48.Cross-sectionformulaeforspeci - PDF document

48.Cross-sectionformulaeforspeci¯cprocesses48.CROSS-SECTIONFORMULAEFORSPECIFICPROCESSESRevisedOctober2009byH.Baer(UniversityofOklahoma)andR.N.Cahn(LBNL).PARTI:STANDARDMODELPROCESSESSettingasideleptoproduction(forwhich,seeSec.16ofthisReview),thecrosssectionsofprimaryinterestarethosewithlightincidentparticles,e+e,°°,q q,gq,gg,etc.,wheregandqrepresentgluonsandlightquarks.Theproducedparticlesincludebothlightparticlesandheavyones-t,W,Z,andtheHiggsbosonH.WeprovidetheproductioncrosssectionscalculatedwithintheStandardModelforseveralsuchprocesses.48.1.ResonanceFormationResonantcrosssectionsaregenerallydescribedbytheBreit-Wignerformula(Sec.19ofthisReview).(E)=2J+1 (2S1+1)(2S2+1)4 k22=4 (EE0)2+¡2=4BinBout;(48:1)whereEisthec.m.energy,Jisthespinoftheresonance,andthenumberofpolarizationstatesofthetwoincidentparticlesare2S1+1and2S2+1.Thec.m.momentumintheinitialstateisk,E0isthec.m.energyattheresonance,and¡isthefullwidthathalfmaximumheightoftheresonance.Thebranchingfractionfortheresonanceintotheinitial-statechannelisBinandintothe¯nal-statechannelisBout.Foranarrowresonance,thefactorinsquarebracketsmaybereplacedby(EE0)=2.48.2.ProductionoflightparticlesTheproductionofpoint-like,spin-1/2fermionsine+eannihilationthroughavirtualphoton,e+e!!f f,atc.m.energysquaredsisgivenbyd¾ d=Nc2 4s1+cos2+(12)sin2Q2f;(48:2)whereisv=cfortheproducedfermionsinthec.m.,isthec.m.scatteringangle,andQfisthechargeofthefermion.ThefactorNcis1forchargedleptonsand3forquarks.Intheultrarelativisticlimit,!1,=NcQ2f4¼®2 3s=NcQ2f86:8nb s(GeV2):(48:3)Thecrosssectionfortheannihilationofaq qpairintoadistinctpairq0 q0throughagluoniscompletelyanalogousuptocolorfactors,withthereplacement!s.Treatingallquarksasmassless,averagingoverthecolorsoftheinitialquarksandde¯ningt=ssin2(µ=2),u=scos2(µ=2),one¯nds[1]K.A.Oliveetal.(PDG),Chin.Phys.C38,090001(2014)(http://pdg.lbl.gov)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesd¾ d(q q!q0 q0)=2s 9st2+u2 s2:(48:4)Crossingsymmetrygivesd¾ d(qq0!qq0)=2s 9ss2+u2 t2:(48:5)Ifthequarksqandq0areidentical,wehaved¾ d(q q!q q)=2s 9st2+u2 s2+s2+u2 t22u2 3st;(48:6)andbycrossingd¾ d(qq!qq)=2s 9st2+s2 u2+s2+u2 t22s2 3ut:(48:7)Annihilationofe+einto°°hasthecrosssectiond¾ d(e+e!°°)=2 2su2+t2 tu:(48:8)TherelatedQCDprocessalsohasatriple-gluoncoupling.Thecrosssectionisd¾ d(q q!gg)=82s 27s(t2+u2)1 tu9 4s2!:(48:9)Thecrossedreactionsared¾ d(qg!qg)=2s 9s(s2+u2)(1 su+9 4t2)(48:10)andd¾ d(gg!q q)=2s 24s(t2+u2)(1 tu9 4s2):(48:11)Finally,d¾ d(gg!gg)=92s 8s(3ut s2su t2st u2):(48:12)Lepton-quarkscatteringisanalogous(neglectingZexchange)d¾ d(eq!eq)=2 2se2qs2+u2 t2:(48:13)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesseswhereeqisthechargeofthequark.Forneutrinoscatteringwiththefour-Fermiinteractiond¾ d(ºd!`u)=G2Fs 42;(48:14)wheretheCabibboanglesuppressionisignored.Similarlyd¾ d( u!` d)=G2Fs 42(1+cos)2 4:(48:15)Toobtaintheformulaefordeepinelasticscattering(presentedinmoredetailinSection16)weconsiderquarksoftypeicarryingafractionx=Q2=(2Mº)ofthenucleon'senergy,where=EE0istheenergylostbytheleptoninthenucleonrestframe.Withy=º=Ewehavethecorrespondences1+cos!2(1y);dcm!4¼fi(x)dxdy;(48:16)wherethelatterincorporatesthequarkdistribution,fi(x).Inthiswaywe¯ndd¾ dxdy(eN!eX)=4¼®2xs Q41 2h1+(1y)2ih4 9(u(x)+ u(x)+:::)+1 9(d(x)+ d(x)+:::)i(48:17)wherenows=2MEisthecmenergysquaredfortheelectron-nucleoncollisionandwehavesuppressedcontributionsfromhighermassquarks.Similarly,d¾ dxdy(ºN!`X)=G2Fxs [(d(x)+:::)+(1y)2( u(x)+:::)](48:18)andd¾ dxdy( ºN!`+X)=G2Fxs [( d(x)+:::)+(1y)2(u(x)+:::)]:(48:19)Quasi-elasticneutrinoscattering(n!p, p!