QCD [large proceedings presentations]

115 Pages·2000·1.86 MB·English

by  S. Alekhin

QCD
Convenors: S.Catani,M.Dittmar,D.Soper,W.J.Stirling, S.Tapprogge.
Contributingauthors: S.Alekhin, P.Aurenche, C.Bala´zs,R.D.Ball,G.Battistoni, E.L.Berger,
T.Binoth,R.Brock,D.Casey,G.Corcella,V.DelDuca,A.DelFabbro,A.DeRoeckC.Ewerz,
D.deFlorian,M.Fontannaz, S.Frixione,W.T.Giele,M.Grazzini, J.P.Guillet,G.Heinrich, J.Huston,
J.Kalk,A.L.Kataev,K.Kato,S.Keller,M.Klasen,D.A.Kosower,A.Kulesza, Z.Kunszt,A.Kupco,
V.A.Ilyin,L.Magnea,M.L.Mangano,A.D.Martin,K.Mazumdar,Ph.Mine´,M.Moretti,
W.L.vanNeerven,G.Parente,D.Perret-Gallix, E.Pilon,A.E.Pukhov,I.Puljak,J.Pumplin,
0
0 E.Richter-Was,R.G.Roberts,G.P.Salam,M.H.Seymour, N.Skachkov, A.V.Sidorov, H.Stenzel,
0 D.Stump,R.S.Thorne,D.Treleani,W.K.Tung,A.Vogt,B.R.Webber,M.Werlen,S.Zmouchko.
2

y
a
Abstract
M
We discuss issues of QCD at the LHC including parton distributions, Monte

3 Carlo event generators, the available next-to-leading order calculations, re-
summation, photon production, smallxphysics, doubleparton scattering, and

1 backgrounds toHiggsproduction.
v

5
1. INTRODUCTION
2
0
It is well known that precision QCD calculations and their experimental tests at a proton–proton col-
5
0 lider are inherently difficult. “Unfortunately”, essentially all physics aspects of the LHC, from particle
0 searches beyond theStandard Model(SM)toelectroweak precision measurements andstudies ofheavy
0
quarksareconnectedtotheinteractionsofquarksandgluonsatlargetransferredmomentum. Anoptimal
/
h exploitation of the LHCis thus unimaginable without the solid understanding of many aspects of QCD
p
andtheirimplementation inaccurate MonteCarloprograms.
-
p
This review on QCD aspects relevant for the LHC gives an overview of today’s knowledge, of
e
h ongoingtheoreticaleffortsandofsomeexperimentalfeasibilitystudiesfortheLHC.Moreaspectsrelated
:
v to the experimental feasibility and an overview of possible measurements, classified according to final
i state properties, can be found in Chapter 15 of Ref. [1]. It was impossible, within the time-scale of
X
this Workshop, to provide accurate and quantitative answers to all the needs for LHC measurements.
r
a Moreover, owing to the foreseen theoretical and experimental progress, detailed quantitative studies of
QCDwillhavenecessarilytobeupdatedjustbeforethestartoftheLHCexperimentalprogram. Theaim
ofthisreviewistoupdateRef.[2]andtoprovidereferenceworkfortheactivitiesrequiredinpreparation
oftheLHCprogram inthecomingyears.
Especially relevant for essentially all possible measurements at the LHCand their theoretical in-
terpretation is the knowledge of the parton (quark, anti-quark and gluon) distribution functions (pdf’s),
discussed in Sect. 2. Today’s knowledge about quark and anti-quark distribution functions comes from
lepton-hadron deep-inelastic scattering (DIS) experiments and from Drell-Yan (DY) lepton-pair pro-
duction in hadron collisions. Most information about the gluon distribution function is extracted from
hadron–hadron interactions withphotons inthefinalstate. Thetheoretical interpretation ofalargenum-
ber ofexperiments has resulted in various sets ofpdf’s which are thebasis for cross section predictions
at the LHC. Although these pdf’s are widely used for LHC simulations, their uncertainties are difficult
toestimateandvariousquantitative methodsarebeingdeveloped now(seeSects. 2.1 2.4).
