Elements of QCD for hadron colliders

1.7 MB·English

by  Gavin P. Salam

Elements of QCD for hadron colliders
∗
GavinP.Salam
LPTHE,CNRSUMR7589, UPMCUniv.Paris6,Paris,France
†
Abstract
The aim of these lectures is to provide students with an introduction to some
ofthecoreconceptsandmethodsofQCDthatarerelevantinanLHCcontext.
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∗Lecturesgivenatthe2009EuropeanSchoolofHighEnergyPhysics,Bautzen,Germany,June2009. Theoriginalversion
ofthiswriteupwascompletedinApril2010andpublishedinCERNYellowReportCERN-2010-002,pp.45–100. ThisarXiv
versioncontainsafewsmallupdates.
†Current address: CERN, Department of Physics, Theory Unit, CH-1211 Geneva 23, Switzerland and Department of
Physics,PrincetonUniversity,Princeton,NJ08544,USA.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1 TheLagrangian andcolour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 ‘SolvingQCD’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Considering e+e hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
−
→
2.1 Softandcollinearlimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Thetotalcrosssection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Thefinalstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Partondistribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 DeepInelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Initial-state partonsplitting, DGLAPevolution . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Globalfits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 PredictivemethodsforLHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Fixed-order predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 MonteCarloparton-shower programs . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Comparingfixed-orderandparton-shower programs . . . . . . . . . . . . . . . . . . . . 38
4.4 Combiningfixed-order andparton-shower methods . . . . . . . . . . . . . . . . . . . . 40
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1 Jetdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Conealgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Sequential-recombination algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4 Usingjets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1 Introduction
Quantum Chromodynamics, QCD,is the theory of quarks, gluons and their interactions. It is central to
allmoderncolliders. And,forthemostpart,itiswhatwearemadeof.
QCD bears a number of similarities to Quantum Electrodynamics (QED). Just as electrons carry
the QED charge, i.e., electric charge, quarks carry the QCD charge, known as colour charge. Whereas
there is only one kind of electric charge, colour charge comes in three varieties, sometimes labelled
red, green and blue. Anti-quarks have corresponding anti-colour. The gluons in QCD are a bit like the
photons of QED. But while photons are electrically neutral, gluons are not colour neutral. They can
be thought of as carrying both colour charge and anti-colour charge. There are eight possible different
combinations of(anti)colour forgluons. Another difference between QCDandQEDliesinitscoupling
α . InQCDitisonly moderately small, ittends tozero athigh momentum scales (asymptotic freedom,
s
QEDdoestheopposite), itblowsupatsmallscales,andinbetweenitsevolution withscaleisquitefast:
at the LHC its value will range from α = 0.08 at a scale of 5TeV, to α 1 at a scale of 0.5GeV.
s s
∼
Thesedifferences betweenQCDandQEDcontribute tomakingQCDamuchrichertheory.
IntheselecturesIwillattempttogiveyouafeelforhowQCDworksathighmomentumscales,and
for the variety of techniques used by theorists in order to handle QCDat today’s high-energy colliders.
2
The hope is that these basics will come in useful for day-to-day work with the QCD facets of hadron
collider physics. In the fifty or so pages of these lectures, it will be impossible to give full treatment of
any of the topics wewill encounter. Forthat the reader is referred to any of the classic textbooks about
QCDatcolliders [1–3].
1.1 TheLagrangianandcolour
Letusstartwithabriefreminderofthecomponents oftheQCDLagrangian. Thissectionwillberather
dense, but wewillreturn tosomeofthe points inmoredetail later. Asalready mentioned, quarks come
inthreecolours. Soratherthanrepresentingthemwithasinglespinorψ,wewillneedthespinortocarry
alsoacolourindexa,whichrunsfrom1...3,
ψ
1
ψ = ψ . (1)
a 2
 
ψ
3
 
ThequarkpartoftheLagrangian (forasingleflavour)canbewritten
= ψ¯ (iγµ∂ δ g γµtC C m)ψ , (2)
Lq a µ ab− s abAµ − b
wheretheγµ aretheusual Diracmatrices; the C aregluonfields, withaLorentz indexµandacolour
Aµ
indexC thatgoesfrom1...8. QuarksareinthefundamentalrepresentationoftheSU(3)(colour)group,
while gluons are in the adjoint representation. Each of the eight gluon fields acts on the quark colour
through oneofthe‘generator’ matricesoftheSU(3)group, thetC factorinEq.(2). Oneconvention for
ab
writingthematricesistA 1λA with
≡ 2
0 1 0 0 i 0 1 0 0 0 0 1
−
λ1 = 1 0 0 , λ2 = i 0 0 , λ3 = 0 1 0 , λ4 = 0 0 0 ,
     −   
0 0 0 0 0 0 0 0 0 1 0 0
       
1 0 0
0 0 i 0 0 0 0 0 0
√3
λ5 = 0 0 −0 , λ6 = 0 0 1 , λ7 = 0 0 i , λ8 = 0 1 0 .
