly resolved, and hadron liadron. known as doubly resolved, collisions. These an- discussed in detail below. All these types of 77-evonts are characterized by low C'.o.M. energies and sizeable longitudinal momentum imbalances. Often these events constitute a hadronic background to the events of real interest in an experiment. We also ment ion one further complication. At very high energies it is necessary to use beams with very small transverse sizes in order to increase t he luminosity and compensate for the falling cross section. This gives very high charge density particle bunches whose intense electromagnetic fields can induce radiation in one another as they approach each other, the so-called beamstrahlung. The details of I his depend on t he specifics of t he beam profile, but can lie treated in a similar vein to breinsstrahlung (Palmer. 1!)!)0). if.2.2

Lepton

hadron

scattering

l.epton hadron scattering is a traditional method of probing the structure of hadrons. Since hadrons are now known to be composite particles with partonic. (nnti)quark and gluon. constituents, such collisions are more complex to describe than lepton lepton collisions. In essence, we view the observed scattering, / h —• I'X. as a manifestation of the hard subprocess fq —> Pq'. The advantage of lepton probes is that they undergo experimentally and theoretically clean, pointlike interact ions which are describable in terms of the exchange of a single, virtual, gauge boson. Multiple boson exchange, whilst possible, is suppressed by additional factors of the electroweak couplings, o 2 m or C y . A basic classification of t he event s is based on t he nature of the boson exchanged by t he initial lepton and quark. In neutral current events, characterized by / = ('. a photon or Z is exchanged. Whilst in charged current events, characterized by C. = e. //. r and / ' = v,.. v„, u r or vice-versa, a W ± is exchanged. For charged leptons the exchanged particle is predominantly a photon. Weak boson. Z or W ± , exchange is observed, however, at low Q~ KS< It IT I I O N (>1

MASK ' UKA ,,

i FA f

+(F\-\

FrM-v

aT,r,,

T

+ {F,

+ Fr,

+ . Empirically, the two form factors are both described well by the dipole formula which corres|)onds to a spherically symmetric, exponentially falling charge distribution. One has K-\r\/o

p(r)

=

S~f7

:1

1

f,(.3. where also the meaning of the plus-prescription is explained. Since the virtualities involved in this initial s t a t e evolution are negative t hese are the space-like splitting functions. Equations very much like cqn (3.4!)) control the / a. parton u will carry a fraction in the range c.c I ; of its parent, lis. nioineutuiu. and any other products. , a fraction 1 trictly, the branching probability densities are given by the distribution functions KS( 'HI I T U JIM < )!•' I I A S K ' | { i : A < ' H O N S

17

li is the application of tills sum rule which provides compelling evidence for the existence of ghions. currying over ,r)0% of the momentum, in a proton (LlewellynSmith. I!)72). Equation (3.49) allows us to calculate how the p.d.f.s change between the scales //(> and //. However, since t he /(.T, //o) ( / = q. {[> = ]>'' —I»'1' ~ Xip/^'. and fi wit h the fraction of the object's momentum carried by the struck constituent. Thus x. the constituents moinent.um fraction with respect to the incoming hadron. is given by x = .rip/l Both .rip and ¡1 lie in the range [0. 1). T h e four-fold differential, diffractive DIS cross section is given by

,

d.riP . - . U M )

r

-

E

i

J

.

.-J r p

g

z

, . (3.I)

Here ///.- is t he factorization scale (we have suppressed the renormalization scale is the usual DIS structure function describing a photon scat/in) and F-\'f\z) tering off a parton / carrying a fraction z of its parent hadron's momentum. T h e remaining terms are the new diffractive parton density functions (Berera and Sopor. 1994), also known as (extended) fracture functions (Trentaduo and Veneziano, 1994). T h e diffract ive p.d.f.s satisfy t he usual D G L A P evolut ion equations. At tempts have been made to go beyond eqn (3.til) using Regge factorization. This assumes that the Pomoron is a real object, whose coupling to the parent