+n)isdirectlyrelatedtothecrossedreaction,neutrondecay.Theformulaforthedi®erentialcrosssectionispresented,forexample,inN.J.Bakeretal.,Phys.Rev.D23,2499(1981).August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesses48.3.HadroproductionofheavyquarksForhadroproductionofheavyquarksQ=c;b;t,itisimportanttoincludemasse®ectsintheformulae.Forqq!QQ,onehasd¾ d(qq!QQ)=2s 9s3s 14m2Q sh(m2Qt)2+(m2Qu)2+2m2Qsi;(48:20)whileforgg!QQonehasd¾ d(gg!QQ)=2s 32ss 14m2Q s"6 s2(m2Qt)(m2Qu)m2Q(s4m2Q) 3(m2Qt)(m2Qu)+4 3(m2Qt)(m2Qu)2m2Q(m2Q+t) (m2Qt)2+4 3(m2Qt)(m2Qu)2m2Q(m2Q+u) (m2Qu)23(m2Qt)(m2Qu)+m2Q(ut) s(m2Qt)3(m2Qt)(m2Qu)+m2Q(tu) s(m2Qu)#:(48:21)48.4.ProductionofWeakGaugeBosons48.4.1.WandZresonantproduction:ResonantproductionofasingleWorZisgovernedbythepartialwidths¡(W!`i i)=p 2GFm3W 12(48:22)¡(W!qi qj)=3p 2GFjVijj2m3W 12(48:23)¡(Z!f f)=Ncp 2GFm3Z 6h(T3Qfsin2W)2+(Qfsin2W)2i:(48:24)TheweakmixingangleisW.TheCKMmatrixelementsareindicatedbyVijandNcis3forq q¯nalstatesand1forleptonic¯nalstates.August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesThefulldi®erentialcrosssectionforfi fj!(W;Z)!fi0 fj0isgivenbyd¾ d=Nfc Nic1 2562ss2 (sM2)2+s2h(L2+R2)(L02+R02)(1+cos2)+(L2R2)(L02R02)2cosµi(48:25)whereMisthemassoftheWorZ.ThecouplingsfortheWareL=(8GFm2W=p 2)1=2Vij=p 2;R=0whereVijisthecorrespondingCKMmatrixel-ement,withananalogousexpressionforL0andR0.ForZ,thecouplingsareL=(8GFm2Z=p 2)1=2(T3sin2WQ);R=(8GFm2Z=p 2)1=2sin2WQ,whereT3istheweakisospinoftheinitialleft-handedfermionandQistheinitialfermion'selectriccharge.TheexpressionsforL0andR0areanalogous.ThecolorfactorsNi;fcare3forinitialor¯nalquarksand1forinitialor¯nalleptons.48.4.2.Productionofpairsofweakgaugebosons:Thecrosssectionforf f!W+Wisgivenintermofthecouplingsoftheleft-handedandright-handedfermionf,`=2(T3QxW),r=2QxW,whereT3isthethirdcomponentofweakisospinfortheleft-handedf,Qisitselectriccharge(inunitsoftheprotoncharge),andxW=sin2W:d¾ dt=2¼®2 Ncs2("ÃQ+`+r 4xWs sm2Z!2+`r 4xWs sm2Z!2#A(s;t;u)+1 2xWQ+` 2xWs sm2Z!(£(Q)I(s;t;u)£(Q)I(s;u;t))+1 8x2W(£(Q)E(s;t;u)+£(Q)E(s;u;t)));(48:26)where£(x)is1forx�0and0forx0,andwhereA(s;t;u)=tu m4W1!Ã1 4m2W s+3m4W s2!+s m2W4;I(s;t;u)=tu m4W1!Ã1 4m2W 2sm4W st!+s m2W2+2m2W t;E(s;t;u)=tu m4W1!Ã1 4+m4W t2!+s m2W;(48:27)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesands;t;uaretheusualMandelstamvariableswiths=(pf+p f)2;t=(pfpW)2;u=(pfpW+)2.ThefactorNcis3forquarksand1forleptons.Theanalogouscross-sectionforqi qj!WZ0isd¾ dt=¼®2jVijj2 6s2x2W(Ã1 sm2W!2"98xW 4utm2Wm2Z+(8xW6)sm2W+m2Z#+"utm2Wm2Zs(m2W+m2Z) sm2W#`j t`i u+utm2Wm2Z 4(1xW)"`2j t2+`2i u2#+s(m2W+m2Z) 2(1xW)`i`j tu);(48:28)where`iand`jarethecouplingsoftheleft-handedqiandqjasde¯nedabove.TheCKMmatrixelementbetweenqiandqjisVij.Thecrosssectionforqi qi!Z0Z0isd¾ dt=¼®2 96`4i+r4i x2W(1x2W)2s2"t u+u t+4m2Zs tum4Z1 t2+1 u2#:(48:29)48.5.ProductionofHiggsBosons48.5.1.ResonantProduction:TheHiggsbosonoftheStandardModelcanbeproducedresonantlyinthecollisionsofquarks,leptons,WorZbosons,gluons,orphotons.TheproductioncrosssectionisthuscontrolledbythepartialwidthoftheHiggsbosonintotheentrancechannelanditstotalwidth.ThebranchingfractionsfortheStandardModelHiggsbosonareshowninFig.1ofthe\SearchesforHiggsbosons"reviewintheParticleListingssection,asafunctionoftheHiggsbosonmass.Thepartialwidthsaregivenbytherelations¡(H!f f)=GFm2fmHNc 4p 214m2f=m2H3=2;(48:30)¡(H!W+W)=GFm3HW 32p 244aW+3a2W;(48:31)¡(H!ZZ)=GFm3HZ 64p 244aZ+3a2Z;(48:32)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesseswhereNcis3forquarksand1forleptonsandwhereaW=12W=4m2W=m2HandaZ=12Z=4m2Z=m2H.Thedecaytotwogluonsproceedsthroughquarkloops,withthetquarkdominating[2].Explicitly,¡(H!gg)=2sGFm3H 363p 2XqI(m2q=m2H)2;(48:33)whereI(z)iscomplexforz1=4.Forz2103,jI(z)jissmallsothelightquarkscontributenegligibly.FormH2mt,z.89;ऐ1=4andI(z)=3"2z+2z(14z)sin11 2p z2#;(48:34)whichhasthelimitI(z)!1asz!1.48.5.2.HiggsBosonProductioninWandZdecay:TheStandardModelHiggsbosoncanbeproducedinthedecayofavirtualWorZ(\Higgstrahlung")[3,4]:Inparticular,ifkisthec.m.momentumoftheHiggsboson,(qi qj!WH)=¼®2jVijj2 36sin4W2k p sk2+3m2W (sm2W)2(48:35)(f f!ZH)=2¼®2(`2f+r2f) 48Ncsin4Wcos4W2k p sk2+3m2Z (sm2Z)2;(48:36)where`andrarede¯nedasabove.48.5.3.WandZFusion:Justashigh-energyelectronscanberegardedassourcesofvirtualphotonbeams,atveryhighenergiestheyaresourcesofvirtualWandZbeams.ForHiggsbosonproduction,itisthelongitudinalcomponentsoftheWsandZsthatareimportant[5].ThedistributionoflongitudinalWscarryingafractionyoftheelectron'senergyis[6]f(y)=g2 1621y y;(48:37)whereg=e=sinW.InthelimitsmHmW,thepartialdecayrateis¡(H!WLWL)=(g2=64)(m3H=m2W)andintheequivalentWapproximation[7](e+e! eeH)=1 16m2W sin2W3"Ã1+m2H s!logs m2H2+2m2H s#:(48:38)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesTherearesigni¯cantcorrectionstothisrelationwhenmHisnotlargecomparedtomW[8].FormH=150GeV,theestimateistoohighby51%forp s=1000GeV,32%toohighatp s=2000GeV,and22%toohighatp s=4000GeV.FusionofZZtomakeaHiggsbosoncanbetreatedsimilarly.IdenticalformulaeapplyforHiggsproductioninthecollisionsofquarkswhosechargespermittheemissionofaW+andaW,exceptthatQCDcorrectionsandCKMmatrixelementsarerequired.EvenintheabsenceofQCDcorrections,the¯ne-structureconstantoughttobeevaluatedatthescaleofthecollision,saymW.AllquarkscontributetotheZZfusionprocess.48.6.InclusivehadronicreactionsOne-particleinclusivecrosssectionsEd3¾=d3pfortheproductionofaparticleofmomentumpareconvenientlyexpressedintermsofrapidityy(seeabove)andthemomentumpTtransversetothebeamdirection(inthec.m.):Ed3 d3p=d3 dÁdypTdp2T:(48:39)Inappropriatecircumstances,thecrosssectionmaybedecomposedasapartoniccrosssectionmultipliedbytheprobabilitiesof¯ndingpartonsoftheprescribedmomenta:hadronic=XijZdx1dx2fi(x1)fj(x2)dbpartonic;(48:40)Theprobabilitythatapartonoftypeicarriesafractionoftheincidentparticle'sthatliesbetweenx1andx1+dx1isfi(x1)dx1andsimilarlyforpartonsintheotherincidentparticle.Thepartoniccollisionisspeci¯edbyitsc.m.energysquared^s=x1x2sandthemomentumtransfersquared^t.The¯nalhadronicstateismoreconvenientlyspeci¯edbytherapiditiesy1;y2ofthetwojetsresultingfromthecollisionandthetransversemomentumpT.Theconnectionbetweenthedi®erentialsisdx1dx2d^t=dy1dy2^s sdp2T;(48:41)sothatd3 dy1dy2dp2T=^s sfi(x1)fj(x2)d^ d^t(^s;^t;^u)+fi(x2)fj(x1)d^ d^t(^s;^u;^t);(48:42)wherewehavetakenintoaccountthepossibilitythattheincidentpartontypesmightarisefromeitherincidentparticle.Thesecondtermshouldbedroppedifthetypesareidentical:i=j.August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesses48.7.Two-photonprocessesIntheWeizsÄacker-Williamspicture,ahigh-energyelectronbeamisaccompaniedbyaspectrumofvirtualphotonsofenergies!andinvariant-masssquaredq2=Q2,forwhichthephotonnumberdensityisdn= 1! E+!2 E2m2e!2 Q2E2d! !dQ2 Q2;(48:43)whereEistheenergyoftheelectronbeam.Thecrosssectionfore+e!e+eXisthen[9]d¾e+e!e+eX(s)=dn1dn2d¾°°!X(W2);(48:44)whereW2=m2X.IntegratingfromthelowerlimitQ2=m2e!2i Ei(Ei!i)toamaximumQ2givese+e!e+eX(s)=2 2Z1zthdz z"lnQ2max zm2e12f(z)+1 3(lnz)3#°°!X(zs);(48:45)wheref(z)=1+1 2z2ln(1=z)1 2(1z)(3+z):(48:46)TheappropriatevalueofQ2maxdependsonthepropertiesoftheproducedsystemX.Forproductionofhadronicsystems,Q2maxm2,whileforlepton-pairproduction,Q2W2.ForproductionofaresonancewithspinJ=1,wehavee+e!e+eR(s)=(2J+1)82R!°° m3R"f(m2R=s)lnm2Vs m2em2R1!21 3lns M2R!3#;(48:47)wheremVisthemassthatentersintotheformfactorforthe°°!Rtransition,typicallym.August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesPARTII:PROCESSESBEYONDTHESTANDARDMODEL48.8.ProductionofsupersymmetricparticlesInsupersymmetric(SUSY)theories(seeSupersymmetricParticleSearchesinthisReview),everybosonhasafermionicsuperpartner,andeveryfermionhasabosonicsuperpartner.TheminimalsupersymmetricStandardModel(MSSM)isadirectsupersymmetrizationoftheStandardModel(SM),althoughasecondHiggsdoubletisneededtoavoidtriangleanomalies[10].UndersoftSUSYbreaking,superpartnermassesareliftedabovetheSMparticlemasses.InweakscaleSUSY,thesuperpartnersareinvokedtostabilizetheweakscaleunderradiativecorrections,sothesuperpartnersareexpectedtohavemassesofordertheTeVscale.48.8.1.Gluinoandsquarkproduction:Thesuperpartnersofgluonsarethecoloroctet,spin1 2gluinos(~g),whileeachhelicitycomponentofquark°avorhasaspin-0squarkpartner,e.g.