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PublishedforSISSAbySpringerReceived:April12,2011Accepted:June17,2011Published:July5,2011Completeresummationofchirally-enhancedloop-eectsintheMSSMwithnon-minimalsourcesofavor-violationAndreasCrivellin,aLarsHoferbandJanuszRosiekcaAlbertEinsteinCenterforFundamentalPhysics,InstituteforTheoreticalPhysics,UniversityofBern,CH-3012Bern,SwitzerlandbInstitutf¨urTheoretischePhysikundAstrophysik,Universit¨atW¨urzburg,D-97074W¨urzburg,GermanycInstituteofTheoreticalPhysics,PhysicsDepartment,UniversityofWarsaw,Ho˙za69,00-681Warsaw,PolandE-mail:crivellin@itp.
unibe.
ch,lars.
hofer@physik.
uni-wuerzburg.
de,Janusz.
Rosiek@fuw.
edu.
plAbstract:Inthisarticlewepresentthecompleteresummationoftheleadingchirally-enhancedcorrectionsstemmingfromgluino-squark,chargino-sfermionandneutralino-sfer-mionloopsintheMSSMwithnon-minimalsourcesofavor-violation.
WecomputetheniterenormalizationoffermionmassesandtheCKMmatrixinducedbychirality-ippingself-energies.
InthedecouplinglimitMSUSYv,whichisanexcellentapproximationtothefulltheory,wegiveanalyticresultsfortheeectivegaugino(higgsino)-fermion-sfermionandtheHiggs-fermion-fermionvertices.
UsingtheseverticesaseectiveFeynmanrules,allleadingchirally-enhancedcorrectionscanconsistentlybeincludedintoperturbativecalculationsofFeynmanamplitudes.
WealsogiveageneralizedparametrizationforthebareCKMmatrixwhichextendstheclassicWolfensteinparametrizationtothecaseofcomplexparametersλandA.
Keywords:SupersymmetricStandardModel,BeyondStandardModel,B-PhysicsOpenAccessdoi:10.
1007/JHEP07(2011)017Contents1Introduction12Chirally-enhancedcontributionstoself-energies42.
1Generalremarks42.
2Explicitexpressionsfortheself-energies73Renormalization103.
1RenormalizationoffermionmassesandYukawacouplings113.
2Fermionwave-functionrenormalization123.
3RenormalizationoftheCKMmatrix133.
4Properrenormalizationsequence144Eectivefermionvertices154.
1Eectivegaugino-fermion-sfermionandhiggsino-fermion-sfermionvertices164.
2EectiveHiggs-fermion-fermionvertices165Conclusions20AGeneralizedWolfensteinparametrization21BCKMrenormalizationinthecaseofCKM-dependentup-quarkself-energies23CTree-levelFeynmanrules25DLoopintegrals261IntroductionIntheStandardmodel(SM)left-andright-handedfermioneldsfLandfRtransformdierentlyundertheSU(2)Lgaugesymmetry.
Thus,therequirementofgaugeinvarianceforbidsexplicitmass-terms.
InsteadtheseeldsacquiremassesviatheHiggsmechanism.
TheHiggseldH(whichisitselfaSU(2)Ldoublet)couplesleft-handedfermionstoright-handedoneswithcouplingstrengthYfi(idenotesthegenerationofthefermion),sothatthenon-vanishingvacuumexpectationvalue1(vev)vofHtheninducesfermionmassesmfi=Yfiv.
ExperimentalmeasurementsrevealedmfivforallthefermionsexceptforthetopquarkimplyingYfi1(fi=t).
SincetheYukawacouplingsYfi(fi=t)arethussmallcomparedtothegaugecouplings,theirvaluesareinprinciplesensitivetoloopcorrectionsifsuchhigher-ordercontributionsmanagetoescapetheYfi-suppression.
1WedeneH=v(withoutafactor√2),sothatv≈174GeV.
–1–However,anyloopcorrectiontoYfihastoinvolveachirality-ipandsinceintheSMtheYukawacouplingsaretheonlysourcesofchirality-violationtheloopmustbeproportionaltoYfiitself,sothattheYfi-suppressioncannotbeavoided.
IntheMinimalSupersymmetricStandardModel(MSSM)thesituationisdierent.
Firstly,itcontainstwoHiggsdoubletsHuandHdcouplingtoup-anddown-typequark(lepton)superelds,respectively.
TheneutralcomponentsoftheseHiggseldsacquirevevsvuandvdwithv2u+v2d=v2.
Ifthereisahierarchyvdvu,onefacesenhancedcorrectionstoFeynmanamplitudesinwhichthetree-levelcontributionissuppressedbythesmallvevvdwhiletheloopcorrectioninvolvesvuinstead.
Inthiscasetheratioofone-looptotree-levelcontributionreceivesanenhancementfactortanβ≡vu/vd[1].
Secondly,theMSSMoersanothersourceofchirality-ips,namelythesoftSUSY-breakingtrilinearHiggs-sfermioncouplingsAf(A-terms)withmassdimensionone.
2WhereasonehasAf∝YfinascenarioofMinimalFlavorViolation(MFV)[2],inthegeneralMSSMthesecouplingsareindependentfreeparameters.
Thus,enhancedcorrectionstoFeynmanamplitudesinwhichthetree-levelcontributionissuppressedbyasmallYfwhiletheloopcorrectioninvolvesAfinsteadarepossible.
Insuchacasetheratiooftheone-looptothetree-levelcontributionreceivesanenhancementfactorAfij/(YfijMSUSY),whereMSUSYisatypicalSUSY-mass.
Inbothcasestherespectiveenhancementfactor(tanβorAfij/(YfijMSUSY))cancom-pensatefortheloopsuppression.
Thereforesuchahigherloopcorrectioncanbeofthesamesize,orevenlarger,astheleadingorderdiagram3andperturbativecalculations(us-ingtheusualcountinginpowersofαs,α1,2)shouldthusbesupplementedbyanall-orderresummationoftheenhancedcorrections.
Innearlyallcasesthiscanbeachievedbyus-ingeectiveFeynmanruleswhichincorporatetheresummedcorrections.
Sucheectiveruleshavealreadybeencalculatedintheliteratureforseveralspecialcasesandvertices(S0=H0,A0,h0):S0bbandH+tbverticesforAb=0[1,7].
S0didjandH+uidjverticesintheMSSMwithMFV[8–13].
S0didjandH+uidjverticesintheMSSMwithMFVandadditionalsourcesofCPviolation[14–16].
4S0fifjandH+fif′jverticesforquarksandleptonsinthegeneralMSSMinthelimitAf=0[17,18].
S0bbvertexforAb=0includingNNLOQCDcorrections[19].
2Strictlyspeaking,theipoffermionchiralityisprovidedbyagauginopropagatorinthecorrespondingloopdiagram.
However,theA-termschangetheSU(2)L-chargeonthesfermion-lineandinthissensetheyarealsonecessaryinordertomediatethechirality-ipofthefermions.
3Sinceself-energydiagramsinvolvingAf-termscanbeofthesameorderasthelightfermionmasses,theycanevengeneratethementirelyinascenariowithloop-inducedsoftYukawacouplings[3–6].
4Ref.
[16]alsoextendstheanalysistogeneralsoft-SUSY-breakingtermsbyexpandingthemintermsoftheYukawacouplings.
–2–gdidj,χ+fif′j,χ0fifjverticesforquarksandleptonsinthegeneralMSSMforAf=0[20].
Thismethodrequiresiterativeresummation.
CompletesetofS0fifj,H+fif′j,gdidj,χ+uidj,χdiuj,χ0didjverticesintheMSSMwithMFVbeyondthedecouplinglimitMSUSYv[21].
Eectiveχ+iνjandχ0ij,verticesinthegeneralMSSM(withA=0)beyondthedecouplinglimitMSUSYv[22].
gdidj,χ+uidj,χdiujverticesinthegeneralMSSMbeyondthedecouplinglimitMSUSYv(correctionsfromgluino-squarkloopsonly)[23,24].
S0didjandH+uidjverticesinthegeneralMSSMincludingA-terms,A′termsbeyondleadingorderinv/MSUSY(correctionsfromgluino-squarkloopsonly)[25].
However,acompletelistofthegaugino(higgsino)-fermion-sfermionandHiggs-fermion-fermionverticesincludingthefullsetofchirallyenhancedcorrectionsisstillmissing.
Inthisarticlewedeliverthemissingpiecestakingintoaccountenhancedcontributionsfromgluino-squark,chargino-sfermionandneutralino-sfermionloopsinthegeneralMSSM.
Fortheresummationwerelyonthemethodsdevelopedinrefs.
[7,21,23,24],whichcanbeappliedforanarbitraryvalueoftheSUSYmassscaleMSUSY,inparticularbeyondthedecouplinglimitMSUSYv.
Ingeneral,however,theresummationofself-energycorrectionsrequiresiterativeproceduresandtheenhancedvertexcorrectionstotheHiggs-fermion-fermionvertexcannotbeabsorbedintoaneectivecoupling.
Thesecomplicationsdonotoccurifcontributionswhicharesubleadinginv/MSUSYareneglected.