I III'! T I I K O H Y O K

QCI)

liadron is described by a function of.in- and I mid whose parlon content is then deserilied by functions of Q and Q 2 . T h i s unproven assumption gives = /..,„(**,/.)F2D(2> ( f i = — . ( A . \ lip J

da: 0 >d/

(3.02)

where fw/\, is often referred to as the Pomcron flux factor. Measurements to date suggest t hat, t he Pomcron has a high gluon content and that there is a significant probability that, a gluon carries nearly all of its momentum. Such a picture has also been promoted in the context of hadron liadron collisions (Ingelmau and Schlcin. 198.r>). Unfortunately, it appears that the same Rcggc factorized structure fnnctious as measured in DIS will not be applicable, without at least, some modification, in the description of hadron hadron collisions (Collins et til.. 1!)!):$). Historically, dilfractive events have long been known in hadrou hadron collisions where a well developed phenomenology has arisen. Indeed, this was used to predict that sizeable dilfractive cross sections would occur at IIKHA (Dounachie and Landshoff, 1987). However, the discovery of such events at IIKHA (ZKUS Collab.. 1993: II1 Collab.. 1994) still came as surprise to many people and it has led to a resurgence of interest in the nature of diffraction. Deep inelastic scattering events, whether dilfractive in nature or not. are characterized by large values of Q2 i i (3 GeV)". There also exist events in which an incoming charged lepton emits via breinsstrahlung a quasi-real. Q~ ~ 0. photon which interacts with the incoming hadron: the so-called photo-product ion events. As mentioned earlier, such photons appear to have a rich structure and variety of behaviours. They may behave as a hadron. giving effectively a hadron hadron scattering. This in turn could be elastic, here meaning ->*h —> Vh with V a vector meson, dilfractive, soft, inelast ic or hard inelast ic. All these categories are elaborated below. T h e hard inelastic events are viewed as due to the scattering of (anti)quark or gluon constituents within both the hadron and photon. Thus we require p.d.f.s to describe even the photon. Of course, it is also possible that th.i»gl e difl'raciivc

60

PP ^double iliffr'.K'liVC

40

20

0 10

10 2

I0 3

V7 (GeV)

.'{.ij. Measured cross sections in hadron hadron collisions as a function of the C.o.M. energy. D a t a are taken from t h e Review of Particle Properties ( P D G . 2000) and from the D u r h a m reactions d a t a b a s e http://durpdg.dur. ac.uk/hepdata/roac.htinl.

I III-) < ¿ ( 1 » I »!•.•;« n i l ' I l< >N O l -

ItASK

HKA(' I IONS

I»' 10o % o o

10

10

10--'

10-1

A ISK. pp. \'s=30.4GeV 10-'

A ISK. pp. \'s = 52.8GeV •

5

to-

P

SI'S WA7. pp. p, i h = 30GcV/c

O SI'S WA7. pp. p,

I

a|)=50GcV/c

tf ti

O GDI-", pp. V s = l 8 0 0 G e V 10"

—I I » »1111 10-

10-7

10

10

1(1

l/l (Gl-V/C)

FIG. .'5.(i. T h e measured elilfcrential e-ross section in pp and pp collisions ) suggest values n (i.8. 1(1 for meson meson, meson baryon and baryon barvon scat tering, respectively, though more explicit calculations based on gluon exchange between the constituent quarks modify these simple exponents (LandshoH", 1!)74). Figure 3.(i shows the /--dependence! of pp anel pp scattering. Approximate forms for the elilferential cross section in the elilfrae-tive region are given by

d 3.0 G e V 2

(3.0-1)

(

lie dip ni struct ure can be described by interference between wo exponentials. The -distribution can be related, via a Fourier cssol I ransloriuat ion. to the impact parameter space distribution of the scattering centres in the liadron. exp vu 4 exp b,.\flr), where h is the impact parameter. efining an effective slope by

( .( ,) the ineasureinents show that ,.ir increases for large s. Thus the liadron shrin s at higher energies. The very forward pea ed nature of the elastic scattering cross section indicates that low momentum transfers are dominant. This essentially straight through behaviour means that speciali ed low angle detectors, usually in conunction with low luminosity, are re uired to measure this large cross section. Interestingly the optical theorem provides a highly non-trivial connection between this forward f I differential cross section and the total cross section sec Ex. -11 .

F i e . . . ingle dilfractive dissociation, double dilfractive dissociation and central diffraction. Experimentally these events are characteri ed by a forward et separated by a rapidity gap from an intact , scattered, incoming liadron or two forward ets separated by a central rapidity gap or central activity separated by two rapidity gaps from t he scattered, incoming hadrons. The next important class of reactions involve dilfractive dissociation processes in which there is some brea -up of the scattering hadrons. possible way to view these events is as the -channel exchange of a colour singlet ob ect called a omeron. see Fig. . . nli e the case of elastic scattering, in a single double dilfractive dissociation event one both of t he hadrons is left in an excited state which then brea s up into a low multiplicity system of hadrons et , for example, p n;r . Typically the mass of t he excited hadronlc system is distributed as drr i . whilst, the -dependence of the cross section falls away exponentially with a coefficient which decreases as i n c r e a s e s . Experimentally the ey signat ure of these events is t he lac of any particle production in bet ween

