!levisla Mexicana de Fúil'll 36 Suplemento 1 (1990) .';20-564

Introduction to perturbative QCDM.H. Reno

IJcprH'tarllCIl/o de Fisica, Centm de lrw('.o¡/igación y Estudios A I'(m::ados,

Instituto J1olitécnú', Nacional, Apartado l}/),';tal 14-740, 07000 Jlhim D,F,

Abstract. The intent of tll('s(' lectuH's is to preseJll a p<,dagogica!inlroduclion lo perturhative QCD. wilh an emphasis on ('a]culalionaitl'chniqlll's rathror lhan formal dl'riv<ltions. Firsl, lhe slandard QCO La-grangian, dimensional regularizatioll and renorma!ization ar(' r('vicwed.'fhen, dC'ep inelastic scattering in tlit' ,onlext of tlj(' naive I,<trtan mude!is df'srrihf'd, and the qc[).illlpro\'l'd parlon model is 1lI0tivatt'd, TheAllarf'lli-l'arisi equatiolls for tlt!' partoll distribution fUll<:t.iolls<lre de-riv('d, ¡¡lid the experimental f'xtradioll of lhe' distriblltiol1 fUIl,:' JIlS isdisCllS!wd. Tite first order Qen radiali\'l' fOrrections to dl'ep inela.slic:sc:attC'ring are lIsed to illustraf¡' factorization. lIigher ordl'f rorrc\tionsto U' alld Z prodllctioll ill hadrollir ('ollid('rs are presenled,

1. Introduction

This series oC lecturl's is I1ll.endcd as in inlroductioll lo perturhatiw calclllatiolls int¡llillllum chrornodynamics (QCD) ami as an aic! 1,0 Illldcrstanding how SOIlH' ('xper-illl('nls intcrprct thcir mea.surcmenls in ll'rJns of lIw QCD-irnpro\'ed parloTl l1Iodel.The sludenl is assllmcd lo have sorne backgroulId wilh Feynrnall diagrallls and per-lurbati\"e QED. Ba.."ic introduclions may be founc! in Hds. [1]+1]. For introdllct iOlls tothe standard model of slrong, weak and e1('ctrolllagnl'tic inl{'ractions and the partonmodel, see, for exarnple, Hcfs. [5J and 16]. :\ particlllarly good undergraduat(' le\'eltcxtbook is Re£. [7J. Discussions more spccific lo QCD are Cound in Hds, [8]-/12].

The prograrn for these Icctures is to begin wilh abrid re\"iew of lhe QCD La-grangian, dimensional rl'glllarizalion, and tile minilllal sllbtraction rt'normalizationschemc (~ts), \Vith llwse tools, we compute the ("olllller Icrm al arder 0,, for lhe<juark self cncrgy. Ncxl, we review the rcnonllalizatioll group cquations, II\(' rllnTlingof lhe coupling constant ilnd the inlroductioll of AQCI)' Stepping back fl'olll QCD astiJe theory oC strong interactions, we re\'iew 11](' nai\'(, <juark rnodel in c1t'('p inf'!a...<;ticscaUering and consider sOllle oC its SllccesSt'S aud dl'fects. \Ve liJen look al lheQCD-irnprO\'ed parto n Illodcl, and heuristically d{'riw the QED equi\'all'uls of theAltarclli.Parisi equations. A discussion of t11l' ('x¡H'rinwntal determination oC thequark and gluoll distrihutioll functiollS follo\\'s, Thl' property of fnctorization isillustrateu with the order n,' corrcction to d('('p illl'laslic scattering, Finall)'. higherorder corrections to IV alld X production tire pres('llted,

lnlroduclion lo perlurbative QCD 521

Particles U(I)y

Fermions

(~L.(:t(:), 1/3

UR, eR, iR 4/3

dn, SR, bR -2/3

(~,),t;),,(V;), -1

eH, IIR, TR -2Spin-l Basaos

11¡, O

IV;, i= 1,. o 0,3 O

A~I, A = 1, . .. ,8 O

Spin-O Bosotls

(::)

SU(2),

2

2

3

2

SU(3)c

3

3

3

8

TAALE ¡.IA. SU(:l) x SU(2) x U(l) charges of lhe fundamental particles. Elcctric charge equalsQ = y /2 + T whcrc T = :::1:1/2ror SU(2)L doublels alld T = :i::l, O ror lriplets, andzero olherwise.

1.1 TllE ST,,~n"Rn MOIlEL

Befare \'iC ¡ook specifically al QCD, \Ve review the standard modeL The standardmodel is gallge lheory based 00 lhe gauge group 5U(3) x 5U(2) x U(I). 5pontaneoussymmelry brcaking in the cleclroweak sector yields finally the elcdroweak theory.The fundamental particlcs of the thcory are the gaugc particles: gluons, ~V::l:, ZOand the photon; amI thc ferrnions: quarks and leptons. A summary of their trans-fonnation propcrties under the gauge group and their masses (including symmetrybreaking) are shown in Tables t.lA and B.

FrolIl Tables 1.1A and LIB, \'le see tilat the gluon (9) field AIJ and the quarks(e.g., 1t and d) are the ollly pa.rticlcs that have non-trivial transformations under thecolor gauge group. ExperimentalIy, fradionally elcctrica.lly charged particles havenot be observed, II5] so our bclicf is that color is confined: physical particles, likemesons amI baryons, are color neutral.

This rnakcs the tcsting of QCD as the thcory of strong interactions a moresubtle task tIJan, for example, the electl'oweak theory. Thcrc are severa.l argumentsof a nllmerical nature that favor the SU(3) gauge grollp. Fir,t, SU(3) has complexreprescntations, so the quarks representations (3) are distinguishablc from antiqua.rkrcprcscntation (3). Thereforc, color singlet spin-O mcsons can be made from qq pairsand spin-lj2 baryons from qqq triplcts. This by itself doesn't fix the gauge group toSU(3), bul lhe calculations of r.0 ~ 2/ alld ¡¡ = O'(e+e- ~ hadrons)/O'(e+e- -+

522 M,ll, Reno

ParticlesFermiolls

"d

e,

b

11;, i = e,JI,T

e

l'

T

Bosons

1W:t

ZO

911'

~1a.<;("l'; [GeV)

.....,6 X 10-3

.... 10 X 10-3

- 13.....,200 x 10-3

> 0(50)- .\

O0,511 X 10-3

105 X 10-3

1.7811

O

810" 1392.4" 18

O> 3.9

TABLE 1.1 R. Partide mas.ses in the spolltancollsly hrokcu thf"OfY. F:lectromagnetism and lhe colorgroup are assumed lo be ullbrokcn. For light quark l1IaSS('5, sce, for example, Ref. (13].For olher ma.'lses, sec Hef. 11-1J.

Jl+Jl-) distinctly favor quark triplets. '1'0 lowest Ordl'f, lhe ratio I? dl'pends on thesum oC the electric charges squared titiles the IIllllliH'r of replicas, that is lile = 3,the numLcr oC colors: /lo = Nc Li q;. Abovc lhe b quark thrcshold, this givcsRo = 11/3. Expcrimcntally, tIJe dala lic vcry closc lo this valuc. Al ..¡; = 29 GeV,for exarnple, the MAC collaboration [16] lIleasures n = 3.96" 0,09, Thi, agrcc, wellw¡th the thcoretical value for R wlwll higher o("(lerQCD ami c1ectroweak correctionsare included. For a sUlllmary of tile data aIHI thc theorctical predictions, scc thePartide Data Book [14].

1.2 QCD LAGRANGIAN

Recal! in QED, tile masslcss fermion Lagrange dellsity is

" '1' 1'1'" ,7.('~ '"),,, 11' 1"'" J'{),I,1...-=-4 '/111' +'f'l¥,+u:tfl 'f'=-;¡ '/11I' +'f'l 'f', (1.1)

for F~II = u¡jAII - uIIA¡¡ and rermioll TjJ with Ilnit charge. Becausc of thc covariantdcrivativc D~ = u¡; + cA¡;, when ~t -+ cxp(icO(.r))~', if A¡; -+ A¡; - u1JO(x), theLagrangian is invariant. The single cOIlt.illllOUSparall\('lt'r family of transformationsrepresent, the grou!, U( 1).

IlIlr(Nillctioll lo lJf'rturootit'c QCD 523

In thc SU(3) tbeory, the fcrmions trallsfofm as a triplet. Under a gauge transfor-matiollljJ -+ exp(ig Lt\=1...8 Ot\(x)tA)t/' fol' lile param('Í('rs OA(x) aIHI the generatorsof SU(3) in the fundamental represcntation: lA. Fol' a covariant derivative and ficldstrcngth tellsor dcfined by

(D¡J)ab = D¡,flab + 9 L(lA A;})ab,\

a, b = 1 ... 3

(1.2 )

the Lagrangian with J) -+ O and F -10 F ill Eq. (1.1) is invadant under SU(3)gauge lransformatiOlls.

Just as in QED, a gauge fixing lerm is il('('(kd lo define lhe gaugc bosan prapaga-I.or. \Ve work in covarianl gauge: (J¡IAII = O. Furlherlllore, ghosl lenns are requiredto remon~ the unphysical polarization of the gluons in loops. (Ghosts are not ncededin QED because the U(l) transforlllation is Ahdian.) A concrete illustralion for thencccssity of ghosts to preserve unitarily [I iJ is scen in a wlIlparison of lile calculationof qij -+ 99 llIatrix delllenl sc¡uared and qq -+ qij rnatrix clernent al. one loop. TheLagrange d(,llsity, including a qllark lIlass 111, !lO\\' r('(lds

el FA F/wA -( en ) 1 (') l/.A)' + o AI(D" Ii)= -:¡ I'V + q lp -11/ q - 2X (¡I/ (}u'l ;tlJ'l, (1.3 )

wherc the Slllll over A, IJ is illlplicit, <tud '1 is a cOlllplt.x scalar ghosl field (\','ithFcrmi statistics) with a covariallt derivative

1)" 0<>' ' ('I'Cl"c)A/J=(/OAlJ+1.IJ / A/J, ( 1.'1)

in tel"ms of the gCllcralors in tiJe adjoint reprcs(,lltatioll. \\'hether in tIJe fundamentalor adjoinl basis, thc gcuerators satisfy tlw SU(:l) anti-nlllllTIutation relations

whcrc ¡AlJC are tlw struclure COIlstallts. st\ch tlJat

('I,A) __ '¡AHClJC - I

(1.5)

( 1.6)

In lllultiplicatin:l)' renorlllalizable tllt'ori('s, the barc fields Illa)' be written intcnns of rClIormalizcd field tilllcs (l rCllorlllalizatioll constant. 1)(,llote bare fieldsby subscript O and cOllventiollally \\Tite. for ('xampl('. :10 = Z~/2:1. The standardprocedurc is lo \Hilc the bare Lagrangian in tcrms of relloTmalized fields and prod-ucts of ,%'.factors, t}WIl writc I.he Z f<lCtor as, (.9., .%':1= (23 - 1) + 1. The counterterm Lagrallgian is the differcncl' l)('tw('(,1I tll(' bare Lagrangian and the renormal-

524 .11./1.lleno

izcd Lagrangian. Tú compute lhe eGunter t('[11I5, a regularization prescriptioll andrenormalizalion schcmc are requircd.

\Vc choase dimensional rcgularization lo rcgulate lhe ultraviolet infinitiL'S [18).An integral

is lag Jivergcnt in lhe ultraviolct, howc\'cr

J~1(2~)" k'

( 1.7)

( 1.8)

is finitc foc H < 4. Far convcllicllce, wc pick 11;;; .1- 2(. 'fhe logarithmic din~rg(,II«-,'Sappear as poles in L Dimensional regularization is a cOllvcnient schemc becausc itpreserves the identitics a..•sociated with gaugc invariance.