     −   √3 
i 0 0 0 1 0 0 i 0 0 0 2
 √−3
     
 
Bylookingatthefirstofthese,togetherwiththetC Cψ termof ,onecanimmediatelygetafeelfor
abAµ b LQ
whatgluons do: agluonwith(adjoint) colour indexC = 1actsonquarks through thematrixt1 = 1λ1.
2
Thatmatrixtakes green quarks (b = 2)andturns them into redquarks (a = 1), andviceversa. Inother
words, when a gluon interacts with a quark it repaints the colour of the quark, taking away one colour
andreplacingitwithanother. Thelikelihoodwithwhichthishappensisgovernedbythestrongcoupling
constant g . Notethattherepainting analogy islessevidentforsomeoftheother colour matrices, butit
s
stillremainsessentially correct.
ThesecondpartoftheQCDLagrangian ispurelygluonic
1
= FµνFAµν (3)
LG −4 A
wherethegluonfieldtensorFA isgivenby
µν
FA = ∂ A ∂ A g f B C [tA,tB] = if tC, (4)
µν µAν − νAν − s ABCAµAν ABC
where the f are the structure constants of SU(3) (defined through the commutators of the tA ma-
ABC
trices). Note the major difference with QED here, namely the presence of a term g f B C with
s ABCAµAν
twogluonfields. Thepresence ofsuchatermisoneofthemajordifferences withQED,and,aswewill
3
Fig. 1: The measured spectrum of hadron masses, compared to a lattice calculation [9]. The open blue circles
are the hadron masses that have been used to fix the three parametersof the calculation: the value of the QCD
coupling,theaverageoftheupanddownquarkmasses(takenequal)andthestrange-quarkmass. Allotherpoints
areresultsofthecalculation.
discuss in moredetail below, itwillbe responsible forthe fact that gluons interact directly withgluons.
Fornow,notesimplythatithastobethereinorderforthetheorytobegaugeinvariantunderlocalSU(3)
transformations:
ψa eiθC(x)tCabψb (5)
→
1
CtC eiθD(x)tD CtC ∂ θC(x)tC e iθE(x)tE (6)
µ −
A → A − g
(cid:18) s (cid:19)
where, in the second line, we have dropped the explicit subscript ab indices, and the θC(x) are eight
arbitrary realfunctions ofthespace-time positionx.
1.2 ‘SolvingQCD’
Therearetwomainfirst-principles approaches tosolving QCD:latticeQCDandperturbative QCD.1
1.2.1 LatticeQCD
The most complete approach is lattice QCD. It involves discretizing space-time, and considering the
values of the quark and gluon fields at all the vertices/edges of the resulting 4-dimensional lattice (with
imaginary time). Through a suitable Monte Carlo sampling over all possible field configurations, one
essentiallydeterminestherelativelikelihoodofdifferentfieldconfigurations,andthisprovidesasolution
to QCD. This method is particularly suited to the calculation of static quantities in QCD such as the
hadron mass spectrum. The results of such a lattice calculation are illustrated in Fig. 1, showing very
goodagreement.
