The FSI contribution to the observed $B_s$ decays into $K^+K^-$ and $π^+K^-$

0.15 MB·English

by  Xu-Hao Yuan

The FSI contribution tothe observed B decays into K+K− and π+K−
s
Xu-Hao Yuan1, Hong-Wei Ke2 , Xiang Liu3, and Xue-Qian Li4
∗ †‡ § ¶
1Center for High Energy Physics, Tsinghua University, Beijing 100084, China
2School of Science, Tianjin University, Tianjin 300072, China
3School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
4School of Physics, Nankai University, Tianjin 300071, China
BecauseatthetreelevelBs → K+K− isCabibbo-triplesuppressed,soitsbranchingratioshouldbesmaller
thanthatofBs → π+K−. ThemeasurementspresentareversedratioasR = B(Bs → π+K−)/B(Bs →
K+K−) ∼ 4.9/33. Therefore, IthasbeensuggestedthatthetransitionBs → K+K− isdominatedbythe
penguin mechanism, whichisproportional toV V∗. Inthiswork, weshow that anextracontributionfrom
cb cs
thefinalstateinteraction(FSI)toBs →K+K−viasequentialprocessesBs →D((s∗))D¯s(∗) →K+K−isalso
2 substantialandshouldbesuperposedonthepenguincontribution.Indeed,takingintoaccountoftheFSIeffects,
1 thetheoreticalpredictiononRiswellconsistentwiththedata.
0
2

t
c
O I. INTRODUCTION

6
2 ItisgenerallybelievedthatthehierarchyoftheCabibbo-Kabayashi-Maskawa(CKM)matrixelementsdeterminesthemag-
nitudes of weak decays, and off-diagonal matrix elements would bring up an order suppression. For example, the ratio

] of (B D+π )/ (B D K ) = 3.2 10 3/3.0 10 4 has been measured [1, 2], and it is roughly deter-
h B s → s − B s → s± ∓ × − × −
mined by the ratio of the CKM matrix elements V /V . Thus one usually categorizes the weak decays according to their
p ud us
CKM structures as the Cabibbo-favored; Cabibbo-suppressed and even the Cabibbo-double or triple-suppressed. However,
-
p the newly observed modes B K+K and B π+K obviously do not follow the rule, namely at first look, the
s − s −
e branching ratio of B K+K→should be smaller→than that of B π+K by V /V 2, by contraries, the datum is
h s → − s → − | us ud|
= (B π+K )/ (B K+K ) = 4.9 10 6/3.3 10 5 [3–5]. Ifonlythetreediagramsaretakenintoaccount,
[ tRhis reBverssio→n would−comBposes →an “anom−aly”. Des×cotes−-Genon×et al−. [6] carefully analyzed the transitions of B K0K¯0
3 andK+K throughflavorsymmetriesandQCDfactorization,andpointedapotentialconflictbetweentheQCDpsre→dictionon
−
v B K+K and data. By analyzingthe transition mechanism, Cheng and Chua [7] determinedthatthe main contribution
s −
6 →
to B K+K comesfromthe penguindiagram,which is proportionalto V V . Ali et al. also calculatedthe branching
8 ratiosso→fB π−+K andB K+K intermsofthePerturbativeQCD(PQcbCDc∗s)toLOandNLO[8,9]andtheauthorsof
6 s − s −
→ →
3 Ref. [10]didcalculationsinSCET.Theirresultsroughlywereconsistentwiththedataavailablethen,sothe“conflict”seemed
. resolved. Eventhoughthetheoreticaluncertaintiesin allthe calculationsare notfullycontrolled,itisnotedthattheobtained
0
central values are not sufficiently large to make up the data. It implies that there must be some mechanisms to remarkably
1
enhancethebranchingratioofB K+K .
2 s −
→
1 Lookingatthecentralvaluestheyobtainedandtheresultantratioof (Bs π+K−)/ (Bs K+K−),onecanfindthat
: the calculated (B π+K ) is close to the data, however, the cenBtral va→lues of (BB K→+K ) calculated in various
v B s → − B s → −
approachesaresmallerthanthenewlymeasureddata[4,5].
i
X Based onthisobservation,wesuggestthatthefinalstate interaction(FSI)inB decaysmaygreatlyenhancethebranching
s
r ratioofBs K+K−butnotmuchforBs π+K−.Infact,intheenergyregionsofb-quarkandc-quarkmostsuchanomalies
a canbenatu→rallyexplainedbyconsideringth→eroleofFSI.Forexample,asimplequarkdiagram-analysistellsthatthebranching
ratioofD0 K0K¯0isalmostzero,butitsmeasuredvalueiscomparablewiththatofD0 K+K ,whichislarge.Thiscan
−
beunderstoo→dbyconsideringthesequentialprocessD0 K+K K0K¯0andthelater→stepisahadronicscattering[11].
