Study of $B \to K_0^*(1430)K^{(*)}$ decays in QCD Factorization Approach

0.18 MB·English

by  Ying Li

Study of B K∗(1430)K(∗) decays in QCD Factorization Approach
→ 0
Ying Li∗, Hong-Yan Zhang, Ye Xing, Zuo-Hong Li
Department of Physics, Yantai University, Yantai 264005, China
Cai-Dian Lu¨
Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,CAS, Beijing 100049, China
(Dated: January 19, 2015)
Within the QCD factorization approach, we calculate the branching fractions and CP asymme-
tryparameters of 12 B→K∗(1430)K(∗) decay modes undertheassumption that thescalar meson
0
K∗(1430) is the first excited state or the lowest lying ground state in the quark model. We find
0
that the decay modes with the scalar meson emitted, have large branching fractions due to the
5 enhancement of large chiral factor rχK0∗. The branching fractions of decays with the vector meson
1 emitted, become much smaller owing to the smaller factor rK∗. Moreover, the annihilation type
0 χ
diagram will induce large uncertainties because of the extra free parameters dealing with the end-
2
pointsingularity. Forthepureannihilation typedecays,ourpredictionsaresmaller thanthatfrom

n PQCD approach by 2-3 orders of magnitudes. These results will be tested by the ongoing LHCb
a experiment,forthcoming Belle-II experiment and theproposing circular electron-positron collider.
J

6
1 I. INTRODUCTION Very recently, the decaysB K∗(1430)K(∗) dominated
→ 0
byb dpenguinoperators,havebeencalculatedwithin

] the P→QCD approach [6], however they have not been
h Although the quark model has made great success in
p touchedinQCDFframe. Tocomplete,wethereforeshall
- describing most of the hadronic states, the lowest lying calculate the branching fractions of B K∗(1430)K(∗)
p scalar mesons are too light to fit in the quark model. → 0
decaysinQCDF,aswellastheirCP asymmetryparam-
e Two possible scenarios have been proposed on the ba-
eters. In the experimental side, these observables are
h sis of whether the scalars lower than 1 GeV belong to
[ too small to be measured at current experiments, but
the four-quark states or the classical two quark states. they are hopeful to be measured in the future experi-
1 They are controversial for decades [1]. In scenario-1
ments such as the updated LHCb experiment, the high
v (S1), the scalarssuch as κ(800), a (980) andf (980)are
0 0 luminosity Belle-II experiment and the proposing high
5 naively seenas the lowestlying qq¯states, andK∗(1430),
6 0 energy circular electron-positron collider.
a (1450), and f (1500) are the first excited states, cor-
8 re0spondingly. In0contrast, K∗(1430), a (1450), f (1500) The paper is organized as follows. In Section 2, we
3 0 0 0 give the analytic formula including the effective Hamil-
are treated as the the qq¯ ground states and their first
0 tonian, the form factor and all corrections to the ampli-
. excited states are about (2.0 2.3) GeV in scenario-2
1 ∼ tudes. Presentation of results and discussions are given
(S2), in which the lightest scalar mesons are the four-
0 in Section 3. At last, we summarize this work in Section
quark bound states.
5 4.
1 In the past decade, many B decay modes involving
: scalar mesons have been reported by BaBar, Belle and
v
i large hadron collider-b (LHCb) experiments. This may
X II. ANALYTIC FORMULA
providea unique feature to distinguish these two scenar-
r ios by the B meson tag [2]. It is thus hoped that the
a
combination of the precise experimental measurements In this section, we shall start from the weak effective
and the accurate theoretical predictions might provide Hamiltonian responsible for B → K0∗(1430)K(∗) decays.
us valuable information on the nature of scalar mesons. In the standard model, it could be written as [7]
To achievethis goal,somehadronicB (q =u,d,s)de-
q
cays to scalar mesons have been studied in detail in the = GF V V∗ (C Ou+C Ou)
framework of the QCD factorization (QCDF) [1, 3, 4] Heff √2 ub ud 1 1 2 2
(cid:20)
and the perturbative QCD approach(PQCD) [5, 6]. Us- 10
ingtheQCDFapproach,H-YChenget.alhadcalculated V V∗ C O +C O +C O +h.c. (1)
− tb td i i 7γ 7γ 8g 8g
the branching fractions and direct CP asymmetries of i=3 (cid:21)
(cid:0)X (cid:1)
most decay modes in refs. [3, 4], such as B f K,
0
B → K0∗(1430)φ(ρ) and B → K0∗(1430)π, whe→re most The explicit form of the operators Oi and the corre-
results accommodate the data with large uncertainties. spondingWilsonCoefficientsCi atdifferentscaleµcould
be found in ref.[7]. V and V are the Cabibbo-
u(t)b u(t)d
Kabayashi-Maskawa(CKM) matrix elements. Note that
O1,2 are tree operators and others O3−10,7γ,8g are pen-
∗Email: [email protected] guin ones.
