Weak annihilation and new physics in charmless B → MM decays

34 Pages·2015·4.16 MB·English

Eur.Phys.J.C(2015)75:340
DOI10.1140/epjc/s10052-015-3535-1
RegularArticle-TheoreticalPhysics
→
Weak annihilation and new physics in charmless B MM decays
ChristophBobeth1,MartinGorbahn2,a,StefanVickers3
1TechnischeUniversitätMünchen,InstituteforAdvancedStudy,Lichtenbergstraße2a,85748Garching,Germany
2DepartmentofMathematicalSciences,UniversityofLiverpool,LiverpoolL693BX,UK
3TechnischeUniversitätMünchen,ExcellenceClusterUniverse,Boltzmannstraße2,85748Garching,Germany
Received:6October2014/Accepted:23June2015/Publishedonline:22July2015
©TheAuthor(s)2015.ThisarticleispublishedwithopenaccessatSpringerlink.com
Abstract We use currently available data of nonleptonic viewofthis,strategieshavebeendevelopedtoconstructtests
charmless2-bodyB → MMdecays(MM=PP,PV,VV) of the weak phases of the Cabibbo–Kobayashi–Maskawa
that are mediated by b → (d,s) QCD- and QED-penguin (CKM)quark-mixingmatrixoftheSMwherethehadronic
operatorstostudyweakannihilationandnew-physicseffects matrixelementsaredeterminedfromdata,usuallyinvolving
in the framework of QCD factorization. In particular we additionalassumptionsof SU(2)and/or SU(3)flavorsym-
introduceoneweak-annihilationparameterfordecaysrelated metries.Althoughthisallowsonetotesttheconsistencyof
by (u ↔ d) quark interchange and test this universal- weak phases extracted in tree- and loop-induced processes
ity assumption. Within the standard model, the data sup- in the framework of the SM, no other detailed information
portsthisassumptionwiththeonlyexceptionsinthe B → canbeobtainedonparticularshort-distancecouplingsofthe
Kπ system, which exhibits the well-known “ΔA puz- involvedQCD-andQED-penguinoperators.
CP
zle”, and some tensions in B → K∗φ. Beyond the stan- In this respect, systematic expansions in the heavy bot-
dardmodel,wesimultaneouslydetermineweak-annihilation tom quark mass,m ,yield at leading order in 1/m a sim-
b b
and new-physics parameters from data, employing model- plifiedrepresentationofhadronic2-bodymatrixelementsin
independentscenariosthataddressthe“ΔA puzzle”,such termsofratherwell-knownheavy-to-lightformfactorsand
CP
as QED-penguins and b → su¯u current-current operators. distributionamplitudes(DA)oftheinvolvedmesons.These
We discuss also possibilities that allow further tests of our approaches,QCDfactorization(QCDF)[1–6],soft-collinear
assumption once improved measurements from LHCb and effectivetheory(SCET)[7–10]orperturbativeQCD(pQCD)
BelleIIbecomeavailable. [11–14], provide predictions at leading order in 1/m that
b
allow one in principle to test short-distance couplings with
data.
1 Introduction Weakannihilation(WA)contributionsareformallyofsub-
leadingorderin1/m ,butanadditionalchiralenhancement
b
Nonleptonic charmless 2-body decays B → MM, with makes them phenomenologically relevant for a consistent
final state mesons MM = (PP, PV, VV), form a large description of experimental data in the SM and scenarios
classofdecaysthatallowonetotestinprincipletheunder- beyond.InQCDFandSCET,theyareplaguedbynonfactor-
lying tree and penguin topologies at the parton level, as izabledivergences,whicharepresentinendpointregionsof
predicted by the standard model (SM). Further, the sub- convolutionsofmesonDAs.InQCDF,thesedivergencesare
classofQCD-andQED-penguin-dominateddecaysaresen- frequently parameterized by a phenomenological complex
sitive to new physics (NP) beyond the SM, as any other parameter[3]andhencearemodel-dependent.Inparticular,
b→(d,s)flavor-changingneutral-current(FCNC)process, theassociatedstrongphasegovernsthesizeofCPasymme-
which makes them valuable probes of the according short- tries.Inpracticethisleadstolargetheoreticaluncertainties
distancecouplings. inthepredictionofobservables[5,15].Althoughbranching
Themajorobstacletoconstrainingtheshort-distancecou- fractions and CP asymmetries are sensitive to new-physics
plingswithdataistheevaluationofhadronicmatrixelements effects,themodel-dependenceandthearisinguncertainties
in 2-body B-meson decays beyond naive factorization. In due to the involved strong phases raise the question how
reliable information can be extracted on the short-distance
couplings.
ae-mail:[email protected]
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340 Page 2 of 34 Eur.Phys.J.C(2015)75 :340
Here we determine the model-dependence in the frame- is governed by the decay rates RH,L ≡ Γ[BH,L → f] of
f D
workofQCDFfromdata,admittingonephenomenological theheavyandlightmasseigenstates BH,L,whichyieldthe
parameterfordecays B → M M thatarerelatedby(u ↔ D
a b averagedandtime-integratedbranchingfraction
d)quarkexchange.Thetheoreticaluncertaintiesofallother (cid:4) (cid:5) (cid:6)
ilnatpeudtapnadrahmaveetebrese(nseinecAlupdpeednidnitxotAh)ealirkeeltirheoatoeddaassexupnlcaoinrreed- B[BD → f]= 21 dt RLf e−ΓDLt + RHf e−ΓDHt
(cid:7) (cid:8)
inAppendixB.WeusedataofmostlyQCD-penguindom- 1 RL RH
inated Bu,d decays into PP = Kπ, Kη((cid:5)), KK or PV = = 2 ΓLf + ΓHf , (2.2)
Kρ, Kφ, Kω, K∗π, K∗η((cid:5)) or VV = K∗ρ, K∗φ, K∗ω, D D
K∗K∗,andfurther B decaysinto PP =ππ, KK, Kπ or
VV =φφ, K∗φ, K∗sK∗finalstates.Wedeterminealsothe withtheirrespectivelifetimesΓDH,Linthetwoexponentials.