Wepresentanalyticalresummationformulaeinthislimit,whichforrealisticvaluesofSUSYmassesturnouttobeanexcellentapproximationtothefullresult:accordingtothenewresultsoftheCMScollaboration[26]andtheAtlasexperiment[27],squarksandgluinosmustberatherheavysothatdecouplingeectsinsquark-mixingcanbeneglectedtoagoodapproximation.
SincemτMW.
Thesefactssupportourstatementthatthedecouplinglimitisalmostalwaysanexcellentapproximation.
Thepaperisorganizedasfollows.
Insection2wecalculatethechirally-enhancedpartsofthequarkandleptonself-energiesintheMSSM.
Section3isdevotedtotherenormalizationofYukawacouplings,fermionwave-functionsandtheCKMmatrixinthepresenceofchirally-enhancedcorrections.
Ourmainresult,theeectivegaugino(higgsino)-fermion-sfermionandHiggs-fermion-fermionverticesarepresentedinsection4.
1andsec-tion4.
2.
Weconcludeinsection5.
OurconventionsandageneralizationoftheWolfensteinparametrizationtothecaseofcomplexλandAparametersaregivenintheappendix.
–3–fjfiiΣfjiFigure1.
Self-energyinducingwave-functionrotationinavor-space.
2Chirally-enhancedcontributionstoself-energiesInthissectionwecalculateallchirally-enhancedcontributionsfromfermionself-energiesinthegeneralMSSM.
Werstgivethecompleteformulaeandthenextracttheleadingorderinv/MSUSY,uptowhichwewillbeabletogiveanalyticresultsfortheeectivevertices.
2.
1GeneralremarksIngeneral,itispossibletodecomposeanyself-energy(seegure1)intochirality-ippingandchirality-conservingpartsinthefollowingway(inwhatfollowswedenotetheavoroftheincoming(outgoingor"nal")fermionbyi(jorf),respectively):Σfji(p)=ΣfLRji(p2)+p/ΣfRRji(p2)PR+ΣfRLji(p2)+p/ΣfLLji(p2)PL(2.
1)Notethatthechirality-changingpartsΣfLRjiandΣfRLjihavemassdimension1andarerelatedthroughΣfLRji(p2)=ΣfRLij(p2),(2.
2)whilethehermitianchirality-conservingpartsΣfLLji=ΣfLLijandΣfRRji=ΣfRRijaredimensionlessandingeneralnotrelatedtoeachother.
Anyloopcontributiontothefermionself-energyinvolvingsfermionsandgluinos,charginosorneutralinoscanbewrittenasΣfλLRji(p2)=116π26s=1NI=1mλIΓλILfjfsΓλIRfifsB0p2;m2λI,m2fs,ΣfλRLji(p2)=116π26s=1NI=1mλIΓλIRfjfsΓλILfifsB0p2;m2λI,m2fs,ΣfλLLji(p2)=116π26s=1NI=1ΓλILfjfsΓλILfifsB1p2;m2λk,m2fs,ΣfλRRji(p2)=116π26s=1NI=1ΓλIRfjfsΓλIRfifsB1p2;m2λI,m2fs.
(2.
3)–4–HereλstandsfortheSUSYfermions(g,χ0,χ±)andNdenotestheircorrespondingnumber(2forcharginos,4forneutralinosand8forgluinos).
ThecouplingcoecientsΓλIL(R)fifsandtheloopfunctionsB0andB1aredenedinCandinD.
Forlow-energydecayswithp2m2fM2SUSY,itispossibletoexpandtheloopfunctioninthesmallparameterp2/M2SUSY:B0p2;m21,m22=B0m21,m22+p2m22D0m21,m22,m22,m22+.
.
.
B1p2;m21,m22=12C2m21,m22,m22+m2p2E2m21,m22,m22,m22,m222.
4)Formostprocesses,itissucienttoevaluatetheself-energiesatvanishingexternalmo-mentum.
Further,onlythechirality-ippingpartofaself-energy(ΣfLRji,ΣfRLji)canbeenhancedintheMSSMeitherbyafactortanβ[1]orbyafactorAfij/(YfijMSUSY)[23].
Therefore,weneglectthechirality-conservingpartsΣfLL,RRjiinthefollowing.
Weparametrizethe6*6sfermionmixingmatricesasM2f=fLLfLRfLRfRR(2.
5)withfXYbeing3*3matricesinavor-space.
ThenumericalvaluesforthefXYijdependonthechosenbasisforthesfermionelds.
ItiscommontochooseforthequarkeldsthebasisinwhichtheYukawacouplingsaredioagonaland,inordertohavemanifestsupersymmetryinthesuperpotential,tosubjectthesquarkstothesamerotationsasthequarks.
Theresultingbasisforthesuper-eldsiscalledsuper-CKMbasis.
Wechoosethesuper-CKMbasisforthesquarkmassmatricesbyrequiringthatthefundamentalbareYukawacouplingsYq(0)inthesuperpotentialarediagonalinavorspace.
Asdiscussedinref.
[24],suchadenitionofthesuper-CKMbasishasseveraladvantagescomparedtoan"on-shell"denitioninwhichthephysicalquarkmassesarediagonalinstead:ThedenitionoftheqXYijdoesnotdependontherenormalizationschemeusedforthefermionmassmatricesmqij.
ThebasisfortheqXYijisdenedatthelevelofbarequantities,sothattheirdef-initionremainsvalidtoallordersinperturbationtheory.
AchoiceofbasiswiththerenormalizedYukawacouplingsYqbeingdiagonal,ontheotherhand,requiresaredenitionoftheqXYijateveryorderinperturbationtheory.
Inoursuper-CKMbasisthesquarkmassmatricesarediagonalinascenarioofavor-blindSUSYbreakingterms.
Ifanon-shelldenitionisusedinstead,thebareYukawacouplingsYq(0)enteringthesquarkmassmatricesarenotdiagonalanymoreandthesquarkmassmatricesdevelopavoro-diagonalentriesevenincaseofavor-blindSUSYbreakingterms.
–5–TheelementsuLRij=vuAuijvdA′uijvdYui(0)δij,dLRij=vdAdijvuA′dijvuYdi(0)δij,LRij=vdAijvuA′ijvuYi(0)δij(2.
6)andfRLij=fLRjiipthe"chiralities".
Appearingingluino-squark,chargino-sfermionorneutralino-sfermioncontributionstofermionself-energies,theygeneratechirality-enhancedeectswithrespecttothetree-levelmassesiftheyinvolvethelargevevvu(tanβ-enhance-mentfordown-quark/leptonself-energies)oratrilinearA(′)f-term(A(′)fij/(YfijMSUSY)-enhancement).
ThecouplingsΓλIL(R)fifsineq.
(2.
3)dependonthecorrespondingsfermionmixingmatrixandthusontheelementsfLRijvMSUSYenteringthesfermionmassmatrices.
Asnon-polynomialfunctionsoftheseterms,theΓλIL(R)fifscontainallordersin(v/MSUSY)n(n=0,1,2,However,inthelimitMSUSYvthispowerseriesrapidlyconvergesandonlythersttermsintheexpansionarerelevant.
TheassumptionMSUSYvisanexcellentapproximationtothefulltheoryassoonasonetakesintoaccountboundsfromdirectSUSYsearches[25].
Qualitativelythiscanbeunderstoodasfollows:theo-diagonalmass-insertiontermsinduceasplittingofthesfermionmassesoftheformm2f1,2M2SUSY±vMSUSY.
Therefore,inordertoestablishsfermionmasseswhichrespectthelowerboundsfromdirectsearches,ahierarchyMSUSYvisneededtoacertaindegree.
Inpracticeitisthensucienttoworktoleadingorderinv/MSUSY.
5Thissimpliestheexpressionsfortheself-energiesandwilllaterallowustogiveanalyticformulaefortheeectivevertices.
Theleadingtermsintheexpansionoftheself-energiesinv/MSUSYdonotvanishinthelimitofinnitelyheavySUSYmasses(ifalldimensionfulSUSYparametersarerescaledsimultaneously).
Werefertotheapproximationinwhichonlysuchnon-decouplingtermsarekeptas"thedecouplinglimit".
Note,however,thatevenwhenworkingonlytoleadingorderinv/MSUSYwedonotintegrateouttheSUSYparticles.
Weratherworkintheframeworkofrefs.
[7,21,24,25]inwhichtheSUSY-particlesarekeptasdynamicaldegreesoffreedomandwhichthuspermitsaconsistentformulationofeectivecouplingsinvolvingtheseparticles.
Toleadingorderinv/MSUSY,thechirality-ippingelementsfLRcanbeneglectedinthedeterminationofsfermionmixingmatrices.
Thesfermionmassmatricesarethen5OnlyinthecaseofverylightSUSYmasses,negative(whichisdisfavoredbytheanomalousmagneticmomentofthemuon)andlargetanβ,bigcorrections(comparedtothedecouplinglimit)intherelationbetweenthebottom-quarkYukawacouplinganditsmassarepossible.
–6–block-diagonalanddiagonalizedbythemixingmatricesWf:WdM2dWd=diag(mqL1,mqL2,mqL3,mdR1,mdR2,mdR3),Wd=WdL00WdR,WuM2uWu=diag(mqL1,mqL2,mqL3,muR1,muR2,muR3),Wu=WuL00WuR,WM2W=diag(mL1,mL2,mL3,mR1,mR2,mR3),W=WL00WR.
