Table Of ContentNNLO real corrections to gluon scattering
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JoaoPires∗,E.W.N.Glover
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2 InstituteforParticlePhysicsPhenomenology
DepartmentofPhysics
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UniversityofDurham
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p DH13LE
- UK
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e E-mail: [email protected],[email protected]
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In this talk we describe a procedure for isolating the infrared singularities present in gluonic
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v scatteringamplitudesatnext-to-next-to-leadingorder.Weusetheantennasubtractionframework
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which has been successfully applied to the calculation of NNLO corrections to the 3-jet cross
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7 sectionandrelatedeventshapedistributionsinelectron-positronannihilation. Hereweconsider
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processeswithcolouredparticlesintheinitialstate,andinparticulartwo-jetproductionathadron
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colliderssuchastheLargeHadronCollider(LHC).Weconstructasubtractiontermthatdescribes
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0 the single and doubleunresolvedcontributionsfrom gluonicprocesses using antenna functions
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withinitialstate partonsandshownumericallythatthe subtractiontermcorrectlyapproximates
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thematrixelementsinthevarioussingleanddoubleunresolvedconfigurations.
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RADCOR2009-9thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumField
TheorytoPhenomenology)
October25-302009
Ascona,Switzerland
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NNLOrealcorrectionstogluonscattering JoaoPires
1. Introduction
In hadronic collisions, the most basic form of the strong interaction at short distances is the
scattering ofacolouredpartonoffanothercolouredparton. Experimentally, suchscatteringcanbe
observed via the production of one or more jets of hadrons with with large transverse energy. In
QCD,thescattering crosssection hastheperturbative expansion,
ds =(cid:229) dsˆLO+ a s dsˆNLO+ a s 2dsˆNNLO+O(a 3) f(x )f (x )dx dx (1.1)
Z (cid:20) ij 2p ij 2p ij s (cid:21) i 1 j 2 1 2
i,j (cid:16) (cid:17) (cid:16) (cid:17)
wherethesumrunsoverthepossible parton typesiand j. Thesingle-jet inclusive anddi-jetcross
sections have been studied at next-to-leading order (NLO) [1] and successfully compared with
datafromtheTEVATRON.
The theoretical prediction may be improved by including the next-to-next-to-leading order
(NNLO) perturbative predictions. This has the effect of (a) reducing the renormalisation scale
dependence and (b) improving the matching of the parton level theoretical jet algorithm with the
hadronlevelexperimentaljetalgorithmbecausethejetstructurecanbemodeledbythepresenceof
athirdparton. Theresultingtheoretical uncertainty atNNLOisestimatedtobeatthefewper-cent
level[2].
Inthis talk, wewillfocus only on the NNLOcontribution involving gluons and will drop the
partonlabels. AtNNLO,therearethreedistinctcontributionsduetodoublerealradiationradiation
ds R , mixed real-virtual radiation ds V,1 and double virtual radiation ds V,2 , that are given
NNLO NNLO NNLO
by
dsˆ = dsˆR + dsˆV,1 + dsˆV,2 (1.2)
NNLO ZdF m+2 NNLO ZdF m+1 NNLO ZdF m NNLO
where the integration is over the appropriate N-particle final state subject to the constraint that
precisely m-jetsareobserved,
= dF J(N). (1.3)
N m
ZdF N Z
As usual the individual contributions in the (m+2), (m+3) and (m+4)-parton channels are all
separately infrared divergent although, after renormalisation and factorisation, their sum is finite.