In dimensional regul<tri;t,ation, lhe fidds and the coupling constant in lhe [eg-ulatcd theory pick up fractional dimcllsiolls. Ilecal] that lhe action should be di-menJionlcss, ami d4x _d"r. Then lhe feguiated Lagrangl' d('nsily eH must haV('dirnensioIl5 of mass [.\tj". From the kindic terJns. we se(' tbat q'" [.\I](n-l)/2 afl(lA '" [.M](n-2)/2. The gallge coupling lo lhe f('nnions gi\'('s 9 '" ¡.l/F-Jl/2 = 1.\1](.so the coupling constant canuot be dilll(,llsionlcss iu n f ,1 dimcnsions. It is con-ventional Lo rewritc lhe dillwlIsionful coupli!lg constan! in t¡'rtllS of a dimcll.'iiorl!csscoupling constant 9 lilllcs a !lew (arbitr(U"y) ("()l\stant 1/, wiH're 11has dimctlsions ofmass,

( 1.9)

The Larc alld rcnoflllalized quantitics ¡Uf' rdakd by

1\0 = Z'/2 1 HIO = ;{'" 11I~.1 I

, F I/'.!'\0 Z,,\'111 = /:2 q , =

,~ 1/2Zg91"'In = /:.1 TJ , 90 =

Now lhe bare Lagrangi<lll <lppears (jS

Co(Ao, 'lo, ~o, "'O,go, "o) = C( 11, '/, '/, "', '11,' ,,,) + !JC(A, '1, '/, "', 91'" ,X)

( 1.10)

( 1.11)

lntnx/uction fo perturootit'f QCD 525

with the counter ter m Lagrangian

,+ (Z3 - 1)ao'I,Uao'IA + 9~' (Z¡ _ l)JAIIC A~ A~(aO AdA - aB AoA)

_ (91~'f(Z. _ l)JABC JADE A~ II~AoIJ ABE - 91t'(Z{ _ l)qtC,~C q

+i91,'(ZI-1)au"lIt(TCA~)IIA'IA .

(1.12)

The Z's nol defined in Eq. (1.10) are products of Z's appcaring together in thecounter tcrrns. Since go = gll{ Zg is lhe samc COUp)illgconstant for al! gauge cou-plings, the Z's are related by

(1.13)

IThis is thc QCDgeneralization of lhe QEn \Vard idelltity Z¡ = Z2. Thc z's are COITl-puled order by order in perturbation theor)'. The minimai subtraction prescriptionis to absorb inlo (Z - 1) on)y the term:s containing poles in L \Ve procced in theIlext section to compute lile fcrmioll self ellf'rgy illld llwreforc Z[ to one loop order.To do this, we use the Feynrnan rule):;in Table 1.2 and sollle of the relations foundin the Appendix.

1.3 SELF-ENEltGY DIAGRAM

In this section we ca1culatc Z[. SL'eMarciano [l9J for a more complete discussion.\Ve ("ampute the Feynman graph in Fig. 1.2 in H = .1- 2( dirnensions, witlt m = O,b"t taking the fcrmions slightly off shcll in lhe spacclike region. We do the ¡alterto a\'oid mixing the ultra\'iolet singularities (which go into thc counter term) withthe infrarf'd singularities.

The drcssed fcrmion propagator may be writlen as

'ZF .. .. l 1 "2 1 I . 1'SI= ---- = -- = -- + --( -,E(p))-- + ...¡j-m-E(l') ¡j-m ¡j-'" v-m ¡j-m

(1.14)

526 JUl. lleno

Table 1.2

A, Q (OI'''MtttO. B, f3a ) b

A ---)--- B

A, Q

6Gb I

f - m + ie

6AB_'_p2 + ii

A,Q

b

B,f3

e B e

q _g¡ABC[goP(p _ q)' + g~'(q _ el"+ g'O(e _ p)p¡

A, Q e, "1

A,Q B,f3

e, "1

_ig'lfXAC ¡x BD(gopg" - go,gp,)

+ ¡XAD ¡XBC(gopg" - go,9P')

+ ¡XAB ¡XBD(90,9P' - ga,gp,)]

QCD Feynman rules, with A,B,e,D = 1. ..8 and a,b,c = 1. ..3. In the three-gluon vertex, aH of the momenta are in-going.

Introduclion lo perturbativl' QCD 527

P P-R PoR

FIGURE 1.2. fermiotl selr-energy graph.

where

, '" A A J d"k-;E(p) = (-;g/.') L.. tab tb, (2~)" X

A,b

First, we c\'aluate the color sum using the identity [20]

(1.15)

(1.16)

which here gives an overall factor of CF6a, = (Nt - 1)/(2Nc)6a,. To simplify thediscussioll, we split up lhe inlegrand into two pieces so that

where

, !d"k.~ P - ! 1E, =. (2~)"(-g ) 1. (p-k)'+i~ 1~k'+i~'

E~=! d"k(l-A)t p-! !_1_ .• (2~)n (p _ k)' + ;'1 k' + i~

(1.17)

(1.18)

\Ve focus our attention on E'} and merely will quote lhe result for Ei. The pro-cedure is lhe following. First, using the gamma malrix identities of the Appendix,

528 .IUI. lleno

evaluale the conlraction of lhe gamma matrices, then Fcynman paramelerize lhedenominalor

" j dnk ¡' (u - 2)(¡l- f)E = -- dz ,, (2rr)n o [k' + z((p - k)' - k') + iryJ- (1.19)

Because lhe inlegrals are finitc for nonzero t:, wc may shift the mornentllrn k = Q+zpand change the order oC integration lo gel

E', = tdzj dnl (u-2)(¡l(I-z)-@)Jo (2rr)n [Q' - C + ir¡)'

C = z(1 - z)(-p') .

(1.20)

Now inlegrale over Q (again, scc lhe Appcl1dix), using al50 the fael that terms oddin Q vanish, lo gel

E' = (n _ 2) i f(2 - n/2) r' dzjl(1 _ z)Cn/'-', (16rr')n/4 f(2) Jo

i (h)' ¡I= (2 - 2e) -- -, I'(e) ¡l dz (1 - z)I-,z-' .(16rr') -1'- o

(1.21)

Now integrate over lile Feynman parameter z lo gel thc result in terms oC r-functions(using the identity that r(x + 1) = xl'(x))

E' = _i_ ji (~)' 1'(1 + e)I"(1 - e), 16rr' _1" r(I-2e) [

1 (I-e)]e (1 - 2e) . (1.22)

The procedure for E~ is lhe same, and after the final intcgration, wc gel

E' = (1 _ ") _i_ ¡l ( 4rr )' f(1 + ,,)1"(1- e), 16rr' _1'2 f(1-2e)

This gives finally foc the unrcnormalized self encrgy

[1(I-e)]-~ (1 - 2e) , (1.2~)

-iE( ) = ig2

¡l C h (hl")' 1'(1 + e)r2(1 - el

l' 16rr2 F oc _1" 1'(1 _ 2e) [1(I-e)]"e(I-2e) ,

(1.24 )

Al this stage we are free lo specify our renormalizatioJ1 schemc. Dile convcntionis the minimal subtraction scheme (MS) where only the poles in l/E are absorhN.iinto the counter term, however, one may choosc to includc finite pieccs as well. In acalculatioo to aH orders, a physical quantity cannot dcpcnd 00 the rcnormalizatiooscheme choice. lo a finite order calculation, the discrepancy bctwccn theorcticalpredictions using two different renorrnalization schemes is oC next higher order [21].

/lItnxluction lo perturootivc QCD 529

Za = 1+ (g'/16")(1/') [Ne (13/6 - A/2) - 'h./TR/3]

ZI = 1+ (g' /16.')( 1/,) [N e (17/12 - 3A/4) - 4n / TR/3]

Z, = 1+ (g' /16.')(11<) [Ne (2/3 - A) - 4"/TR/3]

Za = 1+ (g' /16.')( 1/,) [Ne (3/4 - A!'I) ]

21 = 1-(9'/16.2)(1/')[NCA/2]

Z{ = 1-(9'/16")(II<)[CFAj

zt = 1- (g' /16.')( 11<)[Ne (:1/4 + A/4) + CrA]

Z, = 1- (g'/16.')(I/,) [Nell/6- 2"/7i</3]

TABLE 1.3. Renormalization constants at olle loop in the MS r(,llormalizatioll scheme. Here Nc =3. CF = (N¿ - 1)/(2Ne) = .1/3 and 7i< = 1/2.

Befe we chaase to use tIJe MS renorlllétlization sclH'llIc. \Ve absorb only the 1/(times its (-illdependcnt coefficicnt iuto Z[, so

( 1.25)

\Vc could instead choose the modified minimai subtraction schemc M5, where con-stants indcpendent of ( are abo subtractcd. In the l\IS schemc, t/i (times its coef.ficicnt) is absorbed into the countcr tenn whcrc

l J-: = - + lag <\ rr - , ., , (1.26)

These constants come from the often occurring cornhillation (as in the self energy)of

~ (1.l',I'(J+,)I"(I-,) 1------. -- - + log-lrr -, + G(,) .l'(J-2,) , (1.27)

For ). = 1 and ~(p) =: ¡l8ac~(1'2), \Ve find for the renormalized self cnergies in the

530 .11./1. 110'"

1\\'0 schclllcs:

,g'CF( (1"))~'lS(P-) = --o -, logl" -) + log -,lh;rr- -1'-

"C' (' )\"' 2 _ 9- -1' W~"S(p ) - --Ir ., log -,. or.- -p-

(1.28)

It is orlen COIL\'('Uielll lo complltl'loop corrcctiollS iJl lhe Landau gaugc where,\ = 0,bccausc ~(Jl'!) Villlishcs al arder o." = 92 ¡'l1r.

Thc Ollt' loop H'sults for tlw rellonnalizatioll COllstants in lhe MS schcmc areShOWIl in Tabl(' 1.:L OBe can \'l'rify tbol lhe relatiolls hf'twl'cn lhe Z's in Eq. (1.13)are satisfied.

2. The running coupling and renorma1ization group

'Ve have 8("("11that dimensional [j'gularizatioll IH'Cl'ssitates tll(' introduction of anarbitrary ma.'iS scalc which we ha\'(' ealled JI. Physintl qllnntitics canuo! df'pend onthe valuc of JI. alld this invariallcc gellerales lhe rellorlllillization group equations.

Our cxalllple of a perturhatiw spries in n i:-; tllt' <juantity H ;::. a(e+e- _hadrons)/a(!'+f- _ Jl+J1-). '!'his llH'asurable quantity can be computed theoreti-cally as itn (,XP;IIlSioll in O' ;::. g'1. ¡'17r alld it equals

(2.1 )

where CI. c'1., de. dppeno on ¡l. O' and 5, the (ellter of mass energy squared. Thercsults for ('1 aplH'ar in {{eL [12]. fol' C2, in HeL [2:3j. anl! for C3 in Hd. [2.1]. For afixed bare coupling constant éllld fix('d f, 9 depcnds 011 JI hy 9¡1(Zy ;::.l¡/( Z~, so weare intcrcst.cd jo know ho\\' 11](' ('xplicit /( dCPClLdl'Il(,(' 011 ¡l is collllH'nsated by theimplicit depell<!t'lIce of 9 (01' 0') 011 ¡J. Sincc [( is dilllensiollless, it can <lepend onl)'on t ;::.log8/¡/' aud n: R(t,n). Tlw independellcc of f{ fram 1I implics

dR (¡J ¡Jo '¡J)-- = - - + -- - li(l (l) = Od log Ji2 Di a log ¡l2 ¡Jo . .