−
The accuracy of this traditional approach to describe proton–proton interactions islimited by the
possible knowledge of the proton–proton luminosity at the LHC. Alternatively, much more precise in-
formationmighteventuallybeobtainedfromanapproachwhichconsiderstheLHCdirectlyasaparton–
parton collider at large transferred momentum. Following this approach, the experimentally cleanest
and theoretically best understood reactions would be used tonormalize directly the LHCparton–parton
luminosities to estimate various other reactions. Today’s feasibility studies indicate that this approach
might eventually lead to cross section accuracies, due to experimental uncertainties, of about 1%.
±
Suchaccuraciesrequirethatinordertoprofit,thecorresponding theoreticaluncertaintieshavetobecon-
trolledatasimilarlevelusingperturbative calculations andthecorresponding MonteCarlosimulations.
As examples, the one-jet inclusive cross section and the rapidity dependence of W and Z production
are known at next-to-leading order, implying atheoretical accuracy of about 10 %. Toimprove further,
higherordercorrections havetobecalculated.
Section3addressestheimplementation ofQCDcalculations inMonteCarloprograms, whichare
an essential tool in the preparation of physics data analyses. Monte Carlo programs are composed of
several building blocks, related to various stages in the interaction: the hard scattering, the production
of additional parton radiation and the hadronization. Progress is being made in the improvement and
extensionofmatrixelementgeneratorsandinthepredictionforthetransversemomentumdistributionin
bosonproduction. Besidestheissuesofpartondistributions andhadronization, anothernon-perturbative
piece in a Monte Carlo generator is the treatment of the minimum bias and underlying events. One of
theimportant issuediscussed inthesection onMonteCarlogenerators istheconsistent matching ofthe
variousbuildingblocks. MoredetailedstudiesonMonteCarlogeneratorsfortheLHCwillbeperformed
inaforeseentopicalworkshop.
The status of higher order calculations and prospects for further improvements are presented in
Sect. 4. Asmentioned earlier, one of the essential ingredients for improving the accuracy of theoretical
predictions istheavailability ofhigherordercorrections. Foralmostallprocessesofinterest, containing
a (partially) hadronic final state, the next-to-leading order (NLO) corrections have been computed and
allow to make reliable estimates of production cross sections. However, to obtain an accurate estimate
of the uncertainty, the calculation of the next-to-next-to-leading order (NNLO) corrections is needed.
These calculations are extremely challenging and once performed, they will have to be matched with a
corresponding increaseinaccuracyintheevolution ofthepdf’s.
Section5discussesthesummationsoflogarithmicallyenhancedcontributions inperturbationthe-
ory. Examples of such contributions occur in the inclusive production of a final-state system which
carries a large fraction of the available center-of-mass energy (“threshold resummation”) or in case of
theproductionofasystemwithhighmassatsmalltransversemomentum(“p resummation”). Incaseof
T
threshold resummations, thetheoretical calculations formostprocesses ofinterest havebeen performed
atnext-to-leading logarithmic accuracy. Their importance istwo-fold: firstly, the cross sections atLHC
mightbedirectlyaffected;secondly,theextractionofpdf’sfromotherreactionsmightbeinfluencedand
thus the cross sections at LHC are modified indirectly. For transverse momentum resummations, two
analytical methodsarediscussed.
The production of prompt photons (as discussed in Sect. 6) can be used to put constraints on the
gluondensityintheprotonandpossiblytoobtainmeasurementsofthestrongcouplingconstantatLHC.
The definition of a photon usually involves some isolation criteria (against hadrons produced close in
phasespace). Thisrequirementistheoretically desirable, asitreducesthedependence ofobservables on
thefragmentationcontributiontophotonproduction. Atthesametime,itisusefulfromtheexperimental
point of view as the background due to jets faking a photon signature can be further reduced. A new
schemeforisolation isabletoeliminatethefragmentation contribution.
InSect. 7the issue ofQCDdynamics inthe region ofsmall xisdiscussed. Forsemi-hard strong
interactions, whicharecharacterized bytwolarge,differentscales,thecrosssectionscontainlargeloga-
rithms. Theresummation of these atleading logarithmic (LL)accuracy can beperformed by theBFKL
equation. AvailableexperimentaldataarehowevernotdescribedbytheLLBFKL,indicatingthepresent
of large sub-leading contributions and the need to include next-to-leading corrections. Studies of QCD
dynamicsinthisregimecanbemadenotonlybyusinginclusiveobservables, butalsothroughthestudy
offinalstateproperties. Theseincludetheproduction ofdi-jetsatlargerapidityseparation (studyingthe
azimuthaldecorrelationbetweenthetwojets)ortheproductionofmini-jets(studyingtheirmultiplicity).