I III 1 St n i l ' t I O N O l ' H A S H • H I ' V

I IONS

I lie scattered/dissociated Imdrons. (¡(invent ionally. I his gap is quoted in units of rapidity. II the final state hadrnns have masses M t and A/j. then they have a rapidity gap of A;/ ln(.s/j\/| Since the size of the gap is usually there is a minimum C.o.M. energy , / s ^ 4 . 5 C l e V required for these events to occur. A related class of events, known as cent ral diffraction events, show two large rapidity gaps separating centrally produced jets from forward/backward going hadrons (UAH Collab.. 1988). These can be interpreted as the interaction of two Pomerons. as shown in Fig. 3.7. They are of particular interest because the jet activity indicates the presence of a hard scale and the possibility to apply pQCD to t heir description. At low energies the cross section for all these rapidity gap reactions equals approximately the elastic cross section, with the ratio of single to double dill'ractive events found t o be % 4 : 1. As the C.o.M. energy dependence of the cross cetioti for a fixed excited state is flat, the growth of the total dissociation cross cction with y/s can be attributed to new excitation channels opening up. Experimentally the double diffractive dissociation cross section grows faster than the single diffractive dissociation cross section. This is in accord with the naive expectation oc. a new dominant, contribution must be included. This is the I'omeron. It. has the quantum numbers of the vacuum and the Regge parameters au»(0) « 1.08

and

oj,. « 0.25 .

(3.71)

These values are derived from successful fits to a remarkably wide range of data (Donnachie and Landshoff 1902: 1991). This trajectory does not correspond to any presently known particles, though it lias been conjectured that it is related to t he predicted glueballs of Q C D . Actually, since o B »(0) > 1. t he Pomeron is 'supercritical' and. unless eqn (3.68) is modified, will lead to a violation of unitarity in the .s- —> oo limit . More apparent, is the absurdity T.-i/rr,( ^ (.s/A/j-)" n ' ( , , ) _ l > 1 for s sufficiently large. T h e inclusion of the necessary multiple Pomeron exchanges and unitarization corrections leads to a more complex theory (Khoze r.t. ul.. 2 0 0 0 ) .

It is important to remember that Regge theory has not. been derived from (¿CD. One should therefore be wary of regarding it as doing anything more than providing an accurate and economical, phcuomcnological framework for describing data in the Regge limit.. It also act s as a guide in framing the questions addressed in an experiment. T h a t said, p Q C D has been applied to the region .s » |/| > AQCI), leading to t he development of a hard Pomeron with an intercept significantly above one and a small slope. To distinguish it. the usual Pomeron is now often referred to as the soft Pomeron. This hard Pomeron is associated with the summation of leading logarithms of the form n s l n ( . s / / ) (Kuraev ct a I.. 1977: Balitskv and Lipatov. 1978). T h e simplest model for such an ob ject is the /-channel exchange of two glnons (Low, 197">: Nussinov. 197.r>) (or one 'Roggeized" glnon). which is suggestive of a glueball interpretation. T h e hard Pomeron also manifests itself in the small-.»' behaviour of structure functions where it sums leading [(vsln('l/:»:))" logarit hms. However, a word of caution should be sounded. As the hard Pomeron theory implies a rapid growth in t he number of partons then non-perturbative methods will be required ultimately. T h e search for the predicted hard Pomeron is an active topic of research. Finally, we turn to the rare, hard events which shall be our main focus of interest. By experimentally requiring an event to contain a large momentum scale we raise the possibility of applying p Q C D to its description. Furt hermore, t he short-dist ance scales suggest, working wit h t he quark and glnon constituent s rather than the colliding hadrons themselves. T h e situation is analogous to DIS and again a lact.orized formalism can be applied. This is illust rated in Fig. 3.9.