By introducillg the running couplillg constan! o(.~)SllC]¡ that, for 0-;::' O(ll2),

1=1(1("') ~o p(.r)'

(2.2)

(2.3)

il is possible lo show lhal I!(t,l.l) = H(O,o(,)) is a ,o[ulion lo Eq. (2.2). So incomputing R, cOlllpllted as a IH'rtllr1J,¡\iv(' series in 0, l.Ilf' rt'piaccllIt'lIt of t _ OandO:'_ 0(8) satiRfie..o.;the renonnalizatioll group equatioll. Therc is a gencralization to

/r¡tfY)(lllctioll In lwrturootirf: QC[) 531

account for wave function rCllormalizat.ioll fa~tor d('I)(,lIdell(,c on Jt, but befare wcconsider this more compliratcd case. let us f1rst look g<'rteral1y al tlw {3 functioll¡lself.

The {3 function, defiBcd to be

Do 9 iJ¡1(0) '" -a o ; -1'09 .logJI~ ,Ir. (JI'

(2..l )

is for fixed 90 and (, where 90;; 91/ Zg. The corrections lo 9 tbat del)(,lId on Jl areof order gl, so the first lerrn in the cxpansioll of ¡3 iJl \('rlllS of íl: is al s('cond arderlJl 0::

¡1(0); -bu'(! +1,'0 + '''). (2.5)

Gross and \Vilczck [25] and Politzcr [26] COlllputt'd ¡, aud fOlllld il. positi\'('. Therefore.if olle begins with a slIflicielltly small vallle of 0, tlll'll fOl"¡t proccss characterizcdby an encrgy scale Q, 0'(Q2) is a mOllotollical1y df'n('é\sing fllnrtioll of Q2 andperturbation theory gets better and bcttel".

Substituling lhe expansion of j3 inlo Eq. (2.:3). \n' gl't

! ! , (0(1))bl ; - - - + b log - + tJ( o) .0(1) o o

(2.6)

where t ;; logQ'!/!I'!. HCH~ we note tilal o(t ;; O) ;; o(Q:."! ;; JI'!) := o. To firstapproximatioll, we can sel b' ami SUbS('(II11'llt coeiliciellls 1.0 zero alld solvc for o(t)lo get

(1) o Io (1);--;----1+ o¡'¡ blog(Q'j,Ii.L)

whefe we have chosen lo define

log ( ,;' ) _ I'\LI, 10

(2.7)

(2.8)

This is called tile leading log approximatioll lH'causc the expansion in o of o(t) is aseries 0(1 + Li Ci(o:t)i), tbat is, there are as Illany powers of t as therc are oC 0:.

The nexl approximation is to includc f¡'

o [( I 1,' )]-10(-1(1); b ob + I - ¡,log(o(l)ja)

( (b' o .¡) 3

"" 00 t) 1 - ¡,oo(t)log(log(Q-j,\-) + 0(0),

(2.9)

532 .11.//.lleno

wherc naw our conventional definitioll of ,\ is t hrough

(1") 1 b'log - .= - + - log boA2 !Jo IJ

and oQ(t) is defincd

1<>0(1) = blog(Q'/A')

(2.10)

(2.11 )

that ¡5, oo( t) has lhe same form as lbe leading log l'xpreSSioll, but él,(~ffcrcnt valucof A. Noticc thal lhe fOflll of Ec¡. (2.9) is such tha" nol ollly lhe leading logs arerctaincd in lhe cxpansioIl, hut also sublt.ading logs.

Befare continuing OH lo a sketch of the dcri\'alion of lhe value of b, we makea fe\\' conUllents on A alld its sensitivity lo variollS choiees. FOI"a more completedisClIssioll and general n'view, sce DUKe and Bohcrts [27]. Thc first comment istllat with the introductioll of A. \Ve llave trad('d lhe fundamcntal para meter o =o(t = O) for a diffcrcnt fundamental paramelel' .\. lJnfortullately. A is sClIsitivc tov<l.rious choiees of schenU's. \Ve begin \\'ith the o!>s('l"\.ation that /\U. f:. A, in fad,!\.2 = AL.exp(-b'jblogbn). Furtiwflllore, a sltift in AL/J is comp<lrablf' to termsthat have been neglected, so calculatiolls Leyond lile leadillg log an' ('ssential fol'pilluing down the parametel's of the tbeory.

;\"cxl, 1\ is l'cnormalization schcllle depelldeut. In tbe ~1S amI ~IS l'{'spective!y,lhe rCIIormalization colIstallts are

Z'lS=l_o~~fJ t 2

ZMS (1 1 ) b~fJ = 1 - () ~ + og .111" - "t 2(2.12)

Sincc thc bare coupling cOllslant and ( are fixed, tile renormalized o 's computed forthe two schclllcs diffcr:

(2.13)

induding oIlly lbe Im\'cst order corredioll. Frolll tlle dcfinition of, for cxarnple, ALLin Eq. (2.8),

\ MS _ \ ~IS{.(log .1lr-¡ ){2 _ ') 66 \ MSj L/. - I L/. . - _. 'LL. (2.11 )

Not an insigllificant factor!

Finally, wc shall see tbat b = b( 11 / ), so regardk"Ss of w hich choice for /l., there isall n/ dependcnce to A. Convcntionally what is quoted is the 4-flavor valuc. As an

!flil'uducti01l io lJfrturimtiv(' QCD 533

FIGUHE ~.l. TILe lowl'sl order alld Dril' loop con,'cliolllo th,' quark-gluoll vI'rlex.

l'x;unple, note tllat

Al.:L = 2(iO ~lcV~IS

Al.:L = I(iO ~lcV~IS

,\~= 175 ~lcV;~ts

)\(5) = 100 ~lcV ,MS(2.15 )

\'alul's for ,\~ witb <I(Q') df'fined in EII. (:l.!..!) are lIll'itsurcd in a varil'lv of e+e-~IS . ,II(HlrOIl~hi"ldroIlami decp inclast.ic scatterillg eXperilll<'uts. Thc particlc dala book('('ntral v;¡luc [14] alld rangc frolll a comhillation of eX¡lt'riments is A~~~::::::200~~~o

~Ie\', when'as frOlrl ¡jeep inc1astic scatterillg alone. a \",,111(' of j\~~~::::::238::l:::,13Me\'is f)uote<i.

2.1 TIIE (J FU:-:CTIO;-';

For lhe cOlllpulatioll of lIJe {3fllllction. we follo\\' Allarelli [8]. Considcr lhe quark-gluon wrtex correctioll in Fig. 2,1. According to lbe renormalizalion conslantsdcfined in Eqs. (1.7) and (1.12), the urJ1"cllonnalized \'crtex, through one loop, is

(2.16)

534 ,\/,//, lleno

whefe

(2.17)

and ¡\qqG. JJ'lqG are constanls inciepclI<!cnt of JI. '1'0 compute the f3 function, wesolyc the renofmalization grollp cc¡uatioll for rh"G. BI'cause there are external quarkand gluon !ill(,:", t.ltl' gcncralization of Eq. (2.2) is

__ d_ 1'''(; _ "F ",'/2 [~ '().!!.... d logZ{ ~ d lag Z,] I,,,G _ O? ) - '.J? .f...,1 ? + /} n + ., + 'J -.dlogJj. l ~ Dlog/l~ Do dlog¡l- 2dlog¡t- R

(2,18)Define tile ilnolllalolls dimcllsiolls ;'/ illld 1(; ])Y

d log X{ _ (1)1'1 = 11 -; ¡'I n + ...( og 1[.

d log Z] _ (11A/c.' = ---') = -,(' (l + .

d log 11- '

Then, combining Eqs. (2.17-19), w(' gel tll<l\. tlw low('sl ol'<1cr contrilllltioll is

,(' ,2r. (1) I (1)gn/PU' + /"((1)- + gOl" + -:j!Jtli(; ::;::O .

!J -

The first. tcrm in the cxpansioll of /~ [El[. (2,.5)] is

(2.19)

(2.20)

(2.21 )

Tú compute the anomalolls dillH'llsiollS. \\"l' 111<1.\'IIse 1lit' <jtlilrk sl'lf cnergy and

gluan .polarizalion funclions" r~~~.....,(1 - ~ ,d/l.!)) Sil! i:-.fi,'s llu: rl'llonnalizalian groupequallOn

[a iJ] ,1'1'-----1 2 + {1(o)" + '1, 1'111= o .U ogJI. OC!

(2.22)

The expansion of 13 begins al ordl'l' o:.? howc\"er ~Jr Iwgills al arder o. so 11 dropsout of Ec¡. (2.22) in lowesl arder. FrolTl our reslll! in Eq. (1.28), w(' gC'l

(1) l.'Y, = --(."

.1 r.(2.23)

lnlroduclioll lo ]Jfrttl.rbativc QCD 535

A similar compulalion of rg1 yields

(1) 1 [\' 5 1 '[']IG = - 1 'c 3" - 31l f R4r.(2.24 )

Now it remains to compute JjqqG. \Ve will not do the computation Itere, rathcrwc compute the color faetors for Pigs. 2.1h and 2,lc. Labcling the incoming gluoncharge by A and externa! quark colors by a a.ml b, the diagram in 2.1 b yields a colorfactor of

\Ve have useJ Eq. (1.16) lo simplify the color SUIll. Similarly,

C = i '" ¡ABC lB IC = '" [lA ¡BI lBlc L (lC ch L ' ch acB,C,c LJ.C,c

The standard reslIlt, inclllding al! filctors, is

(2.25)

(2.26)

G 1 [( Nc)Bqq =- CF--47r 2

COllscqucntly,

3NC]+2 . (2.27)

1b = -,-(IINc - 'l"fTR) .12rr

(2.28)

Por another more straightfor\"lard derivation of b, see Ellis' TASI Leetures [12J.For rcference, the cxpress ion for b' is [28]

b' = ~ _(1_7_A_'c_-_I_O_n_f_7_'R_)i_V_c_-_6_"_fT_R.CF.27r llI\lc - '11lfTU

(2.29)

Shown in Fig. 2.2 is o: as a function of Q2 fl'ol11the definition of Q in Eq. (2.9).

536 M.ll. /leila

0.2

o'"

0.15 10 20

Q(GeV)50 100

FIGURE 2.2. 'rhe strollg coupling constan! as a fllllctioll of Q2, followillg Eq. (2.9).

3. Parton Distribution Functions

In principie, lhe Qel) tlwory of strong interactiolls cxplains ho\\' quarks and gluonsare bound togethcr inlo hadrons. In practicc, becausc of difficulties ealculating Ollt-

side of lhe perturoativc regimc, calculations of lhe baryoll spectrum, for eXil.mplc,are fal' from conclusivc. Ncvertheless, it is an experimental (ael that hadrolls appearlo be cornposcd of cOllstitucnt particles. gencrically ealled parton:'>. \\'c bcgin lhissectioll with a cOllsidcratioll of these partans withoul specifying lhe fields that bindthcIIl jnto hadrolls. This motivatcs the introductioll of partan distributiotl functions.After dcscribing sOllle successes, liJen failures. of the naivc parton modcl, we describethe QCD-irnpro\'cd parton rnodel and comment on ils regioll of applicabilily. Finally,we sketch the general procedure for extracling distribution fUllctiolls from <le('p-inela."tic scattering data. For intro<luctions to the partan lIlodel in <leer inc!a."ticscattering, see, for example. Hefs. [.51 and [6]. \Ve follo\\' herc the prescntalion ofClose 15).

3.1 TIIE :-L\IVE QUAHh: ~fODEL IN DEEr I:\ELASTIC SCATTEHING

'1'0 motivate lhe picturc of "he protoll as cornprised of constituent partons, pointparticles cach cOlltaining a fradion of tl)(' parellt proton 's lIlolIl('llllllll, we will iookal the cxample or e1edroll scatterillg. First \\'e compute lhe di/Tt'l"('ntia! !'icattNingcross section ror e1ectron-muon scattering and note lile !'icaling bchavior. Then wecompare with the general forlll for inclastic e1ectron-proton scatterillg in tcrm5 ofform factors. Early cp experiments exhibiled so-calleu Bjorkcn-scaling of the formfactors consistent with spin-l/2 parton conslit uents, so we write the form factorsin lerms of parton distribution funclions f¡(:r), lhe weighting of lhe eledron-parton

!lIlmfiurliotl lo pcrtlJrbatit!e QCD 837

e

(q

e

p

(q

"

F'IGURE 3.1. (a) Electroll-muon scattering, alld (b) c1ectron-protoll inelastic scattering.

matrix e1ement squarcJ for a partan of typc i with 1lI0rncntllITIfraction x of theproton 1ll0mcntlllTI.

