Next-to-leading order multi-leg processes for the Large Hadron Collider

0.13 MB·English

by  T. Binoth

Next-to-leading order multi-leg processes for the
Large Hadron Collider
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ThomasBinoth∗, ThomasReiter†

n
SchoolofPhysics,TheUniversityofEdinburgh,EdinburghEH93JZ,UK.
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J E-mail: [email protected],[email protected]

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1 JeppeR. Andersen
CERN,CH-1211Geneva,Switzerland.
]
h E-mail: [email protected]
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- GudrunHeinrich,JenniferM. Smillie
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e IPPP,UniversityofDurham,DurhamDH13LE,UK.
h E-mail: [email protected],[email protected]
[

Jean-PhilippeGuillet,GregorySanguinetti
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v LAPTH,9,ChemindeBellevueBP110,74941AnnecyleVieux,France.
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E-mail: [email protected],[email protected]
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Stefan Karg
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. InstituteforTheoreticalPhysicsE,RWTHAachen,D-52056Aachen,Germany.
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0 E-mail: [email protected]
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0 NikolasKauer
: InstituteforTheoreticalPhysics,UniversityofWürzburg,D-97074Würzburg,Germany.
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i E-mail: [email protected]
X
r
a In this talk we discuss recent progress concerning precise predictions for the LHC. We give a
statusreportofthreeapplicationsofourmethodtodealwithmulti-legone-loopamplitudes:The
interferenceterm ofHiggsproductionbygluon-andweak bosonfusionto orderO(a 2a 3) and
s
thenext-to-leadingordercorrectionstothetwoprocesses pp→ZZjetanduu¯→dd¯ss¯. Thelatter
isasubprocessofthefourjetcrosssectionattheLHC.
8thInternationalSymposiumonRadiativeCorrections(RADCOR)
October1-52007
Florence,Italy
∗Speaker.
†ThisworkwassupportedbytheBritishScienceandTechnologyFacilitiesCouncil(STFC),theScottishUniversi-
tiesPhysicsAlliance(SUPA)andtheDeutscheForschungsgemeinschaft(DFG,BI-1050/1).
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
NLOprocessesforLHC ThomasBinoth
1. Introduction
The Large Hadron Collider (LHC) at CERN will probe our understanding of electroweak
symmetry breaking and explore physics in the TeV region. A detailed theoretical knowledge of
various kinds ofStandard Model backgrounds isindispensable forthese studies. Especially inthe
startup-phase whenitisnecessary tocalibrate detectors usingmulti-jetsignatures, preferably with
identifiable leptons, Standard Model processes will play an important role. Precise predictions
forsuchmulti-partonic crosssections areonlypossible byincluding higher ordercorrections such
that renormalisation andfactorisation scale dependencies aretamed. While corrections atnext-to-
leadingorderinthestrongcouplingconstanta areknowntobasicallyallrelevant2→2processes,
s
the situation for 2→N processes whith N ≥3 is less satisfactory, although tremendous progress
hasbeenmadeinthelastfewyears. For2→3processes,variousmethodshavebeenusedrecently
to obtain NLOQCD predictions for multi-boson production pp→ZZZ,WWZ,HHH [1, 2, 3, 4],
processes in the context of weak boson fusion, like pp→WW jj,WZjj [5, 6], pp→ Hjj with
effectivegluon-Higgs couplings [7],gg→Hqq¯[8],and pp→tt¯j[9].
Manydifferenttechniquesareusedfortheevaluationofmulti-particleprocesses,asaresultof
recent developments whichweretriggered bytheobservation thatthestandard Passarino-Veltman
reductioningeneraldoesnotleadtonumericallystableamplituderepresentations forthereduction
offive-pointintegrals. Apartfrom Feynmandiagrammatic approaches whichapplynewreduction
techniques for some or all scalar and tensor integrals [10, 11, 12] the evaluation of one loop am-
plitudes byextracting thecoefficientsofacertain basissetofscalarintegrals usingunitarity based
methods[13]bothinalgebraic [14,15,16]andnumerical[17,18,19]variantshasseensubstantial
progressrecently. PurelynumericalapproachesbasedonFeynmandiagrams,whichdonotuseany
reduction to“basisintegrals” arealsoviable[1,20,21,22].