−
→ →
TheParticleDataGroup(PDG)[3]tellsusthattheDs(∗)+Ds(∗)− arethedominanthadronicdecaymodesofBs,thereforeit
implies that the sequentialdecays Bs Ds(∗)+Ds(∗)− K+K− would compose an importantcontributionto the observed
→ →
Bs K+K−. The firststep of Bs Ds(∗)+Ds(∗)− is onlysuppressedbyVbc andthe processdoesnotsufferforma color-
→ →
suppression.Moreoveritisalsoisospin-conservedmode,eventhoughtheweakinteractiondoesnotrequireisospinconservation,
itstillmaybemorefavorablethantheisospinviolatedones. Thus, onecanunderstandwhyBs Ds(∗)+Ds(∗)− isdominant.
→
†Correspondingauthor
∗Electronicaddress:[email protected]
‡Electronicaddress:[email protected]
§Electronicaddress:[email protected]
¶Electronicaddress:[email protected]
2
Thenletuslookatthenextstep.TheFSIisahadronicscatteringprocess,whichiscompletelygovernedbystronginteraction,so
thattheisospinmustbeconserved.TheisospinofDs(∗)±iszero,thus,thefinalstateofthescatteringisrequiredtobezero.The
isospinofK-mesonsis1/2,thustheKK¯-statescanbeeitherisospin0or1, thereforetheinelasticscatteringDs(∗)+Ds(∗)−
K+K is allowed. By contraries, isospin of pion is 1, thus, the system of π+K doesnot containan isospin 0 compone→nt,
− −
thus the scattering Ds(∗)+Ds(∗)− π+K− is forbidden. Definitely, one can expect a substantial contribution from Bs
→ →
Ds(∗)+Ds(∗)− K+K−,whichcanmuchenhancethebranchingratioofBs K+K− incomparisonwithBs π+K−.
→ → →
Itisworthpointingoutthatontheotherhand,thedecayB π+K alsoreceivesacontributionfromtheFSIviaB
s − s
→ →
D(∗)+Ds(∗)− → π+K−. However, the first step process Bs → D(∗)+Ds(∗)− is Cabibbo-doublesuppressed by VcbVc∗d, so is
muchlessthanBs Ds(∗)+Ds(∗)−,thereforetheFSIdoesnotcontributemuchtothebranchingratioofBs π+K−.
→ →
Noticethat,thedirectprocessBs K+K− viapenguindiagramsandthesequentialprocesswithFSIBs Ds(∗)Ds(∗)
→ → →
K+K havethe same initial andfinal states, moreovertheir amplitudesare of the same orderof magnitude,so the two con-
−
tributionsinterfere. Infact,theircontributionsarenotdirectlyexperimentally,buttheoreticallydistinguishable. Moreover,the
strongscatteringwouldhavearealandanimaginaryparts(seebelow,forthecalculationsofthetrianglediagrams),thusaphase
isresulted. InmostcalculationsoftheFSIeffects,onlytheabsorptive(imaginary)partiskept,thereasonisthatonemayargue
thattheabsorptivepartmightbedominantoratmostthedispersiveandabsorptivepartshaveclosemagnitudes.Inthatcase,the
absorptivepartisimaginarywhilethepenguincontributionisreal,sothatthetwocontributionscanbeaddedupattheratelevel
(amplitudesquare). Ourfinalresultsindicatethatwhiletakingintoaccountthenewcontribution,thetheoreticalpredictionsare
indeedclosetothenewdata.
In this work, we calculate the amplitudes of the decay channels of Bs Ds(∗)(D(∗))D¯s(∗) and the direct decay channels
of B K+(π+)K by the factorization approach [12, 13]. Then w→e use the effective SU(4) Lagrangian [14, 15] to
s −
→
determine the vector-pseudoscalar-pseudoscalarand vector-vector-pseudoscalarvertices for calculating the FSI amplitude of
Ds(∗)(D(∗))D¯s(∗) K+(π+)K−. Thepaperisorganizedasfollows,aftertheintroduction,weformulatetheweakdecaysand
→
there-scatteringprocessesinsectionII,ournumericalresultsarepresentedalongwithallnecessaryinputsin sectionIII.The
lastsectionisdevotedtodiscussionandconclusion.
II. THETHEORETICALEVALUATIONSOFTHEBRANCHINGRATIOS
A. Weakdecaysinfactorizationapproach
Because the measurements on the decay widths of Bs Ds(∗)+Ds(∗)− and Bs D(∗)+Ds(∗)−(Bs D(∗)−Ds(∗)+) are
→ → →
notaccurateyet, as formostofthe abovechannelsthere areonlyupperbounds,so thatwe are goingto directlycalculate the
transitionamplitudesbasedonthequarkdiagrams. Eventhoughthisstrategymightbringupcertaintheoreticaluncertainties,it
doesnotbreakourqualitativeconclusionatall.
For calculating the transition amplitudes of Bs Ds(∗)+(D(∗)+)Ds(∗)−, one needs to employ the effective hamiltonian at
→
the quarklevel. With the operatorproductexpansion(OPE), the effectiveHamiltonianwas explicitlypresented in Ref. [16].