The mass eigenstates are related to the flavor eigenstates
relativemagnitudeofsubleadingWAamplitudescompared
|BL(cid:9)= p|B (cid:9)+q|B¯ (cid:9)and|BH(cid:9)= p|B (cid:9)−q|B¯ (cid:9)defin-
totherelevantleadingorderamplitudes.Theresultswithin D D D D D D
ing q and p. On the other hand, theoretical predictions are
the context of the SM are presented in Sect. 4. Given the
madefortheflavoreigenstates,implyingt =0,
current data, a simultaneous fit of the WA parameters and
(cid:2) (cid:3)
theshort-distancecouplingsispursuedinSect.5forgeneric 1 Γ[B¯ → f]+Γ[B → f]
NPextensionsoftheSMinordertoexploretheconstraining B[B → f]= 2 D (cid:2) (cid:3)D (2.3)
D 1 ΓL+ΓH
powerofthesedecays.Beforepresentingourresultsofthe 2 D D
fit, we review the observables and collect the experimental
input of charmless 2-body decays in Sect. 2. The relevant withtheaveragelifetimeτBD = (ΓD)−1 ≡ 2/(ΓDL +ΓDH).
details of QCDF and the definition of the phenomenologi- Bothbranchingfractionsarerelatedvia[16]
calparameteraresummarizedinSect.3.Variousappendices B[B → f]=B[B → f]1+yDHf (2.4)
collectadditionalmaterialonnumericalinputinAppendixA D D 1−y2
D
andthestatisticaltreatmentofexperimentalandtheoretical where y isproportionaltothewidthdifferenceΔΓ
D D
uncertainties as well as determination of pull values and p
valuesinAppendixB. y = ΔΓD ≡ ΓDL−ΓDH. (2.5)
D 2ΓD ΓDL+ΓDH
TheCPasymmetryduetononvanishingwidthdifference,
2 B → MM observablesanddata H = RHf−RLf, (2.6)
f RH+RL
f f
The 2-body decays of B mesons into final states f =
is an independent observable and provides complemen-
PP, PV, V V with light charmless pseudoscalar (P)
h h tary tests of physics beyond the SM. It can be obtained
and/or vector (V ) mesons with polarization mode h =
h from measurements of effective lifetimes in untagged, but
L,⊥,(cid:7)providevariousobservablesintime-integrated,time-
time-dependentratemeasurements[17]ortogetherwiththe
dependent,andalsoangularanalyses.Thesearereviewedin
mixing-inducedCPasymmetry S andthedirectCPasym-
f
the first part of this section, whereas in the second part the metryC =−A ofatime-dependentanalysis
f CP
accordingavailableexperimentaldataislistedthathasbeen
usedinthefits. ACP[BD → f](t)= Scofsshi(cid:5)n(ΔΔΓ2mDDt(cid:6)t−)−HCffsicnohs(cid:5)(ΔΔmΓ2DDtt(cid:6)), (2.7)
2.1 Observables where the mass difference of the heavy and light mass
eigenstates is denoted as Δm = mH − mL > 0. The
D D D
The most important observables for decays of charged B three CP asymmetries are not independent of each other,
u
mesons into a final state f are the CP-averaged branching |Sf|2 + |Cf|2 + |Hf|2 = 1, and they are given in terms
fractionandthe(direct)CPasymmetry ofonecomplexquantity
B[Bu → f]= τ2Bu (cid:2)Γ[B¯u → f¯]+Γ[Bu → f](cid:3), λf = qp AAff , (2.8)
Γ[B¯ → f¯]−Γ[B → f] asfollows:
ACP[Bu → f]= Γ[B¯uu → f¯]+Γ[Buu → f]. (2.1) S = 2Im(cid:9)(λf(cid:9)), H = 2Re(cid:9)(λf(cid:9)),
f 1+(cid:9)λ (cid:9)2 f 1+(cid:9)λ (cid:9)2
f f
ConcerningdecaysofneutralB mesons(D =d,s)into (cid:9) (cid:9)
D ¯ 1−(cid:9)λ (cid:9)2
acommonfinalstate f forbothflavoreigenstatesBDandBD, C = (cid:9) f(cid:9) . (2.9)
the simplest measurements are untagged rates. The decay f 1+(cid:9)λ (cid:9)2
f
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Eur.Phys.J.C(2015)75 :340 Page 3 of 34 340
In Eq. (2.8) A = A[B¯ → f] and A = A[B → f] polarizationfractionsandthreeCPasymmetries
f D f D
denotethedecayamplitudesandinSect.3wereviewtheir
calculationinQCDF. f = 1(cid:5)fB¯ + fB(cid:6), A = fhB¯ − fhB. (2.12)
In the limit ΔΓD → 0 one obtains B[BD → f] = h 2 h h CPh fB¯ + fB
h h
B[B → f]. This is the case for D = d to a very good
D
approximation in the SM where y | = (0.21±0.04)· Concerningthephases,thefollowingtwoCP-averagedand
d SM
10−2 (cid:10) 1 [18]. Currently, the precision of experimen- CP-violatingobservablescanbeconstructedforh =((cid:7),⊥):
tal results does not yet allow one to test the SM predic- (cid:5) (cid:6)
tion. Measurements are available from B-factories |y | = φ = 1 φB¯ +φB
(0.7±0.9)·10−2 [19]andarecentdeterminationofLdHCb h 2 h (cid:5) h (cid:6) (cid:5) (cid:6)
fromeffectivelifetimesy =(−2.2±1.4)·10−2[20],which −πsgn φB¯ +φB θ φB¯ −φB −π ,
d h h h h
assumestheSMresultfor H .Model-independentanalysis (cid:5) (cid:6) (cid:5) (cid:6)
f 1 ¯ ¯
ofeffectsofNPinΔΓd showthatthereisstillroomforhuge Δφh = 2 φhB −φhB −πθ φhB +φhB . (2.13)
nonstandard contributions [21]. We use the approximation
yd = 0 in all our predictions, which is well justified in the Thisconventionimpliesφh = Δφh = 0atleadingorderin
SMandalsotheconsideredNPscenarios. QCDF,whereallstrongphasesarezero[23]andmightdiffer
Ontheotherhand,ΔΓs isnotnegligibleandthecurrent forthesignof AL relativeto A(cid:7),⊥ adoptedbyexperimental
worldaveragefrom Bs → J/ψφanalysesaloneis[19] collaborations.
y =(5.8±1.0)·10−2, (2.10) InthecaseofBs → VV decays,againacorrectionfactor
s Eq.(2.4)dueto y (cid:13)=0applies,however,now
s
whichwillbeusedinouranalysis.Ingeneral−1≤ H ≤1,
f
and thereforethecorrection factoronther.h.s.ofEq.(2.4) (cid:11) 1+y H
can become of O(10 %) for final states f that are CP B[BD → f]= B[BD → fh] 1−Dy2 fh,
h=L,(cid:7),⊥ D
eigenstates, as has been found for some cases [16]. Other
asivse,rathgeeseftfaekcetivinetolifaectcimouentmBeassu→remJe/nψtπofπBang→ularKa+naKly−- f¯h = (cid:11)B[BBD[B→→fh]f ] 1+1−yDyH2 fh. (2.14)
s D h D
and flavor-specific Bs lifetime averages, which involve h=L,(cid:7),⊥
additional assumptions in the potential presence of new
physics.TheyyieldaslightlylargervaluethaninEq.(2.10), Here Hfh is defined as in Eq. (2.9) and the quantity λf is
ys = (6.2 ±0.9)·10−2 [19], being consistent within the evaluatedwith Af → Afh.
