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¯ ¯ NLO QCD corrections to Wbb and Zbb production 8 0 0 2 n a J FernandoFebresCordero 5 UniversityofCalifornia,LosAngeles 1 E-mail: [email protected] ] h LauraReina∗ p FloridaStateUniversity - p E-mail: [email protected] e h DoreenWackeroth [ UniversityatBuffalo-TheStateUniversityofNewYork 1 E-mail: [email protected] v 4 7 WepresentNLOQCDresultsforWbb¯ andZbb¯ productionattheTevatronincludingfullbottom- 3 quark mass effects. We study the impact of QCD corrections on both total cross-section and 2 . invariant mass distribution of the bottom-quark pair. Including NLO QCD corrections greatly 1 0 reduces the dependence of the tree-level cross-section on the renormalizationand factorization 8 scales. We also compareourcalculationto a calculationthatconsidersmassless bottomquarks 0 : andfindthatthebottom-quarkmasseffectsamounttoabout8-10%ofthetotalNLOQCDcross- v i sectionandcanimpacttheshapeofthebottom-quarkpairinvariantmassdistribution,inparticular X inthelowinvariantmassregion. r a 8thInternationalSymposiumonRadiativeCorrections October1-5,2007 Florence,Italy ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina 1. Introduction Precisetheoreticalpredictions fortheproduction ofaweakgaugeboson(W/Z)inassociation with a bb¯ pair are very important both in the search for a light Standard Model (SM)-like Higgs boson andinthesearch forsingle-top production athadron colliders, inparticular attheTevatron. Indeed, W/Z+bb¯ represent the major irreducible backgrounds to the associated production of a lightHiggsbosonwithaweakgaugeboson(WH andZH withH→bb¯),whicharethemainsearch modes for a SM-like Higgs boson at the Tevatron [1, 2]. At the same time,Wbb¯ is an irreducible background forsingle-top production, whichisbeingmeasuredforthefirsttimeattheTevatronas pp¯→tb¯,t¯bwitht(t¯)→b(b¯)W [3,4]. The signal cross sections for both WH/ZH and single-top production are known including higher order QCD and electroweak corrections. It is therefore important to also control the back- ground to a good level of theoretical precision. In the present experimental analyses the effects ofNext-to-Leading (NLO)QCDcorrections onthetotalcross-section andthedijetinvariant mass distribution of the Wbb¯ and Zbb¯ background processes have been taken into account by using the MCFM package [5], which implements the zero bottom-quark mass (m = 0) approxima- b tion[6,7,8]. In this proceedings we present results for the NLO QCDcross-sections and bb¯-pair invariant mass distributions at the Tevatron, including full bottom-quark mass effects [9]. We separately studytheimpactofNLOQCDcorrections andofnon-zerobottom-quark masseffects. NLOQCD corrections stabilize the theoretical prediction of total cross-sections and distributions, reducing the dependence on the renormalization and factorization scales. On the other hand, the presence of a non-zero bottom-quark mass mainly affect the cross-section in the region where the bb¯-pair invariant massissmall,bothatLeadingOrder(LO)andatNLOinQCD,andamounttoanoverall 8-10% differencewithrespecttothezerobottom-quark massapproximation. 2. NLO calculation The hadronic production of a W boson with a bb¯ pair occurs at tree level in QCD via the qq¯′ →Wbb¯ partonic process. On the other hand, the hadronic production of a Z boson with a bb¯ pairconsists, atthetreelevelinQCD,oftwopartonicchannels, namelyqq¯→Zbb¯ andgg→Zbb¯. TheNLOQCDcorrections tothetreelevelpartoniccross-section consistsofbothone-loop O(a ) s virtual corrections and O(a ) real corrections corresponding to the emission of one extra parton s from the tree level parton processes. The NLO hadronic cross-section is obtained by convoluting theparton-level NLOcross-sections withNLOPartonDistribution Functions(PDF). The O(a ) virtual corrections to the partonic cross-section contain ultraviolet (UV) and in- s frared (IR)singularities. TheUVsingularities arecalculated usingdimensional regularization and cancelledbyintroducingasuitablesetofcounterterms(seeRef.