In Fig. 3.1a anJ 3.1b, we show the Feynlllan graphs for eJl anJ ep scattering,the only differencc bcing that in ep scattering, the final state is unspecified and roayha\'e several particles in addition to the e1cclron (represented by the bold arrow),whercas in the eJl case, it is simply an c1astic scattt'ring. \Ve will work in the labframe where the proton is stationary aJl(I make the definitions

1 = (E,O,O, E) ,

t == (EI,E'sinO,O,I.;'cosO),(3.1)

l' = (Al,O,O,O) ,

q=I-I'.

whefe in Fig. 3.1a, Al = mil and in Fig. 3.11>,i\l :; mI" \Vith this, we define thevariables

.< = (1+ 1')' ,

Q' = -q' = 2EE'(! - cosO)

v= q~/,= E-E', (3.2)

538 M.H. lleno

Then, ror CI' scattcring, lhe diffcrcntial cross sectioll in terms oC lhe spin i\\'craged

matrix clernent squarcd L~,tJVlV (~l is

This works out lo be simply

<1'0'" 40'£" ( Q'. o ) ( Q')d£,'dO

r;:;: ~ C05

2 O + 2A/2 SIIl- O {¡ V - 2¡\f (3..l )

For inclastic scattering, we are intcrcstcd in a general fOfm ror lV¡w. Sincc lhe finalhadronic mOlJlcnta are intcgratcd, the hadronic llIat.rix ('\clllcnt squared can dcpcndonly on lbe momento. q and P. By writ.ing the most gClwral sYllIlIldric two indexet\tensor using 1II0ITlcnta q and P, thcll applying c1cctromagnetic gauge invariancc(qIlIVJJV = O), we can redil ce IV lo lhe form

, ( ,.") W( P )"( l' )"\VIlV ;:;:IVl _gllII + q? + ~ p _ ~q p - ~q'1" M- '1- '1"

(:l ..5)

\Ve writc only the tcrms syrnmetric in lhe indiccs hecé\lIsc the clectron matrix cI-ernent squarcd is syrnmetric. In general, H'i ma)' depelld on q2 and p. q (in otherwords, on ti and Q2). It is a simple algebraic exercise to sllow that the inc1asticeleelron pro ton cross scction is

(:l.6)

Then, ir the praton were a structureless point particle like the muon, \\'(' woule! ha\'e

P' o , ( Q' )IV (v Q") = u v - -2 ' '2¡\1 '

2IV,P'(I>,Q')= Q',ó(,,- (2')2M" 2M

(U)

Tbc comhinations vlVt == Ft and .H\Vt == rr' dep(,lld 0111)' 011 tlle <iilll(,llsionlessquantity Q2J(2Mv);::: Q2J(2p.q) == x aHd Ft(J');::: 2J.rr\r), in<i('¡)('Il<iPlltof scale.Frorn clastic pro ton scattcring, wc kilO\\', of course, that lV¡ :f t'l/t, ami it couldharell)" be tIte case rol' inclastic scattering. lIow('wr. tlw cady SL\C- ~t1T illdasticproton scattering data pi] do show tllat F2 is <tpproximat.e1y illc!epencknt of Q2

lntrodl.l.ction to perturbative QCD 539

for Q' '" 1 - 10 Gey2 Furthcrmore, experimentally [321 for Q' between 5 and 15GeY', and x > 0.4, F,(x) = 2xF¡(x). The relationship betwcen the F's is calledthe Callan.Gross relation, and the structure function independence of Q2 is knownas I1jorken scaling.

On the basis of the scaling behavior of thc form factors, one is led to hypothesizethat the hadronic matrix element squared may be rcprescnted by an incoherent sumof spin-l/2 parton matrix c1cments squared appropriatcly weighted by the partandistribution in the proton. Namely, using ¡¡(Xi), tlle distribution of partoos of typei \•...ith mornentum X¡p,

F,(x) = ¿ ¡d.,., r¡"",lVi"(";",Q')f;(x;j,

Jo ( Q')= 2;: d.T¡ c~.r¡JJb .T¡V - 2i\1 ¡¡(Xi),",,'= ~ r,xl,Cr) = 2.1F¡(.r) .

(3.8)

Thc distributioll functiolls can he thought oí as part.on number densities. This isstridly true on1y in tIJe infillite lIlomcnlulll frame where the hard scattering timeseale is long eomparcd to the time scale of tiJe "llIeasurcment" of the partan densi-tieso

Fo!"futurc rcfcrcncc, we writc tIJe inelastic elJ noss scetioll in tcrms of X and y

(3.9)

TiJc spin-l/2 partons are idcntificd wit.h qllarks, and we caH ¡¡(x) the quarkdistribut.ion flllletions labcled by tiwir fla\"OI",('.!J., u(.1'), <I(x), etc. The fundioosF¡ we shall ca]] struelme flllletiolls. Tlley can he \vriUen in terms oí the distribu-tiolls functions according t.o the naiv(' parlon Illodel. This parton model approach isequiva.lent to begillning with a mat.rix e1ellwllt s<¡llare<1for quark-electron seattering,tiJeJl int.egrating wit.h the parton distribution fl1Uctiolls lo find the c1ectron-protoneross sed ion

a'P(q, 1')= ¿ J d.l' qi(.r)Ó"Q'(q, ,Y),

(3.10)

The quark distribllt.iolls t\rc split into \"(1.1('11("(' aud s('a quark distributions. ThevalcJlcc quarks in the pro ton guarantcc that Q(p) = 1 and that the strong isospinof the ¡)foton is + 1/2. Then for isospin + 1/2 and charge +2/3 up quarks ti andisospin -1/2, charge -1/3 clown quarks <1, \\'C have the sum rule for the valence

540 M. JI. lIerlO

quarks that

1, (3.11 )

and similarly, therc is an e<¡uation for lile [lcutron, lhe isospin partner oC lhe proton,whcre Uv = dr" := dv ami dí', = U~T ;: uv.

The sea Cjuarks are just liJe qllark-anti-quark pairs crcatcd by \'acuum nuctua-tions. For lhe mOlllcnt, we élSSllllW that us(.r) = üs(.r) = ds(x) = ds(x) = ss(x) =.~s(.r) = qs(:r} indllding tlw third (strangc) <¡uark. Tilen we could write lhe sumrules abo\'c using ('¡lher qv or q - ¡¡ wherf' q == qV + qS and ij = iis.

Thc ratio oC da( nl)1 da( cp) as a fUllction oC T is just F~n / F~P according lo theCallan-Gross relatioll, and in tCfiTlS oC <¡uark distrihutions.- -

¡,'f:",-¡,'fJ',

UF + 4dv + 12q5

d\' + .tU\! + 12q8(:1.12)

Expcrilllcllt.ally [:t~],[:n], OUI' ohserves tba! al slllaU ;r. lile ratio is ncar1y Iluity. sowe hclic\'c t.ha! Ihe ~('a (lll<trks domillille at smitll :ro As:r inerenses, the quantity inEq. (:Ll'2) dC(TC'as('s, so it aplH'ars th<lt ti\(' valcllCc <]lIarks hegin 1.0 llave a iargerrole, I~ strong isospin sYlllllldry were exact" t.!H'1l l/v = 2dF and in ¡,he ahsensc oCsea qllarks, the ratio wOllld {'qual 2/:L Al large J, howevcr, the ratio de(Teases lo<Ipproximat.e1y 1/'1. This can he anolltllcd for by q5; ~ O as ,T -+ 1 and dF « TlFal. Jilrge l'.

QBe sign lh;ü t.he Ililin' pict.urt' cOllld llol be I.be complete pictllrc comcs IrolTlcOllsidering

¡''" ¡"I' 5 ( .. I ¡- .'( -))'~ + '~ = liJ II + 11 + ( + ( + 5 ,~+ ..•

~lonlf'nt.ulIl cOllservalioll tells liS t.hat

L r' d." ,"I,(x) = 1 ,, Jo (3.11 )

SO in tile limit tllat lhe sl.rilllge <juarks can 1)(' Ileglected, we wuuld cxpcct that theintegral oC Eq, p.lJ) O\'('r.r wOllld ('qllal ;)/~, 111fact. experimentally [3-1]. thc valueoC thc integral is (0..15) , ;'/9. At this stage. tlU' dis('[epancy cOllld in principie beaccounted Cor by the strange ~ea. hOWI'\'f'r illlplausibly, but it would have disastcrollstheoretical con~f'qllelln's for tlw pn-dictl'd Ilcutrino production of charm. A moresatisCactory explallatioll is that roughly half of lhe protoll momentum is carried notby spin-l/2 quarks, !lut hy spill-1. 1'1('("ri('aHy lIelltral gluOTls, as the)' appear inQCO.

Dne Cllfther e\'id(,lIn~ oC 111('f••ilml' of ti\{' llaive qll<lrk lTIo(kl is the violatioTl oCscaling in deep ill('la:,tic ...•cattl'fillg dala. TI\{' structllf<' fllllctions ror d('('p inelastic

/lItroduction to perturootil'c QCD 541

lepton.nucleon scallering, ror fixed x, show,slow ,"arialions in Q' as Q' rangesbetwecn ,....,5 - 100 GeV

2[35}. The QCD.impro\'ed parton model can account for

scaling violations, and it is to this that \vc tuen ncxl.

3.2 QCD ¡"PIlOVED PAIlTON MODEL

The idea or lhe QCD-improved <jllark dislriblllion fllnctions conceplnally Oflgl-nates with the \Veizsacker. \VillialllS dfcctivc photoll approximation in e1ectrody.Ilarnics [36J. Thc idea oC the cffectivc phot.oll approximation is that, say, elcctron.protOfl scattering can be approximated by

(:J.15 )

\Ve will show tbat W(' C¿¡tI define tIJe splitting function I\_t(z) such that

I1ere, z is the ratio oC the photon !Il0llwntulll compared to the initial e1ectronmOlllcntum (the fraction oC thc original 1Il0mentum carried by the photon). Thequantity dP.., __e(.:)dz is Llw probability oC finding the photon carrring fraction z ofthe parent c1ectron's 1II01lH'nt.HIJI.Tlw graph with tll(' virtual photon is dorninatedb.y smalI q2, so tIJe virt.u,¡J photoll is almost I"('al ('12,...., O).

\\'IJat \Ve shall sho\V 1H'lo\\' is t.ltat llll' !jllalltity dP.., __e(z)dz al so dcpends 011Q,the charactcristic encrgy seal!' oC thr' illl('l"ilCtioll. '1'111'('ffectivc pIJoton approxima-tion USL'S QED to compute t.ill' prol,ahilit,\. of itlt ekctroll splitting into an elcctronand a piloton as a fU/lctioll of : ,llld (l, alld Illtilllatc!y, the photon distributionfunction in the e1ectrolt. In t/w QCI) illlpro\"(:'d parton lIIode!, we use the thcory ofQCD to compute the prollallilil ies of Cjllarks splittillg into glllOlls and Cjuarks, hencecorrcctions to the di3tributioll fllllctiolls f¡(J'). 'lb simplify our discussion below, weremain in the el('ctrod~'Il<IIJ1ic case, 'llld gelll'ralize later to the non.Ahelian case,

(3.16)

wbere T :::: log Q2. \Ve can writc ,(x, T), the number density of photons with rno-rnentuIll fraction between x and x + d.r, .so that

<h(x, T) <> J.' 1.1

---=- dy dzó(zy-x)e(y,T)I\_,(z)dT 2r. o o (3.17)

where e(y, T) is the llumber dCllsity of c1ectrons, Irltcgrating aYer z givcs the e1ec.