An important topic at the LHC is multiple (especially double) parton scattering (described in
Sect. 8), i.e. the simultaneous occurrence of two independent hard scattering in the same interaction.
Extrapolations to LHCenergies, based on measurements at theTevatron show the importance oftaking
thisprocess intoaccount whensmalltransverse momentaareinvolved. Manifestations ofdouble parton
scattering are expected in the production of four jet final states and in the production of a lepton in
association withtwob-quarks (wherethelatterisusedasafinalstateforHiggssearches).
The last section (Sect. 9) addresses the issue of the present knowledge of background for Higgs
searches, for final states containing two photons or multi-leptons. For the case of di-photon final states
(used for Higgs searches with 90 < m < 140 GeV), studies of the irreducible background are per-
H
formed by calculating the (single and double) fragmentation contributions to NLO accuracy and by
studying the effects ofsoftgluon emission. Theproduction ofrarefivelepton finalstates could provide
valuable information ontheHiggscouplings form > 200GeV,awaitingfurther studiesonimproving
H
theunderstanding ofthebackgrounds.
During the workshop, no studies of diffractive scattering at the LHC have been performed. This
topicischallengingbothfromthetheoreticalandtheexperimentalpointofview. Thestudyofdiffractive
processes (withatypical signature ofaleading proton and/or alarge rapidity gap)should leadtoanim-
provedunderstanding ofthetransitionbetweensoftandhardprocessandofthenon-perturbative aspects
ofQCD.Fromtheexperimental pointofview,thedetection ofleading protons intheLHCenvironment
ischallengingandrequiresaddingadditionaldetectorstoATLASandCMS.Ifharddiffractivescattering
(leading proton(s) together with e.g. jets as signature for a hard scattering) is to be studied with decent
statistical accuracy at large p , most of the luminosity delivered under normal running conditions has
T
to be utilized. A few more details can be found in Chapter 15 of Ref. [1], some ideas for detectors in
Ref.[3]. Muchmoreworkremains tobedone, including adetailed assessment ofthecapabilities ofthe
additional detectors.
1.1 OverviewofQCDtools
Allofthe processes to beinvestigated at theLHCinvolve QCDto someextent. It cannot beotherwise,
since the colliding quarks and gluons carry the QCD color charge. One can use perturbation theory to
describe thecrosssectionforaninclusivehard-scattering process,
h (p )+h (p ) H(Q, ... )+X . (1)
1 1 2 2
→ { }
Here the colliding hadrons h and h have momenta p and p , H denotes the triggered hard probe
1 2 1 2
(vector bosons, jets, heavy quarks, Higgs bosons, SUSY particles and so on) and X stands for any
unobserved particles produced by the collision. The typical scale Q of the scattering process is set by
the invariant mass or the transverse momentum of the hard probe and the notation ... stands for any
{ }
other measured kinematic variable oftheprocess. Forexample, thehardprocess maybetheproduction
of a Z boson. Then Q = M and wecan take ... = y, where y is the rapidity of the Z boson. One
Z
{ }
canalso measure the transverse momentum Q ofthetheZ boson. Thenthesimple analysis described
T
below applies if Q M . In the cases Q M and M Q , there are two hard scales in the
T Z T Z Z T
∼ ≪ ≪
processandamorecomplicatedanalysisisneeded. ThecaseQ M isofparticularimportanceand
T Z
≪
isdiscussed inSects.3.3,3.4and5.3.
Thecrosssectionfortheprocess(1)iscomputedbyusingthefactorization formula[4,5]
σ(p ,p ;Q, ... ) = dx dx f (x ,Q2)f (x ,Q2) σˆ (x p ,x p ;Q, ... ;α (Q))
1 2 { } 1 2 a/h1 1 b/h2 2 ab 1 1 2 2 { } S
a,b Z
X
+ ((Λ /Q)p) . (2)
QCD
O
Here the indices a,b denote parton flavors, g,u,u¯,d,d¯,... . The factorization formula (2) involves
{ }
the convolution of the partonic cross section σˆ and the parton distribution functions f (x,Q2) of
ab a/h
the colliding hadrons. The term ((Λ /Q)p) on the right-hand side of Eq. (2) generically denotes
QCD
O
non-perturbative contributions (hadronization effects, multiparton interactions, contributions of the soft
underlying eventandsoon).