2
In dealing with the nonleptonic charmless B decays, with i = 1, ,10. The upper (lower) signs apply when
the decay amplitude is usually separated into the emis- i is odd (ev··e·n), C = (N2 1)/(2N ) with N = 3.
F c − c c
sionpartandthe annihilationpartintermsofthe struc- The quantities V (M ) account for vertex corrections,
i 2
ture of the topological diagrams. According to the fac- H (M M ) for hard spectator interactions with a hard
i 1 2
torizationapproximationbasedontheheavyquarklimit, gluonexchangebetweentheemittedmesonandthespec-
the former partcouldbe written asthe productofdecay tatorquarkofthe B mesonandP (M )forpenguincon-
i 2
constant and form factor. While for the latter one, it is tractions. Similarly, the annihilation contributions are
alwaysregardedasbeingpowersuppressed. InQCDF[8] described by the terms b , and b , which have the
i i,EW
based on collinear factorization, the contribution of the expressions
non-perturbativesectorisdominatedbytheformfactors,
andthe non-factorizableeffectinthe hadronicmatrixel- b = CFC Ai,
ements is controlled by hard gluon exchange. Thus, the 1 N2 1 1
c
hadronic matrix elements of the decay can be written as C
b = FC Ai,
2 N2 2 1
1 c
hM1M2|Oi|Bi= j FjB→M1Z0 dxTiIj(x)ΦM2(x) b3 = CNF2 C3Ai1+C5(Ai3+Af3)+NcC6Af3 ,
X c
1 1 1 C h i
+ dξ dx dyTiII(ξ,x,y)ΦB(ξ)ΦM1(x)ΦM2(y), b4 = NF2 C4Ai1+C6Af2 ,
Z0 Z0 Z0 c h i
(2) C
b = F C Ai +C (Ai +Af)+N C Ai ,
3,EW N2 9 1 7 3 3 c 8 3
where TI and TII denote the perturbative short- c
ij i C h i
distance interactions, which can be calculated perturba- b = F C Ai +C Ai , (5)
tively. Φ (x) (X = B,M ,M ) are the universal non- 4,EW N2 10 1 8 2
X 1 2 c
perturbative light-cone distribution amplitudes that can (cid:2) (cid:3)
beestimatedbythelight-coneQCDsumrules. Theform where the subscripts 1,2,3 of Ain,f stand for the an-
factors of B meson to the pseudoscalar (P), the vector nihilation amplitudes induced from (V A)(V A),
− −
(V) and scalar (S) mesons transitions FB→M1 being a (V A)(V +A) and (S P)(S +P) operators, respec-
− −
part of the nonperturbative sector of QCD, lack a pre- tively. The superscripts i and f refer to gluon emission
cise solution. For the B P and B V transition from the initial and final-state quarks, respectively. It
formfactors,we will emplo→y the resultso→f the QCDsum should be stressed that the decays B0 K∗+K(∗)− and
→ 0
rule method [9], since they havebeen usedincalculating B0 K(∗)+K∗− are only induced by the annihilations
B PP,PV and VV modes widely. As the form fac- type→diagrams.0 Hereafter, the order of the ap(M M )
tor→ofFB→S as concerned,to the best of our knowledge, i 1 2
coefficients is dictated by the subscript M M , where
1 2
a number of approaches had been advocated to calcu-
M is the emitted meson and M shares the same spec-
2 1
late them, such as QCD sum rule [10], light-cone QCD
tator quark with the B meson. For the b (M M ) of
i 1 2
sumrule[11],PQCD[12]andcovariantlightfrontquark
the annihilation, M means the one containing an anti-
1
model (cLFQM) [13]. In this work, we use the results
quark from the weak vertex, while M contains a quark
2
obtained by cLFQM [13] for keeping consistent with the
from the weak vertex. The explicit expressions of V ,
i
results of other B SP(V) decay modes [3].
H ,P , b , b and their inner functions could be found
→ i i i i,EW
Following the standardprocedure of QCDF approach,
in refs.[1, 3].