~qLand~qR.Thirdgenerationleft-andright-squarksareexpectedtohavelargemixing,resultinginmasseigenstates~q1and~q2,withm~q1m~q2(here,qdenotesanyoftheSM°avorsofquarksand~qithecorresponding°avorandtype(i=L;Ror1;2)ofsquark).Gluinopairproduction(~g~g)takesplaceviaeitherglue-glueorquark-antiquarkannihilation[11].Thesubprocesscrosssectionsareusuallypresentedasdi®erentialdistributionsintheMandelstamvariabless,tandu.Notethatfora2!2scatteringsubprocessab!cd,theMandelstamvariables=(pa+pb)2=(pc+pd)2,wherepaisthe4-momentumofparticlea,andsoforth.Thevariablet=(pcpa)2,wherecandaaretakenconventionallytobethemostsimilarparticlesinthesubprocess.Thevariableuwouldthenbeequalto(pdpa)2.Notethatsinces,tanduaresquaresof4-vectors,theyareinvariantsinanyinertialreferenceframe.Gluinopairproductionathadroncollidersisdescribedby:d¾ dt(gg!~g~g)=9¼®2s 4s2(2(m2~gt)(m2~gu) s2+(m2~gt)(m2~gu)2m2~g(m2~g+t) (m2~gt)2+(m2~gt)(m2~gu)2m2~g(m2~g+u) (m2~gu)2+m2~g(s4m2~g) (m2~gt)(m2~gu)(m2~gt)(m2~gu)+m2~g(ut) s(m2~gt)(m2~gt)(m2~gu)+m2~g(tu) s(m2~gu));(48:48)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesseswheresisthestrong¯nestructureconstant.Also,d¾ dt(qq!~g~g)=8¼®2s 9s28:4 3m2~gt m2~qt!2+4 3m2~gu m2~qu!2+3 s2h(m2~gt)2+(m2~gu)2+2m2~gsi3h(m2~gt)2+m2~gsi s(m2~qt)3h(m2~gu)2+m2~gsi s(m2~qu)+1 3m2~gs (m2~qt)(m2~qu)9=;:(48:49)Gluinoscanalsobeproducedinassociationwithsquarks:~g~qiproduction,where~qirepresentsanyofthevarioustypes(left-,right-ormixed)and°avorsofsquarks.Thesubprocesscrosssectionisindependentofwhetherthesquarkistheright-,left-ormixedtype:d¾ dt(gq!~g~qi)=¼®2s 24s2h16 3(s2+(m2~qiu)2)+4 3s(m2~qiu)i s(m2~gt)(m2~qiu)2(m2~gu)2+(m2~qim2~g)2+2sm2~g(m2~qim2~g) (m2~gt)!:(48:50)Therearemanydi®erentsubprocessesforproductionofsquarkpairs.Sinceleft-andright-squarksgenerallyhavedi®erentmassesanddi®erentdecaypatterns,wepresentthedi®erentialcrosssectionforeachsubprocessof~qi(i=L;Ror1;2)separately.(Inearlyliterature,thefollowingformulaewereoftencombinedintoasingleequationwhichdidn'tdi®erentiatethevarioussquarktypes.)Theresultforgg!~qi~qiis:d¾ dt(gg!~qi~qi)=¼®2s 4s28:1 3m2~q+t m2~qt!2+1 3m2~q+u m2~qu!2+3 32s28s(4m2~qs)+4(ut)2+7 121 48(4m2~qs)2 (m2~qt)(m2~qu)+3 32h(tu)(4m2~q+4ts)2(m2~qu)(6m2~q+2ts)i s(m2~qt)+3 32h(ut)(4m2~q+4us)2(m2~qt)(6m2~q+2us)i s(m2~qu)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesses+7 96h4m2~q+4tsi m2~qt+7 96h4m2~q+4usi m2~qu9=;;(48:51)whichhasanobviousu$tsymmetry.Forqq!~qi~qiwiththesameinitialand¯nalstate°avors,wehaved¾ dt(qq!~qi~qi)=2¼®2s 9s2(1 (tm2~g)2+2 s22=3 s(tm2~g))hst(tm2~qi)2i;(48:52)whileifinitialand¯nalstate°avorsaredi®erent(qq!~q0i~q0i)weinsteadhaved¾ dt(qq!~q0i~q0i)=4¼®2s 9s4hst(tm2~q0i)2i:(48:53)Ifthetwoinitialstatequarksareofdi®erent°avors,thenwehaved¾ dt(qq0!~qi~q0i)=2¼®2s 9s2st(tm2~qi)2 (tm2~g)2:(48:54)Iftheinitialquarksareofdi®erent°avorand¯nalstatesquarksareofdi®erenttype(i=j)thend¾ dt(qq0!~qi~q0j)=2¼®2s 9s2m2~gs (tm2~g)2:(48:55)Forsame-°avorinitialstatequarks,but¯nalstateunlike-typesquarks,wealsohaved¾ dt(qq!~qi~qj)=2¼®2s 9s2m2~gs (tm2~g)2:(48:56)Therealsoexistcrosssectionsforquark-quarkannihilationtosquarkpairs.Forsame°avorquark-quarkannihilationtosame°avor/sametype¯nalstatesquarks,d¾ dt(qq!~qi~qi)==¼®2s 9s2m2~gs(1 (tm2~g)2+1 (um2~g)22=3 (tm2~g)(um2~g));(48:57)whileifthe¯naltypesquarksaredi®erent(i=j),wehaved¾ dt(qq!~qi~qj)=August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesses2¼®2s 9s28:[st(tm2~qi)(tm2~qj)] (tm2~g)+[su(um2~qi)(um2~qj)] (um2~g)9=;:(48:58)Ifinitial/¯nalstate°avorsaredi®erent,but¯nalstatesquarktypesarethesame,thend¾ dt(qq0!~qi~q0i)=2¼®2s 9s2m2~gs (tm2~g)2:(48:59)Ifinitialquark°avorsaredi®erentand¯nalsquarktypesaredi®erent,thend¾ dt(qq0!