Eur. Phys. J. C (2019) 79:960https://doi.org/10.1140/epjc/s10052-019-7484-y

Regular Article - Theoretical Physics

Calculation of the B → K∗0,2(1430) f0(980)/σ decays in the

perturbative QCD approach

Qi-Xin Li1, Lei Yang1, Zhi-Tian Zou1, Ying Li1,2,a, Xin Liu3

1 Department of Physics, Yantai University, Yantai 264005, China2 Center for High Energy Physics, Peking University, Beijing 100871, China3 Department of Physics, Jiangsu Normal University, Xuzhou 221116, China

Received: 29 October 2019 / Accepted: 11 November 2019 / Published online: 22 November 2019© The Author(s) 2019

Abstract Motivated by the observations of the decaysB0 → K ∗

0 (1430)0 f0(980) and B0 → K ∗2 (1430)0 f0(980)

from BaBar collaboration, we study the B0(+) →K ∗

0,2(1430)0(+) f0(980)/σ decays in the perturbative QCDapproach for the first time. In the absence of reliable nonper-turbative wave functions we only assume the scalar mesonf0(980) and σ are two-quark ground states. In our calcula-tions, these decays are all dominated by the hard-scatteringemission and annihilation diagrams, while the factorizableemission diagrams are forbidden or suppressed heavily by thevector decay constants. Furthermore, the branching fractionsare sensitive to the mixing between f0(980) and σ . Compar-ing our results with the experimental data, a large mixingangle θ is favored. Taking θ = 145◦, the orders of branch-ing fractions of B → K ∗

0 (1430)0σ , B → K ∗2 (1430)0σ and

B → K ∗0,2(1430)0 f0(980) are predicted to be 10−4, 10−5

and 10−6, respectively, which can be measured in the currentexperiments such as LHCb and Belle-2. In addition, althoughthese decays are penguin dominant, the mixing also leads tolarge direct CP asymmetries in these decays. With the pre-cise data in future, our results could shed light on the innerstructure of the scalar mesons and can be used to determinethe mixing angle of the σ − f0(980) system.

1 Introduction

The rare B meson decays have been viewed as an importantplace for testing the standard model [1] and searching for thepossible effects of new physics beyond the standard model[2]. In past few years, much attentions had been paid onthe B → PP, PV and VV decays, where P and V arepseudoscalar and vector mesons. With the development ofhigh energy and high luminosity experiments, the studies of

a e-mail: liying@ytu.edu.cn

B decays with scalar, axial vector and tensor particles becameavailable.

In 2002, the decay B → f0(980)K with large branch-ing fraction was firstly observed in Belle experiment [3],and was confirmed subsequently by BaBar [4] in 2004.Since then, more and more B decays involving a light scalarmeson in final states have been observed in both Belle [5–8], BaBar [9–16] and LHCb [17–19] experiments, whichprovided us another perspective for the study of the scalarmesons, since their underlying structure have not been estab-lished well by studying their decays. In the theoretical side,it is well accepted by most of us that the scalar below ornear 1 GeV including σ , κ , a0(980) and f0(980), form oneSU(3)nonet, while the a0(1450), K ∗

0 (1430), f0(1370), andf0(1500)/ f0(1710) with the mass above 1 GeV are groupedinto another SU(3) nonet, though there is controversy aroundthis classification. The following question is how to under-stand and differentiate these two nonets. For this purpose,on the basis of answering which nonet is the lowest two-quark states, two scenarios have been proposed [20,21]. Inthe first scenario (S1), the mesons below or near 1 GeV aretreated as the lowest qq bound states, and those above 1 GeVare the first excited two-quark states. On the contrary, in theanother scenario (S2), the mesons near 1.5 GeV are viewedas the ground two-quark states, while the lighter mesons areidentified as the predominant qqqq states with a possiblemixing with glueball states. For instance, f0(980) is the low-est two-quark state in S1, while it is a four-quark state inS2. Similarly, the heavy scalar K ∗