(2.
7)The3*3-matricesWfL,R(f=u,d,)takeintoaccounttheavormixingintheleft-andright-sectorofsfermions,respectively.
NotethatSU(2)L-invarianceenforcesuLL=V(0)dLLV(0).
HereV(0)denotesthebareCKMmatrixappearinginthediagonalizationofthefundamentalYukawacouplingsYu(0),Yd(0).
Asaconsequence,themassesmqiLofleft-handedsquarksarethesameintheup-anddown-sectorandthecorrespondingmixingmatricesarerelatedtoeachotherviatheCKMmatrixV(0):WdL=WqL,WuL=V(0)WqL.
(2.
8)ItisfurtherconvenienttointroducetheabbreviationsΛfLLmij=(WfL)im(WfL)jm,(f=u,d,q,),ΛfRRmij=(WfR)im(WfR)jm,(f=u,d,),(2.
9)wherei,j,m=1,2,3andwhereindexmisnotsummedover.
Left-right-mixingofsfermions,ontheotherhand,isnotdescribedbyamixingmatrixbutrathertreatedperturbativelyintheformoftwo-pointfRi-fLjverticesgovernedbythecouplingsfLRji.
2.
2Explicitexpressionsfortheself-energiesToleadingorderinv/MSUSY,theself-energywithagluinoandasquarkasvirtualparticlesisproportionaltooneelementqLRjkofthesquarkmixingmatrix(notethattheself-energyscaleslikeqLRjk/MSUSYandthusthecombinationisnon-decoupling).
WehaveΣdgLRfi=2αs3πmg3j,k,j′,f′=13m,n=1ΛqLLmfjdLRjkΛdRRnkiC0m2g,m2qLm,m2dRn,ΣugLRfi=2αs3πmg3j,k,j′,f′=13m,n=1V(0)ff′ΛqLLmf′j′V(0)jj′uLRjkΛuRRnkiC0m2g,m2qLm,m2uRn.
(2.
10)ThematricesΛqLL,RRmki(q=u,d)takeintoaccountallpowersofchirality-conservingavorchangesinducedthroughtheo-diagonalelementsqLL,RRij.
ForexampleΣdLR11alsocontainsacontributionwhich,inthemassinsertionapproximation,wouldbe∝–7–dLL13dLR33dRR31.
Therefore,eq.
(2.
10)isexactinthedecoupling-limit.
Thecorre-spondingself-energywithippedchiralitiesisdeterminedthrougheq.
(2.
2).
Fortheneutralino-sfermioncontributionstotheleptonandquarkself-energieswegetΣχ0LRfi=116π23j,k=13m,n=1g21M1ΛLLmfjLRjkΛRRnkiC0|M1|2,m2Lm,m2Rn+3m=11√2MWsinβYf(0)ΛLLmfig22M2C0|M2|2,||2,m2Lm+g21M1C0|M1|2,||2,m2Lmg21√2MWsinβM1Yi(0)ΛRRmijC0|M1|2,||2,m2Rm,Σdχ0LRfi=116π23j,k=13m,n=119g21M1ΛqLLmfjdLRjkΛqRRnkiC0|M1|2,m2qLm,m2dRn+3m=11√2MWsinβYdi(0)ΛqLLmfig22M2C0|M2|2,||2,m2qLm+g213M1C0|M1|2,||2,m2qLm+13g21√2MWsinβM1Ydf(0)ΛdRRmfiC0|M1|2,||2,m2dRm,Σuχ0LRfi=116π23m,n=129g21M1V(0)jj′ΛqLLmf′j′V(0)ff′uLRjkΛuRRnkiC0|M1|2,m2qLm,m2uRn.
(2.
11)Finallythechargino-sfermioncontributionstotheleptonanddown-quarkself-energyaregivenbyΣdχ±LRfi=Ydi(0)16π2δi3Yu3(0)3m,n=1V(0)3fΛqLLm33V(0)33uLR33ΛuRRn33C0||2,m2qLm,m2uRn√2g2sinβMWM23m=1ΛqLLmfiC0m2qLm,||2,|M2|2,Σχ±LRfi=√2Yi(0)16π2g2sinβMWM23m=1ΛLLmfiC0m2Lm,||2,|M2|2,(2.
12)wherewehavefurtherneglectedthesmallup-typeYukawacouplingsofthersttwogenera-tionsandmultipleavor-changes.
Charginocontributionstoup-quarkself-energiescannotbechirallyenhanced:atanβ-enhancementisnotpossibleforup-typeself-energiessincethetree-levelup-quarkmassesarenotsuppressedbycosβ(incontrasttothedown-quarkones).
AnA(′)dij/(YuijMSUSY)-enhancement,ontheotherhand,isneitherpossibleforthethirdgeneration,wherethelargetopYukawacouplingpreventssuchaneect,norforthersttwogenerations,wherethecontributionissuppressedbyasmalldown-typecoupling–8–Ydi(i=1,2).
Notefurtherthatwehaveneglectedtermsproportionaltocotβinthecharginoandneutralinomassmatrices.
6WedenotethesumofallcontributionsasΣuLRfi=ΣugLRfi+Σuχ0LRfi,ΣdLRfi=ΣdgLRfi+Σdχ0LRfi+Σdχ±LRfi,ΣLRfi=Σχ0LRfi+Σχ±LRfi.
(2.
13)Inordertosimplifythenotationitisusefultodenethequantityσfji=ΣfLRjimax{mfj,mfi}.
(2.
14)HeremfiistheMSrenormalizedquarkmassextractedfromexperimentusingtheSMprescription.
Ithastobeevaluatedatthesamescaleastheself-energyΣfLRji.
Theratioσfjiisameasureofthechiralenhancementoftheself-energieswithrespecttocorrespondingquarkmasses.
FortherenormalizationoftheYukawacouplingsandtheCKMmatrixitisimportanttodistinguishbetweenthepartsofΣfLRjiwhichcontainaYukawacouplingand/orCKMelementandthosewhichdonot.
Furthermore,forthedeterminationoftheeectiveHiggs-fermion-fermionverticesonehastodistinguishbetweenpartsofΣfLRjiproportionaltodierentHiggsvev's(wecalltermsinΣd(u)LRjiproportionaltovd(u)tobe"holomorphic",whereastermsinΣd(u)LRjiproportionaltovu(d)arecalled"non-holomorphic").
ThereforewewilldeneseveralcorrespondingdecompositionsofΣfLRji(orσfji).
Intheexpressions(2.
10)–(2.
12)eachterminthedown-quark(lepton)self-energyΣd()LRfiinvolvesatmostonepowerofthecorrespondingYukawa-couplingYd().
Theup-quarkself-energyΣuLRfi,ontheotherhand,isapproximatlyindependentofYuasitisalwaysmultipliedbycotβandcanbeneglectedifonetakesintoaccountonlychirally-enhancedcontributions.
WemaketheYd()-dependenceoftheavor-conservingself-energyΣd()LRiiexplicitbydecomposingitasΣd()LRii=Σd()LRiiYi+d()ivuYdi(i)(0).
(2.
15)Inasimilarwaywedecomposetheavor-changingself-energiesΣqLRfi(q=u,d)withrespecttoCKMelements.
Concerningthedown-typequarks,onlyΣdLRf3(f=1,2)dependson(o-diagonal)CKMelementsintheapproximationinwhichweneglectsmallmassratiosandmultipleavor-changes.
Forf=iwewritetheenhancementfactorsσdfiasσdfi=σdf3+dFCV(0)3fV(0)33,i=3σdfi,i=1,2,(2.
16)6Ifonewouldkeepthecotβ-suppressedtermsinthecharginoandneutralinomassmatrices,theself-energieswouldbedivergentandonewouldhavetogothroughtheprocedureofinniterenormalization.
Inaddition,onewouldhavetoconsideralsothechirallyconservingself-energiespΣfLL,RRij(0)sincetheygenerate,afterapplicationoftheDiracequation,fermionmasstermsofthesameorderinmb/MSUSYandintanβasthecotβ-suppressedpartsofΣfLRij.
–9–sothattheσdfidonotdependon(o-diagonal)CKMelementsandεdFC=116π2Yd3(0)md33m,n=1Yu3(0)ΛqLLm33uLR33ΛuRRn33C0||2,m2qLm,m2uRn.
(2.
17)Fortheup-quarkself-energyΣuLRfithesituationismoreinvolved.
ItdependsontheCKMmatrixthroughΛuLL,whichisrelatedtoΛqLLviatheSU(2)relationΛuLL=V(0)ΛqLLV(0)inthedecouplinglimit.
Therefore,thebareCKMmatrixentersthegluino-andneutralino-contributionstoΣuLRfiineqs.
(2.
10)and(2.
11).
However,thereareseveralreasonswhyitseectisusuallyverysmall.
Firstly,aself-energydiagramwithanexternaltopquarkcannotbesignicantlychirallyenhancedasithastobecomparedtothelargetopquarkmass.
Furthermore,eectsoftheCKMmatrixinΣuLRfiareproportionaltothemasssplittingofleft-handedsquarks(andcancelcompletelyiftheleft-handedsquarkmassesaredegenerate7).