the emissionpartofdecayamplitude couldbe writtenas
When dealing with the hard-scattering spectator and
the weak annihilation contributions, we suffer from in-
AS(B0 →M1M2) frared endpoint singularities X = 1dx/(1 x), that
G 0 −
= F V V∗ap(µ) M M O B . (3) cannotbecalculatedfromthefirstprincipleintheQCDF
√2 pb pd i h 1 2| i| iF approachandonlybeestimatedphenRomenologicallywith
p=u,c i
X X a large uncertainty. Following the arguments of ref.[8],
M1M2 Oi B F isthefactorizablematrixelement,which we also parameterize these kinds of contributions by the
h | | i
can be factorized into a form factor times a decay con- complex quantities, X and X namely,
H A
stant, as stated before. The effective parameters ap can
i m
becalculatedperturbatively,whoseexpressionsaregiven X = 1+ρ eiφH ln B, (6)
H H
by Λ
m
X = (cid:0)1+ρ eiφA(cid:1)ln B , (7)
ap(M M ) = C + Ci±1 N (M ) A A Λ
i 1 2 i N i 2 (cid:0) (cid:1)
(cid:18) c (cid:19) where Λ=0.5GeV. ρ , ρ are real parameters,and φ
+ CNi±c1C4Fπαs Vi(M2)+ 4Nπc2Hi(M1M2) aAnlldtφhAeaarbeofvreeefosturronpgHarpahmaAseetserisnsthhoeurladngbee[−fix1e8d0◦,b1y80th◦H]e.
+ Pp(M ), h (i4) experimental data, such as branching fractions and CP
i 2
3
asymmetries. In the so-called PQCD approach [14], one Including the emissionand annihilation contributions,
caneliminate these end-pointsingularitiesbykeepingall the decay amplitude can be finally given as
small transverse momenta of gluons and inner quarks.
A(B− →K−K0∗0) = G√F2 pX=u,cλ(pd)((cid:18)ap4− 12ap10−(ap6− 21ap8)rχK0∗(cid:19)K−K0∗0fK0∗F0B→K(m2K0∗)(m2B −m2K)
+fBfK0∗fK b2δup+b3+b3,EW K−K0∗0), (8)
(cid:0) (cid:1)
A(B− →K0∗−K0) = G√F2 pX=u,cλ(pd)(−(cid:18)ap4− 12ap10−(ap6− 21ap8)rχK(cid:19)K0∗−K0fKF0B→K0∗(m2K)(m2B −m2K0∗)
+fBfK0∗fK b2δup+b3+b3,EW K0∗−K0), (9)
(cid:0) (cid:1)
A(B0 →K0K0∗0) = G√F2 pX=u,cλ(pd)((cid:18)ap4− 12ap10−(ap6− 12ap8)rχK0∗(cid:19)K0K0∗0fK0∗F0B→K(m2K0∗)(m2B −m2K)
1 1
+fBfK∗fK b3+b4 (b3,EW+b4,EW) + b4 b4,EW , (10)
0 (cid:18) − 2 (cid:19)K0K0∗0 (cid:18) − 2 (cid:19)K0∗0K0 )
(cid:2) (cid:3)
A(B0 →K∗00K0) = G√F2 pX=u,cλ(pd)(−(cid:18)ap4− 21ap10−(ap6− 21ap8)rχK(cid:19)K∗00K0fKF0B→K0∗(m2K)(m2B −m2K0∗)
1 1
+fBfK∗fK b3+b4 (b3,EW+b4,EW) + b4 b4,EW , (11)
0 (cid:18) − 2 (cid:19)K∗00K0 (cid:18) − 2 (cid:19)K0K∗00 )
(cid:2) (cid:3)
A(B0 →K0∗+K−) = G√F2 pX=u,cλ(pd)(fBfK0∗fK(cid:2)(b1δup+b4+b4,EW)K0∗+K− +(cid:18)b4− 12b4,EW(cid:19)K−K0∗+(cid:3)), (12)
A(B0 →K+K0∗−) = G√F2 pX=u,cλ(pd)(fBfK0∗fK(cid:2)(b1δup+b4+b4,EW)K+K0∗− +(cid:18)b4− 12b4,EW(cid:19)K0∗−K+(cid:3)), (13)
A(B− →K∗−K0∗0) = G√F2 pX=u,cλ(pd)(−(cid:18)ap4− 12ap10+(ap6− 21ap8)rχK0∗(cid:19)K∗−K0∗02fK0∗A0B→K∗(m2K0∗)mBpc
−fBfK0∗fK∗ b2δup+b3+b3,EW K∗−K0∗0), (14)
(cid:0) (cid:1)
A(B− →K0∗−K∗0) = G√F2 pX=u,cλ(pd)((cid:18)ap4− 12ap10−(ap6− 21ap8)rχK∗(cid:19)K0∗−K∗02fK∗F1B→K0∗(m2K∗)mBpc
−fBfK0∗fK∗ b2δup+b3+b3,EW K0∗−K∗0), (15)
(cid:0) (cid:1)
4
A(B0 →K∗0K0∗0) = G√F2 pX=u,cλ(pd)(−(cid:18)ap4− 21ap10+(ap6− 21ap8)rχK0∗(cid:19)K∗0K0∗02fK0∗A0B→K∗(m2K0∗)mBpc