Evidently, the pdf’s are of great importance to making predictions for the LHC. These functions
are determined from experiments. Some of the issues relating to this determination are discussed in
Sect. 2. In particular, there are discussions of the question of error analysis in the determination of the
pdf’sandthereisadiscussion oftheprospects fordetermining thepdf’sfromLHCexperiments.
The partonic cross section σˆ is computable as a power series expansion in the QCD coupling
ab
α (Q):
S
σˆ (p ,p ;Q, ... ;α (Q))=αk(Q) σˆ(LO)(p ,p ;Q, ... )
ab 1 2 { } S S ab 1 2 { }
n (NLO)
+α (Q)σˆ (p ,p ;Q, ... )
S ab 1 2 { }
+α2(Q) σˆ(NNLO)(p ,p ;Q, ... )+ . (3)
S ab 1 2 { } ···
o
The lowest (or leading) order (LO) term σˆ(LO) gives only a rough estimate of the cross section. Thus
one needs the next-to-leading order (NLO)term, which is available for most cases of interest. A list of
the available calculations isgiven inSect. 4.1. Cross sections at NNLOare not available at present, but
theprospects arediscussed inSect.4.2.
The simple formula (2) applies when the cross section being measured is “infrared safe.” This
means that the cross section does not change if one high energy strongly interacting light particle in
the final state divides into two particles moving in the same direction or if one such particle emits a
light particle carrying very small momentum. Thus in order to have a simple theoretical formula one
does not typically measure the cross section to find a single high-p pion, say, but rather one measures
T
the cross section to have a collimated jet of particles with a given total transverse momentum p . If,
T
instead, a single high-p pion (or, more generally, a high-p hadron H) is measured, the factorization
T T
formula has to include an additional convolution with the corresponding parton fragmentation function
d (z,Q2). Anexample ofacase whereoneneeds amorecomplicated treatment istheproduction of
a/H
high-p photons. Thiscaseisdiscussed inSect.6.
T
AsanexampleofaNLOcalculation,wedisplayinFig.1thepredictedcrosssectiondσ/dE dyat
T
theLHCfortheinclusiveproduction ofajetwithtransverseenergyE andrapidityyaveragedoverthe
T
rapidityinterval 1 < y < 1. ThecalculationusestheprograminRef.[6]andthepdfsetCTEQ5M[7].
−
As mentioned above, the “jets” must be defined with an infrared safe algorithm. Here we use the k
T
algorithm [8,9]withajoiningparameter R = 1. Thek algorithm hasbettertheoretical properties than
T
theconealgorithm thathasoftenbeenusedinhadroncollider experiments.
In Eq. (2) there are integrations over the parton momentum fractions x and x . The values of
1 2
x and x that dominate the integral are controlled by the kinematics of the hard-scattering process. In
1 2
the case of the production of a heavy particle of mass M and rapidity y, the dominant values of the
momentumfractions arex (Me±y)/√s,wheres = (p +p )2 isthesquare ofthecentre-of-mass
1,2 1 2
∼
energy of the collision. Thus, varying M and y at fixed √s, we are sensitive to partons with different
momentumfractions. Increasing√sthepdf’sareprobedinakinematicrangethatextendstowardslarger
valuesofQandsmallervaluesofx . Thisisillustrated inFig.2. AttheLHC,x canbequitesmall.
1,2 1,2
Thus small x effects that go beyond the simple formula (2) could be important. These are discussed in
Sect.7.
In Fig.3weplot NLOcross sections for aselection of hard processes versus √s. Thecurves for
the lower values of √s are for pp¯collisions, as at the Tevatron, while the curves for the higher values
of √s are for pp collisions, as at the LHC. An approximation (based on an extrapolation of a standard
Regge parametrization) to the total cross section is also displayed. We see that the cross sections for
production of objects with a fixed mass or jets with a fixed transverse energy E rise with √s. This is
T
Fig.1: JetcrosssectionattheLHC,averagedovertherapidityinterval 1< y < 1. ThecrosssectioniscalculatedatNLO
−
usingCTEQ5Mpartonswiththerenormalizationandfactorizationscalessettoµ = µ = E /2. Representativevaluesat
R F T
E =0.5,1,2,3and4TeVare(6.2 103,8.3 101,4.0 10−1,5.1 10−3,5.9 10−5)fb/GeVwithabout3%statistical
T
× × × × ×
errors.
because theimportantx valuesdecrease, asdiscussed above,andtherearemorepartons atsmallerx.
