SUSY-QCD corrections to MSSM Higgs boson 0 production via gluon fusion 1 0 2 n a J 9 1 MargareteMühlleitner ∗ InstituteforTheoreticalPhysics,ITP(KIT) ] h E-mail: [email protected] p - HeidiRzehak p InstituteforTheoreticalPhysics,ITP(KIT) e h E-mail: [email protected] [ MichaelSpira 1 v PaulScherrerInstitute,PSI(VilligenPSI) 4 E-mail: [email protected] 1 2 3 In the MSSM scalar h,H productionis mediatedby heavyquarkand squarkloops. Thehigher . 1 order QCD correctionshave been obtainedsome time ago and turned out to be large. The full 0 0 SUSY QCD correctionshave been obained recently including the full mass dependenceof the 1 loopparticles. We describeourcalculationandpresentfirstnumericalresults. We also address : v thequestionofthepropertreatmentofthelargegluinomasslimit,i.e. theconsistentdecoupling i X ofheavygluinoeffects,andpresenttheeffectiveLagrangianfordecoupledgluinos. r a RADCOR2009-9thInternationalSymposiumonRadiativeCorrections(ApplicationsofQuantumField TheorytoPhenomenology), October25-302009 Ascona,Switzerland Speaker. ∗ (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ SUSY-QCDcorrectionstoMSSMHiggsbosonproductionviagluonfusion MargareteMühlleitner 1. Introduction One of the major goals at the LHC is the production of Higgs boson(s) [1]. In the Minimal SupersymmetricExtensionoftheStandardModel(MSSM)twocomplexHiggsdoubletsareintro- duced to give masses to up- and down-type fermions [2]. After electroweak symmetry breaking there are five physcial Higgs states, two CP-even neutral Higgs bosons h,H, one neutral CP-odd Higgs state A and two charged Higgs bosons H . At tree level, the Higgs sector can be param- ± eterized by two independent parameters, the pseudoscalar Higgs boson mass M and the ratio of A thetwovacuumexpectation values(VEV)ofthetwocomplexHiggsdoublets, tanb =v /v . The 2 1 Higgscouplings toquarksandgaugebosons aremodifiedwithsinandcosofthemixinganglesa andb withrespecttotheStandardModel(SM)couplings, wherea denotestheh,H mixingangle. Thebottom(top)Yukawacouplingsareenhanced(suppressed)forlargevaluesoftanb ,sothattop Yukawacouplings playadominant roleatsmallandmoderatevaluesoftanb . AttheLHCandTevatronneutralHiggsbosonsarecopiouslyproducedviagluonfusiongg → h,H,A, which is mediated in the case of h,H by (s)top and (s)bottom loops [3]. The pure QCD corrections to the (s)quark loops have been obtained including the full Higgs and (s)quark mass dependences and increase the cross sections by 100% [4]. This result can be approximated ∼ byveryheavy top(s)quarks with 20 30% accuracy fortanb <5[5]. Inthislimitthenext-to- ∼ − ∼ leadingorder(NLO)QCD[6]andlaterthenext-to-next-to-leadingorder(NNLO)QCDcorrections [7] have been obtained, the latter leading to a moderate increase of 20-30%. Finite top mass effects at NNLO have been discussed in [8]. Finally, the estimate of the next-to-next-to-next-to- leading order effects [9] indicates improved perturbative convergence. The full supersymmetric (SUSY)QCDcorrections havebeenobtainedinthelimitofheavySUSYparticle masses[10]and more recently including the full mass dependence [11]. The electroweak loop effects have been calculated in [12]. In this article we will describe in Section 2 the calculation of the full SUSY- QCD corrections in gluon fusion to h,H, and we will present for the first time numerical results for the total cross section. In Section 3 we will discuss the consistent derivation of the effective Lagrangian forthescalarHiggscouplings togluonsafterthegluinodecoupling. 