1 1
fBfK∗fK∗ b3+b4 (b3,EW+b4,EW) + b4 b4,EW , (16)
− 0 (cid:18) − 2 (cid:19)K∗0K0∗0 (cid:18) − 2 (cid:19)K0∗0K∗0
(cid:2) (cid:3)
A(B0 →K∗00K∗0) = G√F2 pX=u,cλ(pd)((cid:18)ap4− 21ap10−(ap6− 12ap8)rχK∗(cid:19)K∗00K∗02fK∗F1B→K0∗(m2K∗)mBpc
1 1
fBfK∗fK∗ b3+b4 (b3,EW+b4,EW) + b4 b4,EW , (17)
− 0 (cid:18) − 2 (cid:19)K∗00K∗0 (cid:18) − 2 (cid:19)K∗0K∗00 )
(cid:2) (cid:3)
A(B0 →K0∗+K∗−) = G√F2 pX=u,cλ(pd)(−fBfK0∗fK(cid:2)(b1δup+b4+b4,EW)K0∗+K∗− +(cid:18)b4− 12b4,EW(cid:19)K∗−K0∗+(cid:3)),(18)
A(B0 →K∗+K0∗−) = G√F2 pX=u,cλ(pd)(−fBfK0∗fK(cid:2)(b1δup+b4+b4,EW)K∗+K0∗− +(cid:18)b4− 12b4,EW(cid:19)K0∗−K∗+(cid:3)),(19)
where of K∗(1430) have the signs flipped from S1 to S2. The
0
definition of the form factors for the B S transitions
2m2 →
rK(µ) = K , are given by [13]
χ m (µ)[m (µ)+m (µ)]
b u s
rχK0∗(µ) = mb(µ)[m2qm(µ2K)0∗−ms(µ)], hS(p′)|Aµ|B(p)i = −i"(cid:18)Pµ− m2Bq−2m2S qµ(cid:19)F1BS(q2)
rK∗(µ) = 2mK∗ fK⊥∗(µ). (20) m2 m2
χ mb(µ) fK∗ + Bq−2 SqµF0BS(q2) , (23)
#
′ ′
where P = (p+p) and q = (p p) . In cLFQM ,
III. NUMERICAL RESULTS AND DISCUSSION µ µ µ − µ
the momentum dependence of the form factor could be
parameterized in a di-pole model form [13],
Toproceed,weshallpresentnumericalresultsobtained
from the formulas given in the previous section. Firstly,
F(0)
we introduce the adopted parameters in our calculation. F(q2)= . (24)
1 a(q2/m2)+b(q4/m4)
Secondly,wegivethenumericalresultsandshowthethe- − B B
oreticalerrorsduetouncertaintyofsomeparameters. At
Togetherwiththedecayconstants,theparametersF(0),
last, some discussions and comparisons will be added.
a and b for B S transitions in the different scenarios
Thescalarmeson,unlikethepseudoscalarone,hastwo →
are summarized in Table I.
kinds of decay constants, namely the vector decay con-
The twist-2 light-conedistribution amplitude (LCDA)
stant f and the scale-dependent scalar decay constant
f¯S, theSdefinitions of which are give by: ΦKS∗(xm)aadnedotfwthiset-q3uΦasSrk(sx)sda¯nadreΦgσSiv(xen)fboyrthescalarmeson
∗ 0
hK0(p)|s¯γµd|0i=fK0∗pµ,
∗ 1
hK0(p)|s¯d|0i=mK0∗f¯K0∗. (21) hK∗0(p)|s¯(z2)γµd(z1)|0i=pµ dxei(xp·z2+x¯p·z1)ΦS(x),
The two decay constants satisfy Z0
1
fK0∗ = ms(µm)−K∗md(µ)f¯K0∗, (22) hKK∗0∗((pp))|ss¯¯((zz2))dσ(z1d)(|z0i)=0mK∗0Z0 dxei(xp·z2+x¯p·z1)ΦsS(x),
0 h 0 | 2 µν 1 | i
mwhaesrseesm. sI(tµs)hoaunlddmbed(sµt)reasrseedthtehartunthneingdeccuaryrecnotnsqtuaanrtks =−mK∗0(pµzν −pνzµ)Z01dxei(xp·z2+x¯p·z1)ΦσS6(x), (25)
5
∗
TABLE I: The Parameters of theK (1430) in different scenarios
0
F0B→K0∗(0) a b F1B→K0∗(0) a b f¯K0∗(MeV) B1 B3
S1 0.26 0.44 0.05 0.26 1.52 0.64 −300±30 0.39±0.05 −0.70±0.05
S2 0.21 0.44 0.05 0.21 1.52 0.64 445±50 −0.39±0.09 −0.25±0.13
TABLE II: Summary of input parameters
λ A Ru γ ΛM(fS=4) τB0 τB− λB αe
◦
0.225 0.818 0.376 68 250MeV 1.54ps 1.67ps 0.35 1/132
fB mB fK fK∗ fK⊥∗ mK mK∗ mK0∗ F0B→K A0B→K∗