0 (1430) is the excited twoquark state in S1, and in S2 it is viewed as the ground state. Ofcourse, each scenario has its own physical picture. Taking Bdecays with f0(980) as an example, in S2 the light energeticf0(980) dominated by four-quark configuration requires topick up the energetic quark–antiquark pair to form a fast four-quark state, which means that a wave function describing theinteractions among four quarks are needed in the theoreti-

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960 Page 2 of 14 Eur. Phys. J. C (2019) 79 :960

cal calculations [22]. However, the reliable four-quark wavefunctions of scalar mesons are still absent till now. There-fore, we will study some particular decays in the two-quarkassumption in this work. By comparing to experimental data,we hope that our results based on two-quark picture couldshed light on the inner structure and characters of the scalarmesons.

In S1,the lighter scalars are regarded as the ground two-quark states. Because the f0(980)1 is the heaviest and theσ is the lightest one, the ideal mixing is usually adopted,and is also supported by the measurements of D+

s → f0πand φ → f0γ , which illustrates that the f0 is the pure ssstate. However, the observed relation �(J/ψ → f0ω) �12�(J/ψ → f0φ) [23] implies that f0 has uu and dd com-ponents. Moreover, the width of f0 is dominated by theππ mode, which is very similar to the case of a0(980). Allthe above phenomena suggest that in two-quark picture theσ and f0 should be the mixing states of nn and ss withnn = 1√

2(uu+dd), and the mixing matrix can be defined as

(σ

f0

)=

(cos θ − sin θ

sin θ cos θ

) (nnss

)(1)

For the σ − f0 mixing angle θ , it can be constrained bythe existed experimental data. For example, using the ratiobetween the branching fractions of J/ψ → f0ω and thatof J/ψ → f0φ, the mixing angle can be obtained to be(34 ± 6)◦

⋃(146 ± 6)◦ [24]. In Ref. [25], based on the

measurements of the ratio of the coupling of f0 decayinginto ππ and KK the authors obtained the mixing angle tobe (25.1 ± 0.5)◦

⋃(164.3 ± 0.2)◦ with data [26–28], and

(42.3+8.3−5.5)

◦ ⋃(158±2)◦ with data [29]. In addition, the phe-

nomenological analysis of the radiative decay φ → f0γ andf0 → γ γ implied that the obtuse angle θ = (138 ± 6)◦is more preferred. More detailed discussions about the mix-ing angle can be found in Ref. [25]. In short, it is still notclear whether there exists a universal mixing angle θ whichaccommodates simultaneously to all the experimental mea-surements. Conservatively, we set the mixing angle to be afree parameter in this work.

In 2012, BaBar collaboration reported their measure-ments on the decays B0 → K ∗

0 (1430)0 f0(980) and B0 →K ∗

2 (1430)0 f0(980) [30]. It is only the scalar mesons and thetensor mesons that are involved in these decays, which arespecial in contrast to other decays with the pseudoscalar orthe vector meson. When one scalar meson is produced inB decays, its vector decay constant is about zero due to theconjugation invariance, and small values are caused by theviolation of the SU(3) symmetry. Meanwhile, in terms ofthe lorentz invariance, the tensor meson cannot be producedthrough the (V ± A) and (S ± P) currents. Therefore, this

1 For the sake of simplicity, we ignore the (980) and (1430) in thefollowing context unless special statement.

kind of decays are highly suppressed or forbidden in naivefactorization. So, in order to calculate these decays reliably,we should go beyond the naive factorization and evaluatethe contributions from the nonfactorizable and annihilationtype diagrams. In the past few years, the decays involvinga scalar meson or a tensor meson in final states have beenalready explored in different approaches, such as the general-ized factorization approach [31], QCD factorization (QCDF)[22,32–38] and perturbative QCD approach (PQCD) [39–57]. Based on the researching achievements of processes,and stimulated by the experimental data, in this work ,weshall extend our studies to the B0(+) → K ∗0(+)

0 f0/σ and

B0(+) → K ∗0(+)2 f0/σ decays in PQCD approach, and try to

provide new understanding to the mixing angle of the σ − f0mixing.