Therefore,inmostcasesitisanexcellentapproximationtoassumethattheup-quarkself-energiesdonotdependon(bare)CKMelementsandonecansettheCKMelementsV(0)ijineqs.
(2.
10)and(2.
11)totheirphysicalvaluesVij.
Wemakethisapproximationexplicitbywritingσufi≈σufi(2.
18)whereσufiisunderstoodtobeindependentof(o-diagonal)bareCKMelements.
ForcompletenessinBwegiveanalyticexpressionsfortheCKMmatrixrenormalizationwhichtakeintoaccountthedependenceoftheup-squarksectorontheCKMelements.
ForthediscussionoftheeectiveHiggsverticesinsection4.
2wealsoneedadecom-positionofΣfLRjiintoitsholomorphicandnon-holomorphicparts,asmentionedabove.
Inthedecouplinglimitallholomorphicself-energiesareproportionaltoA-terms.
ThuswedenotetheholomorphicpartasΣfLRjiA,whilethenon-holomorphicpartisdenotedasΣ′fLRji.
ThenwehaveΣfLRji=ΣfLRjiA+Σ′fLRji,(2.
19)andthecorrespondingequationforσfLRji.
Forthedecompositionoftheself-energieswehaveassumedthattheA-termsandthebilinearsoftsquarkmasstermsdonotdependonCKMelementsorYukawacouplings.
Forexampleinsymmetry-basedMFV[2]thisisnotthecaseandthoseparameterscarryanadditionaldependenceonCKMelementsandYukawacouplings.
Thentheself-energiesarenolongerlinearintheYukawacouplingsandananalyticresummation,aswewillperforminthefollowingsection,isimpossible.
InsuchcasesonehastorelyonaniterativeprocedureinordertodeterminethebareYukawacouplingsandbareCKMelements.
83RenormalizationInthissectionweconsiderthegeneraleectsofthenitechirally-enhancedself-energiesonmassandwave-functionrenormalizationoffermionsandontherenormalizationoftheCKMmatrix.
WedonotconsidertherenormalizationofthePMNSmatrixbecausetherenormalizationeectsareknowntobeverysmall[22,29].
7Seeref.
[28]foradiscussionofthepossibilityofnon-degeneratesquarkmasses.
8Iterationisalsoneedediftheresultsofref.
[16]areappliedtothegeneralMSSMbecausein[16]thesoftSUSY-breakingtermsareparametrizedintermsofYukawacouplings.
–10–3.
1RenormalizationoffermionmassesandYukawacouplingsChirally-enhancedself-energiesmodifytherelationbetweenthebareYukawacouplingsYfi(0)andthecorrespondingphysicalfermionmassesmfi.
Inourdiscussionweconcentrateonthequarkcasepostponingconclusionsfortheleptoncasetotheendofthissection.
Consideringonlychirally-enhancedcorrections,thephysicalquarkmassisgivenbymqi=vqYqi(0)+ΣqLRii,(q=u,d).
(3.
1)Eq.
(3.
1)implicitlydeterminesthebareYukawacouplingsYqi(0)foragivensetofSUSYparameters.
TheactualvaluesandphysicalmeaningoftherenormalizedYqidepend,ofcourse,ontherenormalizationschemechosenforYqi.
Thus,toniteorderinperturbationtheory,theFeynmanamplitudeforagivenprocesswoulddependonthechosenscheme.
However,inall-orderresummedexpressionstheschemedependencedropsoutandthenalresultsonlydependonthe(nite)bareYukawacouplingsYqi(0),whichareschemeindependent.
9Theself-energyontheright-handsideofeq.
(3.
1)caninprinciplecontainarbitrarilymanypowersofYukawacouplings.
Therefore,ananalyticsolutionofeq.
(3.
1)forYqi(0)isnotpossibleinthegeneralcase.
However,sincethetermsinΣqLRiiwithhigherpowersofYqi(0)aresuppressedbyhigherpowersofv/MSUSY,anumericalsolutionofeq.
(3.
1)canbeeasilyachievedusingiterativemethods.
ForadetaileddescriptionofsuchprocedureintheMFVcasewerefertoref.
[21].
Itisobviouslystillusefultohaveanapproximateanalyticformulaathand,andwederiveitusingthedecouplinglimit.
Intheup-quarksectortheenhancedtermsintheself-energyΣuLRiiareindependentofYui(0).
Thereforeeq.
(3.
1)iseasilysolvedforYui(0)andonendsYui(0)=muiΣuLRii/vu.
(3.
2)Inthedown-quarksector,ifwerestrictourselvestothedecouplinglimitwherewehavetermsproportionaltoonepowerofYdi(0)atmost,werecoverthewell-knownresummationformulafortanβ-enhancedcorrections,withanextracorrectionduetotheA-terms.
TheresummationformulaisgivenbyYdi(0)=mdiΣdLRiiYivd1+tanβεdi(3.
3)withdiandΣdLRiiYidenedthrougheq.
(2.
15).
Finally,wenotethatallstatementsofthissectionconcerningdown-quarkscandirectlybetransferredtotheleptonsector.
InparticulartheYukawacouplingYi(0)isobtainedfromeq.
(3.
3)byreplacingfermionindexdfor,exceptforthevev.
9EventhoughthebareYukawacouplingsYqi(0)areindependentoftherenormalizationschemeappliedtoYqi,theirvaluesdependonthechoiceoftheSUSYinputparameters,i.
e.
ontherenormalizationschemechoseninthesquarksector[21].
–11–3.
2Fermionwave-functionrenormalizationTheavor-changingself-energiesΣfLRfiinducewave-functionrotationsψfLi→UfLijψfLj,ψfRi→UfRijψfRj(3.
4)inavor-spacewhichhavetobeappliedtoallexternalfermionelds.
WedecomposeUfL,RijasUL,Rij=δij+UqL,R(1)ij+UqL,R(2)ij3.
5)wherethesuperscriptsdenotetherespectivelooporder.
Attheone-looplevelUqLfiisgivenby[23]UfL(1)=0σf12+mf1mf2σf21σf13+mf1mf3σf31σf12mf1mf2σf210σf23+mf2mf3σf32σf13mf1mf3σf31σf23mf2mf3σf320,(3.
6)wherewehaveneglectedtermswhicharequadraticorofhigherorderinsmallquarkmassratios.
However,fortransitionsbetweenthethirdandtherstgenerationalsotwo-loopcorrectionsareimportant[23,29].
TheyreadUfL(2)=12σf12212σf132mf3mf2σf13σf32mf2mf3σf12σf32mf3mf2σf13σf3212σf12212σf232mf2mf3σf21σf31σf12σf23σf12σf1312σf13212σf232.
(3.
7)Hereonlytheleadingorderintheexpansioninsmallquarkmassratioshasbeentakenintoaccount.
RespectingnaturalnessconstraintsfortheCKMhierarchy,onlythe31elementineq.
(3.
7)canbenumericallyimportant.
ToleadingorderinthequarkmassratiosthefullUfLthenreadsUfL=1σf12σf13σf121σf23σf13σf12σf23σf231.
(3.
8)ThecorrespondingexpressionsforUfRareobtainedfromtheonesforUfLbyreplacingσfji→σfij.
–12–3.
3RenormalizationoftheCKMmatrixApplicationoftherotationsineq.
(3.
8)totheuidjW+-vertexrenormalizestheCKMelementsVij.
ThebareCKMmatrixV(0)(stemmingfromthemisalignmentbetweentheYukawamatricesYu(0)andYd(0))canbecalculatedintermsofthephysicalCKMmatrixVasV(0)=UuLVUdL.
(3.
9)Intheabsenceoflargeunnaturalcancellations,therotationsUuLandUdLpreservethehierarchyofVsothatV(0)hasthesamehierarchyasV.
However,theconventionalWolfen-steinparametrizationisnotsucienttodescribeUuL,UdLandV(0)sincethesematricescanhaveadditionalcomplexphasescomparedtothephysicalCKMmatrixV(inthecaseofVsuchphasesareabsorbedbyproperredenitionofthequarkelds).
Therefore,weex-tendtheclassicWolfensteinparametrizationinA.
IntermsofourgeneralizedWolfensteinparametrization,denedineq.
(A.
2),wehaveV=Vλ,λ2A,λ3A(ρiη),0≡V(v12,v23,v13,0)(3.
10)andUqL=UqL(σq12,σq23,σq13,0)(q=u,d).
(3.
11)WeparametrizeV(0)accordinglyasV(0)=v(0)12,v(0)23,v(0)13,v(0)Im.
(3.
12)Usingtheapproximation(2.
18),therotationmatrixUuLisindependentofV(0).
WemakethisexplicitbywritingUuL=UuL=UuL(σu12,σu23,σu13,0).
(3.
13)ThematrixUdL,ontheotherhand,consistsofaCKM-dependentandaCKM-independentpartsincetheσdjienteringeq.
(3.
8)decomposeaccordingtoeq.
(2.
16).
WetransferthisdecompositiontoUdLwriting(inwhatfollowsweneglecttermsO(λ4)andhigher)UdL=UdLCKMUdL.
(3.
14)TheCKM-independentpartUdLisdenedbyreplacingσdji→σdjiineq.
(3.