In [11] we have proposed a framework for the evaluation of one-loop multi-leg amplitudes,
based on reduction formulas in Feynman parameter space [23, 24] in the context of dimensional
regularisation. The reduction of rank R N-point to rank R-1 (N-1)-point integrals can be obtained
algebraically forN >5. ForN ≤5weprovide form factor representations whichareexpressed in
termsof3-and4-pointscalarintegralswithFeynmanparametersinthenumerator. Thedimension-
alityoftheboxfunctionsissuchthatallIRdivergences,i.e. polesin1/(n−4),areisolatedintothe
triangle functions. Forall IRdivergent triangle functions explicit representations can beobtained.
WehavecodedallformulasuptoN=6formasslessinternalkinematicsintoaFORTRAN90code,
called golem90 [32]. The code allows to switch between a semi-numerical and completely nu-
mericalevaluationofthebasisfunctions. Thelatterispreferableinexceptionalphasespaceregions
where certain basis integrals canbecome linearly dependent. Inthiswaytheproblems ofinstabil-
ities due to inverse Gram determinants can be tamed. Optionally, our reduction formalism also
allowsareductiontoascalarintegralbasisof1-,2-,3-pointfunctionsinndimensionsand4-point
functionsinn+2dimensions,denotedbyIn,In,In In+2. Thisfullyalgebraicreductionistobeused
1 2 3 4
awayfromexceptional phasespacepoints, whereitisfastandreliable.
In the following we will discuss three applications of our method relevant for LHC phe-
nomenology.
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NLOprocessesforLHC ThomasBinoth
b] b]
a150 a150
[jj [jj
f f
D/d100 D/d100
sd sd
50 50
0 u-u- 0
u-u+
-50 uu--dd-+ -50
Sum
-100 -100 d-u-
d-u+
d-d-
-150 -150 d-d+
Sum
-3 -2 -1 0 1 2 D3f -3 -2 -1 0 1 2 D3f
jj jj
Figure 1: The D f -distributionfor various flavour and helicity-configurations. The purple histogram la-
jj
belled “Sum” indicates the sum over the four contributionsshown. The sum over all flavour and helicity
assignmentsincludingallseaflavoursisshownintheblackhistogram.WBFcutshavebeenused[27].
2. Interference term for pp→Hjj atorder O(a 2a 3)
s
Weak boson fusion is one of the most promising discovery channels for the Higgs boson. As
such it deserves a careful consideration of higher order effects. Very recently, the electroweak
corrections, including a recalculation of QCD corrections have been evaluated in [25]. A Born
levelinterference termbetween thegluon fusionandweakboson fusion isonlyallowedbycolour
conservationifthein-andoutgoingquarksarecrossedinthet↔u-channel,whichiskinematically
disfavoured. Such interference terms are included in the calculations of [26, 25]. However, the
exchange of an extra gluon between the quark lines opens up a viable colour channel. We have
obtained a fully analytic result of this one-loop interference term between gluon fusion and weak
bosonfusion[27],andimplemented theevaluationinaflexibleC++MonteCarloprogramme.
Figure 1 displays the contribution to the distribution in D f from the interference terms for
jj
various helicityandflavourconfigurations foraHiggsbosonmassof115GeV.
NotethattheintegraloftheabsolutevalueoftheD f distribution,
jj
p ds
dD f | | ,
Z−p jj dD f jj
is a useful measure of the impact of the interference effect on the extraction of the ZZH-vertex.
This integral evaluates to9.1 ab. Thetotal integral overthe absolute value of thefully differential
cross section leads to 29.6 ab. We conclude that the interference term can be safely neglected in
phenomenological studies, butthisneededtobechecked bydoingtheexplicitcalculation.
3. The virtual O(a ) corrections to pp→ZZj
s
During the Les Houches 2005 workshop the process pp→VV j was identified as one of the
most important missing NLOcalculations [28]. Theprocess with acharged vector boson pair has
beenevaluatedveryrecentlybytwoindependent groups[29,30]. TheevaluationforZZplusjetis
stillmissing. Theprocessiscomposed ofthreepartonicreactions
qq¯→ZZg, gq→ZZq, gq¯→ZZq¯, (3.1)
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NLOprocessesforLHC ThomasBinoth
16 24
LO 2 LO 2
14 LO+virt 1.6 LO+virt 1.6
1.2 20 1.2
GeV] 1120 00..084 dσdLσOL+Ovirt GeV] 16 00..084 dσdLσOL+Ovirt
b/ 200 300 400 b/ 0 50 100
[f 8 [f 12
Z Z
dσ/dMZ 64 dσ/dpT, 8
4
2
0 0
150 200 250 300 350 400 0 20 40 60 80 100
MZZ[GeV] pT,Z[GeV]
Figure2: ThefinitevirtualNLOcontributiontothehelicitycomponent−−+++ofthepartonicprocess