542 M.H. Reno

tromagnetic equivalents oC lIJe coupled Altarelli-Parisi c\'olution cquations

d¡(x, T) _ ~ 11dy ( )1' (:.)_ e y, í j'_e •

dí 211" 1" Y Y

de (x, T) Q rl dy [ (x) > (x)]dT = h lr 11 e(y, T)P,_, y + ¡(y, T)I ,-, Y

We can also write equivalently the positron dislrihutioll fundioo.

(3.18)

First we sketch lhe derivation oC thc expression Coc the splitting fundiDoP,_e(z). Because oC conservation oC mOlllcntum. and conscrvalion oC probability,we can exhibit relations betwccn splitting functions. \Vc thcn outline what changesare needed lo adapt lo the case oC QCD.

Now we look al the exprcssion Cor dP1_e(z, Q)dz in the I¡mil oC ma.sslessparticles. Altarelli and Parisi did this in thc conlcxt oC QCD using old-fa.shioncdperturhation theory (29]. Anothcr approach is lo compare, using the usual Feynmanrules,

da(e/' ~ eX)

dah/' ~ X)(3.19)

and extract d'P1_e(z, Q)dz from its definition in Eq. (3.1.5). The steps are outlinedin detail in Berestetskii el al. [4]. Schematically, we can obtain the standard resultusing

This gives

'1 l' 3dP ( )

d _ '~1P e Ve ....•t>1+1 d' Pe'1-t> Z Z - 4 (3'

S'I' /', 2K) (2E)

(3.20)

(3.21 )

The matrix elcment squan.~l indudes tbe spin average. Use the mornentulll assign-

ltltroduction lo paturootil'e QCD 543

rncnls

p, = (1'; O, P)

p, = ([(zl')2 + q;.]'/2;qT,ZP)

p,' = ([(1 - z)2 },2 + qW/2; -qT, (1 - z)l')

expandcd to sccond order in qr/ ¡> to compute the factors

d3pe' 11" dz2E' '" '2 (1 - z)dq}

Wcget

dP ( )d _ ~ z(1 - z) IV'-"+,I' di 2 d¡-e Z Z - • 2 ogqr z.h 2 qT

So, thc value for lhc splitting functioll is

(3.22)

(3.23)

(3.24 )

z(1 - z)I\_,(z) = 2 (3.25 )

A few cornments are in order1 namely, why incorporate lhe powcrs of qr in thedCllominator. The reason is that V ex qT. This can be scen from angular momen.tUITlargullIcnts. A spin 1/2 elcctroll with. say, positive hclicity canuol emit a spin1 particlc collinearly (qr -+ O) with a hclicity conserving vcrtex, so the matrixc1cITlcntsquarcd should go likc lhe square of the transvcrse mornentllm. Also, justby cOllsitierillg momentum conscrvatioll at the vertex for A --t [J + C. one knowsthat PlJ_A(Z) = PC-A(1 - z) so the prdactor illcllldes z(1 - z).

'1'0 gel thc splitting fundioo I\_e, use the ITlornenturn assignments fromEq. (:1.22) and substitute in

IV 12 1 '" ,. ,.e-e' +"'1 = 2' ~ t. ( (3.26)

EXJ)and thc momenta through order Q~./ [>2, l'\ote that in evaluatine: th •..nol¡:¡rizat.ion

544 M.H. Reno

SUID, we musl use lhe physical gauge

ir Jl, v = oothcnvise.

(3.27)

This is because we are extracting lhe initial slale radiation. After sorne arithmclic,we find,

(3.28)

As advertised, lhe matrix elernenl squared IS proportional to q}. Now usingEq. (3.25), we determine

P,_,(z) = _1_+_(_I_-_z_)2z

Momenlum conservation al the vertex gives liS

1 + z'P,_,(z) = -- .1 - z

(3.29)

(3.30)

Eq. (3.30) is valid as long as z t- 1, whcre lhe exprcssion is singular. To addrcssthe problem of z = 1, it is lisdul lo define lhe distribulion 1/(1 - z)+ such that

rI /(z) = rI /(2) - /(1)Jo (1 - z)+ Jo (1 - z)

(3.31 )

This makes lhe integral over the distribution finite. The law of conservation ofreernionnumber is then applied lo constrain Pe_e(z;::: 1)

(3.32)

To satisfy Eq. (3.32), add a delta function. It can be shown that

(3.33)

satisfies Eq. (3.32).Anolher conservation law regards conservation oí morncntum. Ir we consider the

!ntrociuclion lo perturoolire QCD 545

[1+ ,'3 ]p (-) ~ C,' --- + -6(1 - ,¡q•.....q~ (l-z)+ 2

TABLE 3.1. QCD splitting functions at lowcst order.

QCD case (fOf a general treatrncnl), thcn 1ll0IllcntUIl1 cOllscrvatioll is cxpres.sed

Jd ("dq'(X,r) "di¡i(X,T) dG(X,T))=

x x ¿ I + ¿ d + d 0,. (T . T T, . (3.34 )

where i = 1 ... n f for n f equal to the number of qUilrk flavors. This implics that

¡1dzz(l'q_,(z) + I'c_,(z)) = °1.1 dz Z(211fl'q_dz) + Pr;_e(z)) = 0,

(3.3.5 )

At this point, Jet U5 addrc.ss the question of tile extcnsion of tIJe QED calculationto QCD. First and forell1ost, is that PG_(~' 1:- O whercas P ,.._,.. = O. Second is tile factthat there are color faetors. As an cxample, look at Pqo--q. Eq. (3.33) will be ITlodificdby a factor of 1/3 for a color average (imes ((g/2)/e)' LA IdA",1 leading (o anoverall relative factor of 4/3. as/a, so replacc Q - as and Pqo-q(z) = (4/3)Pe_e.A 5urnmary of the quarl;: and gluon splitting funetions appears in Table 3.1. Thecoupled Altarelli-Parisi equations (tile equivalcnts oC Eg. (3.18)] are

~ (q(X'T)) = O(T) t dy (I',-q(~)dr G(X,T) h Jx y I'r;-,(~)

l'q-e( ~)) ( q(y, T) )I'G-G(~) G(y, T)

(3,36)

In QED, where evell at low Q2, o. is small, to a good approxirnatioll, the photondistribu tiOD in the electron is

(3,37)

546 .\/.11. ¡¡eno

that ¡s, using t;(y,T) = c5(y -1) ami c(y,r) = O, w{' call ('vaiuate tite integral ovcr zand T. \Vc int('gra!.c T from a ref('fl'IlCC log Q~ lo log Q2, thc charact('ristic seale oflhe process.

For lIJe case of tIJe giuoll, 0,,::;; 11,,(r) rises rapidly as Q2 -+ m~,sO the integral isnol so trivial. FurthermOff', Wl~han.' (111)' thc firsl ord"f ('xpressioIl fOI"PC-q and Qj

is becoming largc. The non-pcrturi>alivc regime is laking over in this limit. For thisrcason, distribution fllllctiolls are dclel"mincd eX¡H'rillll'lltal1y al a particular \'alueof Q6 > m~, thclI "cvolved" in Q llsillg the coupleci :\ltarclli-Parisi e<¡uations.

3.3 Ha\\' AHE D1STHIBUTION FU;'>;CTIO'\S EXTIL\CTED

The idea of cxtracting dislributioll fuw.:tiolls frolll cxpcrimcntalmcasurclllcnts is tofirsl measure lhe structure fUllctiolls ¡.i as fUllctiolls of .r. y ami Q'!.. Then, usinga combination of results from otl\{'r cxperimcnts and :-;olIle assumptioll:-;, separatcout the variolls components frolll the QCD partoll lllodcl. Therc are a variety ofpararncterijl,i1tions of the di:-;trilllltioll fllllCtiolls [:nH42]. 1 follow here es:-;cntiallythc discllssion of EIILQ in Hef. ¡:17]. They use the ncutrino data frolll the CDIISexperirnent [.la] takell \\'ith narrO\\' banel beams of 200 (11, O) and :mo (v) Ce\'.

In decp inelastic neutrino scattering in the lIaive parton rnodel. thcrc are onl)'t\\'o structurc funclions, Fz ami F3• that is t\\'o eilch for neutrino and anti-ncutrinoscattering. In tcrrns of thc structure functions, the l\cutrino ami antincutrino diffcr-enlial cross scctiolls corresponding to Er¡. (J.9) arc, for "-2 = 2F),

<.I2a1' G}MEx[F2'(X)(1 - y) + Fr(x)y' + f'{y (1 -~)]

<.IIdy ~

G}MEx(3.38)

d'!.aiJ

[F{(x)(1 - y) + Ff(x)y' - r:fy (1 - D]--=dId!! ~

In the parton model, ror ncutrino and allti-ncutrino scattering with isoscalar nucle-ons,

(3.39)

~v '"Fl' = L..,(Q-q)-2s

",here q = tl, d, s in the low IlIomentum region.ParticlIJarly intcrcsting at the outset is F3"~'R= (1'3 + Ff)/2 because it is jllst

the sum of the valcncc distributioll functions. F;Vg can be extractcd by taking thediffclcnce of the neutrino and antincutrino diffcrcntial cross sections. A comparisonof chargcd current interactions for hydrogen and deutcrium targcts yiclds dv/uvas a function of x, and togcther •.....¡th p;vg gives the :-;cparate up and down valencecontributions.

!ntnxluction to perturbative QCD S47

28

I!) 1.611

"q 1.2X:;::- 0.8

><0.4

0.1

-- EHLQ 1\=200 MeV

---- EHLQ 1\=290 MeV

_._- 001\=200 MeV

oOA= 400 MeV

0.2 0.3 0.4 0.5 06 0,7 0.8 0.9

X1.0

FIGURE 3.2. Eichten et al. [37] and Duke and Owens [38) distribution functions: gluon, valen ceup ql,lark and sea quark, at Q2 = 5 GeV2.

Turning now to the sum of the neutrino and antineutrino differential cross sec-tions,

(3.40)

This yields the U and d sea distributions given s from neutrino produetion of charrn,and the valence contributions derived aboye. lt is assurned that us(x) = ds(x).

To obtain the gluon distribution function is more difficult as it cannot be done di.reetly. Instead, we know that from the momentum sum rule, Jol dxx(G(x)+ L:(q(x)+q(x))J = 1, so there is a constraint on the integrated gluon distribution. One maypostulate a shape and normalization at a reference Q5 and check for consistencyat higher values of Q2. At larger values of Q2, because the gluon can split into aquark.antiquark pair, an incorrect gluon distribution at Q5 will rnanifest itself as anineorrect theoreticaI predietion for q(x, Q') and ij(x, Q'). In Fig. 3.2, it is clear thatone of the main differences between various sets of distribution functions is how thegluon is para~eterized.

Uncertainties arise from both experimental and theoretical effects. Of the latter,the largest uncertainties come from radiative corrections, the value of /\. and charmquark correetions. Depending 00 precisely how one treats the radiative corrections,there are different parameterizations of F2. In additioo, the gluon distribution isintimately tied to this question because of the momentum sum rule. EHLQ presenttwo different gluon parameteri~ations for two values of 1\.= 200, 290 MeV. Amongthe experimental uncertainties, there are target effects, beam flux, energy calibrationand acceptance uncertainties. For a thorough discussion, see Tung el al. [44].