(¡(I

I III-: I I I K O H Y . /40d{[,. In general, we do not expect ./'i - x-> so t hat the hard subproccss will be boosted with J — (./-| — x->)/(xi I- x>) with respect to the h|h-j laboratory frame, resulting iu the outgoing particles being thrown to one side or the other. T h e sum is over all partonic subprocesscs which contribute lo the production of c and il. For example, the production of a pair of heavy quarks receives contributions from qq —> QQ and gg — QQ. whilst prompt photon production receives contributions from qg —• q~, and qq • g7. These two-to-two scatterings give the leading. 0 ( < \ i ) and C?(o s o,. l n ). contributions lo the hard subproccss cross sin-tion. Beyond the leading order it is necessary to consider two-to-three, etc. processes, which gives rise to a perturbative expansion a — C i , o o " + C'M.()O" + I + CNNI,(>O" +2 + • • •• A complication arises with the higher order corrections as they contain singularities when two incoming or outgoing partons become collinear. It is the factorization of these singularities, order by order, into the p.d.f.s and fragmentation functions which gives them their calculable /ij. dependencies. This, logarithmically enhanced, near collinear

I'lIK < ¿
I »KSOItll'TION O f

i.l

H A S H ' IIKA( • I I O N S

raeliation is manifested as t lie appearance of init ial and linal s t a l e jets associated with eaeli of the incoming and outgoing partons. 'Flic mix of hard subproeesses which contribute* t o eepi (3.72) depends nont t'iviallv on I lie relative* sizes of hot h the cross sections and the p.el.f.s. The latter are iulluenceel by both the type and energy of the colliding be-ams and any rei|iure>ments placeel on the kinematics e>f the final state. For example. requiring the outgoing particles to be produced in a given rapidity range, perhaps e-ort'espoudillg t o the geometry of a detector element , directly affects the .r-ranges being sampled in the integral: see Ex. (3-13). To go further. we consider heavy quark production at the TEVATIION, a y/s = l.<STeV (now 2 T e V ) pp collider at. FKRMILAH. In the case of cent rally (y = 0) produced bottom qua rks one has x\ w x> ~ 2vi\Jy/s 2 x 5 / 1 8 0 0 = U.005G. whilst for top quarks it is n « r> ~ 2 x 175/1800 = 0.19. At small x gluoiis dominate the p.el.f.s, whilst at large x only valence (anti)quarks are present: this is particularly true at the higher scale appropriate for top production. Q ~ 2/«Q. Thus, bottom ejuark production is dominated bv g g —> bb scattering, whilst top quark production is dominated by the annihilation proc:css e|e| —» tt. Here, we* see- that in a high mass 'annihilation process' it pays to have au antihadron in the initial state. In this result the larger cross section for gg —» QQ is overwhelmeel by the p.d.f. contribution. As a seconel example we consider eli-jet production. In the» abseiie-e of any flavour eleterinillation the outgoing jets may be- seede»d by either a primary (auti)quark e>r gltion so that there» are» many cont ributing hard subproeesses: gg —» gg, ge| —» gq, eje|' —» <jej'. etc. Loosely speaking, the relative hard subprocess cross sections are in the ratio : C,\CF '• Cf.- ( 'tc.. reflecting the colour charges of the collieling partenis. This allows us to e-xpress the integrand in eqn (3.72) in terms of an effect,ive p.el.f. (Ce)inbrielge and Maxwell. 1984). see Ex. (3-14).

/.f

-

m)

with

/" , r (:r) -

<j(x)

Y , ^ J= involve an underlying event arising from the» collision of the two beam remnants. In broad out line the underlying event is like a soft., inelast ic e-ollision bet ween two haehnns of reduced C.o.M. energy squared (1 X[X-i)s. Fortunately, the soft particles produe-cel have limited transverse» moment um and so elei not unduly obscure the high transverse energy partie les produced in the hard subprocess. Observationally there is an increased level of hadronic activity in hard events, even away from any jets, as compared to minimum bias events which are effectively equivalent to normal soft inelast ic collisions. T h i s is the so-calleel pedestal e-ffect.. Thus, more refined models builel in an interplay between the hard suhproccss and the underlying event (Sjostrand auel van Zijl. 19X7). OIK> possibility, which Iwcomcs more likely wit h increasing C.o.M. energy, is that a second hard scattering ex-curs

111 .(

between I In par tons in I lie Iii iiiii remnants. y t rent ing I In- two si at I n s as indei ii li iit the rate of double scattering can be estimated as y , nrif 2 i. i T h e assumption of independence is plausible provided all the momentum fractions remain small. In an experiment, it is necessary to supply a criterion to decide when to initiate the read-out of the detector. Typically, t his trigger condition is based upon nown supposed features of the events which are of interest. This introduces inevitably a bias towards ust such events. Therefore, it is also common to collect an unbiased data sample based upon a minimal trigger condition such as the occurrence of a bunch crossing or the presence of an energy deposit somewhere in the detector. Given the relative cross sections for the hadron hadron scatterings these miuimiun bias events coincide essentially with the soft, inelastic collision events. ince hadron hadron colliders are often viewed as discovery machines searching for very rare events, there is a need to use high luminosities. Given large hadron number densities in the colliding bunches it becomes li ely that more than one pair of hadrons from the colliding bunches may interact, most li ely in soft, inelastic collisions. Thus, even when a hard trigger is satisfied it is uite possible that the detector is seeing an event of interest together with several soft, inelastic events. For example, at nominal luminosity at the planned I.I IC at T . . each hard event is. on average, accompanied by 0( 1(1) simultaneous iinniinmn bias events. Fortunately, t hese extra pile-tip events produce mainly low transverse momentum particles, spread throughout longitudinal phase space, whilst the hard event must have high transverse momentum particles, typically restricted kincmatically to the central (y = 0) region. 3.3