236MeV 5.36GeV 131MeV 221MeV 175MeV 0.49GeV 0.89GeV 1.43GeV 0.35 0.34
with z =z z , x¯=1 x,andtheir normalizationsare used parameters, such as the CKM elements, the decay
2 1
− −
constants of the pseudo-scalar and the vector, and the
1 form factors of B K(∗), are also listed in Table II for
dxΦS(x) = fS, (26) convenience. →
Z0
1 1
dxΦs(x) = dxΦσ(x)=f¯ . (27) Now, we turn to discuss the numerical results of the
S S S concerned decay modes. As we had stated in previous
Z0 Z0
section, when calculating the hardspectator and the an-
The above definitions of LCDAs can be combined into a nihilation contributions, two endpoint singularities are
single matrix element as parameterized phenomenologically by four free parame-
ters, namely ρ ,φ and ρ ,ρ , which should be deter-
H H A A
K∗(p)s¯ (z )d (z )0 = 1 1dxei(xp·z2+x¯p·z1) mined fromthe experimental data. In ref.[3], a globalfit
h 0 | 2β 2 1α 1 | i 4Z0 oafndρAφand=φ8A2◦towtihthe Bχ2→=S8P.3,dasotainindthiciastewsorρkAf=or0t.h15e
Φσ(x) A
×(p/ΦS(x)+mS(cid:18)ΦsS(x)−σµνpµzν S6 (cid:19))αβ. cφeAn,tHra=l v8a2lu◦easr(eoard“odpetfeadu.lt”InreTsaubltles),IIρI,Aw,He =list0t.1h5e acanld-
(28) culated branching fractions of B → K0∗(1430)K(∗) de-
cays, where the first uncertainty is due to the variations
In general, the twist-2 light-cone distribution amplitude of B and f , the second comes from the form factors
1,3 S
Φ has the form and the strange quark mass, and the last one is induced
S
by weak annihilation and hard spectator interactions in
ΦS(x,µ)=f¯S(µ)6x(1−x) ranges ρA,H ∈[0,0.3] and φA,H ∈[0,180◦].
∞ Here, we shall take B− K−K∗0(1430) and B−
×"B0(µ)+ Bm(µ)Cm3/2(2x−1)#, (29) K0∗−(1430)K0asexamplest→oillustra0tetherelativesize→of
m=1 each contribution. The decay amplitude formula of each
X
modehasbeenpresentedineqs.(8)and(9), respectively.
whereB (µ)arescale-dependenceGegenbauermoments
m The first decay mode is characterized by B P transi-
and C3/2(u) are the Gegenbauer polynomials. The B →
m m tion with scalar meson emitted; while the second decay
ofdifferentscenariosarealsopresentedinTableI.Asfor
mode is characterized by B S transition with pseu-
the twist-3 distribution amplitudes, we shall adopt the →
doscalar meson emitted. Because of the small vector de-
asymptΦotsSic(xf)or=mf¯fSo,r simpΦlσSic(ixty),=shfo¯Sw6nxa(1s −x). (30) cdisaeyscuacpyopnwrsetisdasntehtdoorfefslactathilevaerfimtrosetfsoKcnhFaK0Bn0→n∗(eK1l40∗s3(h0mo)u2K,fl)dK.0∗bTFeh0Besr→uepKfop(rrmee,s2Ktshe0∗de)
For the LCDAs of K(∗), we will employ the formulaes comparingwiththesecondone. Thenumericalresultsof
obtained from the QCD sum rules [15, 16]. The other these two decay modes are given as below:
A(B− K−K∗0) V V∗ (0.53+0.14i)+V V∗ (0.58+0.08i)+V V∗ ( 0.03+0.01i)+V V∗ ( 0.03+0.01i),
→ 0 ∝ ub ud cb cd ub ud − cb cd −
emissiondiagrams annihilationdiagrams
(31)
| {z } | {z }
6
TABLE III:The branchingratios of B→K∗(1430)K(∗) in theunite 10−7.