Lattice methods have been successfully used ina range ofcontexts, for example, in recent years,
inhelping extract fundamental quantities suchastheCKMmatrix(and limitsonnewphysics) fromthe
vast array of experimental results on hadron decays and oscillations at flavour factories. Unfortunately
latticecalculationsaren’tsuitableinallcontexts. Letusimagine,briefly,whatwouldberequiredinorder
to carry out lattice calculations for LHC physics: since the centre-of-mass energy is (will be) 14TeV,
1Inaddition,effective-theorymethodsprovidewaysoflookingatQCDthatmakeiteasiertosolve,givencertain‘inputs’
thatgenerallycomefromlatticeorperturbativeQCD(andsometimesalsofromexperimentalmeasurements). Theselectures
won’t discuss effective theory methods, but for more details you may consult thelectures at thisschool by Martin Beneke.
Anothersetofmethodsthathasseenmuchdevelopment inrecentyearsmakesuseofthe‘AdS/CFT’correspondence [4–6],
relatingQCD-likemodelsatstrongcouplingtogravitationalmodelsatweakcoupling(e.g., [7,8]).
4
A,µ A,µ
D,σ A,µ
p
r
C,ρ
a b qB,ν C,ρ B,ν
ig tAγµ g fABC[(p q)ρgµν ig2fXACfXBD[gµνgρσ gµσgνγ]+
− s ba − s − − s −
+(q r)µgνρ (C,γ) (D,ρ)+(B,ν) (C,γ)
− ↔ ↔
+(r p)νgρµ]
−
Fig.2: TheinteractionverticesoftheFeynmanrulesofQCD
A,µ A,µ
p
r
C,ρ
q
b a B,ν
Fig. 3: Schematiccolourflow interpretationof the quark–quark–gluon(tA, left) andtriple-gluon(f , right)
ab ABC
vertices of QCD. These interpretations are only sensible insofar as one imagines that the number of colours in
QCD,N =3,islarge.
c
we need a lattice spacing of order 1/(14TeV) 10 5fm to resolve everything that happens. Non-
−
∼
perturbative dynamics for quarks/hadrons near rest takes place on a timescale t 1 0.4fm/c.
∼ 0.5GeV ∼
ButhadronsatLHChaveaboostfactorofupto104,sotheextentofthelatticeshouldbeabout4000fm.
That tells us that if we are to resolve high-momentum transfer interactions and at the same time follow
the evolution of quark and gluon fields up to the point where they form hadrons, we would need about
4 108 lattice units in each direction, of 3 1034 nodes. Not to mention the problem with high
× ∼ ×
particle multiplicities (current lattice calculations seldom involve more than two or three particles) and
all the issues that relate to the use of imaginary time in lattice calculations. Of course, that’s not to say
that it might not be possible, one day, to find clever tricks that would enable lattice calculations to deal
with high-energy reactions. However, with today’s methods, any lattice calculation of the properties
of LHC proton–proton scattering seems highly unlikely. For this reason, we will not give any further
discussion of lattice QCD here, but instead refer the curious reader to textbooks and reviews for more
details[10–13].
1.2.2 PerturbativeQCD
Perturbative QCD relies on the idea of an order-by-order expansion in a small coupling α = gs2 1.
s 4π ≪
Somegivenobservable f canthenbepredicted as
f = f α +f α2+f α3+... , (7)
1 s 2 s 3 s
where one might calculate just the first one or two terms of the series, with the understanding that
remainingonesshouldbesmall.
The principal technique to calculate the coefficients f of the above series is through the use of
i
Feynmandiagrammatic(orotherrelated)techniques. TheinteractionverticesoftheQCDFeynmanrules
areshowninFig.2(insomegaugesonealsoneedstoconsiderghosts,buttheywillbeirrelevantforour
discussions here).
The qqg interaction in Fig. 2 comes from the ψ¯ g γµtC Cψ term of the Lagrangian. We have
a s abAµ b
5
alreadydiscussedhowthetC factoraffectsthecolourofthequark,andthisisrepresentedinFig.3(left),
ab
withthegluontakingawayonecolourandreplacing itwithanother.