~qi~q0j)=2¼®2s 9s2st(tm2~qi)(tm2~qj) (tm2~g)2:(48:60)48.8.2.Gluinoandsquarkassociatedproduction:IntheMSSM,thechargedspin-1 2winosandhiggsinosmixtomakecharginostates1and2,withm1m2.Thespin1 2neutralbino,winoandhiggsino¯eldsmixtogivefourneutralinomasseigenstates01;2;3;4orderedaccordingtomass.Wesometimesdenotethecharginosandneutralinoscollectivelyas-inosfornotationalsimplicityForgluinoandsquarkproductioninassociationwithcharginosandneutralinos[12],thequark-squark-neutralinocouplingsarede¯nedbytheinteractionLagrangiantermsL~ff~0i=iAf~0i~fyL~0iPLf+iBf~0i~fyR~0iPRf+h:c:,whereAf~0iandBf~0iarecouplingconstantsinvolvinggaugecouplings,neutralinomixingelementsandinthecaseofthirdgenerationfermions,Yukawacouplings.TheirformdependsontheconventionsusedforsettinguptheMSSMLagrangian,andcanbefoundinvariousreviews[13]andtextbooks[14,15].PLandPRaretheusualleft-andright-spinorprojectionoperatorsandfdenotesanyoftheSMfermionsu;d;e;ºe;¢¢¢.Thefermion-sfermion-charginocouplingshavetheformL=iAd~i~uyL ~iPLd+iAu~i~dyL ~ciPLu+h:c:foruanddquarks,wheretheAd~iandAu~icouplingsareagainconvention-dependent,andcanbefoundintextbooks.Thesuperscriptcdenotes\chargeconjugatespinor",de¯nedbycCT.Thesubprocesscrosssectionsforchargino-squarkassociatedproductionoccurviasquarkexchangeandaregivenbyd¾ dt(¹ug!~i~dL)=s 24s2jAu~ij2(m~dL;m~i;t);(48:61) *ThecouplingsAf~0iandBf~0iaregivenexplicitlyinRef.15inEq.(8.87).Also,thecouplingsAd~iandAu~iaregiveninEq.(8.93).ThecouplingsXjiandYjiaregivenbyEq.(8.103),whilethexiandyicouplingsaregiveninEq.(8.100).Finally,thecouplingsWijaregiveninEq.(8.101).August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesd¾ dt(dg!~i~uL)=s 24s2jAd~ij2(m~uL;m~i;t);(48:62)whileneutralino-squarkproductionisgivenbyd¾ dt(qg!~0i~q)=s 24s2jAq~0ij2+jBq~0ij2(m~q;m~0i;t);(48:63)where(m1;m2;t)=s+tm21 2sm21(m22t) (m21t)2+t(m22m21)+m22(sm22+m21) s(m21t):(48:64)Here,thevariabletisgivenbythesquareof\squark-minus-quark"four-momentum.Theneutralino-gluinoassociatedproductioncrosssectionalsooccursviasquarkexchangeandisgivenbyd¾ dt(qq!~0i~g)=s 18s2jAq~0ij2+jBq~0ij224(m2~0it)(m2~gt) (m2~qt)2+(m2~0iu)(m2~gu) (m2~qu)22i~gm~gm~0is (m2~qt)(m2~qu)35;(48:65)whereiisthesignoftheneutralinomasseigenvalueand~gisthesignofthegluinomasseigenvalue.Wealsohavechargino-gluinoassociatedproduction:d¾ dt(¹ud!~i~g)=s 18s224jAu~ij2(m2~it)(m2~gt) (m2~dLt)2+jAd~ij2(m2~iu)(m2~gu) (m2~uLu)2+2~gRe(Au~iAd~i)m~gm~is (m2~dLt)(m2~uLu)35;(48:66)where^t=(~gd)2andinthethirdtermonemusttaketherealpartoftheingeneralcomplexcouplingconstantproduct.August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesses48.8.3.Sleptonandsneutrinoproduction:Thesubprocesscrosssectionfor~`L~`Lproduction(`=eor)occursvias-channelWexchangeandisgivenbyd¾ dt(du!~`L~`L)=g4jDW(s)j2 192¼s2tum2~`Lm2~`L;(48:67)whereDW(s)=1=(sM2W+iMWW)istheW-bosonpropagatordenominator.Theproductionof~1~isgivenasabove,butreplacingm~`L!m~1,m~`L!m~andmultiplyingbyanoverallfactorofcos2(whereisthetau-sleptonmixingangle).Similarsubstitutionsholdfor~2~production,excepttheoverallfactorissin2.Table48.1:TheconstantsfandfthatappearinintheSMneutralcurrentLagrangian.HerettanWandccotW. fqfff `11 4(3tc)1 4(t+c)`01 4(t+c)1 4(t+c)u2 35 12t+1 4c1 4(t+c)d1 31 12t1 4c1 4(t+c) Thesubprocesscrosssectionfor~`L~`Lproductionoccursvias-channelandZexchange,anddependsontheneutralcurrentinteraction,withfermioncouplingstoandZ0givenbyLneutral=eqff°fA+ef°(f+f5)fZ(withvaluesofqf,f,andfgiveninTable48.1.Thesubprocesscrosssectionisgivenbyd¾ dt(qq!~`L~`L)=e4 24¼s2tum4~`L(q2`q2q s2+(``)2(2q+2q)jDZ(s)j2+2q`qqq(``)(sM2Z) sjDZ(s)j2);(48:68)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesseswhereDZ(s)=1=(sM2Z+iMZZ).Thecrosssectionforsneutrinoproductionisgivenbythesameformula,butwith`,`,q`andm~`Lreplacedby,,0andm~L,respectively.Thecrosssectionfor~1~1productionisobtainedbyreplacingm~`L!m~1and`!`cos2.Thecrosssectionfor~`R~`Rproductionisgivenbysubstituting``!`+`andm~`L!m~`Rintheequationabove.Thecrosssectionfor~2~2productionisobtainedfromtheformulafor~`R~`Rproductionbyreplacingm~`R!m~2and`!