Atthetreelevel,fromtheeffectiveLagrangianonecannoticethatB K+K issuppressedbyV V i.etriple-Cabibbo
suppressed,comparingwith B π+K , whichis double-Cabibbossu→ppressed−byV . ThereforeCuhbenu∗gsandChuadecided
s − ub
thatthedirecttransitionB K→+K isobviouslydominatedbythepenguindiagramwhoseCKMstructureisV V . The
s → − cb c∗s
transitionBs → Ds(∗)+D(∗)− isalsodouble-Cabibbosuppressed. Instead,Bs → Ds(∗)+Ds(∗)− isproportionaltoVcbVc∗s andit
isatreeprocess. NowletuscomparethesequentialprocessesBs D(∗)D¯s(∗) K+K− withthepenguincontribution. The
→ →
tworeactionsareofthesameCKM structure,butthepenguinundergoesaloopsuppressionaboutα /π 0.06,whereasthe
s
∼
strongscatteringDs(∗)D¯s(∗) → iXiwhereXistandsasanypossiblefinalstatesallowedbysymmetryandenergy-momentum
conservation. K+K− isonlyoPneofthepossiblechannels,anditsprobabilityisproportionalto Ds(∗)D¯s(∗) Heff KK¯ ,which
h | | i
is what we are going to calculate in this work. This is a suppression factor because the total probabilityto all channelsis 1.
Therefor,roughly,wenoticethatthepenguinisloop-suppressedandthesequentialprocessisalsosuppressedbytheprobability,
thusthetwomodescompeteandmayhavea similarorderofmagnitude. Concretely,weneedtocalculatethem. Theexplicit
calculationonthepenguincontributioncanbefoundin[7–9],thuswewillusetheirnumbersandonlyconsiderthecontribution
fromthesequentialprocesses.
Applyingtheeffectivehamiltonianatthequarkleveltothehadronstates,thehadronicmatrixelementscanbeparameterized
as[17]:
0J P(p ) = if p ,
µ 1 P 1µ
h | | i −
0J V(p ,ǫ) = f ǫ m ,
µ 1 V µ V
h | | i
3
m2 m2 m2 m2
P(p )J B (p) = P Bs − Pq F (q2)+ Bs − Pq F (q2),
h 2 | µ| s i h µ− q2 µi 1 q2 µ 0
iǫν P P
V(p ,ǫ)J B (p) = iǫ PαqβA (q2)+(m +m )2g A (q2) µ ν A (q2)
h 2 | µ| s i mBs +mV n µναβ V Bs V µν 1 − mBs +mV 2
P q
2m (m +m ) ν µ[A (q2) A (q2)] , (1)
− V Bs V q2 3 − 0 o
whereJ =q¯ γ (1 γ )q ,P =(p +p ) andq =(p p ) .
µ 1 µ 5 2 µ 1 2 µ µ 1 2 µ
− −
WithEq. (1),wewritedowntheamplitudesofBs Ds(∗)+Ds(∗)−:
→
iG
(B (p) D+(p )D (p )) = FV V a f (m2 m2 )FBsDs(p2), (2a)
A s → s 1 s− 2 − √2 cb c∗s 1 Ds Bs − Ds 0 1
A(Bs(p)→Ds∗+(p1)Ds∗−(p2)) = G√F2VcbVc∗sa1fDs∗mDs∗mBsig+µνmDs∗niǫµναβ(p1+2p2)αpβ1ABVsDs∗(p21)+(mBs
+mDs∗)2gµνAB1sDs∗(p21)−(p1+2p2)µ(p1+2p2)νAB2sDs∗(p21)− (p1+2pp22)µp1νh(mBs
1
+mDs∗)2AB1sDs∗(p21)−(m2Bs −m2Ds∗)AB2sDs∗(p21)−2mDs∗(mBs +mDs∗)AB0sDs∗(p21)io,
(2b)
andtheamplitudesofBs D(∗)+Ds(∗)−readas
→
iG
(B (p) D+(p )D (p )) = FV V a f (m2 m2 )FBsDs(p2), (2c)
A s → 1 s− 2 − √2 cb c∗d 1 D Bs − Ds 0 1
A(Bs(p)→D∗+(p1)Ds∗−(p2)) = G√F2VcbVc∗da1fD∗mD∗mBsig+µνmDs∗niǫµναβ(p1+2p2)αpβ1ABVsDs∗(p21)+(mBs
+mDs∗)2gµνAB1sDs∗(p21)−(p1+2p2)µ(p1+2p2)νAB2sDs∗(p21)− (p1+2pp22)µp1νh(mBs
1
+mDs∗)2AB1sDs∗(p21)−(m2Bs −m2Ds∗)AB2sDs∗(p21)−2mDs∗(mBs +mDs∗)AB0sDs∗(p21)io,
(2d)
theamplitudeofπ+K atthetreelevelis:
−
iG
Adirect(Bs(p)→π+(p1)K−(p2)) = − √2FVubVu∗da1fπ(m2Bs −m2K)F0BsK(p21), (2e)
where a is a propercombinationof the Wilson coefficientsin the effectivehamiltonian[16, 18, 19]. F (q2) and A (q2)
1 0 V,1,2
are the form factors to be detrmined. In this work, due to lack of accurate data, we use the form factors obtained by fitting
thedata ofthedecaysofB meson. Itisa reasonableapproximationbecausetheprocessesB D andB D havethe
s → s∗ → ∗
sametopologicalstructure,andthe flavorSU(3)symmetryforlightquarks(u, dands) wouldleadtothe sameformfactors,
namelythe differencebetweenthe formfactorsfor differentlight-quarkflavors wouldbe proportionalto an SU(3) breaking,
whichissmallfortheeffectiveverticesaswellknown. Atleastsuchsmalldifferencewouldnotovertaketheerrorscausedby
experimentalmeasurementsandtheoreticaluncertaintiesforevaluatingthenon-perturbtiveeffects.
