Table Of Content0 NNLO QCD correction to vector boson production at
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0 hadron colliders
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n
a
J
2
2
] GiancarloFerrera∗
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p DipartimentodiFisica,UniversitàdiFirenzeandI.N.F.N.SezionediFirenze
- I-50019SestoFiorentino,Florence,Italy
p
e E-mail: [email protected]
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[
We presenta fully-exclusivenext-to-next-to-leadingorder(NNLO) QCD calculation for vector
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v bosonproductioninhadron-hadroncollisions. Thecalculationisimplementedina partonlevel
8 MonteCarloprogram,whichincludesg −Z interference,finite-widtheffects,theleptonicdecay
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9 ofthevectorbosonsandthecorrespondingspincorrelations. Thecodeallowstheusertoapply
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arbitrary(thoughinfraredsafe)kinematicalcutsonthefinal-statesandtocomputedistributions
.
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in the formof bin histograms. We show some illustrativenumericalresultsatthe Tevatronand
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0 theLHC.
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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/
NNLOQCDcorrectiontovectorbosonproductionathadroncolliders GiancarloFerrera
Vectorboson production inhadron collisions, thewellknownDrell-Yan(DY)process[1],has
aspecial roleforphysics studies athadron colliders. Havinglarge production rateswithrelatively
simple experimental signatures, this process is important for detectors calibration, it gives strin-
gent informations on Parton Distribution Functions (PDFs), is a strong test for perturbative QCD
predictions andmaysignals effectsfromphysicsbeyondtheStandardModel.
Itisthereforeessentialtohaveaccuratetheoreticalpredictionsforthevector-bosonproduction
crosssectionsandrelateddistributions,whicharepredictedbyperturbativeQCDasanexpansionin
the strong coupling a . The next-to-next-to-leading order (NNLO)QCDcorrections (i.e. O(a 2))
S S
have been calculated analytically for the total cross section [2] and the rapidity distribution of the
vector boson [3]. The fully exclusive NNLOcalculation has also been performed [4, 5]. Further-
more,electroweakcorrectionsuptoO(a )havebeencomputedforbothW [6]andZproduction[7].
Thecomputation ofhigher-order QCDcorrections tohard-scattering processes isahardtask.
Difficultiesarisefromthepresenceofinfraredsingularitiesatintermediatestagesofthecalculation
thatpreventastraightforward implementationofnumericaltechniques. Fortheabovereason,fully
exclusive cross-sections in hadron collisions have been computed so far only for Higgs boson
production [8,9,10,11]andtheDrell–Yanprocess[4,5].
In this contribution we present a recent fully exclusive NNLO QCD calculation for vector
bosonproductioninhadroncollisions[5]. ThecalculationusestheNNLOextensionofsubtraction
formalism introduced inRef.[10]. Themethodisvalid ingeneral fortheproduction ofcolourless
high-mass systemsinhadroncollisions.
Weconsider thehard-scattering process:
h +h →V(q)+X, (1)
1 2
where the colliding hadrons h and h produce the vector bosonV (V =Z/g ∗,W+ orW−), with
1 2
four-momentum qandhighinvariant mass q2,plusaninclusive finalstateX.
p
FollowingRef.[10],weobservethat,atLO,thetransversemomentumq ofV isexactlyzero.
T
Thismeansthat,aslongasq 6=0,the(N)NLOcontributionsaregivenbythe(N)LOcontributions
T
tothefinalstateV + jet(s)[12]:
dsˆV | = dsˆV+jets . (2)
(N)NLO qT6=0 (N)LO
We compute dsˆV+jets by using the subtraction method at NLO [13, 14] and we treat the remain-
NLO
ing NNLO singularities at q = 0 by the additional subtraction of an universal 1 counter-term
T
dsˆCT constructed by exploiting the universality of the logarithmically-enhanced contributions
(N)LO
tothetransverse momentumdistribution 2. Schematically wehave
dsˆV =HV ⊗dsˆV + dsˆV+jets−dsˆCT , (3)
(N)NLO (N)NLO LO h (N)LO (N)LOi
whereHV isaprocess-dependent coefficientfunctionnecessarytoreproducethecorrectnor-
(N)NLO
malization [16,5].
1Itdependsonlyontheflavouroftheinitial-statepartonsinvolvedintheLOpartonicsubprocess.
2Fortheexplicitformofthecounter-termseeRefs.[10,15].
2
NNLOQCDcorrectiontovectorbosonproductionathadroncolliders GiancarloFerrera
Wehaveencoded ourNNLOcomputation inapartonlevelMonteCarloeventgenerator. The
calculation includes finite-width effects, the g −Z interference, the leptonic decay of the vector
bosonsandthecorresponding spincorrelations. Ournumericalcodeisparticularly suitable forthe
computation ofdistributions intheformofbinhistograms.