uncertainties. Besidestheseobservables,furthercombinationsarecon-
BesidesbranchingfractionsandCPasymmetries,2-body sideredthatinvolvedifferenttypesofchargedandneutralB,
decays B → VV with subsequent decays V → PP pro- Ma,andMbmesons.Theyareeitherratiosofbranchingfrac-
vide additional observables in the full angular analysis of tionsordifferencesofdirectCPasymmetries.Thecomplete
the 4-body final state [22]. The decay can be described in setofratios[15,31]is
terms of three amplitudes, which can be chosen to corre- (cid:12) (cid:13) (cid:12) (cid:13)
slVopawo,hniandVgtb,o,thrbdaenwfisvniteihtreshihtayela=icmitpihelbistuo=dfetsh(:LeA,fi(cid:7)+n,⊥a,l−-=s)tao(teAr,v+aesc±tionAr−tmh)ee/s√foon2ls-. RcB =2BB(cid:12)(cid:12)BB−−→→ MMaa−0MMb−(cid:13)b0(cid:13), RcMa =2BB(cid:12)BB¯−0 →→MMa−a−MMb+b0(cid:13),
ctTiaohnne,btwherheemereemawsauegrnefoidtlulodinwesatahnethddreetewfifnooiltdiroenlaasntig[v2ue3la]pr.hHadseeecnsacyeo,fdfiitvsheteriCbAPuh-- RcMb = BB(cid:12)BB¯(cid:12)−0 →→MMa−a0MMbb−+(cid:13),(cid:13) (cid:12) (cid:13)
TainvhetehrraeegceaadrseeanpwdohlaCerrPiez-aattasigyogmninmfgreatocrtfiicothnoesbsianenirtvdiaarlbelBleas-tflicvaaevnoprbheiassmpeseoasfssouibrrleeBd¯. RnB = 21 BB(cid:12)B¯B¯00→→MMa−a0MMbb+0(cid:13) , RnMa = 21 BB(cid:12)BB¯−0 →→ MMaa00MMbb−0(cid:13) ,
(cid:12) (cid:13)
dfehB¯ca=ys(cid:10)|hA¯|hA¯|2h|2, φ(cid:7)B¯,⊥ =arg A¯A¯(cid:7),L⊥. (2.11) RnMb = BB(cid:12)BB¯−0 →→ MMaa−0MMb0b0(cid:13) , (2.15)
IonneviueswesofthtehebrnaonrcmhainligzaftriaocnticoonnadnitdiotnwofLBo¯f+thef(cid:7)B¯po+larfi⊥zB¯at=ion1 wofhRercMe,naf,aMcbt,ocrsonotfraτrBy0t/oτ[B1−5]a.rIetnisoatninticcliupdaetedditnhathtethdeesfienraittiioons
fractions. In combination with the same quantities from B are measured directly in experimental analyses, such that
decays, replacing A¯ → A , one has three CP-averaged commonexperimentalsystematicerrorscancel.Further,the
h h
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340 Page 4 of 34 Eur.Phys.J.C(2015)75 :340
Table1 ObservablesofB→ PPdecaysmediatedbyb→sandb→dtransitionsthatareusedinthefit
b→s b→d
B→Kπ B→Kη B→Kη(cid:5) Bs →KK [24] Bs →ππ B→KK Bs →Kπ
K0π0:B,C, S K0η:B K0η(cid:5):B,C, S K+K−:B,C, S π+π−:B K0K¯0:B K+π−:B,C
K+π−:B,C K+η:B,C K+η(cid:5):B,C K+K−:B
K+π0:B,C K+K0:B,C
K0π+:B,C
Table2 ObservablesofB→ PV decaysmediatedbyb→stransitionsthatareusedinthefit
b→s
B→K∗π B→Kρ B→K∗η B→K∗η(cid:5) B→Kφ[25–28] B→Kω[29,30]
K∗0π0:B,C K0ρ0:B,C, S K∗0η:B,C K∗0η(cid:5):B,C K0φ:B,C, S K0ω:B,C, S
K∗+π−:B,C K+ρ−:B,C K∗+η:B,C K∗+η(cid:5):B,C K+φ:B,C K+ω:B,C
K∗+π0:B,C K+ρ0:B,C
K∗0π+:B,C K0ρ+:B,C
followingtwodifferencesofdirectCPasymmetriesarefre- whichthereforeexclusivelymeasurestheinterferenceofCP
quentlyconsidered: violationinthedecayandinmixing.
(cid:14) (cid:15) (cid:14) (cid:15)
−ΔC =ΔACP =ACP(cid:14)B−→Ma−Mb0(cid:15)−ACP(cid:14)B¯0→Ma−Mb+(cid:15), 2.2 Data
ΔA0CP =ACP B−→Ma0Mb− −ACP B¯0→ Ma0Mb0 ,
(2.16) WeinvestigatemainlyB → MMdecaysmediatedbyb →s
transitions but will consider also some b → d examples.
inwhichinQCDFacancelationofuncertaintiestakesplace
Thefinal2-mesonstateMMconsistseitheroutoftwopseu-
[32]. doscalars (MM = PP) or one pseudoscalar and one vec-
In order to separate NP effects in decays from those in tor (MM = PV)1 or two vectors (MM = VV), which
B −B mixinginS ,wedefinetheobservables[33,34]
D D f are listed in Tables 1, 2, and 3, respectively, together with
(cid:16) the observables that have been measured. We use the most
ΔSf =−ηfSf − SS((BB¯¯ds →→JJ//ψψφK¯)S) DD ==sd (2.17) mrecixeinntg-vianlduuecseodfCbPraansychminmgetrraietisoAs B a=s −weCllanasddSifrreocmt atnhde
CP
Heavy Flavor Averaging Group (HFAG) 2012 compilation
with η = ±1 the CP eigenvalue of the final state f. The
f andupdatesfrom2013/2014onthewebsite[19].Fordecays
decays B¯ → J/ψ K¯ and B¯ → J/ψφ aredominatedby
d S s intoVV finalstatesweincludealsothedataofpolarization
contributions from charm tree-level operators and CP vio- fractions fL,⊥, the relative phases φ(cid:7),⊥ and CP asymme-
lationinthedecayisbothparametrically(CKM)andtopo-
triesCL,⊥.Meanwhile,someobservableshadbeenupdated
logically(loop)suppressed.WeexpectthattheCP-violating
or measured for the first time from individual experiments
phaseinB −B mixing,φ andφ ,canclearlybeextract
D D Bd Bs and not yet included in the HFAG averages. In these cases
fromthosedecays,eveninthepresenceofmostNPscenarios
wedonotmakeuseofHFAGaverages,butinsteadallmea-
[35]
surementsfromindividualexperimentsenterthelikelihood
functioninEq.(B3)assinglemeasurements.Theaccording
λJ/ψK¯S (cid:14)e−iφBd, S(B¯d → J/ψ K¯S)(cid:14)sin2β, referencesaregivenexplicitlyinthetablesforsuchcases.