Moreover,a symmetryanalysisindicatesthatthe sequentialprocessB D +D (D+D ) K+K is forbiddenby
s → s∗ s− s s∗− → −
angular-momentumconservation.
Takingthethree-parameterform,theformfactorsarewrittenas[17]:
F(0)
F(q2)= , (3)
2
1 a q2 +b q2
− m2B (cid:16)m2B(cid:17)
wherea,bandF(0)arethethreeparametersandtheirvaluesarelistedinTableI.
4
F F(0) a b F F(0) a b
FBD 0.67 0.65 0.00 FBπ 0.25 0.84 0.10
0 0
VBD∗ 0.75 1.29 0.45 ABD∗ 0.64 1.30 0.31
0
ABD∗ 0.63 0.65 0.02 ABD∗ 0.61 1.14 0.52
1 2
TABLEI:TheparametersgiveninRef.[17].
K+ K+ K+
D+ D*+ D*+
s s s
B0 D*0 B0 D0 B0 D*0
s s s
D-s K- D*s- K- D*s- K-
(a) (b) (c)
π+ π+ π+
D+ D*+ D*+
B0 D*0 B0 D0 B0 D*0
s s s
D-s K- D*s- K- D*s- K-
(d) (e) (f)
FIG.1:TheQCDone-particleexchangeprocessinD(∗)+D(∗)− →K+K−.
s s
B. EvaluationofFSIeffects
Nowletusturntoevaluatethelong-distanceeffectsathadronlevel. Fortheeffectivevertices,theflavor-SU(4)symmetryis
assumed.Thecouplingofpseudoscalarandvectormesonis
1
=Tr(∂ φ ∂µφ) Tr(F Fµν), (4)
L0 µ † − 2 µ†ν
whereFµν =∂µVν ∂νVµ,andφandV representthe4 4pseudoscalarandvectormesonmatricesinSU(4),respectively:
− ×
π0 + η + ηc π+ K+ D¯0
√2 √6 √12
 
φ = 1  π− −√π02 + √η6 + √η1c2 K0 D− , (5)
√2 K K¯0 2η+ ηc D 
 − −q3 √12 s− 
 D0 D+ D+ 3ηc ,
 s −√12 
ρ0 + ω′ + J/ψ ρ+ K + D¯ 0
√2 √6 √12 ∗ ∗
V = 1  ρ− −√ρ02 + √ω′6 + √J/1ψ2 K∗0 D∗− . (6)
√2 K K¯ 0 2ω + J/ψ D 
 ∗− ∗ −q3 ′ √12 s∗− 
 D 0 D + D + 3J/ψ.
 ∗ ∗ s∗ − √12 
Forthegaugeinvariance,covariantderivativesreplacetheregularones:
ig
∂ φ φ=∂ φ [V ,φ],
µ µ µ µ
→D − 2
(7)
ig
F ∂ V ∂ V [V ,V ].
µν µ ν ν µ µ ν
→ − − 2
5
Nowwearereadytowritedowntherelevanttermsinthepseudoscalar-vectorcoupling:
= +igTr(∂µφ[φ,Vν])+g ǫ ∂αVβ∂µVνφ+ , (8)
0 ′ αβµν
L L ···
wherethethirdtermontheright-handsideofEq. (8)isinvolvedadditionallyasavector-vector-pseudoscalarcoupling[23]. In
theprocessthatDs(∗)+Ds(∗)− re-scatterintoK+K−,thecorrespondingLagrangianis:
LKDsD∗ =igKDsD∗nDµ−∗h∂µK+Ds−−K+∂µDs−i+D¯0∗hK−∂µDs+−∂µK−Ds+io,
LKDs∗D =igKDs∗DnDs−µ∗hK+∂µD0−∂µK+D0i+Ds+µ∗h∂µK−D¯0−K−∂µD¯0io, (9)
LKDs∗D∗ =gKDs∗D∗ǫαβµνh∂αD¯β0∗∂µDs+ν∗K−+∂αDs−β∗∂µDν0∗K+i,
whereastheeffectiveverticesforπDD andπD D canbefoundinRef. [14].