2. GluonFusion Atleadingorder(LO)thegluonfusionprocessesgg h/H aremediatedbyheavyquarkand → squark triangleloops, cf. Fig.1,thelattercontributing significantly forsquarkmasses <400GeV. ∼ The LO cross section in the narrow-width approximation can be obtained from the h/H gluonic decaywidths,[3,13] dLgg s (pp h/H) = s h/Ht (2.1) LO → 0 h/Hdt h/H p 2 s h/H = G (h/H gg) 0 8M3 LO → h/H 2 s h/H = GFa s2(m R) (cid:229) gh/HAh/H(t )+(cid:229) gh/HAh/H(t ) , (2.2) 0 288√2p (cid:12) Q Q Q Q Q Q (cid:12) (cid:12)Q Q (cid:12) (cid:12) (cid:12) (cid:12) e e e (cid:12) (cid:12) e (cid:12) (cid:12) (cid:12) 2 SUSY-QCDcorrectionstoMSSMHiggsbosonproductionviagluonfusion MargareteMühlleitner ˜ Q g Q g Q˜ g h,H h,H h,H g g g Figure1: Diagramscontributingtogg h,H atleadingorder. → where t =M2 /s with s being the squared hadronic c.m. energy and t =4m2 /M2 . h/H h/H Q/Q˜ Q/Q˜ h/H TheLOformfactorsaregivenby 3 Ah/H(t ) = t [1+(1 t )f(t )] Q 2 − 3 Ah/H(t ) = t [1 t f(t )] (2.3) Q˜ −4 − 1 arcsin2 t 1 √t ≥ f(t ) =  1 1+√1 t 2 . −4 log1 √1−t −ip t <1 (cid:20) − − (cid:21)  Andthegluonluminosity atthefactorization scale m isdefinedas F dLgg 1dx = g(x,m 2)g(t /x,m 2), dt t x F F Z where g(x,m 2) denotes the gluon parton density of the proton. The NLOSUSY-QCDcorrections F consist of the virtual two-loop corrections, cf. Fig.2, and the real corrections due to the radiation processesgg gh/H,gq qh/H andqq¯ gh/H,cf. Fig.3. Thefinalresultforthetotalhadronic → → → g g g t˜,˜b t˜,˜b g h/H t˜,˜b g˜ h/H g h/H g g g Figure2: SomegenericdiagramsforthevirtualNLOSUSY-QCDcorrectionstothesquarkcontributions tothegluonicHiggscouplings. crosssections canbesplitaccordingly intofiveparts, a dLgg s (pp h/H+X)=s h/H 1+Ch/H s t +D s h/H+D s h/H+D s h/H. (2.4) → 0 p h/Hdt gg gq qq¯ h/H h i The strong coupling constant is renormalized in the MS scheme, with the top quark and squark contributionsdecoupledfromthescaledependence. Thequarkandsquarkmassesarerenormalized on-shell. The parton densities are defined in the MS scheme with five active flavors, i.e. the top quarkandthesquarksarenotincludedinthefactorizationscaledependence. Afterrenormalization we are left with collinear divergences in the sum of the virtual and real corrections which are 3 SUSY-QCDcorrectionstoMSSMHiggsbosonproductionviagluonfusion MargareteMühlleitner absorbed intherenormalization ofthepartondensity functions, sothattheresultEq.(2.4)isfinite and depends on the renormalization and factorization scales m and m , respectively. The natural R F scalechoicesturnouttobem =m M . Thenumericalresultsarepresentedforthemodified R F h/H ∼ q g g q q g t˜,˜b t˜,˜b t˜,˜b h,H h,H g g q¯ h,H Figure 3: Typical diagrams for the real NLO QCD corrections to the squark contributions to the gluon fusionprocesses. smalla scenario[14],definedbythefollowingchoicesofMSSMparameters[m =172.6GeV], eff t M = 800GeV tanb = 30 Q˜ M = 1000GeV m = 2TeV (2.5) g˜ M = 500GeV A =A = 1.133TeV. 2 b t − Inthisscenario thesquark massesamountto m = 679GeV m = 935GeV t˜1 t˜2 (2.6) m = 601GeV m = 961GeV. b˜1 b˜2 Fig. 4displaysthegenuineSUSYQCDcorrectionsnormalizedtotheLObottomquarkformfactor, 140 C b (gg → H) Preliminary 120 SUSY small a eff 100 tgb = 30 real part 80 imaginary part 60 D approximation b 40 20 0 -20 -40 100 150 200 250 300 350 400 450 500 M [GeV] H Figure 4: The genuine SUSY QCD corrections normalized to the LO bottom quark form factor. Real corrections:red(lightgray),virtualcorrections:blue(darkgray),comparedtotheD approximation(dashed b