λJ/ψφ (cid:14)e−iφBs, S(B¯s → J/ψφ)(cid:14)sin2βs, (2.18) posIendaodbdsietirovnabwlees,inthveesrtaigtiaotseRthcBe,,nMcoa,mMpble(2m.1e5n)taorfitybroanfcchoimng-
fractionsanddifferencesofCPasymmetriesΔC (2.16).In
inwhichthe(cid:2)angles(cid:3)oftheCKMunita(cid:2)ritytri(cid:3)anglearedefined
as β = arg λd/λd and β = arg λs/λs . This source of the future, it is desirable to have direct experimental deter-
c t s t c
minationsoftheuncertaintiesforthese“composed”observ-
CP violation enters the mixing-induced CP asymmetry of
most decays that are triggered by b → s transition in the
samewayandcanbeeliminatedbytheconstructionofΔSf, 1 HereMM = PV standsforboth,MM = PV andMM =VP.
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Eur.Phys.J.C(2015)75 :340 Page 5 of 34 340
Table3 ObservablesofB→VV decaysmediatedbyb→sandb→dtransitionsthatareusedinthefit
b→s b→d
B→K∗ρ B→K∗φ[36–38] B→K∗ω Bs →φφ Bs →K∗K∗[39] Bs →K∗φ B→K∗K∗
K∗0ρ0:B,C, fL K∗0φ:B,C,CL,⊥, K∗0ω:B,C, fL φφ:B, fL K∗0K∗0:B, fL K∗0K∗0:B, fL
K∗+ρ−:B,C, fL fL,⊥,φ(cid:7),⊥ K∗+ω:B,C, fL K∗+φ:B, fL K∗+K∗0:B, fL
K∗+ρ0:B,C, fL K∗+φ:B,C,CL,⊥,
K∗0ρ+:B,C, fL fL,⊥,φ(cid:7),⊥
ables that already account for the cancelation of common 3 B → MM inQCDfactorization
experimentalsystematicuncertainties,whichareonlyacces-
sibletotheexperimentalcollaborationsthemselves.Thisis Here we revisit the building blocks that arise in QCDF to
important, since usuallyoutsiders arenot intheposition to calculate the final-state-dependent corrections needed for a
account retroactively for cancelations of systematic errors suitablepredictionofthedecayamplitudesλ Eq.(2.8)and
f
andarerestrictedtotheapplicationofrulesoferrorpropaga- detailsoftheirtreatmentinouranalysis.Mostimportantly,
tiontotheuncertaintiesofthemeasurementsoftheinvolved the parametrization of the endpoint divergences arising in
components,whichthenmightresultintooconservativeesti- weak-annihilation(WA)andhard-scattering(HS)contribu-
mates.Ofcoursesuchaprocedureontheexperimentalside tionsaregiven,whichwillbedeterminedfromexperimental
requires that the according decay modes with charged and datainSects.4and5intheframeworkoftheSMandscenar-
neutral initial/final states can be analyzed simultaneously, iosofNP,respectively.Further,wedescribeinSect.3.2the
whichisthecaseforBabar,Belle,andalsoBelleII.Inthis determination of the relative magnitude of WA amplitudes
contextitshouldbenotedthatratiosofgaussiandistributed comparedtotheleadingonesintheSMandNPscenarios,as
quantities are not gaussian distributed, although the differ- theyareformallyofsubleadingorderin1/m ,butchirally
b
encesaresmallaslongasthetailregionsofthedistribution enhanced.
do not contribute. The details of the treatment of Gaussian In our analysis all decay modes are driven by the same
and ratio of Gaussian distributed experimental probability flavortransitionb → D (D = d,s),whichisdescribedby
distributionsofthemeasurementsaregiveninAppendixB. theeffectiveHamiltonianofelectroweakinteractions.Inthe
The Tables 1, 2, and 3 show that the decay systems SM[2,3]
B → Kπ, K∗π, Kρ, K∗ρ(andK∗φ)aretheoneswiththe
(cid:7)
mostmeasuredobservables,allowingtoinvestigatethecom- (cid:11)
G
plementarityoftheconstraintsimposedonthephenomeno- HD = √F λ(D) C Op+C Op
eff p 1 1 2 2
2
logical parameter of WA by branching fractions versus CP p=u,c
(cid:8)
asymmetries versus other observables in VV final states. (cid:11)10
In these cases we can also form the ratios of branching + CiOi +C7γO7γ +C8gO8g +h.c., (3.1)
fractionsEq.(2.15)anddifferencesofCPasymmetriesEq. i=3
(2.16).Wewillperformfitsusingtwodifferentsetsofobserv-
ables for these systems. In the first, called “Set I”, we will where G denotestheFermiconstantandλ(D) ≡ V V∗
F p pb pD
use four branching fractions and four direct CP asymme- are products of elements of the CKM matrix. The flavor-
tries.Inthelackofpreciseexperimentaldataonthemixing- changingoperatorsare
induced CP asymmetry S, we rather prefer to predict them
fAropmpenthdeixreBs.u4ltfsoorfdethtaeilfistotnhetnheinpcrloucdeidnugreth.eSmucihnptrheedifictt;iosnees O1p =(D¯p)V−A(p¯b)V−A,
canbetestedwithmeasurementsofSbyBelleIIandLHCb O2p =(D¯αpβ)V−(cid:11)A(p¯βbα)V−A,
inthenearfuture[40–42]andaregivenfortheSMandsome O3(5) =(D¯b)V−A (q¯q)V∓A,
NP fits in Tables 7 and 10. The second “Set II” contains
q
(cid:11)
tthherefeulrlaytiionsdeRpcBe,,nnMdae,nMtbo,btsherreveabdlierescotfCoPneabsyramnmcheitnrgiefsraCctiaonnd, O4(6) =(D¯αbβ)V−A (q¯βqα)V∓A,
thedifferenceofCPasymmetriesΔC—seeTable5forthe (cid:11)q
explicitlistofobservables.Insummary: O7(9) = 3(D¯b)V−A eq(q¯q)V±A,
2
q
(cid:11)
SetI:(4×B)+(4×C) O8(10) = 3(D¯αbβ)V−A eq(q¯βqα)V±A,
SetII:(1×B)+(3× Rc,n)+(3×C)+ΔC. (2.19) 2 q
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340 Page 6 of 34 Eur.Phys.J.C(2015)75 :340
O7γ =−e8πm2b (D¯ σμν(1+γ5)b)Fμν, smon,wcoheefrfiecaiseαntsdoofesth.EeqwueivaakleHnatmarigltuomnieanntsdiontnhoetfrruamnebweloorwk
O8g =−g8sπm2b(D¯ασμν(1+γ5)Tαaβbβ)Gaμν, (3.2) ofbQThCeDNFNcaLnObsαefocuonrdreicnti[o4n3s].to TI,II are work in progress
s
where(q¯1q2)V±A =q¯1γμ(1±γ5)q2,thesumisoveractive andbynowtheonlylackingpartarecorrectionstoTI from
quarks q = (u,d,s,c,b), with e denoting their electric QCD- and QED-penguin operators i = 3,...,10 as well
q
chargeinfractionsof|e|andα,βdenotingcolorindices.The as the dipole operators i = 7γ,8g. These NNLO correc-
Wilsoncoefficients,C ,areobtainedbyamatchingcalcula- tions are especially important for decays under considera-
i
tionatascaletypicallyoftheorderO(M )oftheW-boson tionherebecausestrongphasesaregeneratedinQCDFonly
W
massandareevolveddowntothelowenergyscaleofO(m ) at NLO and higher order corrections might be large, apart
b
of the bottom-quark mass by means of the renormalization fromthereductionofrenormalizationschemedependences.