`cos2.Finally,thecrosssectionfor~1~2productionoccursonlyviaZexchange,andisgivenbyd¾ dt(qq!~1~2)=d¾ dt(qq!~1~2)=e4 24¼s2(2q+2q)2`sin22jDZ(s)j2(utm2~1m2~2):(48:69)48.8.4.Charginoandneutralinopairproduction:48.8.4.1.~i~0jproduction:Thesubprocesscrosssectionfordu!~i~0jdependsonLagrangiancouplingsLWud=g p 2u°PLdW++h:c:,LW~i~0j=g(i)j ~i[Xji+Yji5]~0jW+h:c:,Lq~q~i=iAd~i~uyL ~iPLd+iAu~i~dyL ~ciPLu+h:c:andLq~q~0j=iAq~0j~qyL ~0jPLq+h:c:.ContributingdiagramsincludeWexchangeandalso~dLand~uLsquarkexchange.TheXjiandYjicouplingsarenew,andagainconvention-dependent:thecrosssectionformulaeworksiftheinteractionLagrangianiswrittenintheaboveform,sothatthecouplingscanbesuitablyextracted.Thetermj=0(1)ifm~0j�0(0);itcomesaboutbecausetheneutralino¯eldmustbere-de¯nedbyai°5transformationifitsmasseigenvalueisnegative[15].Thesubprocesscrosssectionisgivenintermsofdotproductsoffourmomenta,whereparticlelabelsareusedtodenotetheirfour-momenta;notethatallmasstermsinthecrosssectionformulaearepositivede¯nite,sothatthesignsofmasseigenstateshavebeenabsorbedintotheLagrangiancouplings,asforinstanceinRef.[15].Wethenhaved¾ dt(d u!~i~0j)=1 192¼s2"TW+T~dL+T~uL+TW~dL+TW~uL+T~dL~uL#(48:70)whereTW=8g4jDW(s)j2n[Xj2i+Yj2i](~0jd~i u+~0j u~id)+2(XjiYji)(~0jd~i u~0j u~id)+[Xj2iYj2i]m~im~0jd uo;(48:71)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesT~dL=4jAu~ij2jAd~0jj2 [(~i u)2m2~dL]2d~0j~i u;(48:72)T~uL=4jAd~ij2jAu~0jj2 [(~0j u)2m2~uL]2 u~0j~id(48:73)TW~dL=p 2g2Re[Ad~0jAu~i(i)j](sM2W)jDW(s)j2 (~i u)2m2~dL8(Xji+Yji)~0jd u~i+4(XjiYji)m~im~0jd u(48:74)TW~uL=p 2g2Re[Ad~iAu~0j(i)j](sM2W)jDW(s)j2 (~0j u)2m2~uL8(XjiYji)~0j ud~i+4(Xji+Yji)m~im~0jd u(48:75)andT~dL~uL=4Re[Ad~0jAu~iAd~iAu~0j]m~im~0jd u [(~i u)2m2~dL][(~0j u)2m2~uL]:(48:76)48.8.4.2.Charginopairproduction:Thesubprocesscrosssectionfordd!~i~+i(i=1;2)dependsonLagrangiancouplingsL=e ~i~iAecotW ~i(xiyi5)~iZandalsoL3iAd~i~uyL ~iPLd+iAu~i~dyL ~ciPLu+h:c:.Contributingdiagramsincludes-channel°;Z0exchangeandt-channel~uLexchange[16,17].Thecouplingsxiandyiareagainnewandasusualconvention-dependent.Thesubprocesscrosssectionisgivenbyd¾ dt(d d!~i~+i)=1 192¼s2T+TZ+T~uL+T°Z+T~uL+TZ~uL(48:77)whereT=32e4q2d s2d~+i d~i+d~i d~+i+m2~id d(48:78)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesTZ=32e4cot2WjDZ(s)j2((2d+2d)(x2i+y2i)d~+i d~i+d~i d~+i+m2~id d4ddxiyid~+i d~id~i d~+i2y2i(2d+2d)m2~id d);(48:79)T~uL=4jAd~ij4 [(d~i)2m2~uL]2d~i d~+i(48:80)T°Z=64e4cotWqd(sM2Z)jDZ(s)j2 s(dxid~+i d~i+d~i d~+i+m2~id ddyid~i d~+id~+i d~i)(48:81)T~uL=8e2qd sjAd~ij2 [(d~i)2m2~uL]2 d~+id~i+m2~id d(48:82)andTZ~uL=8e2cotWjDZ(s)j2jAd~ij2(sM2Z) [(d~i)2m2~uL](dd)2(xiyi)d~i d~+i+m2~i(xiyi)d d(48:83)usingtheupperofthesignchoices.Thecrosssectionforu u!~+i~icanbeobtainedfromtheabovebyreplacingd!u,d!u,qd!qu,~uL!~dL,Ad~i!Au~i,d! u, d!uandadoptingthelowerofthesignchoiceseverywhere.Thecrosssectionforqq!~1~+2;~+1~2canoccurviaZand~qLexchange.Itisusuallymuchsmallerthan~1;2~+1;2production,sothecrosssectionwillnotbepresentedhere.ItcanbefoundinAppendixAofRef.15.August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesses48.8.4.3.Neutralinopairproduction:Neutralinopairproductionviaqqfusiontakesplacevias-channelZexchangeplust-andu-channelleft-andright-squarkexchange(5diagrams)[17,18].TheLagrangiancouplings(seepreviousfootnote*)neededincludetermsgivenaboveplustermsoftheformL=Wij ~0i(5)i+j+1~0jZ.ThecouplingsWijdependonlyonthehiggsinocomponentsoftheneutralinosiandj.Thesubprocesscrosssectionisgivenby:d¾ dt(qq!