[9]fordetails). IRsingularitiesare isolated using dimensional regularization andcancelled against theanalogous singularities arising intheO(a )realcorrections tothepartonic cross-section. s The O(a ) virtual corrections to the partonic cross-section consist of one-loop self-energy, s vertex, box andpentagon diagrams. Weapply techniques developed intheNLOQCDcalculation of Htt¯[10, 11] for the calculation of scalar and tensor loop-integrals. In particular, we calculate 2 NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina the tensor loop-integrals via Passarino-Veltman reduction (PV) [12]. We encounter instabilities duetothequasi-vanishing oftheGramdeterminant(s) oftheprocess onlyinoneboxdiagram and in several pentagon diagrams. We are able to obtain stable numerical results by combining, when necessary, setsofgaugeinvariant diagrams. The O(a ) real corrections to the partonic cross-section consist of the qq¯′ →Wbb¯+g and s qg(q¯g)→Wbb¯+q(q¯) subprocesses for Wbb¯ production and of the qq¯ →Zbb¯+g, gg→Zbb¯+ g, and qg(q¯g)→Zbb¯+q(q¯) subprocesses for Zbb¯ production. We have extracted both soft and collinear IR singularities by implementing a Phase Space Slicing method with two cutoffs [13] in order to isolate the soft (d ) and collinear (d ) singularities respectively. Both the soft, hard- s c collinear, and hard-non-collinear parts of the cross-section depend on the cutoffs, but their sum is cutoffindependent overalargerangeofvaluesofthecutoffs(seeRef.[9]). Both analytical and numerical results for the NLOhadronic cross-section have been checked withtwoindependent calculations basedondifferent programming languages andpublic/in-house packages. The analytical reduction of the calculation has been obtained using FORM [14] and Maple codes, while the numerical results have been obtained using Fortran and C codes. The FF package [15]has been used to check some ofthe IR-finite scalar and tensor integrals. Finally, the hardnon-collinear realcorrections havebeendouble-checked usingMadgraph [16,17,18]. 3. Numerical results Intheseproceedings wepresentresultsforWbb¯ andZbb¯ production attheTevatronincluding NLO QCD corrections and a non-zero bottom-quark mass fixed at m =4.62 GeV. Results for b Wbb¯ have been published inRef. [9], while results forZbb¯ are currently being cross-checked and shouldbeconsideredaspreliminary. BothW andZ bosonareconsideredon-shellandtheirmasses are taken to be M = 80.41 GeV for the Wbb¯ runs and M = 91.1876 GeV for the Zbb¯ runs, W Z while,ineachcase,theotherweakgaugebosonmassiscalculatedviatherelationM =cosq M W w Z with sin2q = 0.223. The LO results use the 1-loop evolution of a and the CTEQ6L set of w s PDF[19],whiletheNLOresultsusethe2-loopevolutionofa andtheCTEQ6MsetofPDF,with s a NLO(M )=0.118. TheW boson coupling to quarks is proportional to the Cabibbo-Kobayashi- s Z Maskawa (CKM) matrix elements. We takeV =V =0.975 andV =V =0.222, while we ud cs us cd neglectthecontributionofthethirdgeneration,sinceitissuppressedeitherbytheinitialstatequark densities orbythecorresponding CKMmatrixelements. We implement the k jet algorithm [20, 21, 22, 23] with a pseudo-cone size R = 0.7 and T we recombine the parton momenta within a jet using the so called