The outline of the present paper is as follows. In Sect. 2,we introduce the formalism of the PQCD approach and theinput quantities relevant to this work, such as the decay con-stants and the wave functions with the light-cone distributionamplitudes. We will apply the PQCD factorization to studythe B0(+) → K ∗0(+)

0 f0/σ and B0(+) → K ∗0(+)2 f0/σ decays

and present the analytic formulas of the decay amplitudes inSect. 3. The numerical results and the detailed discussionswill be given in Sect. 4, and we will summarize this work inthe last section.

2 Formalism and wave function

In the B meson rest framework, because the B meson is aheavy particle, the two daughter particles are energetic withlarge momenta and move fast. Because the light spectatorquark in the B meson is soft, so in order to form an ener-getic final state, a hard gluon is needed to kick the soft spec-tator quark into a collinear one. As a result, the hard ker-nel is a six-quark interaction. The intrinsic character of thePQCD approach is keeping the transverse momentum kT ofthe valence quarks of the hadrons in the initial and final states.After that, the end-point singularity in the amplitudes will bekilled naturally. Moreover, the kept transverse momenta willintroduce the additional energy scale, which will lead to thedouble logarithms in the QCD corrections. Within the resum-mation technology, these double logarithms will be resumedinto the so-called Sudakov form factor, which can effectivelysuppress the contributions from long distance.

As we have already known, there are many scales in thenonleptonic two-body B meson decays, and the factorizationis usually adopted. In particular, when the scale is higherthan the W boson mass (mW ), the physics can be calculatedperturbatively and get the Wilson coefficients C(mW ) at thescale mW . Using the renormalization group, we can get theWilson coefficients containing the physics between the scale

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Eur. Phys. J. C (2019) 79 :960 Page 3 of 14 960

mW and the b-quark mass scale (mb). The physics betweenthe scale mb and the factorization scale t can be calculatedperturbatively and included in the so-called hard kernel inthe PQCD approach. Finally, the physics below the scale tis soft and nonperturbative, which can be parameterized intothe universal hadronic wave functions of the initial and finalstates. In this way, the decay amplitude in the PQCD approachcan be written as the convolution of the Wilson coefficientsC(t), the hard kernel H(xi , bi , t), and the initial and finalhadronic wave functions [58,59]:

A =∫ 1

0dx1dx2dx3

∫ ∞

0b1db1b2db2b3db3 Tr

[C(t)H(xi , bi , t)

×�B(x1, b1)�2(x2, b2)�3(x3, b3)St (xi )e−S(t)

], (2)

xi (i = 1, 2, 3) denoting the momentum fraction of valencequark in the meson. The bi is the conjugate variable of thetransverse momentum kT . The jet function St (xi ) that isresulted from the resummation of the double logarithm ln2 xican smear the end-point singularity in xi threshold effec-tively. The aforementioned Sudakov form factor e−S(t) aris-ing from the resummation of the double logarithms ln2 kTsuppresses the soft dynamics effectively i.e., the long dis-tance contributions in the large b region [60–63]. The mode-dependent hard kernel H(xi , bi , t) and the relevant effec-tive Hamiltonian He f f are similar to B → PP, VV decays,which have been discussed in detail, for example, in Refs.[64,65].

In our calculations, the most important inputs are the wavefunctions of hadrons. For the B meson, as a heavy-lightsystem, after neglecting the numerically suppressed lorentzstructure, its wave function can be defined as

�B(x1, b1) = i√2Nc

( /PB + mB)γ5φB(x1, b1), (3)

with PB denoting the momentum of B meson. φB(x1, b1)

is the light-cone distribution amplitude (LCDA) and can bedefined as

φB(x1, b1) = NBx21 (1 − x2

1 ) exp[

− m2Bx

21

2ω− ω2b2

1

2

]. (4)

In the above equation, the normalization constant NB can bedetermined by the normalization condition∫ 1

0dx1φB(x1, b1 = 0) = fB

2√

6, (5)

where the fB is decay constant of the B meson. As usual, forthe shape parameter ω in the LCDA and the fB , we take ω =(0.4 ± 0.04)GeV, and fB = (0.19 ± 0.02)GeV [60,64,65].