Inthe following wepresent someillustrative numerical results for Z andW production atthe
Tevatron and the LHC. We consider u,d,s,c,b quarks in the initial state. In the case ofW± pro-
duction, we use the (unitarity constrained) CKM matrix elements V = 0.97419, V = 0.2257,
ud us
V = 0.00359, V = 0.2256, V = 0.97334, V = 0.0415 from the PDG 2008 [17]. We use
ub cd cs cb
the so called Gm scheme for the electroweak couplings, with the following input parameters:
G = 1.16637×10−5 GeV−2, m = 91.1876 GeV, G = 2.4952 GeV, m = 80.398 GeV and
F Z Z W
G =2.141 GeV. As for the PDFs, we use MSTW2008 [18] as default set, evaluating a at each
W S
corresponding order(i.e., weuse(n+1)-loop a atNnLO,withn=0,1,2). Wefixtherenormal-
S
ization (m )andfactorization (m )scalestothemassofthevectorbosonm .
R F V
Figure1: Rapiditydistributionofanon-shellZ bosonattheLHC. ResultsobtainedwiththeMSTW2008
set(leftpanel)arecomparedwiththoseobtainedwiththeMRST2004set(rightpanel).
Westart the presentation of our results by considering the inclusive production ofe+e− pairs
from the decay of an on-shell Z boson at the LHC. In the left panel of Fig. 1 we show the ra-
pidity distribution of the e+e− pair at LO, NLO and NNLO, computed by using the MSTW2008
PDFs[18]. Thecorrespondingcrosssectionsares =1.761±0.001nb,s =2.030±0.001nb
LO NLO
ands =2.089±0.003nb. Thetotalcrosssectionisincreasedbyabout3%ingoingfromNLO
NNLO
to NNLO. In the right panel of Fig. 1 we show the results obtained by using the MRST2002 LO
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NNLOQCDcorrectiontovectorbosonproductionathadroncolliders GiancarloFerrera
[19] and MRST2004 [20] sets of parton distribution functions. The corresponding cross sections
ares =1.629±0.001nb,s =1.992±0.001nbands =1.954±0.003nb. Inthiscase
LO NLO NNLO
thetotalcrosssectionisdecreased byabout2%ingoingfromNLOtoNNLO.
Figure2: Rapiditydistributionofanon-shellW+ bosonattheTevatron. TheNNLOresultobtainedwith
theMSTW2008setarecomparedwiththoseobtainedwiththeJR09VFandABKM09sets.
Wenext consider the production ofan on-shellW+ boson atthe Tevatron. InFig. 2weshow
the rapidity distribution of theW+ atNNLO,computed by using the MSTW2008PDFs[18]. We
also show, for comparison, the NNLO prediction by using the JR09VF [21] and ABKM09 [22]
PDFs. Thecorresponding totalcrosssectionsares (MSTW)=1.349±0.002nb,s (JRVF)=1.338±
NNLO NNLO
0.002nbands (ABKM)=1.391±0.002 nb. Thedifferences betweenthethreeresultscanbereach,
NNLO
inthecentralrapidityregion, thelevelofabout5%.
We finally consider the production of a charged lepton plus missing p through the decay of
T
aW boson (W =W+,W−) at the Tevatron. The charged lepton is selected to have p >20 GeV
T
and |h |<2 and the missing p of the event is required to be larger than 25 GeV. The transverse
T
mass of the event is defined as m = 2pl pmiss(1−cosf ), where f is the angle between the
T q T T
the p of the lepton and the missing p . In Fig. 3 we show the transverse mass distribution at
T T
LO, NLO and NNLO: the accepted cross sections are s =1.161±0.001 nb, s =1.550±
LO NLO
0.001nbands =1.586±0.002nb. SinceatLOtheW bosonisproducedwithzerotransverse
NNLO
momentum, the requirement pmiss > 25 GeV sets m ≥ 50 GeV. As a consequence at LO the
T T
transverse mass distribution has a kinematical boundary at m =50 GeV. Around this boundary
T
there are perturbative instabilities due to (integrable) logarithmic singularities [23]. We also note
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NNLOQCDcorrectiontovectorbosonproductionathadroncolliders GiancarloFerrera
Figure3: TransversemassdistributionforW productionattheTevatron.
that,belowtheboundary,theNNLOcorrectionstotheNLOresultarelarge. Thisisnotunexpected,
sinceinthisregionoftransversemasses,theO(a )resultcorrespondstothecalculationatthefirst
S
perturbative order and, therefore, our O(a 2) result is actually only a calculation at the NLOlevel
S
ofperturbative accuracy.