group(RG)equation.Here,wewillusethemodifiedcounting Inthecaseofthecolor-allowedandcolor-suppressedcurrent-
p
scheme,asdescribedin[3],inwhichthedominantpartofthe current contributions due to O1,2, the NNLO contributions
electroweakpenguinWilsoncoefficientsC (i =7,...,10) toTIcancelinlargepartsforboth,realandimaginaryparts,
i
are treated as a leading order effect. The electroweak pen- [44–46] with the ones to TII [47–49] in the corresponding
guin operators belong to the isospin-violating part of HeDff amplitudesα1,2(MM)[46,50]for MM =ππ, ρπ,leaving
and affect in principle isospin-asymmetric observables, as them close to NLO predictions. This might not be the case
forexampleEqs.(2.15)and(2.16). forfinalstatesconsideredhere.
As it is discussed in detail in the literature [3,5], contri-
3.1 WeakannihilationinQCDF butionsfromHSandWAtopologies,whicharesubleading
in1/m ,eludesofarfromasystematictreatmentinQCDF.
b
AswasestablishedbyBenekeetal.[2,3],thematrixelements However,theycanbechirallyenhancedandcontributesiz-
of theinvolved operators can betreated systematicallyina ablecorrectionsinpredictions.Duetotheignoranceofthe
1/m expansionthathasbecomeknownasQCDfactoriza- respectiveQCDmechanisms,additionalphenomenological
b
tion(QCDF).Atleadingorderthisyields parameterswereintroduced
(cid:4)
(cid:16)M1M2|Oi|B(cid:9)=(cid:11)FjB→M1(m22) 1duTiIj(u)ΦM2(u) Xk =(1+ρk)ln Λmb ,
j 0 QCD
+(M1↔ M2)+(cid:4) 1dξdvduTiII(u,v,ξ)ΦB(ξ)ΦM1(v)ΦM2(u) ρk ≡|ρk|eiφk (3.4)
(cid:17) (cid:18) 0
Λ withthecomplexparametersρ fork = A,H.
+O QCD (3.3) k
mb IntheHStheyoriginatefromtermsinvolvingtwist-3light-
coneDAsΦ (y)withΦ (y) (cid:13)= 0for y → 1inconvolu-
twotermswithhard-scatteringkernels TI,II,whicharecal- m1 m1
tions
culable in perturbation theory to higher orders in the QCD
(cid:4) (cid:4)
tcioounpalimngplαitsu.dTeshe(DyAar)eocfotnhveolliugthetdmweisthonlsig,hdte-ncootneeddaisstΦribMui-, 01 Φm11−(y)ydy ≡Φm1(1)XH + 01 Φ[1m−1(yy)]d+y, (3.5)
andaremultipliedbythecorrespondingheavy-to-lightform
factors FB→Mi in the case of TI and involve an additional whichareregulatedbytheintroductionofthephenomeno-
j
convolutionwiththe B-mesondistributionamplitudeΦB in logicalparameterXH,2representingasoft-gluoninteraction
thecaseofTII.InEq.(3.3),themesonM1inheritsthespec- withthespectatorquark.Asindicatedabove,itisexpected
tatorquarkofthedecaying B meson,anddependingonthe that X ∼ ln(m /Λ ) because it arises in a perturba-
H b QCD
finalstate,thedecayamplitudemightdependalsoonmatrix tive calculation of these soft interactions that are regulated
elementswith M1 ↔ M2;see[3,5]fordetails. in principle latest by a physical scale of order ΛQCD. Nei-
At leading order in 1/mb, the perturbative kernels TI,II thertheadequatedegreesoffreedomnortheirinteractions,
havebeencalculateduptoNLOinstrongcouplingαs [2,3] whichshouldbeusedinaneffectivetheorybelowthisscale
andthroughoutwewillstaywithinthisapproximation.Con- areknown.Itisalsoconceivablethatfactorizationmightbe
trary to previous works [3,5,15], we employ Wilson coef- achievedatsomeintermediatescalebetweenm andΛ .
b QCD
ficients of the weak Hamiltonian evaluated at the scale mb Thefactor(1+ρH)summarizestheremainderofanunknown
eveninWAandHScontributions,onlythest(cid:19)rongcoupling nonperturbativematrixelement,includingthepossibilityof
α isevaluatedatthesemi-hardscaleμ = Λ m .In
s h QCD b
theSCETapproachthisisapparentasasubsequentmatching 2 InprincipleonemightintroduceaseparateXH foreachmesonM1
stepfromQCDtoSCETItakenatμ∼mbsuchthattheWil- andM2aswellasforeachoperatorinsertion.
123
Eur.Phys.J.C(2015)75 :340 Page 7 of 34 340
astrongphase,whichaffectsespeciallythepredictionsofCP which in principle are different for each meson and each
i,f i,f
asymmetries.Thenumericalsizeofthecomplexparameter building block A . Explicit expressions for A in terms
k k
ρ is unknown, however, too large values will give rise to of X aregivenfor MM = PP, PV, VP, VV inthelit-
H A
numericallyenhancedsubleading1/m contributionscom- erature [5,23,51,52], but independently one has Af = 0.