1,2
On the other hand, cross sections for jets with transverse momentum that is a fixed fraction of √s fall
with√s. Thisis(mostly)because thepartonic crosssectionsσˆ fallwithE likeE−2.
T T
The perturbative evaluation of the factorization formula (2) is based on performing power series
expansions in the QCD coupling α (Q). The dependence of α on the scale Q is logarithmic and it is
S S
givenbytherenormalization groupequation [4]
dα (Q)
Q2 S = β(α (Q)) = b α2(Q) b α3(Q)+ , (4)
dQ2 S − 0 S − 1 S ···
wherethefirsttwoperturbative coefficients are
33 2N 153 19N
f f
b = − , b = − , (5)
0 12π 1 24π2
andN isthenumberofflavoursoflightquarks(quarks whosemassismuchsmallerthanthescaleQ).
f
Thethirdandfourth coefficients b andb oftheβ-function arealsoknown[11,12]. Ifweinclude only
2 3
theLOterm,Eq.(4)hastheexactanalytical solution
1
α (Q)= , (6)
S b ln(Q2/Λ2 )
0 QCD
wheretheintegration constant Λ fixestheabsolute sizeoftheQCDcoupling. FromEq.(6)wecan
QCD
seethatachangeofthescaleQbyanarbitraryfactoroforderunity(say,Q Q/2)inducesavariation
→
in α that is of the order of α2. This variation in uncontrollable because it is beyond the accuracy at
S S
which Eq. (6) is valid. Therefore, in LO of perturbation theory the size of α is not unambiguously
S
defined.
The QCD coupling α (Q) can be precisely defined only starting from the NLO in perturbation
S
theory. Tothis order, the renormalization group equation (4) has no exact analytical solution. Different
approximatesolutionscandifferbyhigher-ordercorrectionsandsome(arbitrary)choicehastobemade.
Different choices can eventually be related to the definition of different renormalization schemes. The
LHC parton kinematics
109
x = (M/14 TeV) exp(– y)
1,2
108 Q = M M = 10 TeV
107
106 M = 1 TeV
) 105
2V
e
G
2Q ( 104 M = 100 GeV
103
y = 6 4 2 0 2 4 6
102
M = 10 GeV
fixed
101 HERA
target
100
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
x
Fig.2:ValuesofxandQ2probedintheproductionofanobjectofmassM andrapidityyattheLHC,√s=14TeV.
mostpopularchoice[13]istousetheMS-schemetodefinerenormalizationandthentousethefollowing
approximate solution ofthetwoloopevolution equation todefineΛ :
QCD
b ln[ln(Q2/Λ2 )] ln2[ln(Q2/Λ2 )]
1 1 MS MS
α (Q) = 1 + . (7)
S b0ln(Q2/Λ2 ) " − b0ln(Q2/Λ2 ) O ln2(Q2/Λ2 ) !#
MS MS MS
Herethedefinition ofΛ (Λ = Λ )iscontained inthefact that there isnoterm proportional
QCD QCD MS
to 1/ln2(Q2/Λ2 ). In this expression there are N light quarks. Depending on the value of Q, one
QCD f
maywanttousedifferentvaluesforthenumberofquarksthatareconsideredlight. Thenonemustmatch
between different renormalization schemes, andcorrespondingly change thevalue ofΛ asdiscussed
MS
in Ref. [13]. The constant Λ is the one fundamental constant of QCDthat must be determined from
MS
experiments. Equivalently, experiments can be used to determine the value of α at a fixed reference
S
scale Q = µ . It has become standard to choose µ = M . The most recent determinations of α
0 0 Z S
lead [13] to the world average α (M ) = 0.119 0.002. In present applications to hadron collisions,
S Z
±
the value of α is often varied in the wider range α (M ) = 0.113 0.123 to conservatively estimate
S S Z
−
theoretical uncertainties.