8),whatamountstothegeneralizedWolfensteinparametrizationUdL=UdLσd12,σd23,σd13,0.
(3.
15)TheCKM-dependentpartUdLCKMisthengivenbyUdLCKM=UdLUdL=10V(0)31V(0)33εdFC01V(0)32V(0)33εdFCV(0)31V(0)33εdFCV(0)32V(0)33εdFC1=UdLCKM0,V(0)32V(0)33εdFC,V(0)31V(0)33εdFC,0.
(3.
16)–13–Insertingthedecomposition(3.
14)intoeq.
(3.
9)weobtainV(0)=UuLVUdLUdLCKM.
(3.
17)InordertodetermineV(0),wehavetosolveeq.
(3.
17).
Theright-handsideimplic-itlydependsonV(0)throughUdLCKM.
Asarststepwesolveeq.
(3.
17)inthespe-cialcaseUdLCKM≡1,i.
e.
intheabsenceofthecontributionsgovernedbyεdFC.
ThismeanswecalculateV=UuLVUdL.
(3.
18)Exploitingthemultiplicationrule(A.
7)forgeneralizedWolfensteinmatriceswegetV=(v12,v23,v13,vIm)(3.
19)withv12=v12+σu12σd12,v23=v23+σu23σd23,v13=v13+σu13σd13+σu12v23+σd12σu12σd23v12σd23,vIm=v12Imσu12+σd12Imσu12σd12.
(3.
20)SwitchingonUdLCKM=1inasecondstepandsolvingeq.
(3.
17)forV(0),wenallyndV(0)=V(0)v12,v231εdFC,v231εdFC,vIm.
(3.
21)Explicitlywrittendown,thismatrixreadsV(0)=1|v12|22+ivImv12v131εdFCv121|v12|22ivImv231εdFCv12v23v131εdFCv231εdFC1.
(3.
22)Weseethattheelementsv13andv23ineq.
(3.
22)arescaledbyafactor1/(1εdFC).
Thisgeneralizestheobservationofref.
[21],whereithasbeenfoundthattheWolfensteinparameterAisscaledbythisfactorintheMFVversionoftheMSSM,tothecaseofgeneralavorviolation.
3.
4ProperrenormalizationsequenceThedeterminationofthebareYukawacouplingsandbareCKMmatrixiscomplicatedbythefactthatthecorrespondingequations,denedintheprevioussections,areentangled.
Wegivehereadetailedrecipeonhowtodeterminethesequantitiesstepbystep.
–14–1.
OneshouldstartfromthecalculationofthebareYukawacouplings.
a)Calculatetheavor-conservingself-energiesΣuLRiiintheup-sectorfromeq.
(2.
10)andeq.
(2.
11).
NotethatΣuLRiiisindependentofanyYukawacouplingsinceweneglecttermsproportionaltocotβ.
Determinethebareup-quarkYukawacouplingsYui(0)viaeq.
(3.
2).
b)Havingthebaretop-quarkYukawacouplingathand,extractthecharginocon-tributiontod3(denedineq.
(2.
15))fromeq.
(2.
12).
Calculatealsoallothercontributionstod3aswellastotheYd3(0)-independentpartΣdLR33Yd3oftheavor-conservingself-energyΣdLR33fromeqs.
(2.
10)-(2.
12).
Forthissteponecanne-glectsmallcontributionsproportionaltothestrange-ordown-Yukawacouplings,whicharestillundetermined.
CalculatethebarebottomYukawacouplingYd3(0)fromeq.
(3.
3).
c)Calculated2andΣdLR22Yd2forthestrangequarkanalogouslytostep1b)forthebottomquark.
Inthecalculation,Yd3(0)shouldbesettothevaluedeterminedinstep1b),whileYd1(0)canagainbeneglected.
ComputeYd2(0)accordingtoeq.
(3.
3).
d)ProceedinthesamewayforYd1(0)usingthealreadydeterminedvaluesforYd2(0),Yd3(0).
2.
Inthenextstep,thebareCKMmatrixandtheeldrotationmatricescanbedetermined.
a)UsethevalueofthebareYukawacouplingstodeterminedFCandtheCKM-independentself-energyparametersσuijandσdijfromeqs.
(2.
10)-(2.
12)accord-ingtothedecompositions(2.
16),(2.
18).
ThisallowstocomputethebareCKMmatrixV(0)ijwithhelpofeqs.
(3.
20)and(3.
22).
b)Next,insertthebareYukawacouplingsandV(0)ijintoeq.
(2.
14)inordertocomputethefullσdij.
AlsoσuijshouldberecalculatedusingthebareV(0)ij(insteadoftheVijwhichhavebeenusedinthecalculationoftheσuij).
Withtheσuijandσdijathand,onecalculatestherotationmatricesUqL,Rij(q=u,d)ineq.
(3.
8).
Theprocedureusedforthedownquarksappliestothechargedleptonsaswell.
4EectivefermionverticesHavingdeterminedintheprevioussectionthebareYukawacouplingsandthebareCKMmatrixwearenowinapositiontocalculatetheeectivegaugino(higgsino)-fermion-sferm-ionandtheeectiveHiggs-fermion-fermionverticesinthegeneralMSSM.
–15–4.
1Eectivegaugino-fermion-sfermionandhiggsino-fermion-sfermionverticesInordertocalculatetheeectivegaugino(higgsino)-fermion-sfermionvertices,onehastotaketheFeynman-rulesgiveninCandsubstituteinthecouplingsΓλL,Rfjfsthetree-levelYukawacouplingsandtheCKMmatrixbythecorrespondingbarequantities(sincetheFeynman-rulesinCgobeyondthedecouplinglimitapproximation,oneshouldalsore-calculatethesfermionmassesandsfermionmixingmatriceswiththeuseofbarequan-tities).
Inaddition,onehastoapplythewave-functionrotationstothefermioneldsreplacingΓλL,RfjfsbyΓλLefifs=UfLjiΓλLfjfs,ΓλRefifs=UfRjiΓλRfjfs.
(4.
1)Ifthemomentumpowingthroughthefermionlinesatisesp2M2SUSY,therotationsUfL,Rjitakeintoaccounttheeectsofavor-changingchirally-enhancedself-energycor-rections(toleadingorderinp2/M2SUSY).
Forp2M2SUSY,ontheotherhand,nochiralenhancementoccursandtherotationsUfL,Rjidropoutfrominternalfermionlines.
There-foreoureectiveverticescanbeappliedirrespectiveofthemomentumowingthroughthefermionline.
TheappearanceoftherotationsUfL,Rjiingaugino(higgsino)-fermion-sfermionverticesisaconsequenceofthefactthatoursuper-CKMbasisisdenedatthelevelofthebareYukawacouplingsYq(0).
Therefore,itisnaturaltoaskwhethertheseeectscanbeab-sorbedintothedenitionofthesquarkmasstermsifanon-shelldenitionforthesuper-CKMbasisisused.
Note,however,thatatleastforthehiggsino-partsofthechargino-andneutralino-vertices,thisisimpossible:ifanon-shelldenitionforthesuper-CKMbasisisused,thebareYukawacouplingsYq(0)ijdevelopo-diagonalentrieswhicharere-latedtotherotationmatricesUfL,Rji.
Inthiswaythephysicaleectsoftheserotationswouldreappearinthehiggsino-fermion-sfermioncoupling.
Notefurtherthatanabsorp-tionoftheeectsingaugino-fermion-sfermionvertices,isonlypossibleaslongasthebilinearSUSYbreakingtermsareindependentfreeparameters.
AssoonasastructureresultingfromaSUSYbreakingmechanism(likegravity-mediationorgauge-mediation)isassumedforthem,anarbitraryredenitionisnotpossibleanymoreandtheeectsofUfL,Rjibecomephysicalhereaswell.
4.
2EectiveHiggs-fermion-fermionverticesAlsoHiggs-fermion-fermioncouplingsreceivechirally-enhancedcorrectionsfromtheYukawa-andCKM-renormalizationandfromthefermionwave-functionrotations.
Inaddition,wefaceanewclassofchirally-enhancedeects:theHiggscouplingitselfinvolvesaYukawacouplingYfwithYf1forf=t.
ThereforeagenuinevertexcorrectionwhichavoidstheYf-suppressionbycouplingtotheHiggsviatheA(′)f-termcanbechirallyenhancedwithrespecttothetree-levelvertex.
TheloopsuppressioncanbealleviatedbyafactorA(′)fij/(YfijMSUSY)inthiscase.
Notethatthistypeofchiralenhancementcannotreplicateitselfathigherordersinperturbationtheory,sothatnoresummationisneeded.
Sinceallcorrectionstogaugino(higgsino)-fermion-sfermionverticeswereduetofermionself-energies,theydidnotdependonthemomentaoftheSUSYparticlesbutonlyonthe–16–H0kqiqfufdiHiΓLRH0kqfqiPR+ΓRLH0kqfqiPLiΓLRHufdiPR+ΓRLHufdiPLFigure2.
Higgs-quarkverticeswiththecorrespondingFeynman-rules.
momentumpofthefermion.
Asshowninrefs.
[7,21],chirally-enhancedeectsonlyoccurforp2M2SUSY.
Therefore,sucheectsarelocalandcanbecastintoeectiveFeynmanruleswithoutanyfurtherassumptions.