548 J/.Jl. lleoo

4. Factorization

So fae, the discussion has implicitly rdied on lhe property of factorization. E\'cn ",,¡lhhighcr arder corrections, we write the <I(.(.pinelastic CfOSS scction as lhe convolutionoC the partan cross section with lile distrihulioll fUTlctions

aJl (q, 1') =L J dx f,(x,Q') á,(q,xi',o,(Q')).,

(4.1 )

This is essentially writing lile cross sectioll as a product oC lhe hard scattering (a)and soft, non.perturhalivc (h) picccs. For hadron-hadron scattering, \Ve write

alfil (1'1,1',) = L J dXI J dx,f,(xl, Q') Ji(x" Q') á" (XII'I, X21'" 0,( Q'))>,j

(4.2)The properly oC faclorization tclls us:

1. Infrarcd singularitics associaled with ill(orning partan ¡¡nes can always be ab-sorbed into the dislribution functions in a consistent way.

2. The dislributioll functions are incicpcII<!Cllt of lhe proccss. '('he Ji'S in eqn. 4.2are the same as in eqn. 4.1.

For proofs of factorization, see Refs. [.15]. A sketch of thc rnaill points of the proofma)' be found in Hef. 1-16). \Ve content ourselves here with the example of factor-ization in deep inelastic scattering, al order o~ in the ~IS sciwm(" ealculated byAltarelli et al. [47] \Ve follow HeL [471 dosely bclow. emphasizing the pole structurerather than the finitc corrections themsdv('s.

4.1 DEEP INELASTIC SCATTERING F,XAMI'LE

Fig. 4.1 shows the graphs contributing to the 0,'1 corrections to <Ieep inelastic scat-tering. They come from the interferenee of the two graphs in Fig . ..t.la and theabsolute squares of Fig. 4.tb and Fig .. Llc. \Ve are intercsted in tbe inclusive "Y.pcross section, so we integrate over the final parton momenta.

There are several types of singularities associated with tbcse diagram."l. Thcreare, of course, the ultraviolet singularities in tlw loop graphs which are cancelled bycounterterms. We may choose to do the calculatioll in the Laudau gauge where thefermion self energy corrections vanish at oue loop [19]. For the real graph5, thereare singularities a.ssociated \•...ith 50ft and collinear gluons (in the massless limit).Fig. 4.2 shows the potential dangcr of the coHincar gIuon.

Since the gluon i5 on shell, with J.:2 = O, and the incident quark is massless,the intermcdiate quark momentum is k'2 = (1' - J.:f = -2poJ.:(1 - cos O), whichvanish~ in the cosO -+ 1 (coHinear) limit, so the intcrmediatc quark propagatordiverges. Thi5 corresponds to the y -+ llimit ror y == (1 +cosO)/2. \Ve shollld note,

Inlroduclion lo ¡>erlurootive QCD 549

/

( O )

( b )

( e )FIGURE .¡ .1. The 0. correcliolls to deep ínelastic scattering.

I

P

p

FIGURE 4.2. Diagram for q"f. - qg.

550 AI.l/. Reno

however, that caHincar gluons are already accounled roc in the partan dislributionfundions. For examplc, \Ve computcd lhe splitting of a <}uark into a quark plus gluonwith ncgligible lransversc momenta. \Vhat we w¡JI fiud is that the singularity willmultiply the quark splitting functían times the lowesl arder cross scction (evaluatcdal a fraction oí the incidenl quark lIlomcntum). This is absorbed w¡lh the naivepartan dislribution fundian lo yield lile QCD improved distribution fUllction.

Dne should remark thal thc measurcd strucl.ure flltlctions F¡(x,Q2) are finite.In terms oí the partan modcl, the Pi's are exprcsscd in tcrms of lhe parlan dislri.bution functions, which are ealculaled lo ha\'c pcrturhative infinitc contributions.Therefore, there is sorne kind of "'physical" regularization in the proton due tobound state effects, which \\'e cannot compute. Our procedure here is to compute inperturhation theory (using dimensional reguiarization to regulate all infinitics) andput aH of the wave fundion (10\.•..energy) effects in the definition of the parton distri-bution functions, which, in perturbatioll lhoor)', include infinite picees. \Ve then uselhe measllrrd distribution functions, appropriately evolved using the Altarelli.Parisiequations, in the numerical calculations.

Following Altarelli el al. [47} we st;\rt. with the naivc quark TlIodcl and QCDperturbation theory. It is convenicllt to define :F¡ in Lerms of the F¡ in Eq. (3.9)such that

l',:F, = -x

(4.3)

so that al lowcst order;:2 = L¡ e?(ql(.r) + </.(x)), where i sums over quark flavors.The Callan-Gross rclation appears as :F) = :F2. \Ve rcwrite:F2 at lo\\'est order as

= [dy [dZÓ(Zy-x)¿el(q,(y)+q,(Y))Ó(l-Z).o O i

(4.4)

Our aim is to compute :F2 through order 0", and \Ve shall define i2q and i2G suchthat

:F, = r' dyó(zy - x) ¿ el [(q;(y) + q,(y))j'q(z,Q') + G(y)j,c(z,Q')j. (4.5)J, •

To lowcst order, jJ~) = 8(1 - z) and jJ~ = O. The quark contribution is just the(appropriately normalized) matrix e1clIlcnl squarcd of the lowest order graph.

htlrotillctio71 (o I'cTtltr/mlil'c. QC/) 551

1'0 derive i2q and i2G, wc write t.lw integratcd matrix c1elI1l'ntsr¡llarcd as

p ( q"qv),J' ( '1 . l' ) ( q' 1') l'lV¡W = -g¡W + q2 n, I + Jl - ---:}2q /1 P - q2 q 11 U'2 (4.6)

in analogy with Eq. (:1.5), where now JI is the porton 1Il0lllcntUTn.Furthcr definez'" -'1'/'1'1' ~ Q'/q '1',11"'11 j, ~ ('1.p/z)l\'{' alld ji ~ 21\'(. I'o(e (hal (heprojections -g/III and P¡IPII gi\.c

l' . ('1 )(' ,)-g'llIH'¡w=(I-t)h- 2"-( 12-fl

"VI\'I' Q' (J' J' )¡r P /111 == 8:2 2 - I

At lowcst order,

1\,(0)1' _ ¡'S (O)¡lll - • ¡.w/III

- I'S .2 I ./- 'l. (-'1 tr 1/1")11.

Tile single body rhase space l'<¡uals

'2r.~-, -8(1-:),2q. P

(U)

(4.8)

(4.9)

for final quark mOlIlentulll 1". First !lote tilat pJjpllw~~ = 0, so the Callan.Grossrclation is satisfied. The cvaluation of gllllu..-,}¡UJ togcthcr with thc phasc spacc factorgives ji~) proportional lo h( 1 - z), wlLerc thc prdaetors go iuto thc normalizationof lhe cross sections.

For thc next order, tbere are two contriblltiolls 1,0:Fz. From q¡. scaltcring wegel ii~)and frolll G,. sc:attcring, we gd jJ~. \Ve look first il1. glllOIl cmission. Bere,

(4.10)

for TI = p- k, r2 = p+q and.'l:::; Ti, 1 = 1.r. Twa body phase space in n dimensions

552 .11./1. Reno

IS

1'5, = J (2~~~~1J (2~;:-1ó(¡i')ó(k')(2~)"ó"(p + q - k - p')

= ~(h)'-" J Ikl"-3dkdfl,,_,ó((p+ q - k)')

\Ve TIlay dlOose lo evaluatc the pha.."c span~ integral in lile frame where

p = (Ipl. [J •...• O.lpl)

q = (qo.O....• O.-lpi)

k = (Ikl •.... Ikl cosO)

(-1.11)

(-1.12)

in which case, lIsing 11 == 4 - 2( and thf' i<!cntities (:\.6) in lhe Appell<!ix, \Ve filld

(4.1 :1)

for 11 = (1 + cosO)/2.

In terms of y and z, s == Q2(1 - :)/:; an£! t = _Q2(1 - y)/z. First. Jlllll'(.¡",':I~qyields a tcrIn proportional to s + Q2. so lhe cvaluatioIl of lhe phase spacc integral

yields a finit.c differcnce fol' ji~) - j:~) in th(' ( -jo O limit. Evaluating lhe g/Iv

conlraction givcs (up to nurncricai foclors)

(.1.11 )

By making lhe s and t substitutiolls in tel'llls of '!J ami z, lhe pole strllctufl' ismallife:-t. Far cxamplc, lf(st) = -.:2/((2"(1 - .:)(1 - y)). The integral ove!, y gi\'csa factor of l/e and al z = 1. there is i\ 1/( cOllling from tile expansion

Z' 1 1 (Iog(l - z)) ,(1 _ z)I+, = -~Ó(l - z) + (l _ z)+ -, 1 _ z + + 0(, ) (.1.15)

lntroduction lo pcrturbatit:e QCD 553

\Ve just quate the answcr for an incollling quark [47] scattering with .,/:

'(1)1 - a, (h¡,2)' f(1 - 'J x12q <eal-hCF Q2 f(1-2,)

[2 1(1+.2) 3 ]-6(1 - z) - • + -6(1 - z) + finitc tcrm,,2 , (1 - z)+ 2,

(4.16)

The first tcrm is thc averlap of soft illld col1inear divcrgences, \,,'hile the sccondcomes fram the collinear gluon, ami tlle third, from a 50ft gluon. In terms of lhesplitting function Pqq(z), (sec Table ;I,I) we may write

'(1)1 a, (I~Jl2)' f(1-,)f -- -- -~-~x2q real - 2rr Q2 r(l - 2()

[?CF 3CF I:...., 6(1 - z) + - 0(1 - z) - - Pqq(z) +(~ ( (

The first order virtual corrections at arder 0.'l are [47J

finitc tcrms] .

(4.17)

[2CF 3CF . ]

- -0- - -- + filllte terms,- ,

'(1)1 = a, (4~¡")'f"'q ? Q'- virt _71" -

['(1 - ,) 61'( I _ 2, J (1 - z) x

(4.18)

so

'(1) _ a, (4~Jl2)' r(1 - ,) [ 1 (1) ]12q - h Q2 1'(1 _ 2,) --;Pqq(z) + IZq (z) (4.19)

wilerc fJ~)(z) is the Slllll of the finite tt'rms in Eqs. (-1.17) and (4.18).

IIcn: we begill to sec the consequetl('(,.'Sof the faetorizatian theorcm. The onlysillgularity rcmailling from ').q scatl.crillg is the coHincar olle, which Olllltiplics aqllalltity indepelldent of thc details of tll<.'process (deep inclastic scattering), namely,the Altarelli.Parisi splitting funetioll lJqq. If \Ve had instead computed order o .•corrections to qij -+ ')-, we would have again found this l/lPqq singularily !47].As we shall sec Lelo\\', this will be absorbed into thc rcnormalized, Q2 dcpendentdistributioll fUlldioll.