b 1,2
pared to the formally leading terms putting to question the Asafurthersimplification,itisassumedintheliteraturethat
validityofthe1/m expansionofQCDF. thereisonlyonephenomenologicalparameter,independent
b
WAisentirelysubleadingin1/m andconsistsinprinci- of meson type and Dirac structure, such that Ai,f(X ) are
pleofsixdifferentbuildingblocks Abi,f (k =1,2,3),which functions of the same parameter. In this contexkt we wAould
k
arecharacterizedbygluonemissionfromtheinitial(i)and like to note that in the most relevant WA amplitude βc the
3
final (f) states and the three possible Dirac structures that buildingblock Af isparametricallyenhancedby N andin
3 c
are involved: k = 1 for (V − A) ⊗ (V − A), k = 2 for theSMalargeWilsoncoefficient.Inconsequenceitscontri-
(V −A)⊗(V +A),andk =3for(−2)(S−P)⊗(S+P). butiondominatesovertheonesofAi .Itshouldbenotedthat
1,3
Theycontributetonon-singletannihilationamplitudeswith theWAamplitudes(3.6)inthelight-conesumrule(LCSR)
specificcombinationsofWilsoncoefficientsofthe4-quark approach exhibit the same dependence on the products of
operators[3,5] Wilsoncoefficientsandbuildingblocks[53],however,inthis
i,f
C approach the calculation of A does not suffer from end-
b = FC Ai, k
1 N2 1 1 pointsingularitiesduetodifferentassumptionsandapprox-
c
imations. With the latter in mind, a more general approach
C
b2 = NF2C2Ai1, wouldbetointerpretthebuildingblocksthemselvesasphe-
c (cid:12) (cid:13) nomenologicalparameters,orequivalentlyintroduceoneXA
C
bp = F C Ai +C (Ai + Af)+N C Af , foreachofthem.Wheninvestigatingnew-physicseffects,it
3 N2 3 1 5 3 3 c 6 3
c (cid:12) (cid:13) is desirable to keep the explicit dependence on the Wilson
C
bp = F C Ai +C Ai , coefficients in (3.6) since they depend on NP parameters,
4 N2 4 1 6 2 including new weak phases. In the case of non-negligible
c (cid:12) (cid:13)
bp = CF C Ai +C (Ai + Af)+N C Af , WAcontributions,theCPasymmetriesandbranchingfrac-
3,EW N2 9 1 7 3 3 c 8 3 tionswillbesensitivetotheinterferenceofthenew-physics
c (cid:12) (cid:13)
bp = CF C Ai +C Ai , (3.6) phasesandthestrongphasesfrom XA.
4,EW N2 10 1 8 2 As already indicated, the phenomenological parameters
c
XA,H areunknownandtheirsizeisconventionallyadjusted
anddependon M1 and M2.Here, Nc =3denotesthenum- withinsomerange|ρA,H|(cid:2)2toreproducedatawhereasthe
ber of colors and the color factor CF = 4/3. In particu- phaseφA,H iskeptarbitraryandvariedfreelytoestimatethe
lar,theycorrespondtotheamplitudesduetocurrent-current uncertainty in theoretical predictions of observables within
(b , b ), QCD-penguin (bp, bp) and electroweak penguin QCDFduetoWAandHS.Thisprocedureshowedthephe-
1 2 3 4
(bp , bp )annihilation.Belowwewillfrequentlyrefer nomenological importance of WA and constitutes a major
3,EW 4,EW
toWAamplitudeswiththenormalization[5] source of theoretical uncertainty in predictions within the
⎧
SM[5]andsearchesbeyond[15]andbelowwewillreferto
⎪⎪⎪⎨ bi(p) for M M = V±V± itas“conventionalQCDF”.
1 2
βi(p) =Nβ ⎪⎪⎪⎩mbi(Mp)2 forallothers (3.7) alsoInρtHhi—swfroormk,dwaeta.arAesgaoicnognstoeqfiuteρncAe—,naondprfeodricBtio→nswKiπll
m bepossibleforthoseobservablesthatareusedinthefit,while
B
the fitted values of ρ depend on the short-distance model
A
wheretheargument M1M2 hasbeensuppressedandNβ ≡ under consideration. Yet, the consistency of the underlying
fB fM1/(mBFB→M1)isindependentonM2.Asinthecaseof short-distance model can be tested. We perform our fits in
HS,theendpointsingularitiesinWAamplitudesareregulated theframeworkoftheSM,andfurtherinnew-physicsscenar-
in a model-dependent fashion. The results are expressed in iossimultaneouslywiththeadditionalNPparameters.Inthe
termsofconvolutionsofhard-scatteringkernelswithDAsof lattercase,thedeterminationoftheNPparameterswilltake
twist-2andchirallyenhancedtwist-3,involvingphenomeno- into account the uncertainty of the WA contribution when
logicalparameters XA marginalizingoverρA.
(cid:4)
1 dy This procedure is different from conventional QCDF in
y → XA, as much as it assumes one universal parameter ρA for all
(cid:4) 0 observablesinonespecificdecaymode.Indeed,inconven-
1 lnydy 1
→− (XA)2, (3.8) tional QCDF the independent variation of ρA,H for each
y 2
0
123
340 Page 8 of 34 Eur.Phys.J.C(2015)75 :340
observableinaspecificdecaycorrespondstoadifferentWA interest to know the relative magnitude of WA to leading
(andHS)parameterforeachobservable.However,sincein amplitudesforthebest-fitregionsofρA(H).Forthispurpose
QCDF the parameters ρA,H are introduced at the level of weintroducethequantities
decay amplitudes one would expect that they are the same
(cid:9) (cid:9)
forallobservablesofaspecificdecaymode.Consequently, (cid:9) β (ρ ) (cid:9)
conventionalQCDFallowsforsituationswhereexperimental ξiA(ρA)=(cid:9)(cid:9)α i A (cid:9)(cid:9), (3.9)
(i+Δi3),I
measurementsandtheorypredictionsfortwoobservablesare
inagreement,althoughforthefirstobservabletheagreement
forWAand
is reached for values of φA,H that might be very different (cid:9) (cid:9)
forbosmervthaobslee.where the agreement is reached for the second ξiH(ρH)=(cid:9)(cid:9)(cid:9)(cid:9)ααit,wI+I−3α(ρtwH−)2(cid:9)(cid:9)(cid:9)(cid:9), (3.10)
In the lack of precise data for most of the decays, we i,I i,II
makethefurtherassumptionofaWAparameterthatiseven
universal for decay modes that are related by the exchange for HS amplitudes. For the latter, αtw−3 denotes the sub-
i,II
of (u ↔ d) quarks. As an example, this allows one to leading,butchirallyenhanced,twist-3contribution,whereas
combine observables of the four decay channels B¯0 → αtw−2 is the leading HS contribution from twist-2 DAs,
K¯0π0, K−π+, and B− → K−π0, K¯0π−, to which we wih,IiIch is free of endpoint divergences. The leading ampli-
refer as “decay system” B → Kπ. All considered decay tudes are splitted into αi = αi,I +αi,II, with the two con-
systems and the according observables have been listed in tributionsfromkernelIandIIintroducedinEq.(3.3).Note
Tables1,2,and3.Thisassumptionismotivatedbythecir- that ξ3A = |β3/α4,I|. This definition is generalized for the
cumstancethatthedominantcontributionstotheamplitude case MM = VV, where ξA is defined as the mean value
inallconsidereddecayscomeactuallyfromthelinearcombi- ofthecorrespondingratiosforthelongitudinalandnegative
nationαˆ (M M )=α (M M )+β (M M ),whichisdue polarizedamplitudesξA ≡(ξA +ξA )/2.