The parton distribution functions f (x,Q2) at any fixed scale Q are not computable in pertur-
a/h
bation theory. However, their scale dependence is perturbatively controlled by the DGLAP evolution
proton - (anti)proton cross sections
109 109
108 s 108
tot
107 107
Tevatron LHC
106 106
105 105
s
b
104 1041
-s
103 103-2m
c
102 s jet(ETjet > (cid:214) s/20) 102330
1
=
s (nb) 111000-011 s jet(ETjet > 100 GessVWZ) 111000-011nts/sec for L
e
v
10-2 10-2e
10-3 s 10-3
t
10-4 s jet(ETjet > (cid:214) s/4) 10-4
10-5 s Higgs(MH = 150 GeV) 10-5
10-6 10-6
s (M = 500 GeV)
Higgs H
10-7 10-7
0.1 1 10
(cid:214) s (TeV)
Fig.3: Crosssectionsforhardscatteringversus√s. Thecrosssectionvaluesat√s = 14TeVare: σ = 99.4mb,σ =
tot b
0.633mb,σ =0.888nb,σ =187nb,σ =55.5nb,σ (M =150GeV)=23.8pb,σ (M =500GeV)=3.82pb,
t W Z H H H H
σ (Ejet > 100 GeV) =1.57 µb,σ (Ejet >√s/20) = 0.133nb,σ (Ejet >√s/4) = 0.10fb. Allexceptthefirstof
jet T jet T jet T
thesearecalculatedusingthelatestMRSTpdf’s[10].
equation [14–17]
df (x,Q2) 1 dz
Q2 a/h = P (α (Q2),z)f (x/z,Q2) . (8)
dQ2 z ab S a/h
b Zx
X
Having determined f (x,Q2) at a given input scale Q = Q , the evolution equation can be used to
a/h 0 0
computethepdf’satdifferentperturbative scalesQandlargervaluesofx.
The kernels P (α ,z) in Eq. (8) are the Altarelli–Parisi (AP) splitting functions. They depend
ab S
on the parton flavours a,b but do not depend on the colliding hadron h and thus they are process-
independent. TheAPsplitting functions canbecomputedasapowerseriesexpansion inα :
S
P (α ,z) = α P(LO)(z)+α2P(NLO)(z)+α3P(NNLO)(z)+ (α4) . (9)
ab S S ab S ab S ab O S
(LO) (NLO)
TheLOandNLOtermsP (z)andP (z)intheexpansion areknown[18–24]. Thesefirsttwo
ab ab
terms (their explicit expressions are collected in Ref. [4]) are used in most of the QCD studies. Partial
(NNLO)
calculations [25,26] of the next-to-next-to-leading order (NNLO) term P (z) are also available
ab
(seeSects.2.5,2.6and4.2).
As in the case of α , the definition and the evolution of the pdf’s depends on how many of the
S
quark flavors are considered to be light in the calculation in which the parton distributions are used.
Again, there are matching conditions that apply. In the currently popular sets of parton distributions
thereisachangeofdefinitionatQ = M,whereM isthemassofaheavyquark.
The factorization on the right-hand side of Eq. (2) in terms of (perturbative) process-dependent
partonic cross sections and (non-perturbative) process-independent pdf’s involves some degree of arbi-
trariness, which is known as factorization-scheme dependence. We can always ‘re-define’ the pdf’s by
multiplying (convoluting) them by some process-independent perturbative function. Thus, we should
always specify the factorization-scheme used to define the pdf’s. The most common scheme is the MS
factorization-scheme [4]. Analternativescheme,knownasDISfactorization-scheme [27],issometimes
used. Ofcourse,physicalquantitiescannotdependonthefactorization scheme. Perturbativecorrections
beyond the LO to partonic cross sections and AP splitting functions are thus factorization-scheme de-
pendent to compensate the corresponding dependence of the pdf’s. In the evaluation of hadronic cross
sections at a given perturbative order, the compensation may not be exact because of the presence of
yetuncalculated higher-order terms. Quantitativestudiesofthefactorization-scheme dependence canbe
usedtosetalowerlimitonthesizeofmissinghigher-order corrections.