~0i~0j)=1 192¼s2TZ+T~qL+T~qR+TZ~qL+TZ~qR(48:84)whereTZ=128e2jWijj2(2q+2q)jDZ(s)j2q~0iq~0j+q~0jq~0iijm~0im~0jqq;(48:85)T~qL=4jAq~0ij2jAq~0jj2(q~0iq~0j [(~0iq)2m2~qL]2+q~0jq~0i [(~0jq)2m2~qL]2ijm~0im~0jqq [(~0iq)2m2~qL][(~0jq)2m2~qL])(48:86)T~qR=4jBq~0ij2jBq~0jj2(q~0iq~0j [(~0iq)2m2~qR]2+q~0jq~0i [(~0jq)2m2~qR]2ijm~0im~0jqq [(~0iq)2m2~qR][(~0jq)2m2~qR])(48:87)TZ~qL=16e(qq)(sM2Z)jDZ(s)j2(Re(WijAq~0iAq~0j) [(~0iq)2m2~qL]2q~0iq~0jijm~0im~0jqq+ijRe(WijAq~0iAq~0j) [(~0jq)2m2~qL]2q~0jq~0iijm~0im~0jqq)(48:88)TZ~qR=16e(q+q)(sM2Z)jDZ(s)j2(Re(WijBq~0iBq~0j) [(~0iq)2m2~qR]2q~0iq~0jijm~0im~0jqqAugust21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesRe(WijBq~0iBq~0j) [(~0jq)2m2~qR]2q~0jq~0iijm~0im~0jqq):(48:89)Asbefore,i=1correspondingtowhethertheneutralinomasseigenvalueispositiveornegative.Wheni=jintheaboveformula,onemustremembertointegrateoverjust2steradiansofsolidangletoavoiddoublecountinginthetotalcrosssection.48.9.UniversalextradimensionsIntheUniversalExtraDimension(UED)modelofRef.[19](seeRef.[20]forareviewofmodelswithextraspacetimedimensions),theStandardModelisembeddedina¯vedimensionaltheory,wherethe¯fthdimensioniscompacti¯edonanS1=Z2orbifold.EachSMchiralitystateisthenthezeromodeofanin¯nitetowerofKaluza-Kleinexcitationslabelledbyn=0¡1.AKKparityisusuallyassumedtohold,whereeachstateisassignedKK-parityP=(1)n.Ifthecompacti¯cationscaleisaroundaTeV,thenthen=1(orevenhigher)KKmodesmaybeaccessibletocollidersearches.Ofinterestforhadroncollidersaretheproductionofmassiven1quarkorgluonpairs.TheseproductioncrosssectionshavebeencalculatedinRef.[21,22].Welisthereresultsforthen=1caseonlywithM1=1=R(Risthecompacti¯cationradius)ands,tanduaretheusualMandelstamvariables;moregeneralformulaecanbefoundinRef.[22].ThesuperscriptstandsforanyKKexcitedstate,whilestandsforleftchiralitystatesandstandsforrightchiralitystates.d¾ dt=1 16¼s2T(48:90)whereT(qq!gg)=2g4s 27"M214s3 t02u02+57s t0u0108 s+20s2 t0u093+108t0u0 s2#(48:91)andT(gg!gg)=9g4s 27"3M41s2+t02+u02 t02u023M21s2+t02+u02 st0u0+1+(s2+t02+u02)3 4s2t02u02t0u0 s2#(48:92)wheret0=tM21andu0=uM21.Also,T(qq!q01q01)=4g4s 9"2M21 s+t02+u02 s2#;August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesT(qq!q1q1)=g42 9"2M214 s+s t021 t0!+23 6+2s2 t02+8s 3t0+6t0 s+8t02 s2#;T(qq!q1q1)=g4s 27"M216t0 u02+6u0 t02s t0u0+23t02 u02+3u02 t02+4s2 t0u05!#;T(gg!q1q1)=g4s"M414 t0u0s2 6t0u03 8+M214 ss2 6t0u03 8+s2 6t0u017 24+3t0u0 4s2#;T(gq!gq1)=g4s 3"5s2 12t02+s3 t02u0+11su0 6t02+5u02 12t02+u03 st02#;T(qq0!q1q01)=g4s 184M41s t02+5+4s2 t02+8s t0;T(qq0!q1q01)=2g4s 9M21s t02+1 4+s2 t02;T(qq!q1q1)=g4s 9M212s3 t02u024s t0u0+2s4 t02u028s2 t0u0+5;T(qq0!q1q01)=g4s 9"2M211 t0+u0 t02+5 2+4u0 t0+2u02 t02#;andT(qq0!q1q01)=g4s 9"2M211 t0+u0 t02+1 2+2u02 t02#:August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocesses48.10.LargeextradimensionsIntheADDtheory[23]withlargeextradimensions(LED),theSMparticlesarecon¯nedtoa3-brane,whilegravitypropagatesinthebulk.Itisassumedthatthenextradimensionsarecompacti¯edonann-dimensionaltorusofvolume(2¼r)n,sothatthefundamental4+ndimensionalPlanckscaleMisrelatedtotheusual4-dimensionalPlanckscaleMPlbyM2Pl=Mn+2(2¼r)n.IfM1TeV,thentheMWMPlhierarchyproblemisjustduetogravitypropagatinginthelargeextradimensions.Inthesetheories,theKK-excitedgravitonstatesGnforn=1¡1canbeproducedatcolliderexperiments.Thegravitoncouplingstomatteraresuppressedby1=MPl,sothatgravitonemissioncrosssectionsd¾=dt1=M2Pl.However,themasssplittingsbetweentheexcitedgravitonstatescanbetiny,sothegravitoneigenstatesareusuallyapproximatedbyacontinuumdistribution.Asummation(integration)overallallowedgravitonemissionsendsupcancellingthe1=M2Plfactor,sothatobservablecrosssectionratescanbeattained.SomeofthefundamentalproductionformulaeforaKKgraviton(denotedG)ofmassmathadroncollidersincludethesubprocessesd¾m dt(ff!°G)=®Q2f 16Nf1 sM2PlF1(t s;m2 s);(48:93)whereQfisthechargeoffermionfandNfisthenumberofQCDcolorsoff.Also,d¾m dt(qq!gG)=s 361 sM2PlF1(t s;m2 s);(48:94)d¾m dt(qg!qG)=s 961 sM2PlF2(t s;m2 s);(48:95)d¾m dt(gg!gG)=3s 161 sM2PlF3(t s;m2 s);(48:96)whereF1(x;y)=1 x(y1x)h4x(1+x)(1+2x+2x2)+y(1+6x+18x2+16x3)6y2x(1+2x)+y3(1+4x)i(48:97)F2(x;y)=(y1x)F1x y1x;y y1x(48:98)andF3(x;y)=1 x(y1x)h1+2x+3x2+2x3+x42y(1+x3)+3y2(1+x2)2y3(1+x)+y4i:(48:99)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesTheseformulaemustthenbemultipliedbythegravitondensityofstatesformuladN=Sn1M2Pl Mn+2mn1dmtogainthecrosssectiond2 dtdm=Sn1M2Pl Mn+2mn1d¾m dt(48:100)whereSn=(2)n=2 ¡(n=2)isthesurfaceareaofann-dimensionalsphereofunitradius.Virtualgravitonprocessescanalsobesearchedforatcolliders.Forinstance,inRef.