∗ ∗ ∗
IntheEq. (9),thevaluesofthecouplingconstantsshouldbeobtainedbyfittingtheexperimentdata. However,itisnoticed
thatinpreviousliteraturethosecouplingconstantsarestillnotwellfixedyet(seeTableII),sointhiswork,weuseanaverage
valueforeachcouplingconstant,andtheuncertaintyisconsideredasasystematicalerror,whichmightbeattributedtoourinput
parameters. Forexample,therearenoavailabledatafordeterminingthecouplingconstantsgKDsD∗ andgKDs∗D,wearegoing
tofixthembasedontheSU(4)symmetry,whichtellsusthat: gKDsD∗ =gKDs∗D =√2gρππ =√3gψDD/2,andbyfittingthe
data,gρππ = 8.8andgψDD = 7.9areset(seeTableII).We mayhavetwodifferentvaluesforgKDsD∗ andgKDs∗D: 6.0and
9.4, butbyourstrategywe adoptanaveragevalueforgKDsD∗ andgKDs∗D as: gKDsD∗ = gKDs∗D = 7.7±1.7. We use the
samemethodtogetthevaluesforotherrelevantcouplingconstantsaslongastherearenodatatodirectlyfixthem,andretain
theerrors.Thenwehave:
√3
gKDsD∗ =gKDs∗D =√2gρππ = 2 gψDD =7.7±1.7,
gKDs∗D∗ = √13gψD∗D =3.2±0.8GeV−1, (10)
gπD∗D∗ =9.0 0.1GeV−1, gπDD∗ =4.4.
±
To evaluate the FSI effects, we calculate the absorptive part (see above arguments) of the one-particle-exchange triangle
diagrams. Fig. 1showsthediagramsofBs Ds(∗)+Ds(∗)− K+K− byexchangingD0(∗). Forcalculatingtheabsorptive
→ →
partofthetriangle,withtheCutkoskycuttingrule[20,21],therelatedamplitudesare:
A1hBs(p)→Ds+(p1)Ds−(p2)→K+(p3)K−(p4)i
1
= 2Z dp˜1dp˜2(2π)4δ(p−p1−p2)AhBs →Ds+(p1)Ds−(p2)i(−gKDsD∗)h−i(p1+p3)µi(−gKDsD∗)h−i(p2+p4)νi
q q i
g + µ ν 2(q2,m2 ), (11a)
×h− µν m2D0∗iq2−m2D0∗F D0∗
B (p) D (p )D (p ) K+(p )K (p )
A2h s → s∗ 1 s∗ 2 → 3 − 4 i
1
= 2Z dp˜1dp˜2(2π)4δ(p−p1−p2)AhBs →Ds+∗(p1)Ds−∗(p2)i(−gKDs∗D)h−i(q+p3)ξi(−gKDs∗D)h−i(q−p4)σi
p p p p i
g + 1µ 1ξ g + 2ν 2σ 2(q2,m2 ), (11b)
×h− µξ m2Ds∗ ih− νσ m2Ds∗ iq2−m2D0F D0
B (p) D +(p )D (p ) K+(p )K (p )
A3h s → s∗ 1 s∗− 2 → 3 − 4 i
1
= 2Z dp˜1dp˜2(2π)4δ(p−p1−p2)AhBs →Ds+∗(p1)Ds−∗(p2)i(igKDs∗D∗)(−ip1σ)(−iqα)ǫσωαβ(igKDs∗D∗)(−ip2ξ)(iqκ)
q q p p p p i
ǫξλκρ g + β ρ g + 1µ 1ω g + 2ν 2λ 2(q2,m2 ), (11c)
× h− βρ m2D0∗ih− µω m2Ds∗ ih− νλ m2Ds∗ iq2−m2D0∗F D∗0
6
instead,forthedecaychannelBs D(∗)+Ds(∗)− π+K−,wehave:
→ →
B (p) D+(p )D (p ) π+(p )K (p )
A1h s → 1 s− 2 → 3 − 4 i
1
= 2Z dp˜1dp˜2(2π)4δ(p−p1−p2)AhBs →D+(p1)Ds−(p2)i(−gπDD∗)h−i(p1+p3)µi(−gKDsD∗)h−i(p2+p4)νi
q q i
g + µ ν 2(q2,m2 ), (11d)
×h− µν m2D0∗iq2−m2D0∗F D0∗
B (p) D +(p )D (p ) π+(p )K (p )
A2h s → ∗ 1 s∗− 2 → 3 − 4 i
1
= 2Z dp˜1dp˜2(2π)4δ(p−p1−p2)AhBs →D+∗(p1)Ds−∗(p2)i(−gπDD∗)h−i(q+p3)ξi(−gKDs∗D)h−i(q−p4)σi
p p p i
g + 1µ 1ξ g + 2νp2σ 2(q2,m2 ), (11e)
×h− µξ m2D∗ ih− νσ m2Ds∗ iq2−m2D0F D0
B (p) D +(p )D (p ) π+(p )K (p )
A3h s → ∗ 1 s∗− 2 → 3 − 4 i
1
= 2Z dp˜1dp˜2(2π)4δ(p−p1−p2)AhBs →D+∗(p1)Ds−∗(p2)i(igπD∗D∗)(−ip1σ)(−iqα)ǫσωαβ(igKDs∗D∗)(−ip2ξ)(iqκ)
q q p p p p i
ǫξλκρ g + β ρ g + 1µ 1ω g + 2ν 2λ 2(q2,m2 ), (11f)
× h− βρ m2D0∗ih− µω m2D∗ ih− νλ m2Ds∗ iq2−m2D0∗F D∗0
wheredp˜= dp3/((2π)32E),qisthemomentumoftheexchangedD0( ) mesonandadipoleformfactor (q2,m2) = (Λ2
∗
m2)2/(Λ2 q2)2withΛ=m+βΛ 1isintroducedtocompensateitsoff-shelleffect[22]. F −
QCD
−
g value g value
gρππ 8.8[14] gDDψ 7.9[22]
gD∗Dψ 4.2GeV−1[22],7.7[14] gπD∗D∗ 8.9GeV−1[22],9.1[24]
gπDD∗ 4.4[23]
TABLEII:Thevaluesofthecouplingconstantsinvolvedinourwork.