Forprocesses withtwopartons inthe initial state, the parton level cross sections arerelated tothe
interference ofM-particlei-loopand j-loopamplitudes [hM(i)|M(j)i] by
M
dsˆR ∼ hM(0)|M(0)i
NNLO
h im+4
dsˆV,1 ∼ hM(0)|M(1)i+hM(1)|M(0)i
NNLO
h im+3
dsˆV,2 ∼ hM(1)|M(1)i+hM(0)|M(2)i+hM(2)|M(0)i (1.4)
NNLO
h im+2
Inthis talk, wespecialise tothe gluonic contributions todijet production. Explicit expressions for
theinterferenceofthefour-gluontree-levelandtwo-loopamplitudesisavailableinRefs.[3],while
the self interference of the four-gluon one-loop amplitude is given in [4]. The one-loop helicity
amplitudesforthefivegluonamplitudearegivenin[5]. Thiscontributioncontainsexplicitinfrared
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NNLOrealcorrectionstogluonscattering JoaoPires
divergences coming from integrating over the loop momenta and implicit poles in the regions of
the phase space where one of the final state partons becomes unresolved. This corresponds to the
soft and collinear regions of the one-loop amplitude that were analyzed in [6]. The double real
six-gluon matrix elements were derived in [7]. Here the singularities occur in the phase space
regionscorresponding totwogluonsbecomingsimultaneously softand/orcollinear. The“double”
unresolved behaviour isuniversalandwasdiscussed in[8,9].
2. The antenna subtraction formalism
There have been several approaches to build ageneral subtraction scheme for the double real
contribution at NNLO [10, 11, 12]. We will follow the antenna subtraction method which was
derived in[13]for NNLOprocesses involving only (massless) finalstate partons. Thisformalism
isbeingextendedtoincludeprocesses witheitheroneortwoinitialstatepartons [14,15,16].
Torenderthecontributions withdifferent finalstatesseparately finite,thegeneralstructure of
thesubtraction termsatNNLOis
dsˆ = dsˆR −dsˆS + dsˆS
NNLO NNLO NNLO NNLO
ZdF m+2(cid:0) (cid:1) ZdF m+2
+ dsˆV,1 −dsˆVS,1 + dsˆVS,1
ZdF m+1(cid:16) NNLO NNLO(cid:17) ZdF m+1 NNLO
+ dsˆV,2 , (2.1)
ZdF m NNLO
wheredsˆS (dsˆVS,1 )isthesubtractiontermforthedoubleradiation(real-virtual)contributions
NNLO NNLO
respectively.
Inthistalk,weconcentrate onthedoubleunresolvedsubtraction termdsˆS relevantforthe
NNLO
six-gluon contribution two-jet production inhadronic collisions. Itismade upof several different
contributions, thatdependonhowtheunresolved partonsareconnected incolourspace,
dsˆS =dsˆS,a +dsˆS,b +dsˆS,c +dsˆS,d +dsˆA
NNLO NNLO NNLO NNLO NNLO NNLO
(a) Oneunresolved parton buttheexperimental observable selects only mjetsfrom the(m+1)
partons.
(b) Twocolour-connected unresolved partons(colour-connected).
(c) Two unresolved partons that are not colour connected but share a common radiator (almost
colour-unconnected).
(d) Twounresolved partons thatare wellseparated from each other inthe colour chain (colour-
unconnected).
(A) Correction fortheoversubtraction oflargeanglesoftradiation.
Contribution (a) is precisely the same subtraction term as used for the NLO (m+1)-jet rate,
and is the product of a three-parton antenna and reduced (m+1)-particle matrix elements. Con-
tributions (b)–(d) are derived from the product of double unresolved factors and reduced (m+2)-
parton matrixelements. Subtraction termsfortheseconfigurations canbeconstructed usingeither
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NNLOrealcorrectionstogluonscattering JoaoPires
6000
double soft limit for ggfigggg xx==1100--45
5000 #PS points=10000 x=10-6
i 1487 outside the plot x=(s-sij)/s
317 outside the plot
4000 59 outside the plot
l # events 3000
1 2 2000
k 1000
0
j 0.99997 0.99998 0.99999 1 1.00001 1.00002 1.00003
R
(a) (b)
Figure1:(a)Exampleconfigurationofadoublesofteventwiths ≈s =s. (b)DistributionofRfor10000
ij 12
doublesoftphasespacepoints.
single four-parton antenna functions [17] or products of three-parton antenna functions. For glu-
onicprocesses, oneencountersthefour-gluonantennaF0forthefirsttime. Finally,thelarge-angle
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soft subtraction terms(A)contains soft antenna functions which precisely cancel the remnant soft
behaviour associated with the antenna phase space mappings for the final-final [18], initial-final
and initial-initial configurations. Explicit formulae for the various contributions to dsˆS are
NNLO
available in[19]
Note that the subtraction terms are also needed in integrated form. When both radiators are
in thefinal state, asneeded for electron-positron annihilation, theintegrated antennae aregiven in
ref.[13]. Forprocesseswithonehardradiatorintheinitialstate,theintegralsareknown[15]while
theworkisstillinprogress forprocesses withtwohadronic initialradiators [16].