The triple-gluon vertex in Fig. 2 comes from the 1FµνFAµν part of the Lagrangian, via the
product of a ∂ term in one Fµν factor with the g f −4 BA C term in the other. It is the fact that
µAν A s ABCAµAν
gluonscarrycolourchargethatmeansthattheymustinteractwithothergluons. Intermsofcolourflows,
wehavetherepetitionoftheideathattheemissionofagluoncanbeseenastakingawaythecolourfrom
thegluon(oranti-colour) andreplacing itwithadifferentone. Becauseofthedoublecolour/anti-colour
charge ofagluon, onecananticipate thatitwillinteract with(oremit)othergluons twiceasstrongly as
does a quark. Before coming to mathematical formulation of that statement, let’s comment also on the
4-gluonvertexofFig.2. Thiscomesfromtheproductoftwog f B C typetermsin 1FµνFAµν
s ABCAµAν −4 A
andisorderg2 whereasthetwootherinteractions areorderg .
s s
Though Fig. 3 gives some idea of how the colour factors tC and f in the Feynman rules are
ab ABC
to be understood, it is useful to see also how they arise in calculations. After squaring an amplitude
and summing over colours of incoming and outgoing particles, they often appear in one or other of the
followingcombinations:
Tr(tAtB) = T δAB, T = 1 A B (8a)
R R
2
tAtA = C δ , C = NC2 −1 = 4 a c (8b)
ab bc F ac F 2N 3
C
A
X
fACDfBCD = C δAB, C = N = 3 A B (8c)
A A C
C,D
X
b a
tAtA = 1δ δ 1 δ δ (Fierz) = 1 −1 (8d)
ab cd 2 bc ad− 2N ab cd 2 2N
C
c d
where N N = 3is the number of colours in QCDand it is useful to express the results for general
C
≡
numbersofcolours (because itissometimesuseful toconsider howresultsdepend onN ,especially in
C
thelimitN ). Eachmathematicalcombinationofcolourfactorshasadiagrammaticinterpretation.
C
→ ∞
Equation(8a)corresponds toagluonsplittingintoqq¯whichthenjoinbackintoagluon;or,thesumover
coloursinthesquaredamplitudeforg qq¯. Equation(8b)corresponds tothesquareofgluonemission
→
from a quark. Equation (8c) arises as the square of gluon emission from a gluon. One sees that there
is almost a factor of 2 between Eqs. (8b) and (8c) (modulo corrections terms 1/N ), which is the
C
∼
mathematical counterpart of our statement above that gluons emit twice as strongly as quarks. Finally
the approximate colour-flow interpretation that we had in Fig. 3(left) can be stated exactly in terms of
theFierzidentity, Eq.(8d).
1.2.3 Therunningcoupling
Most higher-order QCD calculations are carried out with dimensional regularization (the use of 4 ǫ
−
dimensions) in order to handle the ultraviolet divergences that appear in loop diagrams. In the process
of going from 4 to 4 ǫ dimensions, one needs to introduce an arbitrary ‘renormalization’ scale, gen-
−
erally called µ, inorder to keep consistent dimensions (units) for all quantities.2 Thevalue of the QCD
2Therenormalizationprocedureitself,i.e.,theremovalofthe1/ǫdivergences,isusuallycarriedoutinthemodifiedminimal
subtraction(MS)scheme(see,e.g.,Section11.4ofRef.[14]),byfarthemostwidespreadschemeinQCD.
6
coupling, α = gs2, depends on the scale µ at which it is evaluated. That dependence can be expressed
s 4π
intermsofarenormalization groupequation
dα (µ2)
s = β(α (µ2)), β(α ) = α2(b +b α +b α2+...), (9)
dlnµ2 s s − s 0 1 s 2 s
where
11C 2n 17C2 5C n 3C n 153 19n
b = A − f , b = A − A f − F f = − f , (10)
0 12π 1 24π2 24π2
with n being the number of ‘light’ quark flavours, those whose mass is lower than µ. The negative
f
sign in Eq. (9) is the origin of asymptotic freedom, the fact that the coupling becomes weaker at high
momentum scales, i.e., the theory almost becomes a free theory, in which quarks and gluons don’t in-
teract. Conversely at low momentum scales the coupling grows strong, causing quarks and gluons to
be tightly bound into hadrons. Theimportance of the discovery of these features was recognized in the
2004NobelprizetoGross,PolitzerandWilczek. WhydoestheQCDβ-function havetheopposite sign
ofthat inQED?Thefactthatthevector particles (gluons) ofthetheory carry colour charge iscentral to
theresult. However,whiletherehavebeenvariousattempts togivesimplebutaccurate explanations for
thenegative sign [15,16],inpractice they allendupbeing quite involved.3 So,forthe purpose ofthese
lectures, letusjustaccepttheresults.