The g')- seattcrillg contribution t.o j2 is similarly dcrivcd. Thc only singulari-ties are collincar. Nott: that the glllOIl may interaet via an interrnediate quark or

554 M. H. Reno

antiquark. Consequently, there is a factor oC two in lhe expression [47J

j(l)1 = ", (ÜI")' r(l - e) x'G ,,,1 2" Q' r(1 - 2e)

H P,dz) + 2J;2(z)] .\Ve now write :F2 including the first arder exprcssion

F, = l dyl dz6(zy - x) L e1 [[q¡(y) + qi(y)] (6(1 - z)•

", (1 "') ", (1) )+ -P,,(z) -- +, -log47:" -log Q' + - J, (z)211'" (: 21f q

The MS scheme definition of the dislributioIl functions could be

l'd [ '] ( )2 Y 0'$ 1 J1 x

'lMS(x,Q);;q(x)+, y27:" -~+,-log47:"-logQ' q(y)/'" y

l'dy", [ 1 "'] (x)+ -- -- +, -log47:" -log - G(y)/' G -x Y 211" (: Q2 q Y

(4.20)

(4.21 )

(4.22)

This is ane choice, howevcr, it is nol thc only choice which reproduces the Altarclli-Parisi equations. In particular, all oC the finite terms in (4.21) are independentof logQ' /1''. Allarelli, Ellis and Marlinelli [47J propose inslead lo define beyondleading arder:

F,(x,Q') ;; L e1(qDls(X, Q') + qDlS(X, Q'))

'" , J' dy",(Q2) [[ .. , _ ,] (1) (x)+ L. ei i, --?- q.(x, Q ) + qi(X, Q ) 1" -j x Y ~1f Y

+ 2G(y,Q')Ji2 (~)]

(4.23)

This is sometimes rcfcrred lo as the Jeep inelastic scattering (015) factoriziltion. In

Introdudion to perturbative QCD 555

this case, the structure Cunction .rl(X,Q2) is

Fl(X,Q') = Le; [;[.(~ -1) +o,(Q') [i'l (~) -j" (~)]] (q(y,Q'),

+ ¡¡(y,Q')) + [o,(Q') [iGl (~) - iG2 (~) j]C(y,Q')

(4.24)

Theoretically one advantage oC this choice is calculational simplicity. The diC-[ereoees betweeo 'i,,(z) aod i,,(z) aod betweeo iIG(z) aod i,G(z) are fioite at anyorder, since Fll the measurcd structure fundion, is finite.

5. Transverse momentum of IV or Z

The final discussion will focus on the calculation oC the IV or Z transverse momen-tum at hadrno colliders, thrnugh nrder o; [48J, [491. The aoalytie results deseribedbelow also apply to virtual photons, however, we restrict ourselves here lo weakbasan production. For defioitencss, we refer to W production. In hadron colliders,because the partoos participating in the hard scattering carry unknown Cractions oCthe longitudinal momenta of the incoming hadrons, only cooservation oC transversemamen tu m can be experimentally enforced 00 the final state momenta. We presenthere the inclusive distribution of transverse momentum (qr) Cor a single W producedin proton.aotiproton collisioos.

The order 0'; contribution is the next-to-Icading order contribution to the llTdistribution. Figure 5.1 shows Drell-Yan production and the first order corrections.Since the partoos coBide with negligible incident transverse momenta, only whenthere is a "balancing" quark or gluoo will there be transverse momentum Cor theIV, so the qT dislribution begins at arder 0:.,. Consequeotly, it is a good process tostudy QCD, especialIy as eleclroweak bosoos are fairly unambiguously identifiable inhadron colliders. By includiog the 0'; correctioos, we reduce the theoretical errors.

The theoretical errors are not completely under control in this second ordercalculation in the regioo oC smal! lJT. A problem arises because we have more thanooe scale in the problem, here Afw and qT' In the QCD-improvcd parton model, therenormalizalion and Cactorization scales, gcnerically denoled by Al, are chosen toreduce or eliminate logs oC the form lag M /Q. When we have a transversc momenlumdistribulion with qt < Affv, we have two scales which cannot be simultaneouslyreduccd. Chnnsiog M = Mw lcaves us with Ing Mw /'/T. [t eao be shnwo [501 thatthe o:~ corrections at small qT go Iike

n 1 1 m(M?v)0:., 2'" og -,-,qT qT

m::; 2n - 1 15.1)

556 JUl. lleno

_r""""FIGURE 5.1. Drcll- Yan and firsl order correct.ioll to qq productiotl of W's. The lOOPC(l lincs are

gluolls; lile wavy, unlooped tilles are lV's.

so the expansion in higher orders oC Q~ is accompanicd by large (lag) cocfficients.Pcrturbation thcory brcaks clowlI.

For lV ami Z productioll, tiJe leading lag terms have bcen sumnwd lo give aSudakov like form factor [51]. Tile distribution is normalized so that liJe integralover qT reproduces the perturbativc cross scction through arder 0", and is rcquiredlo reproduce the perturbativc qT distrihution roc sufficicntly largc qT. Il has hecnshown [52) thal (or él range of cculee of mass cnergif's. (.¡s = 0.6:1 - 10 TcV) thesurnmed and first arder perturhativc fl-'Sults match [oc QT,2:: 20 Ce\'. On this basis,wc restrict oursclvcs to qr > 20 CeV in the second onler perturbative calculation.

The earlier partial o; rcsult by Ellis, ~lartinclli ilnd Pctronzio [5:l] has recentlybeen completcd {4S]' [49]. Thc procedure is to compute lhe matrix c1clIlcnt squaredat the parton Icvcl and analytically illtegrate over the final parton phase spacc usingdimensional rcgularization to regulale aH singularitics. The result is then factorizedand numerically intcgrated.

The weak boson couplings are distinguishcd from gluon or pholon couplingsby the appcarance of ')'5 couplings. 'fhe 1'5 is not IIl1amhiguously dcfined in n f:. .1dimensions. \Ve 148] have silllplificd the 15 problclll sOlllcwhat by olllitting graphswhich are proportional to quark lIlass splittillgs \•...ithin an clcctrowcak doublet.Furthermorc, following Chanowitz el al. [54} \"'c aSSllIllC:

1. h"~1""'5}= O in n dimcnsions,

2. il = 1,

9. tr(¡51~1v1°I'P) = 4it:~vaP + O(t) ambiguity.

and we also assumc4. tr(¡51'~l'v1nl'.8) is antisyrnlllctric.

!ntrodlJ.ctirm lo pertlJ.rixJtil'e QCD 557

FIGURf; 5.2. Virtual correction diagram'i for gq - gW. The o; corrections come from the inter-ference of V, with [,¡ of Fig. 5.1.

FIGURE 5.3. Diagrams ror gij _ ggW.

FIGURE,5.4. Diagrams for qq _ qqW.

558 M.H. Reno

FIGURE5.5. Diagrams (or qq - qqW.

These assumptions, together with gauge ¡nvafiance arguments, are sufficicnl foc OUT

purposes. For a more complete discussion oí /5 '5 in this context, togcther with lheresults oí lhe omitted diagrams (numcrically small contributions), sec Gonsalves el

al. (491.

The sel oí diagrams fo! nonzero qr al arder a; are shown in Figs. 5.2-5.5. Inaddition lo these diagrams are sorne crosscd diagrams, Cor examplc, lhe qg ---. qtVdiagram which is related lo the qij -+ gH/. Far lhe real graphs, the thrce body pllasespace reduces lo two angular tntegrals:

sdO' 1 J da . "o J d . 1 "odtdu = 28 172 sm- u2 01 sm - ul

'" ,S f " 1LJ IM( 2'~4 (4~) f(l _ 2<).,'(5.2)

where s, t and u are lhe Mande1slam variahles defined hy s = (PI+p,)', t = (q-p¡)'and u = (q - P2)2, and 82, the invarianl mass oí lhe two-parton final state, isS'l = (PI + P2 - q)2 ::;:;;(P3 + P4?' Bere, PI ami P'l are incident parton morncnta andq is the W momentum. The factor SI is thc statistical factor. The angular integralsare evaluated in the rest frame of the two parton final state.

The parameter choices for the following figures are mq ::;::O, NI ::;::5 and the fourflavor value of AQCD = 260 MeV in the modified mínimal subtradion scheme. \Veuse lhe parlide dala book cenlral values [141 oCMw = 81.8 GeV and Mz = 92.6GeV together with the next.to.leading order cvolved structure functions of Diemoz,Ferroni, Longo and Martinelli [42]. For a choice of the renormalization scalc (cqual

1t,troduclion lo perlurbative QC/) 559

10-4 • UAlN UA2'>vCl

" 10-5O-

""-b"O- 10-11

"-b"-

10+7

20 40 60q, (GeVJ

80 100 120

q, (GeVJ120

.>v2-

10-5"O-""-b"" lO-eo-"-b"-

10-7

10-6

20

FlcunE 5.6. Predietions and experiment [551. [56) for W production aL ..¡s = 630 GeV.

lO-J

FIGUlU; 5.7. Predictions for W product.ioll at"¡;;;: 1.8 TeV.

560 M./I. /leno1

8

.6A

F

e

2

o ~

-2

200ISO501

(J lOO'1, [CeV]

FIGURE 5.8. Rclative conlributiollS oC difrNf'lIt pro('('ss{'s to W production al fi = 6:JO \,,'Vusing Af = Mw. First arder rOlllri!Jtll.iOIlS an' (A) qq -- gB' and (U) '19 -- I/n' andqg - tiW. Second onkr wnlrihlltiollS are (e) [qri -- yH' + gyW] + IFI + Fzj2, (O)t¡y -- qW + qgW and ijg -- ijH/ + lJyJr, (E) 9Y - /jqlV, (r) rClll.linillg 'Jlj - I/(jlt',(G) qq -- qqtv and ijij - qqlt' .

.6

5

A4

.3

2

Be

.1O

F

o-------_.--

______~::::::::::::.;=::::::::::==:~7~~~:;;::::::. ~

-.1o 50 lOO

q, (GeV)150 200

FIGURE 5.9. Same as Fig. 5.8 with .¡s = 1.8 TeV.

Inlrrxlu.cliorl lo pcrturbatit'e QCD 561

to tile factorization scale) of JJ~ror A1z, we ,get for proton-antiproton collisions

M=Mz

Js = 1.8 TeV

cr(lV+) = 9.68 nb

cr(Z) = 6.0.) nb

Js = 630 CeV

cr(W+) = 2.80 nb

cr(Z) = Li8 nb

(5.3)

In Fig. 5.6 we see tiJe qT distributioll (normalizetl to 1'1"(7) at tile srs energy oC..fi = 630 CeV, colllpared with the VAl [S51and UA2 ["(;1 data. Fig. S.7 shows thesame distribution at the Tevatron encrgy oC.¡s = 1.8 TeV. '1'0gel an idea oC thc rel-ative importancc oC t1le various graphs, we show in Figs. 5.8 and 5.9 the various com-ponents, normalizcd lo the full differcntial cross section, as a Cunction of qr for SPSand Tevatron energies, respectively. By comparison, it is c1car lilat the Tevatron is amuch belter probe oC the gluon compollcnt of the protoll t1lall the SI'S in this context.Exeepl at the lowcst transverse 1TI0illcnta, wbere lhe qij matrix elerncnt is larger,liJe qg eontribulioll dominatcs at tiJe Tcvatron until qr ~ 150 GeY, where x getssufficicntly large so that the valencc qllark eomponent dominates the cross section.

The uneertainties in the thcoretical calculation at large qT indude the choiceof sea le, the value oC A and thc structurc functions. By considcring ,\1 bctwccnqT and ."/nr, and A = 160-360 ~leY, with the Diemoz el al. struclure functions,we estímate thc following crrors: For .¡s = 1.8 TeV, the error in I/ada/dQT is isapproximately :t1O-15% relative to tiJe mean of i\1 = fJT and Al = Alw. Tlle tolalcross sections were evaluated al II1 = Ala' and i\f = (q.],) 1/2. For .¡s = 630 CeY,at low qT the cstimalcd error is :tl0% ami at high qT, :t35%, again, relative to themean.

The larger error at the srs encrgy for large qT originates from the fact that(q})1/2 '" 10 CeV at ..fi = 630 CeV, whereas (q})1/2 '" 20 CeV at Js = 1.8TeV. At the Tevalron, at qr = 20 GcV, the normalized distribution l/ada/dq-ris larger for .\12 = q], than for 1\12 = Alfr' For QT > A1w, this is revcrsed aso,(q}) < a,(i\lrv)' For the SI'S, howevcl', Cven al qT = 20 GeY, the normalizcddistribulion is largcr for 1\12 = i\frr lIJan for 1\12 = qj.., beeause the normalizingeross section is on the order of 30% larger for the lalter scale (comparcd with _ 15%smaller at the Tcvatron). Again, as qT increascs beyond .Hw, the distribution with¡\l2 = q} drops more rapidly than with .\12 = Ata,. The discrcpaney is compoundcdhy the larger C1'OSS section normalizatioll factor.