qq¯→ZZg. TheinvariantmassoftheZpairisshownontheleftandthe p distributionontheright.Weuse
T
thecut p >20GeVandapartonandbeampipeseparationcutofq >1.50.
T,jet ij
Withourmethods wehaveobtained thevirtual order O(a )corrections forallhelicity amplitudes
s
of both processes. Using spinor helicity methods we have obtained analytical formulas for the
coefficients of all basis scalar integrals. We work in dimensional regularisation and treat g by
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applying the ’t Hooft-Veltman scheme. As an illustration we show the contribution of the virtual
correction to some typical distributions. Only the contributions which are related to finite basis
integrals areplotted. Forthefullresulttherealemissioncorrections remaintobeincluded [31].
4. The amplitude uu¯→dd¯ss¯
Not a single NLO 2 → 4 process relevant for LHC phenomenology has been evaluated so
far. As a test case for our reduction methods we have evaluated the 6-photon amplitude [15] and
have compared our result with an evaluation using a fully numerical approach [21] and unitarity
based methods[18,16]. Thesameset-upcanbeusedtoattack nowprocesses like pp→ jjjj and
pp→bb¯bb¯ which are of relevance for background studies at the LHC. As an example we show
the result of one colour factor of the finite virtual NLO contribution to the uu¯→dd¯ss¯amplitude
in massless QCD. The calculation has been carried out using spinor helicity amplitudes in the
’t Hooft-Veltman scheme. We have chosen a convenient colour basis, which allows to split the
amplitude asfollows
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(cid:229) (cid:229) CiAl (p ,...,p ), (4.1)
i 1 6
l i=1
l
whereA arethehelicityandcoloursub-amplitudes. Inparticular wechosethecolourstructures
c
~C=(d c2d c3d c5,d c2d c5d c3,d c5d c2d c3,d c5d c3d c2,d c3d c5d c2,d c3d c2d c5). (4.2)
c1 c4 c6 c1 c4 c6 c1 c4 c6 c1 c4 c6 c1 c4 c6 c1 c4 c6
Inournotationl isthevector(l ,...,l ),andl =±1isthehelicityoftheparticlewithmomen-
1 6 j
tum p ofwhichthecolourindexisc . Inthesix-quarkamplitudeonecanidentifytwoindependent
j j
helicity configurations, l a=(+,+,+,+,+,+)andl b=(+,+,+,+,−,−).
We reduced the tensor integrals to form factors as outlined above (for more details see [11]),
and deal with the spinor algebra by completing spinor lines to traces. The expressions for the
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NLOprocessesforLHC ThomasBinoth
0.8
la
lb
0.7
0.6
0.5
-3s
aM| 0.4
|
0.3
0.2
0.1
00 p /2 p 3p /2 2p
q
Figure 3: Six-quark amplitude. The finite parts of the two helicity configurations s|Al a|a −3 (solid) and
1 s
s|Al b|a −3 (dashed)areplottedforanarbitrarykinematicalpointwherethefinalstate momentahavebeen
1 s
rotatedaboutthey-axisbyanangleq .
diagrams are transformed into a Fortran90 program. The golem90 library is used for the
numerical evaluation oftheformfactors. Thecodereturnsthesub-amplitudes intheform
g6 1 A B
Al (p ,...,p )= s + +C+O(e ) (4.3)
i 1 6 4p 2 s (cid:18)e 2 e (cid:19)
for each of the six colour structures and forall non-zero helicities, where A, BandC are complex
coefficients. AsanexampleinFig.3weplottheamplitudes|Al |a −3forthecolourstructurec=1
c s
and the two helicity configurations l a and l b. The initial state momenta have been fixed to be
alignedwiththez-axiswhilethefinalstatemomentahavebeenrotatedaboutthey-axisbyanangle
q . Forq =0themomentaarechosenasinRef.[21]. Inthechosenunitstherenormalisation scale
is m =1. The amplitude has been evaluated at 50 successive points between q =0 and q =2p ,
whichtook2.4seconds perpointandhelicity onanIntelPentium4CPU(3.2GHz).
5. Conclusion
The LHC demands next-to-leading order precision for many multi-particle processes. We
havepresentedinthistalksomerecentresultsusingourone-loopreductionmethods,rangingfrom
completely massless amplitudes to ones with several mass scales. The progress which has been
madeinthelastyearsinthecontextofprecisionphenomenology bymanygroupsisgoodnewsfor
theLHC.Manyrelevant predictions beyond leading order areorwillbecomeavailable inthenear
future and there is a clear movement towards largely automated NLO tools as well as combining
NLOamplitudes withpartonshowers.
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