If we ignore all terms on the right of Eq. (9) other than b , and also ignore the subtlety that the
0
numberof‘light’flavoursn dependsonµ,thenthereisasimplesolution forα (µ2):
f s
α (µ2) 1
α (µ2) = s 0 = , (11)
s 1+b α (µ2)ln µ2 b ln µ2
0 s 0 µ20 0 Λ2
where one can either express the result in terms of the value of the coupling at a reference scale µ , or
0
in terms of anon-perturbative constant Λ(also called Λ ), the scale atwhich the coupling diverges.
QCD
Onlyforscales µ Λ,corresponding toα (µ2) 1,isperturbation theory valid. NotethatΛ,sinceit
s
≫ ≪
is essentially a non-perturbative quantity, is not too well defined: for a given α (µ ), its value depends
s 0
on whether we used just b in Eq. (9) or also b , etc. However, its order of magnitude, 200MeV, is
0 1
physically meaningful insofarasitiscloselyconnected withthescaleofhadronmasses.
One question that often arises is how µ, the renormalization scale, should relate to the physical
scale of the process. We will discuss this in detail later (Section 4.1), but for now the following simple
statement is good enough: the strength of the QCD interaction for a process involving a momentum
transfer Qisgivenbyα (µ)withµ Q. Onecanmeasurethestrength ofthatinteraction inarange of
s
∼
processes, at various scales, and Fig. 4 [17] shows a compilation of such measurements, together with
the running of an average over many measurements, α (M ) = 0.1184 0.0007, illustrating the good
s Z
±
consistency ofthemeasurements withtheexpectedrunning.
1.2.4 QCDpredictions andcolliders
Colliders like the Tevatron and the LHC are mainly geared to investigating phenomena involving high-
momentum transfers (more precisely large transverse-momenta), say in the range 50GeV to 5TeV.
There, the QCD coupling is certainly small and we would hope to be able to apply perturbation theory.
Yet, the initial state involves protons, at whose mass scale, m 0.94GeV, the coupling is certainly
p
≃
not weak. And the final states of collider events consist of lots of hadrons. Those aren’t perturbative
either. Andtherearelotsofthem—tenstohundreds. Evenifwewantedtotry, somehow,totreatthem
perturbatively, wewouldbefacedwithcalculations tosomeveryhighorderinα ,atleastashighasthe
s
particle multiplicity, which is far beyond what we can calculate exactly: depending on how you count,
3Youmightstillwanttocheckthesignforyourself: ifso,pickupacopyofPeskinandSchroeder[14],arrangetohavean
afternoonfreeofinterruptions,andworkthroughthederivation.
7
0.5
July 2009
α
(Q)
s
Deep Inelastic Scattering
0.4 e+e– A nnihilation
Heavy Quarkonia
0.3
0.2
0.1
QCD α s (Μ Z ) = 0.1184 ± 0.0007
1 10 100
Q [GeV]
Fig.4:TheQCDcouplingasmeasuredinphysicsprocessesatdifferentscalesQ,togetherwiththebandobtained
byrunningtheworldaverageforαswithinitsuncertainties.FiguretakenfromRef.[17].
at hadron colliders, the best available complete calculation (i.e., all diagrams at a given order), doesn’t
go beyond α2 or α3. Certain subsets of diagrams (e.g., those without loops) can be calculated up α10
s s s
roughly.
Sowearefacedwithaproblem. Exactlatticemethodscan’tdealwiththehighmomentumscales
that matter, exact perturbative methods can’t deal with low momentum scales that inevitably enter the
problem, northe high multiplicities that events have inpractice. Yet,itturns outthat weare reasonably
successfulinmakingpredictionsforcolliderevents. Theselectureswilltrytogiveyouanunderstanding
ofthemethodsandapproximations thatareused.
2 Considering e+e− → hadrons
One simple context in which QCD has been extensively studied over the past 30 years is that of e+e
−
annihilation to hadrons. This process has the theoretical advantage that only the final state involves
QCD.Additionally, hugequantities ofdatahavebeencollected atquiteanumberofcolliders, including
millions of events at the Z mass at LEPand SLC.We therefore start our investigation of the properties
ofQCDbyconsidering thisprocess.