For the scalar mesons, the two decay constants can bedefined as

〈S(p)|qγμq′|0〉 = fS pμ, 〈S(p)|qq ′|0〉 = fSmS . (6)

The vector decay constant fS and the scalar decay constantf S can be related through the equations of motion

f S = μS fS = mS fSm2(μ) − m1(μ)

, (7)

where mS and m1(2) are the scalar meson mass and the run-ning current quark mass, respectively. From the above equa-tion, one can find that, compared to the scalar decay con-stant, the vector decay constant is highly suppressed by thetiny mass difference between the two running current quark.Furthermore, for some neutral scalar mesons, such as theconsidered f0 and σ , their vector decay constants are zerodue to the charge conjugation invariance.

Up to the twist-3, the wave function of the scalar mesoncan be written as [22,32,38].

�S(x) = i√6

[/PφS(x) + mSφ

sS(x) + mS(/n/v − 1)φt

S(x)],

(8)

with the light-like unit vectors n = (1, 0, 0T ) and v =(0, 1, 0T ). Similarly, the twist-2 LCDA φS(x) and twist-3LCDAs φ

s(t)S (x) satisfy the normalization conditions

∫ 1

0dxφS(x) = fS,

∫ 1

0dxφs(t)

S (x) = fS . (9)

The twist-2 LCDA φS(x, μ) can be expanded as the Gegen-bauer polynomials

φS(x) = 3

2√

6x(1 − x)

[fS + f S

∞∑m=1

BmC3/2m (2x − 1)

],

(10)

where scale-dependent Bm are the Gegenbauer moments andC3/2m are the Gegenbauer polynomials. In the case of the two

twist-3 LCDAs, for simplicity, we shall adopt the asymptoticforms [66]

φsS(x) = f S

2√

6, φt

S(x) = fS

2√

6(1 − 2x). (11)

The explicit values of the parameters Bm , fS , and f S arereferred to the Refs. [22,32,38].

In the quark model, the tensor meson with J PC = 2++has the angular momentum L = 1 and spin S = 1. Dueto angular momentum conservation, the polarizations withλ = ±2 vanish in two-body B decays with one tensor meson[34,35]. In this case, the wave function of the tensor mesonis very similar to the vector meson, and can be defined as

�T = 1√6

[mT /ε∗•LφT (x) + /ε∗•L /Pφt

T (x) + m2T

ε• · v

P · v φsT (x)

],

�⊥T = 1√

6

[mT /ε∗•⊥φv

T (x) + /ε∗•⊥ /PφTT (x)

+mT iεμνρσ γ5γμε∗ν•⊥nρvσ φa

T (x)

]. (12)

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960 Page 4 of 14 Eur. Phys. J. C (2019) 79 :960

with ε0123 = 1. The reduced polarization vector ε•μ can be

expressed as ε•μ = εμνvν

P·v , where the εμν is the polarizationtensor of the tensor meson. The expressions of the twist-2and twist-3 LCDAs are given as

φT (x) = fT

2√

6φ‖(x), φt

T (x) = f ⊥T

2√

6ht‖(x),

φsT (x) = ⊥

4√

6

d

dxhs‖(x), φT

T (x) = f ⊥T

2√

6φ⊥(x),

φvT (x) = fT

2√

6gv⊥(x), φa

T (x) = fT

8√

6

d

dxga⊥(x), (13)

with the auxiliary functions

φ‖,⊥(x) = 30x(1 − x)(2x − 1), gv⊥(x) = 5(2x − 1)3,

ht‖(x) = 15

2(2x − 1)(1 − 6x + 6x2),

hs‖(x) = 15x(1 − x)(2x − 1),

ga⊥(x) = 20x(1 − x)(2x − 1). (14)

3 Perturbative calculation

In this section, we shall perform the calculation of the hardkernel H(xi , bi , t), which depends on the specific Feynmandiagram. We start from the common low energy effectivehamiltonian, which are given as [67]