3. Numerical results
Tonumericallytestthatthesubtraction termcorrectlyreproduces thesameinfraredbehaviour
as the matrix element, wegenerate a series of phase space points that approach a given double or
singleunresolved limit. Foreachgenerated pointwecomputetheratio,
dsˆR
R= NNLO (3.1)
dsˆS
NNLO
whichshould approachunityaswegetclosertoanysingularity.
Asanexample,weconsiderthedoublesoftlimit. Adoublesoftconfigurationcanbeobtained
by generating a four particle final state where one of the invariant masses s of two final state
ij
particles takesnearlythefullenergyoftheeventsasillustrated infigure1(a).
Infigure1(b)wegenerated 10000randomdoublesoftphasespacepointsandshowthedistri-
bution of the ratio between the matrix element and the subtraction term. We show three different
valuesofx=(s−s )/s[x=10−4(red),x=10−5(green),x=10−6 (blue)]andwecanseethatfor
ij
smallervaluesofxthedistribution peaksmoresharply aroundunity. Forx=10−6 weobtained an
average of R=0.9999994 and a standard deviation of s =4.02×10−5. Also in the plot wegive
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NNLOrealcorrectionstogluonscattering JoaoPires
5000
4500 triple collinear limit for ggfigggg xx==--1100--89
x=-10-10
4000 #PS points=10000
i 3500 1107170 o ouutstisdide et hthe ep plolott x=s1jk/s
3000 54 outside the plot
j # events 22050000
1 2
k 1500
1000
500
0
l 0.995 0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 1.004 1.005
R
(a) (b)
Figure2:(a)Exampleconfigurationofatriplecollineareventwiths →0.(b)DistributionofRfor10000
1jk
triplecollinearphasespacepoints.
the number of outlier points that lie outside the range of the histogram. As expected this number
decreases asweapproach thesingular region.
As a second example, we perform a similar analysis for the triple collinear limit with three
hard particles sharing a collinear direction as shown in figure 2 (a). In this case the variable that
controls the approach to the triple collinear region is x=s /s. We show results for x=−10−7
1jk
(red), x=−10−8 (green), x=−10−9 (blue). For 10000 phase space points with x=−10−9, we
obtained an average value of R=0.99954 and a standard deviation of s =0.04. As before, the
numberofoutliers systematically decreases asweapproach thetriplecollinear limit.
Similarbehaviour isobtained foralloftheremainingdoubleunresolved limits.
4. Conclusions
In this contribution, we have discussed the application of the antenna subtraction formalism
toconstruct thesubtraction termrelevantforthegluonic double realradiation contribution todijet
production. The subtraction term is constructed using four-parton and three-parton antennae. We
showedthatthesubtractiontermcorrectlydescribesthedoubleunresolvedlimitsofthegg→gggg
process. The construction of similar subtraction terms for processes involving quarks should in
principle be straightforward. Together with the integrated forms of the antenna functions (see
Refs. [15]for the initial-final and Ref. [16] for the initial-initial configurations), these double real
subtraction terms willprovide amajor step towards the NNLOevaluation ofthe dijet observables
athadroncolliders.
5. Acknowledgements
ThisresearchwassupportedinpartbytheUKScienceandTechnologyFacilitiesCounciland
by the European Commission’s Marie-Curie Research Training Network under contract MRTN-
CT-2006-035505 ‘Tools and Precision Calculations for Physics Discoveries at Colliders’. EWNG
gratefully acknowledges the support of the Wolfson Foundation and the Royal Society. JP grate-
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NNLOrealcorrectionstogluonscattering JoaoPires
fully acknowledges the award of a Fundação para a Ciência e Tecnologia (FCT - Portugal) PhD
studentship.
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