covariant E-scheme [21]. We checked that our implementation of the k jet algorithm coincides with the one in MCFM. We T require all events to have a bb¯ jet pair in the final state, with a transverse momentum larger than 15 GeV (pb,b¯ >15 GeV) and a pseudorapidity that satisfies |h b,b¯|<2. We impose the same p T T and|h |cutsalsoontheextrajetthatmayariseduetohardnon-collinear realemissionofaparton, i.e. in the processesW/Zbb¯+gorW/Zbb¯+q(q¯). This hard non-collinear extra parton is treated eitherinclusivelyorexclusively, followingthedefinitionofinclusiveandexclusiveasimplemented in the MCFM code [5]. In the inclusive case we include both two- and three-jet events, while in the exclusive case we require exactly two jets in the event. Two-jet events consist of a bottom- quark jet pair that may also include a final-state light parton (gluon or quark) due to the applied 3 NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina recombinationprocedure. Resultsinthemasslessbottom-quarkapproximationhavebeenobtained usingtheMCFMcode[5]. 5 5 LO LO NLO inclusive NLO inclusive NLO exclusive NLO exclusive 4 4 (pb)total3 (pb)total3 s s 2 2 cuts: p > 15 GeV |h|t < 2 c u t s : p|ht |> < 1 25 GeV 1 R = 0.7 m 0 = Mw/2 + mb 1 R = 0.7 m0 = MZ/2 + mb 0.5 1 2 4 0.5 1 2 4 m /m m /m f 0 f 0 Figure1: DependenceoftheLO(blacksolidband),NLOinclusive(bluedashedband),andNLOexclusive (reddottedband)totalcross-sectionsontherenormalization/factorizationscales,includingfullbottom-quark masseffects. Thel.h.s. plotisfor pp¯→Wbb¯ andther.h.s. plotfor pp¯→Zbb¯ . Thebandsareobtainedby varyingboth m and m independentlybetween m /2and 4m (with m =m +M /2forV =W,Z in the R F 0 0 0 b V pp¯→Wbb¯ and pp¯→Zbb¯ casesrespectively). In Fig. 1 we illustrate the renormalization and factorization scale dependence of the LO and NLO total cross-sections, both in the inclusive and exclusive case. The bands are obtained by varyingbothm andm independently betweenm /2and4m (withm =m +M /2forV =W,Z R F 0 0 0 b V in the pp¯ →Wbb¯ and pp¯ → Zbb¯ cases respectively), including full bottom-quark mass effects. Wenotice thattheNLOcross-sections haveareduced scale dependence overmostoftherangeof scalesshown,andtheexclusive NLOcross-section ismorestablethantheinclusive oneespecially atlowscales. WhiletheLOcross-section stillhasa40%uncertainty duetoscaledependence, this uncertainty is reduced at NLO to about 20% for the inclusive and to about 10% for the exclusive cross-section respectively. This is consistent with the fact that the inclusive NLO cross-section integrates over the entire phase space of the qg(q¯g)→bb¯W/Z+q(q¯) channels that are evaluated withNLOa andNLOPDF,butareactuallytree-levelprocessesandretainthereforeastrongscale s dependence. Intheexclusivecaseonlythe2→3collinearkinematicoftheseprocessesisretained, since3-jetseventsarediscarded,andthismakestheoverallrenormalizationandfactorizationscale dependence milder. This is better illustrated in Fig. 2 for the case of pp¯→Zbb¯, where the r.h.s. plots show the scale dependence of the cross-sections due to the single partonic channels (qq¯, gg and qg+q¯g). The strong residual scale dependence of the inclusive NLO cross-section is clearly drivenbytheqg+q¯gchannel. Similarresultsareobtainedfor pp¯→Wbb¯,asreportedinRef.