III. NUMERICALRESULT
WiththeamplitudesgiveninEq.(11),wecaneasilycalculatethedecaywidthofthesequentialprocessesBs Ds(∗)Ds(∗)
K+K−. We take the relevant parameters from PDG [3] as: mBs = 5.366GeV, mDs = 1.968GeV, mDs∗→= 2.112Ge→V,
mD0 =1.864GeV,mD0∗ = 2.007GeV,mD± =1.869GeV,mD∗± =2.01GeV;andthevalueoftheweakdecayconstants,
suchasf f f f canbefoundinRef. [25];a =1.14issetinournumericalcomputations[13]. Thetotalamplitudesare
π K D Ds 1
(cid:12)A(Bs →K+K−)(cid:12) = (cid:12)Adirect(Bs →K+K−)+A1(Bs →Ds+Ds− →K+K−)+A2(Bs →Ds∗+Ds∗− →K+K−)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) +(cid:12) (B D +D K+K ) =4.1+1.9 10 8, (12)
A3 s → s∗ s∗− → − (cid:12) −1.4× −
(cid:12)
(cid:12)
(B π+K ) = direct(B π+K )+ (B D+D π+K )+ (B D +D π+K )
(cid:12)A s → − (cid:12) (cid:12)A s → − A1 s → s− → − A2 s → ∗ s∗− → −
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) +(cid:12) A3(Bs →D∗+Ds∗− →π+K−)(cid:12)=29.5+−00..11×10−9, (13)
(cid:12)
(cid:12)
1Here,mdenotesthemassoftheexchangedmesonandβisaphenomenologicalparameter.ΛQCDistakenas220MeV.
7
whereAdirect(B K+K )isthepenguincontribution[7]. So,thebranchingratiosare:
s −
→
p
B(Bs →K+K−) = ΓBs→ΓtKot+K− = 32π21Γtot(cid:12)(cid:12)(cid:12)mK2Bs(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Atot(Bs →KK)(cid:12)(cid:12)2dΩ=13.6+−175.7.5×10−6,
(cid:12) (cid:12)
p
Γ 1 K 2
B(Bs →π+K−) = BΓs→totπK = 32π2Γtot(cid:12)(cid:12)(cid:12)m2Bs(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Atot(Bs →Kπ)(cid:12)(cid:12) dΩ=7.3+−00..11×10−6,
(cid:12) (cid:12) (14)
wheretheerrorsaresystematical,originatingfromtheuncertaintyofthecouplingconstants.
TABLEIII:VariousapproachespredictionsonB(Bs →K+K−)(inunitsof10−6).
andtheratioR=B(Bs →π+K−)/B(Bs →K+K−)
QCDF[7] pQCD(LO)[8] pQCD(NLO)[9] SCET[10] FSI FSI+pQCD(NLO) Experiment[4,5]
B 25.2+12.7+12.5 13.6+8.6 15.6+5.1 18.2±6.7±1.1±0.5 13.6+15.5 29.2+15.8 33±9
−7.2−9.1 −5.2 −3.9 −7.7 −8.7
Ra 0.194 0.360 0.314 0.269 0.360 0.167 0.148
aSincethemaincontributionforthetransitionBs → π+K−comesfromthetreediagramandallpredictionsmadeinvariousmodelsonthischannelare
closetoeachother,weusethedataB(Bs→π+K−)=4.9×10−6asinputinourcalculations.
InTableIIIwelistthetheoreticalpredictionson (B K+K )invariousapproaches.Itisnotedthatmostofthepredicted
s −
B →
centalvaluesarelowerthanthenewlymeasuredvalue[7–10]aslongasthecontributionfromFSIisnotincluded. Weaddthe
contributionsfromFSI tothatcalculatedin pQCD (NLO),thenonecanfind thattheresultantcentralvalueisconsistentwith
data[30]: (B K+K )=(3.3 0.9) 10 5within1σ.
s − −
B → ± ×
Thedeviationsofourtheoreticalpredictionfromthedatamightcomefromtheloopholeinourcalculation.Asindicatedabove
weonlyconsidertheabsorptivepartofthehadronictriangle.Indeedthedispersivepartmayalsomakesubstantialcontributions
[26]. Assuggestedinliterature,thedispersivecontributionshouldbesmallerthantheabsorptiveone(itisconsistentwiththe
generalprincipleofthequantumfieldtheory),oratmosthasthesamemagnitudeasthatoftheabsorptivepart.Ifthecontribution
ofthedispersivepartisindeedofthesameorderastheabsorptivepart,thentakingitintoaccount,wemayhavearesult,which
isevenclosertothedata.