AI'I'ENDIX

l. Gamma matrix propertics in n dirnensions:

/1J/1I + /1I/1J = 291J1I

'.'0'" = (2 - nho

562 M.H. Reno

2. Feynman pararncterization:

1 _ I'(r + m + n) 1.1 r-l m+li-la'bmc" - l'(r)l'(m)r(n) o dx(l - x) x X

1.' y"-'(l - y)m-Idy------

O [a + x(b - a) + xy(c - b))'+m+"

l'(¿:7:, ni) 1.' 1.'- l'(n¡)l'(n2) ... l'(nk) o ... o dz,dz2 ...dz

kX

3. Phase space integrals:

(A.2)

J d"Q QpQ. J d"Q gp.(Q2/n)(2~)" (Q2 _ C + ;()m = (2~)" (Q2 - C + i<)m

J d"Q (Q2)' ;(_I)'-m (C . )'-m+"/2 (A.3)(h)" (Q2 _ C + i<)m = (16~2),,/' -lt

r(r + n/2)l'(m - r - n/2)x l'(n/2)l'(m)

4. Useful Gamma function relations:

The heta function B(x,y) is dcfined by

Thc Gamma functions obcy thc following:

r(l + z) = zl'(=)

t2 11'"2lim r(l + ,) = 1 -1' + -(/ + -) + 0(,3){-o 2 6

where i = 0.5772 ... is the Euler constant.

(AA)

(A.S)

lntroliuctiotl to pcrturbative QCD

5. Phase space inlegrals in n = 4 - 2l dimensjons:

J 2,,3/2-, Jdíl._2 = f(3/2 _ e) = dO._2 sin.-3 0._2 ... dO,

563

(A.6)

wheref( !!!.ll )[ dO sinm O 2J7if( mi2) (A.7)

rO) = J7i

References

[l] J.D. Bjorken and S.D. Drell, Ilelalivislic Quanlt17n Afcchanics and Rc1alivislicQuantum Fie/ds, McGraw-Jlill, Inc., New York (1964).

[2] P. Ramond, Field 1'heory: A Afodern Primer, The Benjamin/Curnmings PublishingCo., Massachusetls (1981).

[3] J. Collins, RenormaJizalion, Cambridge University Press, Cambridge, England(l!l84).

[4] V.B. Berestetskii, E.1-1. Lifshitz and L.P. Pitaevskii, transo by J.B. Sykcs and J.5.llell, IleJalivislic Quantum 1'heory, Pa1'l 1, Pergamon Press, New York (1971).

[5] F. Clase, I1n lntroduction lo Qual'ks and Partons, Ac<tdemic Press (1979).[6) C. Quigg, Gauge 1'heories 01 the Stmng, Weak and Eled1'Omagncfic lnteraetions,

The BenjaminjCummings Publishing Company, Mass<tchusetts (1983).[7] D. Griffiths, lnlroduction lo Elemenlm'y Particlcs, Harper and Row. Publishers,

New York (1987).[8] G. Altarelli, Phys. Rep. 81 C (1982) I.[9J S. Narison, Phys. Rep. 84C (1982) 263.{lO] A.JI. Mueller, TASI Lectures (1985).[Il] 1. IIinchliffe, TASI Lectures (1986).{l2] R.K. Ellis, TASI Lectures (1987).[13] C.A. Domínguez and E. de Rafael, Ann. Phys. 174 (1987) 372; J. Gasser and H.

Leutwyler, Phys. Rep. 87 (1982) 77.[14] ParticIe Data Group, G.P. Yost, el al., Phys. Lelt. 204B (1988) I.(15] See, however, G.S. LaRue, \V.M. Fairb<tllk and A.F. Hebard, Phy8. Rev. Lett. 38

(1977) 1011, and G.S. LaRue, W.M. Fairbank and J.D. Phillips, Phy-'. lIev. Le/t. 42(1979) 142. These results remain lIn:::onfirmcdby other cxperiments.

{16] E. Fernandez, et al., Phys. llevo D31 (1985) 1537.[17] R. Feynman, Acta Fisica Polonica and in Pmceedings ollhe 1976 Les l/ouches

Summer School, ed. by R. Balian and C.lI. L1ewellyn Srnith, North Holland. Se(!also Ellis, ReL [12].

[18] G. 't lIooft and M.J.G. Yeltman, Nue/. Phys. 844 (1972) 189; G.G. BoIlini and J.J.Giambiagi, Nuo. Gim. 12B (1972) 20.

[19) W. Marciano, Phys. Rev. D12 (1975) 386I.[20) For examples of color factor computations, scc e.g., G. Kane, in "Color Symmetry

and Quark Confinement," Proceedings 01 the 121h Renconlre de Aforiond, Vol. IlI,Tran Thanh Yan, ed., (1977), p. 9, and Grilliths, ReL [7].

[21] For a review of the renormalization scheme dependence of perturbative quantities (atfinite order), sec, for example, D.W. Duke and R.G. Robert" Phys. Rep. 120 (1985)275.

[22J T. Appelquist and II.D. Politzer, Phys. Rev. D12 (1975) 1404.

564 ,\1./1.Unw

[23] ~I. J)ille anl! J. Sapirsll'ill, I'hy.~. Uf'l'. I,tlt. 43 (IrJ7rJ) ()(j:'{; I~.C. Chetyr~ill, A.L.Kataev ¡¡lid F.V. TkadH'V. I'hys. 'AIt. 85B (1970) '277; W. ('('lmaster and ILo!.Gonsal\'('s,l'/¡ys. Un'. /'111.44 (197U) [)fíO; I'hy,~'.Uf!'. /)21 (t!J:--:O) Ti7'2.

[2.1)S. C:orishIlY. ¡\. Kat;w\' <tlld S. Larill. I'h!l.~. '-tll. 11212 (19~:O;;)'2:IS.[2.')]D.,I. (;ro,..;s aud F. Wilrz,,~, I)"ys. U11'. ',111.• 10 (197:~) 1:11:\; I'hy.~. UII'. /)8 (197:1)

:l6:n.[26]II.n. Politz('r, I'''ys. IhlJ. I,tlt. 30 (1!)7:1) I:~l(j.['ti] n.\\'. Dllkl' and H.t.. Hohnts. Hef. ['21].[28} W. Casw,,]]. J>hys. U"1'. 1,111.33 (I!J7,1) '2,1.1;n.H.T . .I01Il'S. Xuel. I)"ys. 1J75 (197.1)

.5JI.[29] G. :\It,Hf'lli alld G. Parisi, Sud. P/¡y.~. IJIUi (HJ77) 2!):-{.[30) G. :\ltatl'lli. H. K. Ellis and C:. ~lartilH'lIi, Sud Phy.~. 1J157 (1979) Ifll.[:11]5(,l' J.!. Fril'dman ¡¡lid 11.\\'. Kí'ndall, ,.t/lf!. U11'. SIWI. Sri. 22 (1972) 2m. for a

SUlIlllléHYof earl)" exp('rillll'llls.[:12]A. Bod,,~ (t al.. SLAC.r..lIT Collaboratioll, I'hys. Uev. /)20 (1!J79) 1.171.[33] LJ. Auhl'rt fl (¡l .• E~I('.I)lty.~. Let!. 123B (19S:l) ton.[:HJ J. Drees, in I)ror(f(li"g."i of Ihe 1981 IlIltnwliollnJ .'iYlllJ!O,''¡Il1fl Ofl Lrplon and

Il/lOll)lI III/U'actiolls af I/ig" ElIfl~/iI,"i.l'tl. \\'. Pfeil, Bonll (1981). p. -\7,1.[;35]S('('. for ('xalllple. p. 11:) of Ud. [J.lj.[3f1]C.F. mil W('izsack<'t. X. r"y.~.88 (!tI:! 1) Ii! 2; E.J. \\'illiallls. l'h,l/8. Rfl'. 45 (1!nl)

729.[;~i]E. Eidlt(,lI. 1. lIinchlilf", K. talle and C. QHiK~, Uf{l. ,\1o(i. I)hy,<;. 56 (\!),~'1) 7,,)9;

(1:,58 (IU~(¡) 100!j).[38] D. Du~(' alld J. OW(,lls. /'IIY.~. lle.!'. J)JO (I !IS-I) .19.[;!!)] J. OWPllS. I)hys. Ihl'. 1J:1O(198-1) 9-1:1.[.IOJ Gluck " "l., Z. I'hy-,. 03 ( 1(182) IEL(-llJ .-\.D. ~Iartill, U.C:. HoiH'rlS aud \\' ..1. Slirlill~. rhy$. !lc"!'. /)37 (1988) 1161.[.12]~1. Diellloz. F. Ferrolli, E. LOlIgo alld (;. ),tartillelli. X. I'hys. (39 (1988) 21.[.t3]11. Ahratllowicz, d al., 7,.1'11.11:'<.C13 (1982) l!J9; C17 (I!)S;~) 28:3.[-\.t] Wu-Ki TUllg. J. C. r..lorfin, 11. SclH'lilllOlu, S. hUllori, A. Caldwell and F. 0In(,s5,

"5tructllf(' fllllctiolls alld ¡¡arloll distriblltiolls," Ai\L-IIEP-CP-89-01 (1989). to ap-p('ar in tl.(' l'roc('edillgs of tbp 1988 SlllllllH'r Study 011 lIigh EII('fg .••.Physics in the1990's, SlIowlllass. Colorado.

[.t.5]n.l\. Ellis d a/.. Xuel. J'hy.~. 11152 (1979) 2S;); A.II. ~luel1er, t>hy,<;. Nn. [)18 (1978)370.5; D. Alllati el ai., SIId. IJlly.~. /Jl40 (197S) 5.1; A. Efrí'IllO\' anl! A. Hadyushkin,TIIIOI'. Mat/¡. Phys. 44 (1981) 6G-I; .J.C. Collins el ni., Phys. "di. 109B (1982) :188,NI/d. I'hy.~.In23 (19S;Q ;~s1, I'hy.~.¡,dt. l a4 B (198.1) 2G:L

!.16J 11.1>. Politzer, "5tiH QCI>.ing," in IJI'O('('('flillgs of /}¡r I~'ig}¡t}¡ l/alOaii l'opicaJCon/fTC1Icf, lIawaii (1979).

¡.ti) G. ,\ltardli, R. K. E!lis ¡¡nd G. :-'lartilll'lIi. Suelo I'hy$. 1J157 (1979) --161.[.t8) P.B. ,\rllold and 11.11. H(,tlo, SIJeI. Phys. 11319 (1989) :J7. Fur additional numerical

results ..••.('(' P.B. Arnold, H.K. Ellis and ~1.1I. H('no, Phy.~. nevo D40 (1989) 912.(49] RJ. GOllsalves, J. Pawlows~i and Chung.Fai Wai, I'hys. Nev. J)40 (1989) 2'2,15.[!j01 See, c.g., H"ference [81.[51] G. Altardli. R. R, Ellis, 11. Greco alld G. ~léHtin('lIi. Nud. P/¡ys. In46 (198.1) 12.[52} G. Altarplli, IL K. ElIis antl G. ~lartill('lIi. Z. Phy8. C27 (l!)8:l) 617.{53]R.n. Ellis. G. '-Ia.rtinelli alld R. Petrollzio. Xllc/. Phy.~. /1211 (1983) 106.[5-1],,1. Chanowitz,:-'1. Furman and 1. Hinchliffe, Nllel. Phys. 8159 (1979) 225.[55] VAl Collaboration, I'hys. /-etl. 193B (1987) 389.[56] VA2 Collahoration, presented by P. Jcnlli in Pl'oceeding.'i 01 lile /987 InternationaJ

SympO$itlm on Leplon and Photon /lItfracfions al I/igll Energies, Hamhurg( 1987).