2.1 Softandcollinearlimits
There is one QCDapproximation that wewill repeatedly make use of, and that is the soft and collinear
approximation. ‘Soft’impliesthatanemittedgluonhasverylittleenergycomparedtotheparton(quark
or gluon) that emitted it. ‘Collinear’ means that it is emitted very close in angle to another parton in
theevent. Byconsidering gluons that aresoftand/or collinear onecan drastically simplify certain QCD
calculations, whilestillretaining muchofthephysics.
Thesoftandcollinearapproximationissufficientlyimportantthatit’sworth,atleastonce,carrying
out a calculation with it, and we’ll do that in the context of the emission of a gluon from e+e qq¯
−
→
events. Though there are quite a few equations in the page that follows, the manipulations are all quite
simple! We’re interested in the hadronic side of the e+e qq¯amplitude, so let’s first write the QED
−
→
matrix element for a virtual photon γ qq¯(we can always put back the e+e γ and the photon
∗ − ∗
→ →
8
propagator partslaterifweneedto—whichwewon’t):
p
1
ie γµ
= u¯ (p )ie γ δ v (p ) ,
qq¯ a 1 q µ ab b 2
M
p
2
where the diagram illustrates the momentum labelling. Here u¯(p ) and v(p ) are the spinors for the
1 2
outgoingquarkandanti-quark(takenmassless),e isthequark’selectricchargeandtheγ aretheDirac
q µ
matrices. In what follows we shall drop the a,b quark colour indices for compactness and reintroduce
themonlyattheend.
Thecorrespondingamplitudeincludingtheemissionofagluonwithmomentumkandpolarization
vectorǫis
p p
1 1
= ie γµ k,ε + ie γµ (12a)
Mqq¯g k,ε
p p
2 2
i(p/ +/k) i(p/ +/k)
= u¯(p )ig ǫ/tA 1 ie γ v(p )+u¯(p )ie γ 2 ig ǫ/tAv(p ), (12b)
− 1 s (p +k)2 q µ 2 1 q µ(p +k)2 s 2
1 2
with one term for emission from the quark and the other for emission from the anti-quark and use of
the notation p/ = p γ . Let’s concentrate on the first term, collecting the factors of i, and using the
µ µ
anti-commutation relation oftheγ-matrices, A/B/ = 2A.B B/A/,towrite
−
(p/ +/k) [2ǫ.(p +k) (p/ +/k)ǫ/]
iu¯(p )g ǫ/tA 1 e γ v(p ) = ig u¯(p ) 1 − 1 e γ tAv(p ), (13a)
1 s (p +k)2 q µ 2 s 1 (p +k)2 q µ 2
1 1
p .ǫ
ig 1 u¯(p )e γ tAv(p ), (13b)
s 1 q µ 2
≃ p .k
1
where to obtain the second line we have made use of the fact that u¯(p )p/ = 0, p2 = k2 = 0, and
1 1 1
taken the soft approximation k p , which allows us to neglect the terms in the numerator that are
µ µ
≪
proportional tok ratherthanp. Theanswerincluding bothtermsinEq.(12)is
p .ǫ p .ǫ
u¯(p )ie γ tAv(p ) g 1 2 , (14)
qq¯g 1 q µ 2 s
M ≃ · p .k − p .k
(cid:18) 1 2 (cid:19)
where the first factor has the Lorentz structure of the amplitude, i.e., apart from the colour matrix
qq¯
M
tA, is simply proportional to the result. We actually need the squared amplitude, summed
qq¯ qq¯
M M
overpolarizations andcolourstates,
2
p .ǫ p .ǫ
2 u¯ (p )ie γ tAv (p )g 1 2
qq¯g a 1 q µ b 2 s
|M | ≃ p .k − p .k
A,a,b,pol(cid:12) (cid:18) 1 2 (cid:19)(cid:12)
X (cid:12) (cid:12)
(cid:12) (cid:12) 2
(cid:12) = M2 C g2 p1 p2(cid:12) = M2 C g2 2p1.p2 . (15)
−| qq¯| F s p .k − p .k | qq¯| F s(p .k)(p .k)
(cid:18) 1 2 (cid:19) 1 2
We have now explicitly written the quark colour indices a,b again. To obtain the second line we
have made use of the result that tAtA = C N [cf. Eq. (8b)], whereas for M2 we have
A,a,b ab ba F C | qq¯|
δ tA = N . To carry out the sum over gluon polarizations we have exploited the fact that
A,a,b ab ba C P
ǫ (k)ǫ (k) = g , plus terms proportional to k and k that disappear when dotted with the
Ppol µ ∗ν − µν µ ν
amplitudeanditscomplexconjugate.