0
Decay Mode S1 S2
B0 →K∗0(1430)K0 11.12+3.80+7.07+4.91 2.39+1.20+1.95+2.67
0 −2.99−3.57−5.05 −0.85−0.90−2.00
B− →K∗−(1430)K0 5.36+1.73+5.13+1.73 1.14+0.54+1.40+1.17
0 −1.37−2.36−2.30 −0.38−0.56−0.92
B0 →K0K∗0(1430) 32.27+9.41+7.80+5.49 40.47+13.36+6.09+6.06
0 −7.83−5.48−6.30 −10.77−5.38−6.16
B− →K−K∗0(1430) 23.71+6.67+6.73+2.61 33.70+10.33+5.52+3.37
0 −5.60−4.61−3.64 −8.47−4.82−3.94
B0 →K∗+(1430)K− 0.97+0.43+0.01+0.63 0.58+0.45+0.02+0.14
0 −0.31−0.01−0.44 −0.29−0.03−0.05
B0 →K+K∗−(1430) 8.33+3.31+0.07+7.28 1.07+0.72+0.03+2.27
0 −2.55−0.10−4.83 −0.47−0.04−0.97
B0 →K∗0(1430)K∗0 0.08+0.07+0.07+0.09 0.03+0.03+0.02+0.03
0 −0.02−0.04−0.06 −0.00−0.00−0.02
B− →K∗−(1430)K∗0 0.62+0.08+0.21+0.05 0.14+0.06+0.11+0.03
0 −0.08−0.18−0.04 −0.05−0.08−0.03
B0 →K∗0K∗0(1430) 10.86+2.54+0.25+0.53 20.13+5.35+0.50+0.42
0 −2.24−0.24−0.75 −4.49−0.49−0.59
B− →K∗−K∗0(1430) 12.71+3.14+0.26+1.16 21.71+5.51+0.54+0.30
0 −2.72−0.26−1.07 −4.66−0.53−0.24
B0 →K∗+(1430)K∗− 0.33+0.14+0.00+0.22 0.11+0.09+0.00+0.09
0 −0.10−0.00−0.16 −0.06−0.00−0.06
B0 →K∗+K∗−(1430) 14.54+5.75+0.00+12.48 1.84+1.26+0.00+3.94
0 −4.44−0.00−8.31 −0.83−0.00−1.75
A(B− K∗−K0) V V∗ (0.20+0.05i)+V V∗ (0.22+0.03i)+V V∗ (0.08 0.01i)+V V∗ ( 0.04 0.01i).
→ 0 ∝ ub ud cb cd ub ud − cb cd − −
emissiondiagram annihilationdiagrams
(32)
| {z } | {z }
From these two equations, it is apparent that the emis- based on S1 are a bit smaller than those of S2. For
sion diagrams are dominant. However, there is a large K∗0K∗0(1430) and K∗−K∗0(1430), the predicted cen-
0 0
enhancementfromO operatorsduetothefactthatthe tral values of S2 is larger than those of S1 by a factor
6,8
K∗ 2. Although there is difference between the central val-
chiral factor r 0 = 12.3 at µ = 4.2 GeV is much larger
χ
than rK = 1.5 owing to the larger mass of K∗(1430). ues of different scenarios,we cannot distinguish two sce-
χ 0
It follows that (ap 1ap)rK0∗ is much greater than narios due to large uncertainties taken by annihilation
(ap6− 21ap8)rχK and (6ap6−−221a8p8)rχχK∗. This compensate with dwiiatghraanmnsi,hiulantlieosnssoenffeecatipvperlyoainchQwCeDreF.pIrnopporesvedioutsostdueda-l
the suppression of scalar meson decay constant to result ies [3], by comparing the calculated branching ratios of
sincaalalrarmgeesrobnraenmcihttinedg.ratio of B− →K−K0∗0(1430) with Bcon→cluKde0∗dπtahnadt tBhe→S2Kis0∗φprwefietrhede,xpi.ee.rimKe∗ntiaslvdearytal,ikoenlye
0
In the B PP(V) and VV decay modes, the weak the lowest lying state. If so, the branching fractions
→
annihilation contribution is usually expected to be very of K0K∗0(1430) and K−K∗0(1430) could be measured
smallbecauseitbelongstothenextleadingpowercorrec- 0 0
by analyzing the data of K0K+π− and K−K+π−. Un-
tion. However,onecanseefromTableIIIthattheuncer-
tainties induced by the weak annihilation are very large, fortunately, the predicted results of K∗00(1430)K∗0 and
even much larger than the central values in some decay K0∗−(1430)K∗0 istoosmalltobemeasuredwiththecur-
modes, such as B0 K∗0(1430)K0 and pure annihila- rent data of Belle, but they are hopeful to be measured
tion mode B0 →K0∗→+(14300)K∗−. This phenomenon can ifinnatlhestafotretshocfomKi+nKg −Bπel+leπ-−II.by analyzing the four-body
be understood as follow, when discussing the penguin-
Within the framework of PQCD, X. Liu et.al had cal-
induced annihilation diagram of B PP mode, the de-
→ culated these decays [6], and both their results and ours
cay amplitude is helicity suppressed heavily because the
arebelowthe experimentalupper limits. Comparingour
helicity of one of the final-state mesons cannot match
predictionsandtheirs,wefindthatforpureannihilations
with that of its own quarks. On the contrary, this kind
they obtained rather larger branching fractions by keep-
of helicity suppression can be alleviated when the scalar
ing the transverse momenta, which are larger than ours
meson involved due to the nonzero orbital angular mo-
by more than two order of magnitude. For decays with
mentum L of the scalar meson.