He f f = GF√2

{VubV

∗us

[C1(μ)O1(μ) + C2(μ)O2(μ)

]

−VtbV∗ts

10∑i=3

Ci (μ)Oi (μ)

}+ H.c., (15)

whereVub,us,tb,ts are Cabibbo–Kobayashi–Maskawa (CKM)matrix elements. The local four-quark operators Oi (i =1, . . . , 10) are given as:

• current–current (tree) operators

O1 = (uαbβ)V−A(sβuα)V−A,

O2 = (uαbα)V−A(sβuβ)V−A, (16)

• QCD penguin operators

O3 = (sαbα)V−A

∑q ′

(q ′βq

′β)V−A,

O4 = (sβbα)V−A

∑q ′

(q ′αq

′β)V−A, (17)

O5 = (sαbα)V−A

∑q ′

(q ′βq

′β)V+A,

O6 = (sβbα)V−A

∑q ′

(q ′αq

′β)V+A, (18)

• electro-weak penguin operators

O7 = 3

2(sαbα)V−A

∑q ′

eq ′(q ′βq

′β)V+A,

O8 = 3

2(sβbα)V−A

∑q ′

eq ′(q ′αq

′β)V+A, (19)

O9 = 3

2(sαbα)V−A

∑q ′

eq ′(q ′βq

′β)V−A,

O10 = 3

2(sβbα)V−A

∑q ′

eq ′(q ′αq

′β)V−A, (20)

where α and β are color indices and q ′ are the active quarksat the scale mb, i.e. q ′ = (u, d, s, c, b). The left handedcurrent is defined as (q ′

αq′β)V−A = q ′

αγν(1 − γ5)q ′β and the

right handed current (q ′αq

′β)V+A = q ′

αγν(1 + γ5)q ′β . The

combinations ai of Wilson coefficients are defined as usual[68]:

a1 = C2 + C1/3, a2 = C1 + C2/3, a3 = C3 + C4/3,

a4 = C4 + C3/3, a5 = C5 + C6/3,

a6 = C6 + C5/3, a7 = C7 + C8/3, a8 = C8 + C7/3,

a9 = C9 + C10/3, a10 = C10 + C9/3. (21)

In this work, we shall study two types of decays: oneis the B decay with two scalar mesons, while another is Bdecay involving a tensor meson and a scalar meson. In thedecay amplitudes, the subscripts SS and T S represent differ-ent types, respectively. According to the effective Hamilto-nian (15), we can draw the lowest order diagrams of decayswe concerned, and the diagrams of decay B → K ∗0

0 f0/σare shown in Fig. 1 as an example. These Feynman diagramscan be categorized into two classes based on the typologicalstructures: the emission diagrams (a, b, c and d), in whichthe light quark in B meson enter one of the light mesonsas a spectator, and the annihilation diagrams (e, f, g and h),in which both of the two quarks in B meson are involvedin the operators. In PQCD approach, for each diagram withdifferent operator, the whole amplitude is expressed as theconvolution of the hard kernel, the related hard function, andthe wave functions of involved mesons.

We first calculate the usual factorizable emission diagrams(a) and (b). When we insert the (V − A)(V − A) current inthe corresponding vertices, the amplitudes associated to thesecurrents are given as:

F LLSS,S = 8πCFm

4B fS

∫ 1

0dx1dx3

∫ ∞

0b1db1b3db3φB(x1, b1)

{[(1 + x3)φS3(x3)

−r3(2x3 − 1)(φsS3(x3) + φt

S3(x3))]

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Eur. Phys. J. C (2019) 79 :960 Page 5 of 14 960

(b)(a)

(d)(c)

(f)(e)

(h)(g)

B

b

Fig. 1 Leading order Feynman diagrams in PQCD appraoch

Eef (ta)hef [x1, x3(1 − r2), b1, b3[+2r3φ

sS3(x3)Eef (tb)hef [x3, x1(1 − r2

2 ), b3, b1]},

(22)