4 1 2 4 1 2 3 1 2 i,VV i,L i,−
toisospin-conservingQCD-penguinoperators O3,...,6 (3.2). Themostimportantcontributionfrompower-suppressed
The definition of all α ’s can be found in [5], whereas β ’s corrections are clearly obtained from HS in α , which is
i i 2
are given in Eq. (3.7). Other assumptions have been tested enhancedbythelargeWilsoncoefficientC andfromtheWA
1
inthe literatureas for example universal weak annihilation correctionβ inQCD-penguin-dominateddecays.Therefore
3
among B and B decaysintofinalstatescontainingkaons ξH andξAwillplayanimportantroleinthephenomenolog-
s d 2 3
andpions[54]. icalpartofthiswork.
Theprocedurereflectsthegeneralideainherentto1/m In the SM, the ξA-ratios depend exclusively on ρ and
b A
expansions,whichaimatafactorizationintoshort-distance contour lines of constant ξA can easily be obtained in the
anduniversalnonperturbativequantities,wherethelatterare complex ρ -plane. Concerning fits in new-physics scenar-
A
determined from data in the lack of first-principle determi- ios,theξA-ratiosdependinadditiononnew-physicsparam-
nations. Presently, however, factorization theorems are not etersxNP,wheredim(xNP)correspondstotheirnumber.The
yet established at subleading order that would support the dependence is both, explicit in the Wilson coefficients and
existenceofsuchuniversalquantities.Whilestepshavebeen implicit on data via the likelihood. In this case one would
madetoshowthatweakannihilationisfactorizableandreal be interested in the minimal value of ξA(xNP) in the 68 %
i
[55], doubts are cast on the reality condition [56]. In view credibilityregions(CR)ofallNPparameters,butmarginal-
ofthis,ourstudycanaffirmatmostexperimentalevidence ized over ρ . Since the determination of this CRs requires
A
againsttheassumptionofoneuniversalparameterperdecay huge computational efforts when dim(xNP) > 2, we pro-
system. Therefore a positive affirmation may not be over ceeddifferently.Inthecourseofthefit,wehistograminall
interpreted.Finally,itmustalsobenotedthatcontributions two-dimensional subspaces of NP parameters (xNP, xNP),
a b
ofnotincludedNNLOcorrectionscouldbesizeableandin with a (cid:13)= b, those values ξA in each bin (xNP, xNP) that
a b
ourfitstheyareinterpretedaspartofthephenomenological belong to the largest likelihood value when sampling the
WAparameter. complementary subspace of the remaining NP parameters
xNP with c (cid:13)= a and c (cid:13)= b. As a result, in each of
c
3.2 Sizeofpower-suppressedcorrections the 2D-marginalized planes, “labeled” by (a,b), the 68 %
CR will contain a smallest and a largest ξA in one of the
i
Inthiswork,wedeterminethesizeofsubleadingWA(and bins in (xNP, xNP), which all belong to the minimal χ2
a b
HS)contributionsfromdataintheframeworkoftheSMand in the subspace of x . As the final range we choose the
c
NPscenarios.Duetothechiralenhancement,WAcontribu- minimum of the smallest values and the maximum of the
tionsarenotnecessarily1/m suppressednumericallywith largestvaluesfromallthepairs(a,b)inourNPanalysesin
b
respect to the leading order amplitudes. Therefore, it is of Sect.5.
123
Eur.Phys.J.C(2015)75 :340 Page 9 of 34 340
4 Weakannihilationinthestandardmodel Our findings for lower and upper bounds on ρ in the
A
68%CRaresummarizedinTable4forallconsidereddecay
Inthissectionwepresenttheresultsofthedeterminationof systems. It can be seen that data requires nonzero values
the WA parameter ρ from data of various QCD-penguin- of |ρ | to be in agreement with QCDF predictions in the
A A
and WA- dominated nonleptonic charmless B → MM SM. In some cases they are much larger compared to the
decays in the framework of the SM. This includes charac- conventionallyadaptedranges,allowingthusinprinciplefor
teristics of the best-fit regions, the p values at the best-fit abetteragreementoftheoreticalpredictionswithdata.Since
pointandpullvaluesforobservables,aswellastherelative weuseWilsoncoefficientsatthescaleμ∼m inWA(and
b
amountofthesubleadingWAcontributionneededtoexplain HS)contributions,contraryto[5,15],ournumericalvalues
the data, which we quantify by the ratio ξA defined in Eq. of|ρ |areingeneralabitlargercomparedtotheonesknown
3 A
(3.9). in the literature. Representing the size of a nonperturbative
We start with an extensive discussion of the B → Kπ quantity,|ρ |isexpectednaivelytobeoforderone,whereas
A
system,whichshowsthelargestdeviationsfromSMpredic- too large values would put in doubt the convergence of the
tions for the difference of CP asymmetries ΔC(Kπ) (see 1/m expansion.
b
Eq.(2.16)),commonlyknownintheliteratureasthe“ΔA FurtherwelisttheratioξA ofWAamplitudestoleading
CP 3
puzzle”andtoalesserextentintheratioRB(Kπ).Weinves- ones as a measure of the numerical relevance of these for-
n
tigatethe“ΔA puzzle”furtherinasimultaneousfitofthe mallysubleadingbutchiralityenhancedcontributions.While
CP
parameterofWA,ρ ,andHS,ρ ,anddiscusstheimplica- largesubleadingWAamplitudesdonotnecessarilyimplya
A H
tionsonotherCPasymmetriesin Bd,s → Kπ. breakdown of the 1/mb expansion, it is generally assumed
We turn then to the discussion of the decays B → thatthefactorizationachievedatleadingorderprovesuseful
Kρ, K∗ρ, K∗π, which allow also for studies of different onlyifsubleadingWAamplitudesarenotbetoolarge.Using
setsofobservablesinSetIandSetIIduetotherathernumer- the priors for our analysis which result in a wide range for
ousandquiteprecisemeasurements.Subsequently,wedis- ξA, we can explicitly show that data does not require huge
3
cussbrieflytheresultsforotherdecayslistedinTables1,2, WA.Atthebest-fitpointofρ theaccordingvalueisindeed
A
and 3 with some special comments on B → Kω and ξA < 1 for many decay systems. Although at the best-fit
3
B → K∗φ.Foreachdecaysystemwepresentseparatecon- pointξAmightreachvaluesupto2oreven3forsomedecay
3
straintsfrombranchingfractions,CPasymmetries,polariza- systems,onceconsideringthe68%CRinρ ,itispossible
A
tionfractionsandrelativephasesontheWAparameterρ , tohaveagainξA <1(exceptforB → K∗K∗)fortheprice
A 3 s
besidesthecombinedones. ofsometensionamongdataandprediction.Bearinginmind
Apart from the above listed penguin-dominated decays, the chirality enhancement, our fits of the data thus do not
wealsostudydecaysmediatedsolelybyweakannihilation, indicateanomalouslyhugeWAcontributions,whichputthe
suchas B → K+K−and B →π+π−.Beingindependent usefulnessofthe1/m expansionintoquestioninprinciple.
s b
of β and hence Af, these decay modes are sensitive to a By definition, there is no ξA for the two pure WA modes
3 3 3
WA contribution from Ai and provide access to different B → K+K−and B →π+π−.