z emissiondiagrams,theyalsogotthelargerbranchingra-
From Table III, one could notice that our predicted
tios with ratherlargenonfactorizaiondiagramsbasedon
central values for the branching ratios of K∗00(1430)K0 both two different scenarios. We hope the future mea-
and K0∗−(1430)K0 based on S2 are smaller than the re- surements could distinguish two different frameworks.
sults based on S1 by a factor of 4 5. In contrast, TheCP asymmetrieshavenotbeenobservedinanyB
the central values of of K0K∗0(1430)∼and K−K∗0(1430) decays involving a scalar meson. For the charged decay
0 0
7
TABLE IV:The direct CP asymmetries (%) of B →K∗(1430)K(∗)
0
Decay Mode S1 S2
B− →K∗−(1430)K0 −5.27+0.50+0.31+2.59 −22.51+4.90+5.63+19.61
0 −0.59−0.57−2.82 −7.57−9.36−22.86
B− →K−K∗0(1430) −2.06+0.60+0.85+5.46 −2.60+1.61+0.59+3.52
0 −0.65−0.69−6.67 −1.76−0.59−5.47
B− →K∗−(1430)K∗0 −18.30+0.94+0.65+1.27 −31.02+4.67+4.28+5.06
0 −1.12−0.89−0.99 −6.56−7.85−3.80
B− →K∗−K∗0(1430) 5.03+1.30+0.17+11.40 0.64+2.54+0.23+7.57
0 −1.37−0.17−13.52 −2.64−0.22−9.22
modes, according the definition, S appearing in above equations are defined by
f
AdCiPr = AA((BB−− →ff))−+AA((BB++ →ff¯¯)), (33) Cf = ||AA¯ff||22+−||AA¯ff||22, (37)
→ → 2Im(λ )
f
S = , (38)
the predictions of the direct CP asymmetries based on f 1+ A¯ /A 2
f f
two different scenarios of scalar mesons are summarized V V|∗ A¯ |
inTableIV.The resourceofthis asymmetryis the inter- λ = tb td f, (39)
f V∗V A
ferencebetweentheemissiondiagramsandannihilations. tb td f
The large uncertainties in the latter lead to the large er- Replacing f by f¯, we could obtain the formulaes for Cf¯
rorsinthedirectCP asymmetries,asshowninthetable.
and Sf¯, correspondingly. If the experiment could find
moFroerctohmepnliecuattreadlddueecatyo mthoedfeasc,ttthheatsibtuoatthioBn0baencdomBe0s tnheewvpaalureasmoefteCrfs:, Sf, Cf¯andSf¯,wethus canobtainfour
could decay to the same final states. For illustration, we
1 1
wdeilclatyaakme Bpl0it(uBd0e)s,→AfK,∗0A0(f¯1,4A3¯0f)Kan0daAs¯f¯anareexdaemnpolteesd. aFsour C = 21(Cf +Cf¯), ∆C =12(Cf −Cf¯), (40)
S = 2(Sf +Sf¯), ∆S = 2(Sf −Sf¯). (41)
Af =hK0∗K0|B0i, Af¯=hK∗0K0|B0i; Physically, S is the mixing-induced CP asymmetry and
A¯f =hK0∗K0|B0i, A¯f¯=hK∗0K0|B0i. (34) CareisCfrPo-mevtehneudnidreecrtCCPPtarsaynmsfmoremtrayt,iownhλilef ∆→C1a/nλdf¯.∆ISn
Then, we can write down the CP asymmetry as Table.V, we present our estimations of ACP, C, ∆C, S
and ∆S for the final states K∗0K0, K∗0K∗0, K∗+K−
0 0 0
A = |Af|2+|A¯f|2−|Af¯|2−|A¯f¯|2. (35) and K0∗+K∗−, under two different scenarios. It should
CP |Af|2+|A¯f|2+|Af¯|2+|A¯f¯|2 btreiesstrbeescsaeudsethoafttihnePaQbsCenDce[6o],f tthreeereopisernaotoCrsP. Hasoywmevmeer-,
in QCDF, by including the penguin contractions (P ),
Due to B0 B0 mixing, the four time-dependent decay i
the charming penguin namely, an extra strong phase to-
widths are −given by (f =K∗0(1430)K0) getherwithanotheronefromannihilationsmightleadto
0
the large CP asymmetry, as shown in the table. Again,
Γ(B0(t) f) = e−Γt1(A 2+ A¯ 2) thelargeuncertaintiesofannihilationsresultinthelarge
f f
→ 2 | | | | error for the pure annihilation decay modes, especially
[1+C cos∆mt S sin∆mt], underS2. Wehopetheseparameterscanbemeasuredin
f f
−
Γ(B0(t)→f¯) = e−Γt21(|Af¯|2+|A¯f¯|2) fauntdureevecnollhidigehres,rseuncehrgayseh+igeh−lcuomlliindoesri.ty Belle-II, LHC-b
[1−Cfcos∆mt+Sf¯sin∆mt],
1
Γ(B0(t)→f¯) = e−Γt2(|Af¯|2+|A¯f¯|2) IV. SUMMARY
[1+Cfcos∆mt−Sf¯sin∆mt], Inthiswork,westudiedtheB →K0∗(1430)K(∗)decays
Γ(B0(t) f) = e−Γt1(A 2+ A¯ 2) under two different scalar meson scenarios by using the
→ 2 | f| | f| QCD factorization approach. We calculated the branch-
[1 C cos∆mt+S sin∆mt],(36) ing fractions and the CP asymmetry parameters. It is
f f
− found that the decay modes with the scalarmeson emit-
where ∆m stands for the mass difference of two mass ted, have large branching fractions due to the enhance-
eigenstate of B0/B0 meson, and Γ for the averagedecay mentoflargechiralfactorrK0∗,withsomeofthebranch-
χ
width ofthe B meson. The auxiliaryparametersC and ing fractions around the corner of Belle II. In contrast,
f
8
TABLE V: The CP-violating parameters ACP,C, ∆C, S and ∆S (%) of B→K0∗(1430)K(∗).