B o r n level c a l c u l a t i o n s of Q C D c r o s s s e c t i o n s

lu this section, we shall review the calculational techniques required to evaluate basic tree-level processes. We shall concentrate on the process e + e ~ — qq. which is a paradigm for several important processes, together with its lowest. C?(o s ). tree-level. Q C D correction. e + e ~ —> qqg. which we will use in our discussion of the Q C D improved parton model. We will also look at the pure Q C D process qq —» gg which will give us an insight into the nature of gauge invariance. We do assume some previous familiarity wit h Dime spinors and working with Feyinnan diagrams. The interested reader can refresh t heir memory and find more details in any good text book, such as the one bv Ait.chison and llev (1989) or by IVskiu and Schroeder (l!)!).r>). 3.3.1

e + e ~ annihilation

to quarks at 0 ( o " )

T h e basic Feyuman diagram for e + e _ —» qq is given in Fig. .'{.10(a). Strictly speaking, this lowest C?(o") process is more an electroweak than a Q C D interaction. However, it remains of great importance in the description of e + e ~ annihilation to hadrons. and using crossing symmetry, also t o deep inelastic scattering. Fig. .'{. 10(b) and the Drell Van process. Fig. 3.10(c). Furthermore, by replacing t he leptou pair by a new quark pair (q q'). we can learn about di-jet production

IT< >KN MOVI' I C A L C H I

;+

\ I K >NS ( )!•' ( J< 'I ) ( 'IK )SS SI

q. j

' = ( f + |- £")'', t h e f o u r - m o m e n t u m transfer. For simplicity, only photon e x c h a n g e is included, which is appropriate for Q~ My. and we have chosen to work in the covariant, Feynnian g a u g e , ^ = 1. T h e

X (re,,) 2 • Nc • &(

"')(1 I" 7r./)|,, | 8 | p «

|(/> -I-»")(1 F 7r,)],,

(3-79)

The spin polarization state is specified by the space-like four-vector s which is orthogonal to />. s • /> = (I and which for a pure state is normalized such that I. The approximate form is appropriate to the high energy limit, HI <SC E. when the spin vector is parallel/ant ¡parallel to the particle's direction of travel: the so-called helicity basis. Often the incoming particles in a collision are unpolarizcd. that is. they are an equal admixture of all possible polarizations. It is t herefore conventional to include an average over the incoming particle spins: again see Appendix 13. Concentrating on the hadronic part o f e q n (."i.7-r>). making t he spinor indices explicit and assuming no spin sensitive measurements are made on the outgoing quarks this result allows us to write: E " ( ' / ) . '>:, " ( < / h ' ' ( ' / h y ' u « ( ' / ) / = spiiis

( X "('/)"'('/)• K spins

(

X •>(f~ihH

"}

//'""IV {I |

- >r>r

//'"' • I. Tims the

+ 'r>r\ •

Of course, in practical situations a number of tricks (short cuts) can often be used to speed up evaluations, for example. •" • 7/i w i 2 = ¿ r e 2 • T i »th,(r

+ »}

x ( c c q ) - . V r . Tr {(,? + « i „ b " ( j f - '" C U O S S S I P

i,,

H O N S

velocity i»t' the final state quarks. In this C.o.M. frame, with massless loptons I ravelling along the ¿¡-direction and the scattering in the x-z plane, the fourlilomenta are given l>y

=

and

(->/'

'/

=

().(), ± 1 )

Jo2

(:{.<S7)

(1, ±fl' x sin 6>', 0,

cosfl*)

with

I ii* = J 1 -

.

Note that as a Lorentz invariant quantity |A4|- could only depend on the partii |es" four-momenta via their invariants, for example, their scalar products, ]>,-¡ij. In a two-to-two scattering. » + PI,)2 = (Pc T P,l)~

I = (/>« ~ Pr f = (pi, - P