P
One main point of the result here is that in the soft limit, the 2 squared matrix element
qq¯g
|M |
factorizesintotwoterms: the 2matrixelementandapiecewitharathersimpledependence onthe
qq¯
|M |
gluonmomentum.
9
The next ingredient that we need is the the phase space for the qq¯g system, dΦ . In the soft
qq¯g
approximation, wecanwritethis
d3~k
dΦ dΦ , (16)
qq¯g ≃ qq¯2E(2π)3
where E E is the energy of the gluon k. We see that the phase space also factorizes. Thus we can
k
≡
write the full differential cross section for qq¯production plus soft gluon emission as the qq¯production
matrixelementandphasespace, 2dΦ ,multiplied byasoftgluonemissionprobability d ,
qq¯ qq¯
|M | S
2dΦ 2dΦ d , (17)
qq¯g qq¯g qq¯ qq¯
|M | ≃ |M | S
with
dφ 2α C 2p .p
s F 1 2
d = EdEdcosθ , (18)
S 2π · π (2p .k)(2p .k)
1 2
wherewehaveusedd3k = E2dEdcosθdφ,expressingtheresultintermsofthepolar(θ)andazimuthal
(φ) angles of the gluon with respect to the quark (which itself is back-to-back with the antiquark, since
we work in the centre-of-mass frame and there is negligible recoil from the soft gluon). With a little
morealgebra, wegetourfinalresultfortheprobability ofsoftgluonemissionfromtheqq¯system
2α C dE dθ dφ
s F
d = . (19)
S π E sinθ 2π
Thisresulthastwotypesofnon-integrable divergence: one,calledthesoft(orinfrared)divergencewhen
E 0andthe other, acollinear divergence, whenθ 0(orπ), i.e., thegluon becomes collinear with
→ →
thequark(orantiquark)direction. Thoughderivedhereinthespecificcontextofe+e qq¯production,
−
→
thesesoftandcollinear divergences areaverygeneral property ofQCDandappear wheneveragluonis
emittedfromaquark,regardless oftheprocess.
2.2 Thetotalcrosssection
Ifwewanttocalculatethe (α )correctionstothetotalcrosssection,thediagramsincludedinEq.(12)
s
O
arenotsufficient. Wealsoneedtoinclude aone-loop correction (‘virtual’), specifically, theinterference
between one-loop γ qq¯diagrams and the tree-level γ qq¯amplitude, for example acontribution
∗ ∗
→ →
suchas
−ie γµ ieγµ
x
whichhasthesameperturbative order(numberofg factors) asthesquareofEq.(12).
s
The total cross section for the production of hadrons must be finite. The integral over the gluon-
emission correction has two non-integrable, logarithmic divergences. These divergences musttherefore
somehow be cancelled by corresponding divergences in the virtual term. This is the requirement of
unitarity, whichisbasically thestatement thatprobability ofanything happening mustaddupto1. The
most straightforward way of doing the full calculation for the total cross section is to use dimensional
regularization in the phase space integral for the real emission diagram and for the integration over the
loop momentum in the virtual diagram. However, in order just to visualize what is happening one can
alsowrite
2α C dE dθ E 2α C dE dθ E
s F s F
σ = σ 1+ R ,θ V ,θ , (20)
tot qq¯
π E sinθ Q − π E sinθ Q
(cid:18) Z Z (cid:18) (cid:19) Z Z (cid:18) (cid:19)(cid:19)
where the first term, 1, is the ‘Born’ term, i.e., the production of just qq¯, the second term is the real
emission term, and the third term is the loop correction. Since we need to integrate gluon emission
10