We have presented a fully exclusive NNLO QCD calculation for vector boson production in
hadron-hadron collisions. Our calculation is directly implemented in a parton level event gener-
ator. This feature makes it particularly suitable for practical applications to the computation of
distributions intheformofbinhistograms. Forillustrative purpose, wehaveshownsomeselected
numerical distributions attheTevatronandtheLHC.
References
[1] S.D.DrellandT.M.Yan,Phys.Rev.Lett.25(1970)316[Erratum-ibid.25(1970)902].
[2] R.Hamberg,W.L.vanNeervenandT.Matsuura,Nucl.Phys.B359(1991)343[Erratum-ibid.B644
(2002)403];R.V.HarlanderandW.B.Kilgore,Phys.Rev.Lett.88(2002)201801.
[3] C.Anastasiou,L.J.Dixon,K.MelnikovandF.Petriello,Phys.Rev.D69(2004)094008.
[4] K.MelnikovandF.Petriello,Phys.Rev.Lett.96(2006)231803,Phys.Rev.D74(2006)114017.
[5] S.Catani,L.Cieri,G.Ferrera,D.deFlorianandM.Grazzini,Phys.Rev.Lett.103(2009)082001
[arXiv:0903.2120[hep-ph]].
[6] S.DittmaierandM.Kramer,Phys.Rev.D65(2002)073007;U.BaurandD.Wackeroth,Phys.Rev.
D70(2004)073015;V.A.Zykunov,Phys.Atom.Nucl.69(2006)1522[Yad.Fiz.69(2006)1557];
5
NNLOQCDcorrectiontovectorbosonproductionathadroncolliders GiancarloFerrera
A.Arbuzovetal.,Eur.Phys.J.C46(2006)407[Erratum-ibid.C50(2007)505];C.M.Carloni
Calame,G.Montagna,O.NicrosiniandA.Vicini,JHEP0612(2006)016.
[7] U.Baur,O.Brein,W.Hollik,C.SchappacherandD.Wackeroth,Phys.Rev.D65(2002)033007;V.
A.Zykunov,Phys.Rev.D75(2007)073019;C.M.CarloniCalame,G.Montagna,O.Nicrosiniand
A.Vicini,JHEP0710(2007)109;A.Arbuzovetal.,Eur.Phys.J.C54(2008)451.
[8] C.Anastasiou,K.MelnikovandF.Petriello,Phys.Rev.D69(2004)076010.
[9] C.Anastasiou,K.MelnikovandF.Petriello,Phys.Rev.Lett.93(2004)262002,Nucl.Phys.B724
(2005)197;C.Anastasiou,G.DissertoriandF.Stockli,JHEP0709(2007)018.
[10] S.CataniandM.Grazzini,Phys.Rev.Lett.98(2007)222002.
[11] M.Grazzini,JHEP0802(2008)043.
[12] W.T.Giele,E.W.N.GloverandD.A.Kosower,Nucl.Phys.B403(1993)633;J.Campbell,R.K.
Ellis,MCFM-MonteCarloforFeMtobarnprocesses,http://mcfm.fnal.gov.
[13] S.Frixione,Z.KunsztandA.Signer,Nucl.Phys.B467(1996)399;S.Frixione,Nucl.Phys.B507
(1997)295.
[14] S.CataniandM.H.Seymour,Nucl.Phys.B485(1997)291[Erratum-ibid.B510(1997)503].
[15] G.Bozzi,S.Catani,D.deFlorianandM.Grazzini,Nucl.Phys.B737(2006)73,
[16] D.deFlorianandM.Grazzini,Phys.Rev.Lett.85(2000)4678,Nucl.Phys.B616(2001)247.
[17] C.Amsleretal.[ParticleDataGroup],Phys.Lett.B667(2008)1.
[18] A.D.Martin,W.J.Stirling,R.S.ThorneandG.Watt,reportIPPP/08/190[arXiv:0901.0002].
[19] A.D.Martin,R.G.Roberts,W.J.StirlingandR.S.Thorne,Eur.Phys.J.C28(2003)455.
[20] A.D.Martin,R.G.Roberts,W.J.StirlingandR.S.Thorne,Phys.Lett.B604(2004)61.
[21] P.Jimenez-DelgadoandE.Reya,Phys.Rev.D79(2009)074023.
[22] S.Alekhin,J.Blumlein,S.KleinandS.Moch,reportDESY09-102(arXiv:0908.2766[hep-ph]).
[23] S.CataniandB.R.Webber,JHEP9710(1997)005.
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