[9]. A first illustration of the impact of keeping a non-zero bottom-quark mass in the calculation of the NLO QCD cross-section is given in the l.h.s. plots of Fig. 2, where LO and NLO total cross-sections for pp¯ →Zbb¯ are given, both for m =0 and m =4.62 GeV, as functions of the b b renormalizationandfactorizationscales(identifiedforthepurposeofthisplot). Neglectingbottom- quark mass effects overestimate the NLOcross-section by about 8-10%, depending on the choice of the scale. In Fig. 3 we analyze the impact of a non-zero bottom-quark mass on the bb¯-pair 4 NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina NLO massless NLO massive 5 5 _ NLO massive qq initiated 4.5 LO massless gg initiated LO massive 4 qg initiated 4 b) b) (ptotal3.53 (ptotal23 s cuts: p > 15 GeV s Inclusive case 2.5 t |h | < 2 1 2 m = M/2 + m 0 Z b 1.5 R = 0.7 0 0.5 1 2 4 0.5 1 2 4 m /m m /m 0 0 4.5 NLO massless NLO massive NLO massive 4 qq initiated 4 LO massless gg initiated LO massive 3.5 3 qg initiated b) b) p p (total 3 (total2 s 2.5 s Exclusive case cuts: p > 15 GeV t 1 2 |h | < 2 m = M/2 + m 0 Z b R = 0.7 0 1.5 0.5 1 2 4 0.5 1 2 4 m /m m /m 0 0 Figure2: Dependenceofthe LO andNLO inclusive(upperplots) andexclusive (lowerplots)totalcross- section for pp¯→Zbb¯ onthe renormalization/factorizationscale, when m =m . The l.h.s. plotscompare R F bothLOandNLOtotalcross-sectionsforthecaseinwhichthebottomquarkistreatedasmassless(MCFM) ormassive(ourcalculation).Ther.h.s.plotsshowseparately,forthemassivecaseonly,thescaledependence oftheqq¯,ggandqg+q¯gcontributions,aswellastheirsum. invariant-mass (m ) distribution. We give results for both inclusive and exclusive distribution in bb¯ the pp¯→Wbb¯ case. Inbothcasesdistributions mostoftheimpactisinthelowm region,which bb¯ canbeimportantinavarietyofdifferent analyses. References [1] V.M.Abazovetal.,theD0Collaboration[arXiv:0712.0598v1]. [2] A.Abulenciaetal.,theCDFCollaboration,Phys.Rev.Lett.96(2006)081803. [3] V.M.Abazovetal.,theD0CollaborationPhys.Rev.Lett.99(2007)191802 [hep-ex/0702005v1]. [4] D.Acostaetal.,theCDFCollaboration,Phys.Rev.D71(2005)012005,[hep-ex/0410058]. [5] J.CampbellandR.K.Ellis,MCFM-AMonteCarloforFeMtobarnprocessesatHadronColliders, availableat[http://mcfm.fnal.gov]. [6] R.K.EllisandS.Veseli,Phys.Rev.D60(1999)011501(R),[hep-ph/9810489]. [7] J.CampbellandR.K.Ellis,Phys.Rev.D62(2000)114012,[hep-ph/0006304]. [8] J.CampbellandR.K.Ellis,Phys.Rev.D65(2002)113007,[hep-ph/0202176]. 5 NLOQCDcorrectionstoWbb¯ andZbb¯ production LauraReina Figure 3: The inclusive (upperplots) and exclusive (lower plots) distributions, ds /dmbb¯, for pp¯→Wbb¯ derivedfromourcalculation(withm 6=0)andfromMCFM(withm =0). Therighthandsideplotshows b b theratioofthetwodistributions,ds (m 6=0)/ds (m =0). FromRef.[9]. b b [9] F.FebresCordero,L.ReinaandD.Wackeroth,Phys.Rev.D74(2006)034007, [arXiv:hep-ph/0606102]. [10] L.Reina,S.Dawson,andD.Wackeroth,Phys.Rev.D65(2002)053017,[hep-ph/0109066]. [11] S.Dawson,C.Jackson,L.H.Orr,L. Reina,andD.Wackeroth,Phys.Rev.D68(2003)034022, [hep-ph/0305087]. [12] G.PassarinoandM.J.G.Veltman,Phys.Lett.B237(1990)537. [13] B.W.HarrisandJ.F.Owens,Phys.Rev.D65(2002)094032",[hep-ph/0102128]. [14] J.A.M.Vermaseren,“NewfeaturesofFORM"(2000)[math-ph/0010025]. [15] G.J.vanOldenborgh,Comput.Phys.Commun.66(1991)1-15. [16] H.Murayama,I. Watanabe,andK.Hagiwara,“HELAS:HELicityamplitudesubroutinesfor Feynmandiagramevaluations"(1992),KEK-91-11. [17] T.StelzerandW.F.Long,Comput.Phys.Commun.81(1994)357,[hep-ph/9401258]. [18] F.MaltoniandT.Stelzer,JHEP02(2003)27,[hep-ph/0208156]. [19] H.L.Laiandothers,theCTEQcollaboration,Eur.Phys.J.C12(2000)375,[hep-ph/9903282]. [20] S.Catani,Y.L.Dokshitzer,andB.R.Webber,Phys.Lett.B285(1992)291. [21] S.Catani,Y.L.Dokshitzer,M.H.Seymour,andB.R.Webber,Nucl.Phys.B406(1993)187. [22] S.D.EllisandD.E.Soper,Phys.Rev.D48(1993)3160,[hep-ph/9305266]. [23] W.B.KilgoreandW.T.Giele,Phys.Rev.D55(1997)7183,[hep-ph/9610433]. 6

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