lines). A hasbeenrenormalizedintheMSscheme. b i.e. Ah/H(t ) Ah/H(t )(1+Cb a s). The corrections can be sizeable, but can be described b b → b b SUSY p reasonably withtheusualD approximation [15],ifA isrenormalized intheMSscheme. b b 3. Decoupling oftheGluinos In this section we will address the limit of heavy quark, squark and gluino masses, where in addition the gluinos are much heavier than the quarks and squarks. For the derivation of the 4 SUSY-QCDcorrectionstoMSSMHiggsbosonproductionviagluonfusion MargareteMühlleitner effective Lagrangian for the scalar Higgs couplings to gluons weanalyze the relation between the quarkYukawacouplingl andtheHiggscouplingtosquarksl inthelimitoflargegluinomasses. Q Q˜ Wedefinethesecouplings atleading orderinthecaseofvanishing mixing, m m2 v l =gH Q , l =2gH Q =kl 2 , withk =2 , (3.1) Q Q v Q˜ Q v Q gH Q H where g denotes the normalization factor ofthe MSSMHiggs couplings to quark pairs withre- Q specttotheSM.Inthefollowingwewillsketchhowthemodifiedrelationbetweenthesecouplings forscalesbelowthegluino massM isderived. Fordetails, seeRef.[16]. Westartwiththeunbro- g˜ kenrelationbetweentherunningMScouplingsofEq.(3.1)andthecorresponding renormalization group equations (RGE)forscales above M . Ifthescales decrease below M thegluino decouples g˜ g˜ from the RGEs leading to modified RGEs which are different for the two couplings l and kl 2 Q˜ Q so that the two couplings deviate for scales below M . The proper matching at the gluino mass g˜ scale yields a finite threshold contribution for the evolution from the gluino mass scale to smaller scales,whilethelogarithmicstructureofthematchingrelationisgivenbythesolutionoftheRGEs below M . Inorder todecouple consistently thegluino from theRGEforgluino massscales large g˜ comparedtothechosenrenormalizationscale,amomentumsubstractionofthegluinocontribution for vanishing momentum transfer hastobe performed [17]. Werefer thereader to[16]for details and giveheredirectly theresult forthe modifiedrelation between thequark Yukawacoupling and theeffectiveHiggscoupling tosquarks takingintoaccountthepropergluinodecoupling: 2gH m2Q =l¯ (m ) 1+C a s logMg2˜ +3logm2Q˜ +1 , (3.2) Q v Q˜,MO Q˜ F p m2 2 m2 2 ( Q˜ Q !) wherem isthepolemassandMOdenotesthemomentumsubstracted coupling,whichistakenat Q the squark massscale, which is the proper scale choice of the effective Higgs coupling tosquarks andwhichisrelevantforanadditional largegapbetweenthequarkandsquarkmasses. Takingintoaccounttheradiativecorrectionstotherelationbetweentheeffectivecouplingsaf- terdecoupling thegluinosleadstothefollowingeffectiveLagrangianinthelimitofheavysquarks andquarks, H Leff = 1a2sp Gamn Gamn Hv (cid:229) gHQ 1+141ap s +(cid:229) g4Q˜ 1+CSQCDap s +O(a s2) , (3.3) (Q (cid:20) (cid:21) Q˜ h i ) wheregH =vl¯ (m )/m2. ThecofficientC isgivenby Q˜ Q˜,MO Q˜ Q˜ SQCD 37 C = . (3.4) SQCD 6 Itiswell-definedinthelimitoflargegluinomassesandthusfulfillstheconstraintoftheAppelquist– Carazzone decoupling theorem[18]. 4. Conclusions We have presented first results for the NLO SUSY QCD corrections to gluon fusion into CP-even MSSM Higgs bosons, including the full mass dependence of the loop particles. The 5 SUSY-QCDcorrectionstoMSSMHiggsbosonproductionviagluonfusion MargareteMühlleitner genuine SUSY-QCD corrections can be sizeable. We furthermore demonstrated, that the gluino contributions canbedecoupled inthelargeM limitinaccordance withtheAppelquist-Carazzone g˜ theorem. 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