[24]thecrosssectionforDrell-Yanproductionofleptonpairsviagluonfusionwascalculated,whereitisfoundthat,inthecenter-of-masssystemd¾ dz(gg!`+`)=2s3 64¼M8(1z2)(1+z2)(48:101)wherez=cosandisamodel-dependentcouplingconstant1.FormulaeforDrell-YanproductionviaqqfusioncanalsobefoundinRefs.[24,25].48.11.WarpedextradimensionsIntheRandall-Sundrummodel[26]ofwarpedextradimensions,thearenaforphysicsisa5-danti-deSitter(AdS5)spacetime,forwhichanon-factorizablemetricexistswithametricwarpfactore2().Itisassumedthattwooppositetension3-branesexistwithinAdS5atthetwoendsofanS1=Z2orbifoldparametrizedbyco-ordinatewhichrunsfrom0.The4-DsolutionoftheEinsteinequationsyields()=krcjj,wherercisthecompacti¯cationradiusoftheextradimensionandkMPl.The4-De®ectiveactionallowsonetoidentify M2Pl=M3 k(1e2krc),whereMisthe5-DPlanckscale.PhysicalparticlesontheTeVscale(SM)branehavemassm=ekrcm0,wherem0isafundamentalmassoforderthePlanckscale.Thus,theweakscale-Planckscalehierarchyoccursduetotheexistenceoftheexponentialwarpfactorifkrc12.InthesimplestversionsoftheRSmodel,theTeV-scalebranecontainsonlySMparticlesplusatowerofKKgravitons.TheRSgravitonshavemassmn=kxnekrc,wherethexiarerootsofBesselfunctionsJ1(xn)=0,withx1'3:83,x2'7:02etc.WhiletheRSzero-modegravitoncouplingssuppressedby1= MPlandarethusinconsequentialforcollidersearches,then=1andhighermodeshavecouplingssuppressedinsteadby¤=ekrc MPlTeV.Then=1RSgravitonshouldhavewidth¡1=½m1x21(k= MPl)2,whereisaconstantdependingonhowmanydecaymodesareopen.TheformulaefordileptonproductionviavirtualRSgravitonexchangecanbegainedfromtheaboveformulaefortheADDscenarioviathereplacement[27] M4!i2 8¤21Xn=11 sm2n+imnn:(48:102)August21,201413:17 48.Cross-sectionformulaeforspeci¯cprocessesReferences:1.J.F.Owensetal.,Phys.Rev.D18,1501(1978).NotethatcrosssectiongiveninpreviouseditionsofRPPforgg!q qlackedafactorof.2.F.Wilczek,Phys.Rev.Lett.39,1304(1977).3.B.L.Io®eandV.Khoze,LeningradReport274,1976;Sov.J.Nucl.Phys.9,50(1978).4.J.Ellisetal.,Nucl.Phys.B106,292(1976).5.R.N.CahnandS.Dawson,Phys.Lett.B136,196(1984),erratum,Phys.Lett.B138,464(1984).6.S.Dawson,Nucl.Phys.B249,42(1985).7.M.S.ChanowitzandM.K.Gaillard,Phys.Lett.B142,85(1984).8.R.N.Cahn,Nucl.Phys.B255,341(1985).9.Foranexhaustivetreatment,seeV.M.Budnevetal.,Phys.Reports15C,181(1975).10.Seee.g.H.Haber,Supersymmetry,PartI(Theory),thisreview.11.P.R.HarrisonandC.H.LlewellynSmith,Nucl.Phys.B213,223(1983),Erratum-ibid.,B223,542(1983);S.Dawson,E.Eichten,andC.Quigg,Phys.Rev.D31,1581(1985);V.Bargeretal.,Phys.Rev.D31,528(1985);H.BaerandX.Tata,Phys.Lett.B160,159(1985).12.H.Baer,D.Karatas,andX.Tata,Phys.Rev.D42,2259(1990).13.H.HaberandG.Kane,Phys.Rept.117,75(1985).14.TheoryandPhenomenologyofSparticles,M.Drees,R.Godbole,andP.Roy(WorldScienti¯c)2005.15.WeakScaleSupersymmetry:FromSuper¯eldstoScatteringEvents,H.BaerandX.Tata(CambridgeUniversityPress)2006.16.A.Bartl,H.Fraas,andW.Majerotto,Z.Phys.C30,441(1986).17.H.Baeretal.,Int.J.Mod.Phys.A4,4111(1989).18.A.Bartl,H.Fraas,andW.Majerotto,Nucl.Phys.B278,1(1986).19.T.Appelquist,H.C.Cheng,andB.Dobrescu,Phys.Rev.D64,035002(2001).20.Forareviewofmodelswithextraspacetimedimensions,seeG.GiudiceandJ.Wells,ExtraDimensions,thisReview.21.J.M.SmillieandB.R.Webber,JHEP0510,069(2005).22.C.Macesanu,C.D.McMullen,andS.Nandi,Phys.Rev.D66,015009(2002).23.N.Arkani-Hamed,S.Dimopoulos,andG.Dvali,Phys.Lett.B429,263(1998)andPhys.Rev.D59,086004(1999).24.J.L.Hewett,Phys.Rev.Lett.82,4765(1999).25.G.Giudice,R.Rattazzi,andJ.Wells,Nucl.Phys.B544,3(1999);E.A.Mirabelli,M.Perelstein,andM.E.Peskin,Phys.Rev.Lett.82,2236(1999);T.Han,J.Lykken,andR.Zhang,Phys.Rev.D59,105006(1999).26.L.RandallandR.S.Sundrum,Phys.Rev.Lett.83,3370(1999).27.H.Davoudiasl,J.L.Hewett,andT.G.Rizzo,Phys.Rev.Lett.84,2080(2000).August21,201413:17