In general, even though we cannot precisely re-produce the experimental data, we can confirm ourselves that the FSI is
importantandnon-negligibleforunderstandingthe (B K+K )> (B π+K )“conflict”.
s − s −
B → B →
IV. CONCLUSIONANDDISCUSSION
Asexpected,theLHCbisextensivelyaimingonstudyoftheB-physics,especiallytolookforsome”anomalies”inexperi-
ments, which needhighstatistics and precisemeasurements. For B ’s charmlessnon-leptonictwo-bodydecay, the early MC
s
studies show that, nearly 37K B K+K signals will be seen at 2 fb 1 integrated luminosity [27]. On the other hand,
s − −
314 27B π+K signalswer→eobservedbasedonthe2010datawithitsintegratedluminosity0.35fb 1[28]. Webelieve
s − −
that±thestatist→icsoftheB decayintoK+K andπ+K issufficienttodrawadefinitedecisionabouttheirbranchingratios.
s − −
ThelatestresultreportedbytheLHCbcollaborationshowsthatthebranchingratioshavebeenmeasuredas[29]:
B(Hb →F) = fHb′ N(Hb →F) ǫrec(Hb′ →F′)ǫPID(F′), (15)
(H F ) f N(H F ) ǫ (H F) ǫ (F)
B b′ → ′ Hb b′ → ′ rec b → PID
wherethefHb(′) isthepossibilityofbquarkhadronizingintohadronH,N istheobservednumberofsignalsforcertaindecay
modes,ǫ istheefficiencyofthereconstructionexcludingtheparticleidentification(PID)cutsandǫ isjusttheefficiency
rec PID
ofPIDcuts.
It is noted that in Table III, the experimentaldata are taken from Refs. [4, 5], but the 2012 data of PDG indicate that the
branchingratioof B K+K is (2.64 0.28) 10 5 [30], which is smaller than the data in Ref. [4, 5]. Moreover,the
s − −
LHCb collaborationre→portsthat with the 0±.37fb 1×2011 data, the branchingratios of B K+K and B π+K are
− s − s −
→ →
experimentallydeterminedas:
(B K+K )=(23.0 0.7 2.3) 10 6,
s − −
B → ± ± × (16)
(Bs π+K−)=(5.4 0.4 0.6) 10−6,
B → ± ± ×
8
wheretheformeruncertaintiesarestatisticalandthelateroneissystematical,whichincludetheuncertaintiesofPIDcalibration,
finalstateradiationwithsoftgamma,signalshapeusedforfitting,andtheimpactofthebackground:theadditionalthree-body
background2,thecombinatorialbackgroundandthecross-feedbackground3. SinceBs K+K− andBs Ds(∗)+D¯s(∗)−
→ → →
K+K havethesamefinalstates,itmeanswecannotsingleoutthecontributionsofFSIbysimplymeasuringthecrosssection
−
inexperiments.Eventhoughourtheoreticalpredictionon (B K+K )presentedinTableIIIisslightlyabovethevalueof
s −
B →
theLHCbmeasurements,itisstillconsistentwiththedatawithintheexperimentalerrortolerance. Thereforeweareexpecting
moreprecisemeasurementsinthefuture.
ChengandChuasuggestedthatthedecayB K+K isdominatedbythepenguindiagram,whichdoesnotsufferfrom
s −
largeCKMsuppression[7]. Inthatscenariothe→“conflict” (B K+K ) > (B π+K )couldbepartlyexplained.
s − s −
B → B →
In our work, we show that the FSI definitely makes an importantcontribution because Bs Ds(∗)D¯s(∗) are dominant decay
→
modesofBs asconfirmedbythedataandthescatteringDs(∗)+Ds(∗)− K+K− isallowedbyallthesymmetryrequirements.
→
Therefore,wemightconsiderthatthepenguincontributionandtheFSIeffectsareinparalleltocontributetothedecayB
s
K+K .4 Actually, there are several phenomenologicalparameters in the model for evaluating the FSI effects, which we→re
−
obtainedbyfittingtheearlierdataofheavyflavorprocesses.OnethingcanbesurethattheFSIshouldexistandcontributetothe
processB K+K ,buthowsignificantitwouldbeisdeterminedbybothrequirementofexperimentalmeasurementsand
s −
theoreticale→stimate. Ifthedataon (B K+K )areindeedgoingdown,theFSIeffectsforB K+K wouldbeless
s − s −
B → →
significant,bythatsituation,onecanfurtherrestricttheinvolvedphenomenologicalparametersatthisenergyregion. Onother
aspect, the errorsof the measurementsare still too large to make a definite conclusion yet, so we are expectingmore precise
measurementsnotonlyfromLHCb,butalsothefuturesuper-Bfactory.
Definitely the FSI effects also play roles in other similar decay channels, such as B K ω. Thereforecarefulstudies
± ±
→
onsuchmodeswiththesametheoreticalframeworkwouldbehelpfulformoreaccuratelyevaluatingtheFSIeffects. ThePDG
indicatesthat, (B K ω)/ (B π ω)=(6.7 10 6)/(6.9 10 6),soonewouldexpectthat (B Kω)isalso
± ± ± ± − −
B → B → × × B →
enhancedbythe FSI. Moreover,a measurementonthe polarizationofω mightbe helpfulto gainmoreinformationaboutthe
roleoftheFSImechanism,anditwillbestudiedinourcomingwork.