1,2 d s
buildingblocks.
Based on our previous fit results, we discuss finally the 4.1 Resultsfor B → Kπ
assumptionofauniversalWAfor B and B decaysintothe
d s
same final state and investigate in particular consequences Inthissectionwediscussfirstfitsofonlyρ fromB → Kπ
A
for CP asymmetries in B → Kπ in view of the “ΔA observables encountering thereby the well-known “ΔA
s CP CP
puzzle”in B → Kπ. puzzle”.IntheSMitcanbeexplainedonlyreasonablywell
The statistical procedure used in all fits is described in inthepresenceoflargecolor-suppressedcurrent-currenttree
Appendix B. In the SM, we deal mostly with the fit of one contributionsthatarestronglydependentonρ .Therefore
H
complex-valuedparameterρ exceptforthe B → Kπ sys- weperforminasecondstepasimultaneousfitofρ andρ
A A H
tem,wherewealsoperformasimultaneousfitofρ andρ . toamodifiedsetof B → Kπ observables,anddiscussthe
A H
Whenfitted,forbothparametersauniformprior possibility to discern large color-suppressed tree contribu-
tionsinfuturemeasurementsofCPasymmetrieswithK0π0
0≤|ρA,H|≤8, 0≤φA,H ≤2π, (4.1) finalstates.Itshouldbeemphasizedthatingeneralboth,ρA
and ρ should be considered in fits due to their different
H
is assumed and no restriction is imposed on the phases. In contributions to CP-averaged and asymmetric observables.
comparison,inconventionalQCDFthemagnitude|ρA,H|≤ Hence simultaneous fits of ρA and ρH are in principle of
2isusedforuncertaintyestimatesoftheoreticalpredictions. greater interest, but apart from B → Kπ data, the current
In the case that ρ is not fitted, but treated as a nuisance precision of other decay system’s data does not yet permit
H
parameterinstead,weuse|ρ |=1andvary0≤φ ≤2π. thiskindofanalysis.
H H
123
340 Page 10 of 34 Eur.Phys.J.C(2015)75 :340
Table4 Compilationofthepower-suppressedratioξA atthebest-fit fitwiththeobservableSetII.ThepureWAdecay B0 → K+K− is
3
point(BFP)andinthe68and95%CRs,aswellaslowerandupper notincludedinthedecaysystem B → KK andρA-boundsaregiven
boundsonthefitparameter|ρA|inthe68%CRforallrelevantdecay separatelyinparentheses
systems.For B → (Kπ, K∗π, Kρ, K∗ρ),valuescorrespondtothe
B→Kπ B→Kη B→Kη(cid:5) B→KK Bs →KK Bs →ππ Bs →Kπ
MM = PP
ξA
3
BFP 0.39 0.08 1.83 0.58 1.83 – 0.96
68%CR [0.37;0.54] [0.00;−] [0.18;3.25] [0.00;2.07] [0.02;2.09] – [0.56;1.54]
95%CR [0.34;0.69] [0.00;−] [0.16;3.34] [0.00;2.10] [0.00;2.13] – [0.44;1.83]
|ρ |
A
Lower >1.8 >0 >0.9 >0(0.9) >0 >3.4 >2.3
Upper <3.9 – <7.7 <6.1(8.6) <5.5 <10.9 <4.8
B→K∗π B→Kρ B→K∗η B→K∗η(cid:5) B→Kφ B→Kω
MM = PV
ξA
3
BFP 0.89 0.78 2.74 0.48 0.50 2.7
68%CR [0.75;1.40] [0.39;1.55] [0.71;3.77] [0.02;7.84] [0.40;2.41] [0.63;2.88]
95%CR [0.69;1.56] [0.16;2.18] [0.64;5.06] [0.02;8.41] [0.32;2.54] [0.57;2.88]
|ρ |
A
Lower >1.4 >0.8 >1.1 >0 >0.8 >1.3
Upper <3.4 <3.4 <4.4 <6.1 <3.6 <4.3
B→K∗ρ B→K∗φ Bs →K∗φ B→K∗K∗ Bs →K∗K∗ Bs →φφ B→K∗ω
MM =VV
ξA
3
BFP 1.33 0.38 1.53 1.84 3.01 0.50 0.91
68%CR [0.84;1.94] [0.31;0.43] [0.30;2.05] [0.85;2.68] [1.94;3.79] [0.49;1.11] [0.20;1.39]
95%CR [0.56;2.33] [0.25;0.50] [0.10;2.15] [0.09;2.90] [0.96;4.17] [0.41;1.38] [0.09;1.46]
|ρ |
A
Lower >1.0 >0.6 >0.3 >1.2 >1.6 >0.7 >0.3
Upper <2.9 <1.8 <3.2 <3.0 <3.6 <2.3 <2.4
The B → Kπ system offers the most precise measured posed observables in Set II at the best-fit point: −2.8σ for
branchingfractionsandCPasymmetries(seeTable1)among ΔC(Kπ)and−1.9σforRB(Kπ),comparedtoSetI:−2.1σ
n
thedecaysconsideredhere.Inconsequence,wefindstringent forC(B− → K−π0),showingtheimportanceandcomple-
boundsontheWAparameterρKπ.ThiscanbeseeninFig.1a mentarity of composed observables. The large pull values
A
when using the observables in Set I and Fig. 1b for Set II. in CP asymmetries arise from the higher statistical weight
Theallowedregionsfromboth,Band/orRcB,,nMa,Mb (blue)as ofthemorepreciselymeasuredbranchingfractionsintheir
well as C and/or ΔC (green) are very distinct leaving two combinedfit,reflectingthe“ΔA puzzle”inthe B → Kπ
CP
tinyoverlapregions(red)at68%probabilityaroundρKπ ≈ sytem.TheindividualpullvaluesofC(B− → K−π0)and
A
2.1exp(i5.5) and ρKπ ≈ 3.4exp(i2.7), which differ only C(B¯0 → K−π+)inSetIadduptothelargepullofΔC(Kπ)
A
slightly for both Sets I and II. The best-fit points listed in inSetII.
Table5fallintothesolutionwithlarger|ρKπ| ≈ 3.4,butit Inthefollowingwewillelaborateontheconstraintsposed
A
mustbenotedthattheothersolutionprovidesalmostequally byindividualobservables.Forexample,thegeneralshapeof
goodfitsintermsofχ2. thecontourofbranchingfractionscaneasilybeunderstood
As shown in Table 5, the p values at the best-fit points as follows: The leading contribution of the decay ampli-
ofthefitsofSetIandSetIIareverydifferent:0.44versus tude αˆc(Kπ) = αc(Kπ) + βc(Kπ) is given as the sum
4 4 3
0.04,respectively.Thereasonarelargepullvaluesofcom- oftheQCD-penguinandtheρ -dependentWAamplitudes
A
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