F LLT S,S =

√2

3F LLSS,S | φ

(s,t)S3 → φ

(s,t)T , (23)

where CF = 4/3 and ri = mMimB

, with Mi denoting the finalstates. The second term “S” in the subscripts indicates thatthe scalar meson is emitted. The superscript “LL” means the(V − A)(V − A) current. The expressions of the related hardfunctions Eef , hef , and the scale t are the same as those inB → VV decays, which can be found in the Appendix ofRef. [65]. The (V − A)(V + A) current cannot contributeto the decays we considered, so we do not include it here.When the (S − P)(S + P) current, that is arising from theFierz transformation of (V − A)(V + A) current, is inserted,

the amplitudes can be read as

F SPSS,S = −16πCF fSm

4Br2

∫ 1

0dx1dx3

×∫ ∞

0b1db1b3db3φB(x1, b1)

{[φS3(x3)

+r3(φsS3(x3)(2 + x3) − φt

S3(x3)x3)]

×Eef (ta)hef [x1, x3(1 − r22 ), b1, b3]

−2r3φsS3(x3)Eef (tb)hef [x3, x1(1 − r2

2 ), b3, b1]},

(24)

F SPT S,S =

√2

3F SPSS,S | φ

(s,t)S3 → φ

(s,t)T . (25)

Due to the fact that the tensor meson can not producedthrough (V − A) and (S+ P) currents, the factorizable emis-sion diagrams with a tensor meson emitted are forbidden, and

F LLT S,T = F SP

T S,T = 0, (26)

The second row in Fig. 1 are the hard-scattering emis-sion diagrams, whose decay amplitudes involve three mesonwave functions. This means that the decay amplitudes aremore complex than that of factorizable emission diagrams.After the variable b3 is integrated out by the delta functionδ(b1 − b3), the expressions of the amplitudes are presentedas follows

• (V − A)(V − A)

MLLSS,S = −16

√2

3CFπm4

B

∫ 1

0dx1dx2dx3

×∫ ∞

0b1db1b2db2φB(x1, b1)φS2(x2)

×{[

φS3(x3)(x2 − 1)+r3x3(φsS3(x3)−φt

S3(x3))]

×Een f (tc)hen f [α, β1, b1, b2]+

[φS3(x3)(x2 + x3) − r3x3(φs

S3(x3)

+φtS3(x3))

]Een f (td )hen f [α, β2, b1, b2]

}, (27)

MLLT S,S =

√2

3MLL

SS,S | φ(s,t)S3 (x3) → φ

(s,t)T (x3), (28)

MLLT S,T =

√2

3MLL

SS,S | φS2(x2) → φT (x2). (29)

• (V − A)(V + A)

MLRSS,S = 16

√2

3CFπr2m

4B

∫ 1

0dx1dx2dx3

×∫ ∞

0b1db1b2db2φB(x1, b1)

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960 Page 6 of 14 Eur. Phys. J. C (2019) 79 :960

×{[

(x2 − 1)φS3(x3)(φsS2(x2) + φt

S2(x2))

+r3

((1 − x2 + x3)(φt

S2(x2)φtS3(x3)

−φsS2(x2)φs

S3(x3))

+(x2 + x3 − 1)(φtS2(x2)φs

S3(x3)

−φsS2(x2)φt

S3(x3)))]

×Een f (tc)hen f [α, β1, b1, b2]+

[x2φS3(x3)(φs

S2(x2) − φtS2(x2))

+r3

((x3 − x2)(φs

S2(x2)φtS3(x3)

+φtS2(x2)φs

S3(x3))

+(x2 + x3)(φsS2(x2)φs

S3(x3)

+φtS2(x2)φt

S3(x3))]

×Een f (td )hen f [α, β2, b1, b2]}, (30)

MLRT S,S =

√2

3MLR

SS,S | φ(s,t)S3 (x3) → φ

(s,t)T (x3), (31)

MLRT S,T =

√2

3MLR

SS,S | φ(s,t)S2 (x2) → φ

(s,t)T (x2), (32)

• (S − P)(S + P)