Thefactorization-scheme dependence isnottheonlysignal oftheuncertainty related tothecom-
putation of thefactorization formula (2)bytruncating itsperturbative expansion atagiven order. Trun-
cation leads to additional uncertainties and, in particular, to a dependence on the renormalization and
factorization scales. The renormalization scale µ is the scale at which the QCDcoupling α is evalu-
R S
ated. The factorization scale µ is introduced to separate the bound-state effects (which are embodied
F
inthepdf’s)fromtheperturbative interactions (whichareembodied inthepartonic crosssection) ofthe
partons. InEqs.(2)and (3)wetookµ = µ = Q. Onphysical grounds thesescales havetobeofthe
R F
sameorderasQ,buttheirvaluecannotbeunambiguously fixed. Inthegeneral case,theright-hand side
ofEq.(2)ismodifiedbyintroducing explicitdependence onµ ,µ according tothereplacement
R F
f (x ,Q2) f (x ,Q2) σˆ (x p ,x p ;Q, ... ;α (Q))
a/h1 1 a/h2 2 ab 1 1 2 2 { } S
↓
f (x ,µ2) f (x ,µ2) σˆ (x p ,x p ;Q, ... ;µ ,µ ;α (µ )) . (10)
a/h1 1 F a/h2 2 F ab 1 1 2 2 { } R F S R
Thephysicalcrosssectionσ(p ,p ;Q, ... )doesnotdependonthearbitraryscalesµ ,µ ,butparton
1 2 R F
{ }
densities and partonic cross sections separately depend on these scales. The µ ,µ -dependence of the
R F
partonic cross sections appears in their perturbative expansion and compensates the µ dependence of
R
α (µ ) and the µ -dependence of the pdf’s. The compensation would be exact if everything could
S R F
be computed to all orders in perturbation theory. However, when the quantities entering Eq. (10) are
evaluated at,say, then-thperturbative order, theresult exhibits aresidual µ ,µ -dependence, whichis
R F
formally of the (n+1)-th order. That is, the explicit µ ,µ -dependence that still remains reflects the
R F
absence of yet uncalculated higher-order terms. For this reason, the size of the µ ,µ dependence is
R F
oftenusedasameasureofthesizeofatleastsomeoftheuncalculated higher-order termsandthusasan
estimatorofthetheoretical errorcausedbytruncating theperturbative expansion.
As an example, we estimate the theoretical error on the predicted jet cross section in Fig. 1. We
varytherenormalization scaleµ andthefactorization scaleµ . InFig.4,weplot
R F
dσ(µ /E ,µ /E )/dE dy
R T F T T
∆(µ /E ,µ /E ) = h i (11)
R T F T
dσ(0.5,0.5)/dE dy
T
h i
versus E for four values of the pair µ /E ,µ /E , namely 0.25,0.25 , 1.0,0.25 0.25,1.0 ,
T R T F T
{ } { } { } { }
and 1.0,1.0 . We see about a 10% variation in the cross section. This suggests that the theoretical
{ }
uncertainty isatleast10%.
The issue of the scale dependence of the perturbative QCDcalculations has received attention in
theliteratureandvariousrecipeshavebeenproposed tochoose‘optimal’valuesofµ(seethereferences
Fig.4: Variationofthejetcrosssectionwithrenormalizationandfactorizationscale. Weshow∆definedinEq.(11)versus
E forfourchoicesof µ /E ,µ /E .
T R T F T
{ }
in [13]). There is no compelling argument that shows that these ‘optimal’ values reduce the size of
the yet unknown higher-order corrections. These recipes may thus be used to get more confidence on
the central value of the theoretical calculation, but they cannot be used to reduce its theoretical uncer-
tainty asestimated, forinstance, byscale variations around µ Q. Thetheoretical uncertainty ensuing
∼
from the truncation of the perturbative series can only be reduced by actually computing more terms in
perturbation theory.
Wehavesofardiscussed thefactorization formula(2). Weshouldemphasizethatthereisanother
mode of analysis of the theory available, that embodied in Monte Carlo event generator programs. In
this type of analysis, one is limited (at present) to leading order partonic hard scattering cross sections.
However,onesimulatesthecompletephysicalprocess,beginningwiththehardscatteringandproceeding
throughpartonshoweringviarepeatedonepartontotwopartonsplittingsandfinallyendingwithamodel
forhowpartons turnintohadrons. Thisclassofprograms, whichsimulatecomplete eventsaccording to
anapproximation toQCD,arevery important to thedesign andanalysis ofexperiments. Current issues
inMonteCarloeventgenerator andotherrelatedcomputerprogramsarediscussed inSect.3.
2. PARTONDISTRIBUTIONFUNCTIONS1
Parton distributions (pdf’s) play a central role in hard scattering cross sections at the LHC. A precise
knowledge of the pdf’s is absolutely vital for reliable predictions for signal and background cross sec-
tions. In many cases, it is the uncertainty in the input pdf’s that dominates the theoretical error on the
prediction. Such uncertainties can arise both from the starting distributions, obtained from a global fit
to DIS, DY and other data, and from DGLAP evolution to the higher Q2 scales typical of LHC hard
scattering processes.