InthecaseofthegenuinevertexcorrectionstotheHiggs-fermion-fermioncouplings,thesituationisdierent.
Thesecorrectionsarechirallyenhanced,independentlyofthescaleoftheexternalmomenta.
InordertoderiveeectiveFeynmanrulesforthesevertices,however,wehavetoassumethattheexternalmomentaaremuchsmallerthanthemassesofthevirtualSUSYparticlesrunningintheloop.
ThisassumptionlimitstheapplicabilityoftheresultingFeynmanrules:ifmH0,mA0,mH±MSUSY(H0,A0,H±denotetheneutralCP-even,CP-oddandthechargedHiggsboson,respectively),theycanbeusedforallprocessesincludingdiagramswheretheHiggsbosonsareinvolvedinaloop.
Ifthishierarchyisnotsatised,theycanonlybeusedforprocessesinwhichthemomentum-owthroughtheHiggs-fermion-fermionvertexissmallcomparedtoMSUSY.
ImportantexamplesforprocessesofthelatterkindaretheHiggspenguinscontributingtoBd,s→+,B+→τ+νorthedoubleHiggspenguincontributingtoF=2processes.
EectiveHiggs-fermion-fermionverticeshavebeencalculatedinref.
[25],butonlythegluino-squarkcontributionshavebeentakenintoaccount.
Weextendtheresultsofref.
[25]byincludingalsochargino-fermionandneutralino-fermioncorrections.
Inref.
[25]twodierentderivationsoftheeectiveHiggsverticeshavebeenpresented:therstone,usingadiagrammaticmethod,deliversaresultvalidtoallordersinv/MSUSY,whilethesecondone,usinganeectivetheoryapproach,reproducesonlytheleadingorderinv/MSUSY.
Itturnedoutthattheleadingorderinv/MSUSYisanexcellentapproximationtothefullapproachandthereisnoreasonwhythisstatementshouldnotbetrueforthecharginoandneutralinocontributions.
Furthermore,sincewerestrictedourselvestoleadingorderinv/MSUSYintheresummationoftheYukawacouplings(whichentertheHiggscoupling),forconsistencyweshouldrelyonthisapproximationincalculatingthegenuinevertexcorrectionsaswell.
Therefore,wewillusetheeectiveeldtheoryapproachinourstudyoftheHiggs-fermion-fermioncouplingswhichsimpliesthecalculations.
ThismeansthatincontrasttotheprevioussectionswereallyintegrateouttheSUSYparticlesandremovethemasdynamicaldegreesoffreedom,limitingsomewhattheapplicabilityoftheeectiveHiggsverticesasdiscussedinthepreviousparagraph.
–17–TheresultingeectiveYukawa-Lagrangianisthatofageneral2HDMandweparametrizeit(inthesuper-CKMbasis)asLeY=QafL(Ydi(0)δfi+Edfi)abHbdE′dfiHaudiRQafL(Yui(0)δfi+Eufi)abHbu+E′ufiHaduiR.
(4.
2)Herea,bdenoteSU(2)L-indicesandabisthetwo-dimensionalantisymmetrictensorwith12=1.
ApartfromtheYukawa-couplingsYuiandYdi,wehaveintheeectivetheoryloop-inducedholomorphiccouplingsEqfiandnon-holomorphiccouplingsE′qfi(q=u,d).
InthegeneralMSSMthesecouplingscanbeexpressedintermsofthecorrespondingself-energies,whichalsodecomposeintoaholomorphicandanon-holomorphicpartaccordingtoeq.
(2.
19).
WehaveEdij=ΣdLRijAvd,E′dij=Σ′dLRijvuEuij=ΣuLRijAvu,E′uij=Σ′uLRijvd.
(4.
3)Theseeectivecouplingsareinprincipleloop-suppressedcomparedtothetree-levelYdi(0),Yui(0)butachiralenhancementofA(′)qij/(YqijMSUSY)cancompensateforthissuppresion.
Inoureectivetheoryapproach,thewave-functionrotations10modifytheeectiveLagrangianasfollows[25]:LeY=dfLmdivdδfiE′dfitanβH0d+E′dfiH0udiRufLmuivuδfiE′uficotβH0u+E′ufiH0duiR+ufLVfj(cotβtanβ)E′dji+mdivdδjisinβHdiR+dfLVjf(tanβcotβ)E′uji+muivuδjicosβH+uiR(4.
4)withE′qfi=UqLjfE′qjkUqRki≈E′qfiE′qfi,E′q=0σq12E′q22(σq13σq12σq23)E′q33+σq12E′q23E′q22σq210σq23E′q33E′q33(σq31σq32σq21)+E′q32σq21E′q33σq320.
(4.
5)TheeldsH0uandH0ddecomposeintothephysicalcomponentsH0,h0andA0asH0u=1√2H0sinα+h0cosα+iA0cosβ,H0d=1√2H0cosαh0sinα+iA0sinβ.
(4.
6)10Notethateventhoughtheserotationsareidenticaltotheonesineq.
(3.
8)theiroriginisdierentintheeectiveeldtheoryapproach.
ThematricesUqL,Rarenowobtainedbyaperturbativediagonalizationofthe(physical)quarkmassmatrices(seeref.
[25]fordetails).
–18–Withoutthenon-holomorphiccorrectionsE′qijtherotationmatricesUqL,Rwouldsimul-taneouslydiagonalizetheeectivemasstermsandtheneutralHiggscouplingsineq.
(4.
4).
However,inthepresenceofnon-holomorphiccorrectionsthisisnolongerthecaseandapartfromaavor-changingnon-holomorphiccorrectionalsoatermproportionaltoaavor-conservingnon-holomorphiccorrectiontimesaavor-changingself-energyisgenerated.
Itisinstructivetodiscussthiseectalsointhefulltheory.
Thetwodiagramsingure3(bothinvolvingaholomorphicA-terms)haveoppositesignandcancelinthelimit,A′q→0.
However,inthepresenceofnon-holomorphictermsthecancellationisincompleteandapartproportionalto111+btanβ(fordown-quarks)survives(secondtermineq.
(4.
5)).
Eventhoughthistermisformallyofhigherlooporder,itisnumericallyrelevantduetoitschiralenhancement.
Thenon-holomorphicpartsofthefermionself-energy,asdenedineq.
(2.
19),canbeextractedfromeqs.
(2.
10),(2.
11)and(2.
12).
Notethatthewholecharginocontributionisalwaysnon-holomorphicexceptforcotβ-suppressedterms.
Thesameistruefortheneutralinocontributionexceptforthepurebinopartwhichdecomposesinthesamewayasthe(dominant)gluinocontribution(thelattergivenalreadyin[25]).
Usingeq.
(4.
5)andeq.
(4.
6),theeectiveLagrangianineq.
(4.
4)leadstothefollowingeectiveHiggs-fermion-fermionFeynmanrules11(notethattheCKMmatrixVinthechargedHiggscouplingisthephysicalone):ΓH0kLReufui=xkumuivuδfiE′uficotβ+xkdE′ufi,ΓH0kLRedfdi=xkdmdivdδfiE′dfitanβ+xkuE′dfi,ΓH±LReufdi=3j=1sinβVfjmdivdδjiE′djitanβΓH±LRedfui=3j=1cosβVjfmuivuδji+E′ujitanβ,(4.
7)whereforH0k=(H0,h0,A0)thecoecientsxkqaregivenby12xkd=1√2cosα,1√2sinα,i√2sinβ,xku=1√2sinα,1√2cosα,i√2cosβ.
(4.
8)11NotethatsomeoftheHiggs-quark-quarkcouplingsaresuppressedbyafactorcosβorsinαstemmingfromtheHiggsmixingmatrices.
Ifonedecidestokeepthesesuppressedcouplings,oneshouldbeawareofthefactthattheyreceivepropervertexcorrectionsinwhichthesuppressionfactordoesnotoccurandwhicharethustanβ-enhancedwithrespecttothetree-levelcouplings.
SuchenhancedcorrectionstothecouplingofH±toright-handedup-quarksareimportantforb→sγ[30,31].
12InprinciplealsotherenormalizationoftheHiggspotentialshouldbeaddressed.
OurderivationofchirallyenhancedavoreectsdoesnotdependonthespecicrelationsbetweenHiggsself-couplingsandtheirmasses.
Sincenochirally-enhancedeectsoccurintheHiggssector,itisconsistenttousethetree-levelvaluesfortheHiggsparameters.
However,onecanaswellusetheNLOvaluesfortheHiggsmassesandmixingangleswhichmightbeevenbetterfromthenumericalpointofview.
–19–H0kvq3q2ΣqLR23AYq2q3q2H0kq2Aq23Figure3.
Self-energyandgenuinevertexcorrectioninvolvingAq23contributingtotheeectiveHiggscoupling.
Itisimportanttokeepinmindthattheσfijineq.
(4.
5)mustbecalculatedusingthebarequantities(Yf(0)andV(0)).
Fortheleptoncase,thenon-vanishingeectiveHiggsverticesreadΓH0kLRefi=xkdmivdδfiE′fitanβ+xkuE′fi,ΓH±LReνfi=3j=1sinβVPMNSfjmivdδjiE′jitanβ.
(4.