MSPSS,S = −16

√2

3CFπm4

B

∫ 1

0dx1dx2dx3

×∫ ∞

0b1db1b2db2φB(x1, b1)φS2(x2)

×{[

φS3(x3)(−1 + x2 − x3) + r3x3(φsS3(x3)

+φtS3(x3))

]×Een f (tc)hen f [α, β1, b1, b2]+

[φS3(x3)x2 + r3x3(φ

tS3(x3) − φs

S3(x3))]

×Een f (td)hen f [α, β2, b1, b2]}, (33)

MSPT S,S =

√2

3MSP

SS,S | φ(s,t)S3 (x3) → φ

(s,t)T (x3), (34)

MSPT S,T =

√2

3MSP

SS,S | φS2(x2) → φT (x2). (35)

Particularly, when the emitted meson is a pseudoscalar or avector light meson, the total contributions of these nonfac-torizable emission diagrams are suppressed highly, due tothe cancelation between the two diagrams (c and d). Whilefor the current considered decays with a scalar/tensor mesonemitted, because LCDAs are antisymmetric, the contribu-tions between the two diagrams are no longer destructive butconstructive. Therefore, the nonfactorizable emission dia-

grams contributions are not suppressed in those considereddecays.

Now we move to calculate the annihilation diagrams,where two quarks in the initial B meson are involved thefour-quark interaction and qq quarks included in final statesare produced from a hard gluon. In Fig. 1, the diagrams (eand f) in third row are the so-called factorizable annihilationtype diagrams, whose decay amplitudes can be calculated asfollow:

• (V − A)(V ± A) current

ALL(LR)SS,S = 8CF fBπm4

B

∫ 1

0dx2dx3

∫ ∞

0b2db2b3db3

×{[

(x3 − 1)φS2(x2)φS3(x3)

+2r2r3φsS2(x2)(φ

sS3(x3)(x3 − 2) + φt

S3(x3)x3)]

×Ea f (te)ha f [α1, β, b2, b3]+

[− 2r2r3φ

sS3(x3)(φ

sS2(x2)(1 + x2)

+φtS2(x2)(x2 − 1)) + x2φS2(x2)φS3(x3)

]

×Ea f (t f )ha f [α2, β, b2, b3]}, (36)

ALL(LR)T S,S =

√2

3ALL(LR)

SS,S |φ(s,t)S3 (x3) → φ

(s,t)T (x3), (37)

ALL(LR)T S,T =

√2

3ALL(LR)

SS,S |φ(s,t)S2 (x2) → φ

(s,t)T (x2), (38)

• (S − P)(S + P) current

ASPSS,S = −16CF fBπm4

B

∫ 1

0dx2dx3

∫ ∞

0b2db2b3db3

×{[

2r2φS3(x3)φsS2(x2)

+r3(x3 − 1)φS2(x2)(φsS3(x3) + φt

S3(x3))]

×Ea f (te)ha f [α1, β, b2, b3]−

[2r3φS2(x2)φ

sS3(x3) + r2x2φS3(x3)(φ

tS2(x2)

−φsS2(x2))

]Ea f (t f )ha f [α2, β, b2, b3]

}, (39)

ASPT S,S =

√2

3ASP

SS,S|φ(s,t)S3 (x3) → φ

(s,t)T (x3), (40)

ASPT S,T =

√2

3ASP

SS,S|φ(s,t)S2 (x2) → φ

(s,t)T (x2). (41)

In above equations, the related scales te, f , the functions ha fand the inner functions can be found in the Appendix of Ref.[65].