Decay Mode Scenario ACP C ∆C S ∆S
K∗0K0 S1 −0.52−+00..0044+−00..1120+−00..0183 0.05−+00..0000+−00..0000+−00..0011 −0.03−+00..0000+−00..0011+−00..0000 1.00−+00..0000+−00..0011+−00..0000 0.00−+00..0000+−00..0000+−00..0000
0 S2 −0.90+0.03+0.06+0.08 0.05+0.01+0.01+0.03 −0.02+0.01+0.01+0.02 1.00+0.00+0.00+0.00 0.02+0.00+0.00+0.00
−0.02−0.04−0.08 −0.00−0.01−0.01 −0.00−0.01−0.00 −0.00−0.00−0.00 −0.00−0.00−0.00
K∗0K∗0 S1 −0.98−+00..0020+−00..0011+−00..0021 0.33−+00..0017+−00..0003+−00..1048 0.20−+00..0016+−00..0003+−00..1048 0.91−+00..0033+−00..0022+−00..0242 0.02−+00..0032+−00..0032+−00..0232
0 S2 −1.00+0.00+0.00+0.00 0.27+0.09+0.08+0.21 0.18+0.09+0.08+0.20 0.86+0.07+0.06+0.07 −0.13+0.07+0.06+0.07
−0.00−0.00−0.00 −0.24−0.23−0.33 −0.24−0.23−0.33 −0.01−0.00−0.22 −0.01−0.00−0.22
K∗+K− S1 0.79−+00..0011+−00..0000+−00..0025 −0.04−+00..0011+−00..0010+−00..0044 0.06−+00..0012+−00..0000+−00..0076 0.04−+00..0022+−00..0011+−00..0025 −0.84−+00..0033+−00..0011+−00..1004
0 S2 0.29+0.06+0.01+0.35 −0.02+0.03+0.01+0.27 0.25+0.04+0.01+0.29 −0.36+0.05+0.04+0.50 −0.20+0.12+0.01+1.02
−0.01−0.00−0.98 −0.02−0.00−0.10 −0.03−0.01−0.26 −0.05−0.02−0.11 −0.10−0.02−0.35
K∗+K∗− S1 0.96−+00..0000+−00..0000+−00..0001 −0.01−+00..0000+−00..0000+−00..0012 0.02−+00..0010+−00..0010+−00..0032 0.01−+00..0000+−00..0000+−00..0000 −0.98−+00..0000+−00..0000+−00..0021
0 S2 0.88+0.02+0.00+0.05 0.01+0.01+0.00+0.14 0.11+0.03+0.01+0.23 0.02+0.01+0.00+0.48 −0.91+0.02+0.00+0.63
−0.00−0.00−0.66 −0.01−0.00−0.02 −0.02−0.00−0.11 −0.02−0.00−0.02 −0.02−0.00−0.05
the branching fractions of decay modes with the vector Acknowledgments
meson emitted, are much smaller. Moreover, the anni-
hilationcontributionstakelargeuncertaintiesbecauseof
the free parameter from endpoint singularity. For the
pure annihilation type decays, we predicted very small
branching fractions, which are 2-3 orders of magnitudes This workis partly supportedby the NationalScience
smaller than the results from PQCD. Some of the decay Foundation of China (Grants No. 11175151, 11375208
channels are hopeful to be measured in future colliders, and No. 11235005), and the Program for New Century
such as Belle-II, LHC-b and even high energy e+e− col- Excellent Talents in University (NCET) by Ministry of
lider. Education of P. R. China (Grant No. NCET-13-0991).
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