9)5ConclusionsInthegeneralMSSM,chirally-enhancedcorrectionsareinducedbygluino-squark,chargino-sfermionandneutralino-sfermionloopsandcannumericallycompetewith,orevendom-inateover,tree-levelcontributions,duetotheirenhancementbyeithertanβorAfij/(YfijMSUSY).
Inthisarticlewehaveidentiedallpotentialsourcesofchirally-enhancedcorrectionsanddiscussedtheireectsontheniterenormalizationofYukawacouplings,fermionwave-functionsandtheCKMmatrix.
Toleadingorderinv/MSUSY,whichnumer-icallyisaverygoodapproximationforrealisticchoicesofMSSMparameters,weobtainedanalyticresummationformulaeforthesequantities.
FortheCKMresummation,itturnedouttobeusefultodeneageneralizedWolfen-steinparametrization,obtainedbyextendingtheclassicalonetothecaseofcomplexλandAparameters.
ThisparametrizationispresentedinA.
Fortheresummationofthechirally-enhancedcorrectionsinsupersymmetricfermionvertices,wehaveusedthediagrammaticapproachdevelopedinrefs.
[7,21,23,24].
Thismethodallowedustocastchirally-enhancedcorrectionstogaugino(higgsino)-fermion-sfermioncouplingsintoeectivevertices,asdescribedinsection3.
4and4.
1.
–20–Moreover,wehavegivenformulaefortheeectiveHiggs-fermion-fermionvertices,whereweextendedtheresultsof[25]byaddingthecharginoandneutralinocontributions.
OureectiveHiggs-verticescanbeusedinthelimitmH0,mA0,mH±MSUSYasFeynmanrulesinaneectivetheorywiththeSUSYparticlesbeingintegratedout.
However,theyremainstillvalidinthecasemH0,mA0,mH±MSUSYaslongasthemomentaowingthroughtheHiggsverticesaremuchsmallerthanMSUSY.
ThusoureectiveHiggs-fermion-fermionFeynmanrulescane.
g.
beappliedtocalculateHiggspenguinscontributingtoBd,s→+,B+→τ+νorthedoubleHiggspenguincontributingtoF=2processes.
Ifoureectivematterfermion-sfermion-SUSYfermionandHiggs-fermion-fermionFeynmanrulesareusedforthecalculationofanFeynmanamplitudeatleadingorderinperturbationtheory,allkindsofchirally-enhancedeectsareautomaticallyincludedandresummedtoallordersintheresult.
AcknowledgmentsWearegratefultoUlrichNiersteforusefuldiscussionsandforacarefulproofreadingofthemanuscript.
A.
C.
issupportedbytheSwissNationalFoundation.
TheAlbertEinsteinCenterforFundamentalPhysicsissupportedbythe"Innovations-undKooperationspro-jektC-13oftheSchweizerischeUniversit¨atskonferenzSUK/CRUS".
TheworkofL.
H.
hasbeensupportedinpartbytheFederalMinistryofEducationandResearch(BMBF,Ger-many)undercontractNo.
05H09WWEandbytheHelmholtzAlliance"PhysicsattheTerascale".
TheworkofJ.
RhasbeensupportedinpartbytheMinistryofScienceandHigherEducation(Poland)asresearchprojectsNN202230337(2009-12)andNN202103838(2010-12)andbytheEuropeanCommunity'sSeventhFrameworkProgrammeun-dergrantagreementPITN-GA-2009-237920(2009-2013).
L.
H.
andJ.
R.
liketothanktheITPBernforthehospitalityduringtheirvisitsthere.
AGeneralizedWolfensteinparametrizationWhileageneralunitarymatrixisdescribedby3mixinganglesand6complexphases,onlyoneofthosephasesisphysicalinthecaseoftheCKMmatrixV.
Theother5phasesareabsorbedbyproperredenitionofthequarkeldsexploitingtheU(3)3-avorsymmetryofthegaugeinteractions.
AfterapplicationofthisproceduretothephysicalCKMmatrixV,thequarkeldphasesarexed.
Asaconsequence,possibleadditionalphasesinthebareCKMmatrixV(0),whichoriginatefromthediagonalizationofthebareYukawacouplingsYu(0),Yd(0),cannotbeabsorbedanymoreandthustheyarephysical.
1313Inprincipletheadditionalphasescouldbeabsorbedintothewavefunctionsofthebarequarkeldsψ(0).
However,inthiscasetheywouldmodifytherelationbetweenthebareeldsψ(0)andthephysicaleldsψandtheywouldenterFeynmanamplitudesintheformofcomplexCP-violatingwave-functionfactors.
SinceinthiscaseCPviolationinthequarksectorwouldnotberestrictedtotheCKMmatrixanymore,werefrainfromintroducingthiskindofwave-functionrephasing.
–21–ThehierarchicalstructureofthemeasuredCKMmatrixVcanbemadeexplicitbyusingtheWolfensteinparametrizationV≈1λ22λλ3A(ρiη)λ1λ22λ2Aλ3A(1ρiη)λ2A1(A.
1)withthesmallexpansionparameterλ=0.
225.
ThethreemixinganglesandthephaseofVareexpressedviathefourrealparametersλ,A,ρ,η.
Consideringne-tuningarguments,itisreasonabletoassumethatthebareCKMmatrixV(0)inthegeneralMSSMhasasimilarhierarchicalstructureasthephysicalCKMmatrixV.
Thereforeitisdesirabletohaveaparametrizationanalogoustoeq.
(A.
1)butallowingforpossibleadditionalphasesofV(0).
WewillconsiderthefollowinggeneralizationoftheWolfensteinparametrization:U(u12,u23,u13,uIm)=1|u12|22+iuImu12u13u121|u12|22iuImu23(u13u12u23)u231.
(A.
2)Theparametersu12,u23,u13∈CandtheparameteruIm∈RshouldfollowthehierarchicalstructureoftheusualWolfensteinparametrization(A.
1):u12=O(λ),u23=O(λ2),u13=O(λ3),uIm=O(λ2).
(A.
3)Ourparametrizationisclosedunderhermitianconjugationandundermatrixmultiplica-tion.
Wehave(neglectingtermsofO(λ4)andhigher)hermitianconjugation:U(u12,u23,u13,uIm)=U(u12,u23,u13,uIm)(A.
4)whereu12=u12,u23=u23,u13=(u13u12u23),uIm=uIm.
(A.
5)matrixmultiplication:U(u′′12,u′′23,u′′13,u′′Im)=U(u12,u23,u13,uIm)U(u′12,u′23,u′13,u′Im)(A.
6)whereu′′12=u12+u′12,u′′23=u23+u′23,u′′13=u13+u′13+u12u′23,u′′Im=uIm+u′Im+Imu12u′12.
(A.
7)–22–NoteinparticularthattheparameteruImhadtobeintroducedinordertomakethisparametrizationclosedundermultiplication.
Wewillnowdemonstratethatourparametrization,whichallowsfor3mixinganglesand4complexphases,canbeusedtodescribethebareCKMmatrixV(0)intheMSSM.
FirstwerecognizethatdeningV=V(v12,v23,v13,vIm)withv12=λ,v23=Aλ2,v13=λ3A(ρiη),vIm=0(A.
8)werecovertheusualWolfensteinparametrization(A.
1).
Furthermore,alsothematrixUfLgivenineq.
(3.
8)canbedescribedintheformUfL=UfL(ufL12,ufL23,ufL13,ufLIm):ufL12=σf12,ufL23=σf23,ufL13=σf13,ufLIm=0.
(A.
9)Becausetheparametrizationisclosedunderhermitianconjugationandmatrixmultiplica-tion,therelationbetweenthephysicalCKMmatrixandthebareoneineq.
(3.
9)impliesthatV(0)canalsobeparametrizedusingeq.
(A.
2).
Insection3.
3wetookadvantageofthisparametrizationofV(0)inourstudyoftheCKMrenormalization.
BCKMrenormalizationinthecaseofCKM-dependentup-quarkself-energiesInthecaseofnon-degenerateleft-handedsquarkmasses,theup-quarkself-energiesdependonCKMelementsduetotheSU(2)relationbetweenthesoftmassmatricesoftheleft-handedsquarks.
Theup-squarkmixingmatrixWuLentersthegluino-andneutralino-contributionstotheup-quarkself-energythroughΛuLLmij=(WuL)im(WuL)jm.
TheSU(2)relation(2.
8)impliesΛuLLmfi=V(0)fjΛqLLmjkV(0)ikandleadsinthiswaytoaCKM-dependenceofΣuLRfi,whichhasbeenmadeexplicitineqs.
(2.
10),(2.
11).
Ifweassumethattheo-diagonalelementsΛqLLmfiareatmostofthesameorderintheWolfensteinparameterλasthecorrespondingelementsVfioftheCKMmatrix,wehavetoleadingorderinλ:ΛuLLm12=ΛqLLm12+V(0)12V(0)22ΛqLLm22ΛqLLm11,ΛuLLm23=ΛqLLm23+V(0)23V(0)33ΛqLLm33ΛqLLm22,ΛuLLm13=ΛqLLm13+V(0)13V(0)33ΛqLLm33ΛqLLm11+V(0)12V(0)32ΛqLLm22ΛqLLm11+V(0)12ΛqLLm23V(0)33+V(0)11ΛqLL12V(0)32.