To predict LHC cross sections we will need accurate pdf’s over a wide range of x and Q2 (see
Fig. 2). Several groups have made significant contributions to the determination of pdf’s both during
and after the workshop. The MRST and CTEQ global analyses have been updated and refined, and
small numerical problems have been corrected. The ‘central’ pdf sets obtained from these global fits
are, not surprisingly, very similar, and remain the best way to estimate central values for LHC cross
sections. Speciallyconstructed variantsofthecentralfits(exploring, forexample,differentvaluesofα
S
ordifferenttheoretical treatmentsofheavyquarkdistributions) allowthesensitivityofthecrosssections
tosomeoftheinputassumptions.
Arigorousandglobaltreatmentofpdfuncertaintiesremainselusive,buttherehasbeensignificant
progressinthelastfewyears,withseveralgroupsintroducingsophisticatedstatisticalanalysesintoquasi-
global fits. Whilesomeofthemorenovelmethods arestillataratherpreliminary stage, itishoped that
overthenextfewyearstheymaybedeveloped intousefultools.
One can reasonably expect that by LHCstart-up time, the precision pdf determinations willhave
improved from NLO to NNLO. Although the complete NNLO splitting functions have not yet been
1Sectioncoordinators: R.Ball,M.DittmarandW.J.Stirling.
d (exp)
F
2
d (feed-down)
~ S xq
d a ,d g
S
d (higher-twist)
low log Q2 high
Fig. 5: Schematicrepresentation of the various uncertainties contributing to theprediction of astructure function or parton
distributionathighQ2.
calculated, severalstudieshavemadeuseofpartialinformation (moments,x 0,1limitingbehaviour)
→
toassesstheimpactoftheNNLOcorrections.
Atthesametime,accuratemeasurements ofStandardModel(SM)crosssections attheLHCwill
further constrain the pdf’s. The kinematic acceptance of the LHC detectors allows a large range of x
and Q2 to be probed. Furthermore, the wide variety of final states and high parton-parton luminosities
available will allow an accurate determination of the gluon density and flavour decomposition of quark
densities.
All of the above issues are discussed in the individual contributions that follow. Lack of space
hasnecessarilyrestrictedtheamountofinformationthatcanbeincluded, butmoredetailscanalwaysbe
foundintheliterature.
2.1 MRS:pdfuncertaintiesandW andZ productionattheLHC2
There are several reasons why it is very difficult to derive overall ‘one sigma’ errors on parton distri-
butions of the form f δf . In the global fit there are complicated correlations between a particular
i i
±
pdf at different xvalues, and between the different pdf flavours. Forexample, the charm distribution is
correlated withthegluondistribution, thegluondistribution atlowxiscorrelated withthegluonathigh
xviathemomentumsumrule,andsoon. Secondly, manyoftheuncertainties intheinputdataorfitting
procedure are not ‘true’ errors in the probabilistic sense. For example, the uncertainty in the high–x
gluon in the MRSTfits [28] derives from a subjective assessment of the impact of ‘intrinsic k ’ on the
T
prompt photon cross sections included in the global fit. Despite these difficulties, several groups have
attempted to extract meaningful δf pdf errors (see [29,30] and Sects. 2.3,2.4). Typically, these anal-
i
±
yses focus on subsets of the available DIS and other data, which are statistically ‘clean’, i.e. free from
undetermined systematic errors. As a result, various aspects of the pdf’s that are phenomenologically
important, theflavourstructure oftheseaandtheseaandgluondistributions atlargexforexample, are
eitheronlyweaklyconstrained ornotdeterminedatall.
Facedwiththedifficultiesintryingtoformulateglobalpdferrors,onecanadoptamorepragmatic
approach to the problem by making a detailed assessment of the pdf uncertainty for a particular cross
section of interest. This involves determining which partons contribute and at which x and Q2 values,
and then systematically tracing back to the data sets that constrained the distributions in the global fit.
Individual pdfsetscanthenbeconstructed toreflecttheuncertainty intheparticular partons determined
byaparticular dataset.
2Contributingauthors:A.D.Martin,R.G.Roberts,W.J.StirlingandR.S.Thorne.