Fromtheexperimentalaspect,astheFSIisascatteringfullygovernedbythestronginteraction,itspropertimeistooshort
to be measured in the LHCb detector. Although the LHCb can well reconstructthe events B K+(π+)K and identify
s −
→
the π+K purely with the PID cuts [31], it is impossible to distinguish between the direct decays B K+(π+)K and
− s −
→
thesequentialones. ConsideringthegoodabilityofLHCbfortrackingchargedparticles, webelievethat, B K(π) ω is
± ±
anothergoodchannelto study FSI. The decayrates of (B D¯0D( )+) havebeen wellmeasuredas 1.8→0.2 10 2,
∗ −
B → ∼ ± ×
andwecanmakeMont-Carlosimulationsonthere-scatteringofD¯0Ds(∗)+ intoK(π)+ω tocompletethetheoreticalscenario.
The abnormal ratio of the two channels (B K ω)/ (B π ω) = (6.7 10 6)/(6.9 10 6) [30] can also be
± ± ± ± − −
explainedastheFSIcontributionaswell5B. WithB→ DKB(π) m→easurement[32]a×ndconsidering×theefficiency,about0.8K
± ±
→
B+ K(π)+ω can been seen in the 2011 database and it implies thatthe FSI contributionto the polarizationof the vector
→
mesonω mightbedistinguishedfromthatofthedirectdecay. Comparingmoreaccuratedatawithourtheoreticalcalculation,
one can expect to gain more knowledge on the FSI effects, for example how to determine the parameters in the dipole form
factoretc.
Oneobjectcanberecommendedforpiningdowntherole ofthe FSI effects. AswellknownthatthedirectCPviolationis
proportionaltosin(α α ) sin(φ φ ),wherethesubscripts1and2refertotwodifferentroutesandα,φaretheweakand
1 2 1 2
− · −
strongphasesrespectively. Therefore,ifadirectCPviolationexistsinB decays,theremustatleastbetwodifferentroutesto
s
thefinalstateswhichpossessdifferentweakandstrongphases. Infact,thepenguinandtreediagramshavedifferentstrongand
weakphases.TheweakphaseofthetreediagramiscomingfromV V anddoesnotpossesastrongphase.Instead,theweak
ub u∗s
phaseofthepenguinisfromV V andthestrongphaseisduetoexistenceoftheabsorptivepartoftheloop. Inourscenario
cb c∗s
thesequentialprocessesBs(B¯s) → Ds(∗)D¯s(∗) → K+K− whichhavethesameweakphaseVcbVc∗s,butdifferentstrongphase
fromthepenguin.Becauseoftheextracontribution,theamplitudewouldbecome
M =Mtreeeiα1 +Mpenguinei(α2+φ1)+Mseqei(α2+φ2),
wherethelasttermistheFSIcontribution.
Thustheinterferencebetweenthetreediagramwithasumofthepenguinandsequentialprocesseswouldbedifferentfrom
the situation where the tree diagram only interferes with the penguin. Therefore by measuring the CP violation of (B
s
B →
2Missormisidentifythepionorkaoninfinalstate.
3Theuncertaintiesfromthedistributionofsignalindataandsimulation.
4Inthisworkwedonotconsiderthepenguincontribution,butonlythatoftheFSIeffectstothedecay. Weacknowledgethepossiblecontributionfromthe
penguindiagramaswell,furtherstudyinvolvingbothofthemandmoreovertheirinterferencewillbemadeinourlaterworks.
5B(B→K(π)ω)=6.7(6.9)×10−6,atthesametime,theCabibbosuppressiondeterminesthatthebranchingratioofKωfinalstateshouldbesmaller.
9
K+K ) (B¯ K+K ), one can distinguish between the penguin contributions and the FSI effects. But since the
− s −
− B →
experimentalerrorsandtheoreticaluncertaintiesarenotwellcontrolledsofar,wecannotmakeamoredefinitepredictiononthe
CPviolationyet,wewouldwaitformoreprecisedatatobeavailableandthencontinueourtheoreticalcomputations.
Moreover,accordingtotheabovearguments,theredoesnotexistatreediagramforB K0K¯0. Consequently,thedirect
s
transition B K0K¯0 is uniquely determined by the penguin diagram and FSI effects→. Because the tree contribution to
s
B K+K →is very small, the rate of B K0K¯0 must be close to that of B K+K . However the interference
s − s s −
→ → →
between penguin (neglecting the Cabibbo-suppressed penguin diagrams where t-quark and u-quark are intermediate agents)
andFSIdoesnotleadtoadirectCPviolationatall,becausetheyhavethesameweakphase. Thiscanbeconfirmedbyfuture
experiments.
Acknowledgments
We would like to thank Ming-Xing Luo and Cai-Dian Lu for useful discussion. This project is supported by the National
NaturalScienceFoundationofChina(NSFC)underContractNos. 10775073,11005079,11222547,11175073and11035006,
theMinistryofEducationofChina(theSpecialGrantforthePh.D.programunderGrantNo. 20070055037,FANEDDunder
Grant No. 200924, DPFIHE under Grant No. 20090211120029,NCET under Grant No. NCET-10-0442, the Fundamental
ResearchFundsfortheCentralUniversities),andtheFokYing-TongEducationFoundation(No. 131006).
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