(B.
1)FortheCKMrenormalizationitisimportanttodistinguishbetweencontributionstoΣuLRfiwhichdependonV(0)fiandthosewhichdonot.
Tothisendwedecomposeσufiinanalogytoeq.
(2.
16)forσdfiasσufi=σufi+V(0)fiV(0)iiεufi(f=i).
(B.
2)Fori,f=3thequantityσufidoesnotdependonanyo-diagonalCKMelement.
Theparametersσuf3andσu3idependontheCKMelementsV(0)12andV(0)23(orequivalentlyon–23–V(0)21andV(0)32),buttheyneitherdependonV(0)13noronV(0)31.
Theparametersεufiaregivenbyεufi=1max{mui,muf}3m,n=1ΛqLLmiiΛqLLmffuLR33ΛuRRn33*2αs3πmgC0m2g,m2qLm,m2uRn+g2172π2M1C0|M1|2,m2qLm,m2uRn.
(B.
3)ThetermΛqLLmiiΛqLLmffcausesastrongGIMsuppressionofεuficulminatinginεufi=0fordegeneratesquarkmasses.
Thereforeitisagoodapproximationtoneglecthigher-ordereectsrelatedtoεufiintheresummationformulaforV(0),asithasbeendoneineq.
(3.
22)usingtheapproximation(2.
18).
Forcompletenesswederivehereanextendedversionofeq.
(3.
22)resummingtheeectsofεufitoallorders.
Wedecomposethewave-functionrotationmatrixUuLasUuL=UuLCKMUuL(B.
4)inanalogytoeq.
(3.
14)forthedownsector.
TheCKM-independentpartUuLisdenedbyreplacingσuji→σujiineq.
(3.
8),whatamountstothegeneralizedWolfensteinparametriza-tionUuL=UuL(σu12,σu23,σu13,0).
(B.
5)TheCKM-dependentpartUuLCKMisthengivenbyUuLCKM=UuLUuL=UuLCKM(uuLCKM)12,(uuLCKM)23,(uuLCKM)13,(uuLCKM)Im(B.
6)with(uuLCKM)12=V(0)12V(0)22εu12,(uuLCKM)23=V(0)23V(0)33εu23,(uuLCKM)13=V(0)13V(0)33εu13V(0)12V(0)22εu12σu23,(uuLCKM)Im=ImV(0)12V(0)22εu12σu12.
(B.
7)Insertingthedecomposition(3.
14)and(B.
4)intoeq.
(3.
9)weobtainV(0)=UuLCKMVUdLCKM.
(B.
8)ThematrixVisdenedineq.
(3.
18)anditselementsaregivenintermsofgeneralizedWolfensteinparametersineq.
(3.
20).
Solvingeq.
(B.
8)forV(0),wenallygetv(0)12=v121εu12,v(0)23=v231εdFCεu23,v(0)13=11εdFCεu13v13+v12εu12(v23σu23)1εu12+v12v23εdFC(1εu12)1εdFCεu23,v(0)Im=vIm+Imv12εu12(v12σu12)1εu12.
(B.
9)–24–ParameterCurrentpaperRefs.
[32,33]Down-quarkandleptonYukawacouplingsY,YdY,YdHiggsvevsHu(d)=vu(d)Hu(d)=vu(d)/√2LeptonA-termsAij,A′ijAij,A′ijDown-squarkA-termsAdij,A′dijAdij,A′dijUp-squarkA-termsAuij,A′uijAuij,A′uijSquarkmasstermsfLLij,fRRij,fLRijfLLji,fRRji,fLRjiTable1.
DierencesinconventionsfortheMSSMparametersinthecurrentpaperandin[32,33].
Fortheapplicationofeq.
(B.
9)onehastokeepinmindthatσu13dependsonV(0)12andV(0)23.
Thereforeonehastoproceedasfollows:inarststepv(0)12,v(0)23andv(0)Imarecalculatedfromeq.
(B.
9).
Theresultsareusedtodetermineσu13.
Withthehelpofσu13onecanthencalculatev13fromeq.
(3.
20)andnallyv(0)13fromeq.
(B.
9).
CTree-levelFeynmanrulesThetreelevelFeynmanrulesusedthroughoutthepaperarebasedonthoselistedinrefs.
[32,33].
Oneshouldhowevernotefewdierencesinconventions,whichwesummarizeintable1.
Furthermore,thebareYukawacouplingscalculatedinsection3areingeneralcomplex,asshownexplicitlyinverticesdisplayedbelow.
WeusetheconventionthatYf(0)isthecouplingappearinginthePRcomponentofthe(pseudo-)scalarHiggs-fermion-fermionvertex,whereasYf(0)appearsinthePLcomponent.
14BelowwelisttheFeynmanrulesforthegaugino(higgsino)-fermion-sfermionvertices.
Thegeneraldenitionsofsupersymmetricfermionandsfermionmixingmatricesaregivenin[32,33].
Ineq.
(2.
7)weintroducedsquarkmixingmatricesWu,dforthedecouplinglimit.
ThesematricescanbeobtainedfromZU,Din[32,33]substitutingZD=WdL00WdR+OvMSUSYZU=WuL00WuR+OvMSUSY(C.
1)Notethecomplexstarsontheup-squarkmixingmatricesineq.
(C.
1),whichhavebeenaddedinordertostaycompatiblewithconventionsof[32,33].
qiqsgaiΓgLqiβqsαPL+ΓgRqiβqsαPRwithΓgLdiβdsα=gs√2TaαβZisDΓgRdiβdsα=gs√2TaαβZi+3,sDΓgLuiβusα=gs√2TaαβZisUΓgRuiβusα=gs√2TaαβZi+3,sU14AlsoYukawacouplingsintheLRblocksofthesfermionmassmatrices,usedtocalculatesfermionmixingmatrices,shouldbetreatedascomplex.
Tondthecorrectpositionsofcomplexstarsinthesfermionmassmatrices,inourconventionsonecanusethemnemotechnicreplacementruleY→Y(0),Y→Y(0).
–25–qiqsχ0kiΓχ0kLqiqsPL+Γχ0kRqiqsPRwithΓχ0kLdids=1√2ZisD(g2Z2kN13g1Z1kN)Ydi(0)Zi+3,sDZ3kNΓχ0kRdids=g1√23Zi+3,sDZ1kNYdi(0)ZisDZ3kNΓχ0kLuius=1√2ZisU(g2Z2kN+13g1Z1kN)Yui(0)Zi+3,sUZ4kNΓχ0kRuius=2√2g13Zi+3,sUZ1kNYui(0)ZisUZ4kNlilsχ0kiΓχ0kLisPL+Γχ0kRisPRwithΓχ0kLis=1√2ZisL(g1Z1kN+g2Z2kN)Yi(0)Zi+3,sLZ3kNΓχ0kRis=g1√2Zi+3,sLZ1kNYi(0)ZisLZ3jkNdiusχkiΓχ±kLdiusPL+Γχ±kRdiusPRwithΓχ±kLdius=3j=1(g2ZjsUZ1k++Yuj(0)Z(J+3)sUZ2k+)V(0)jiΓχ±kRdius=Ydi(0)3j=1ZjsUZ2kV(0)jiuidsχ+kiΓχ±kLuidsPL+Γχ±kRuidsPRwithΓχ±kLuids=3j=1(g2ZjsDZ1k+Ydi(0)Zj+3,sDZ2k)V(0)ijΓχ±kRuids=3j=1Yui(0)ZjsDZ2k+V(0)ijiνjχkiΓχ±kLiνsPL+Γχ±kRiνsPRwithΓχ±kLiνs=g2Z1k+ZisνΓχ±kRiνs=Yi(0)Z2kZisνDLoopintegralsThemomentumdependentloopfunctionsineq.
(2.
3)aredenedasB0p2;m21,m22=(2π)4diπ2ddk1(k2m21)(kp)2m22,pB1p2;m21,m22=(2π)4diπ2ddkk(k2m21)(kp)2m22.
(D.
1)–26–EvaluatingthefunctionB0forvanishingexternalmomentum,onegetsB0m21,m22=B00;m21,m22=1+m21lnQ2m21m22lnQ2m22m21m22.
(D.
2)Hereadivergentconstant24dγE+log4πhasbeendropped.
Italwayscancelsintheformulaeofthisarticlewhenthesumoverallinternalparticlesisperformed.
ThesameistrueforthearticialscaleQ2.
Theloop-functionsC0andD0aredenedinanalogytoB0butcorrespondtointegralswiththreeandfourpropagators,respectively.
ForvanishingexternalmomentatheyaregivenbyC0m21,m22,m23=B0(m21,m22)B0(m21,m23)m22m23,=m21m22lnm21m22+m22m23lnm22m23+m23m21lnm23m21m21m22m22m23m23m21,D0m21,m22,m23,m24=C0(m21,m22,m23)C0(m21,m22,m24)m23m24.
(D.
3)OpenAccess.
ThisarticleisdistributedunderthetermsoftheCreativeCommonsAttributionNoncommercialLicensewhichpermitsanynoncommercialuse,distribution,andreproductioninanymedium,providedtheoriginalauthor(s)andsourcearecredited.
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