123

Eur. Phys. J. C (2019) 79 :960 Page 7 of 14 960

The amplitude for the nonfactorizable annihilation dia-gram in Fig. 1g, h results in

• (V − A)(V − A)

WLLSS,S = 16

√2

3CFπm4

B

∫ 1

0dx1dx2dx3

×∫ ∞

0b1db1b2db2φB(x1, b1)

×{[

− φS2(x2)φS3(x3)x2

+r2r3

(φtS2(x2)(φ

tS3(x3)(1 − x2 + x3)

+φsS3(x3)(x2 + x3 − 1))

+φsS2(x2)(φ

tS3(x3)(1 − x2 − x3)

+φsS3(x3)(3 + x2 − x3))

)]×Ean f (tg)han f [α, β1, b1, b2]−

[φS2(x2)φS3(x3)(x3 − 1)

+r2r3

(φsS2(x2)(φ

sS3(x3)(1 + x2 − x3)

−φtS3(x3)(1 − x2 − x3))

+φtS2(x2)(φ

sS3(x3)(1 − x2 − x3)

−φtS3(x3)(1 + x2 − x3))

)]

Ean f (th)han f [α, β2, b1, b2]}, (42)

WLLT S,S =

√2

3WLL

SS,S|φ(s,t)S3 (x3) → φ

(s,t)T (x3), (43)

WLLT S,T =

√2

3WLL

SS,S|φ(s,t)S2 (x2) → φ

(s,t)T (x2), (44)

• (V − A)(V + A)

WLRSS,S = 16

√2

3CFπm4

B

∫ 1

0dx1dx2dx3

×∫ ∞

0b1db1b2db2φB(x1, b1)

×{[

r2φS3(x3)(φsS2(x2) + φt

S2(x2))(x2 − 2)

−r3φS2(x2)(φsS3(x3)

−φtS3(x3))(x3 + 1)

]Ean f (tg)han f [α, β1, b1, b2]

+[

− r2x2φS3(x3)(φsS2(x2) + φt

S2(x2))

+r3(x3 − 1)φS2(x2)(φsS3(x3)

−φtS3(x3))

]Ean f (th)han f [α, β2, b1, b2]

}, (45)

WLRT S,S = WLR

SS,S |φ(s,t)S3 (x3) → φ

(s,t)T (x3), (46)

WLRT S,T = WLR

SS,S |φ(s,t)S2 (x2) → φ

(s,t)T (x2), (47)

• (S − P)(S + P)

W SPSS,S = 16

√2

3CFπm4

B

∫ 1

0dx1dx2dx3

×∫ ∞

0b1db1b2db2φB(x1, b1)

×{[

(x3 − 1)φS2(x2)φS3(x3)

+r2r3

(φtS2(x2)(φ

tS3(x3)(1 − x2 + x3)

−φsS3(x3)(x2 + x3 − 1))

+φsS2(x2)(φ

sS3(x3)(3 + x2 − x3)

+φtS3(x3)(x2 + x3 − 1))

)]×Ean f (tg)han f [α, β1, b1, b2]+

[x2φS2(x2)φS3(x3)

−r2r3

(φsS2(x2)(φ

sS3(x3)(1 + x2 − x3)

+φtS3(x3)(1 − x2 − x3))

+φtS2(x2)(φ

tS3(x3)(−1 − x2 + x3)

+φsS3(x3)(x2 + x3 − 1))

)]

×Ean f (th)han f [α, β2, b1, b2]}, (48)

W SPT S,S =

√2

3W SP

SS,S |φ(s,t)S3 (x3) → φ

(s,t)T (x3), (49)

W SPT S,T =

√2

3W SP

SS,S |φ(s,t)S2 (x2) → φ

(s,t)T (x2). (50)

The related functions and the scales(tg and th) can be referredin the Ref. [65]. From the Eq. (36), it is obvious that thereexist large cancellations between the two annihilation typediagrams (e and f), thus the annihilation diagrams is viewedas power suppressed. This picture is consistent with the naiveargument about the neglect of the annihilation type diagrams[69,70]. However, although these diagrams are power sup-pressed, they can provide a large strong phase, which is usedto explain the CP asymmetry in B decays [60–64].

Finally, the total amplitude of B → K ∗+0 S can be written

as

A(B+ → K ∗+0 S(nn))

= GF

2

{V ∗ubVus

[a1(F LL

SS,S + ALLSS,S)

+C1(MLLSS,S + WLL

SS,S)]

−V ∗tbVts

[(a4 + a10)(F LL

SS,S + ALLSS,S)

+(a6 + a8)(F SPSS,S + ASP

SS,S)

+(C3 + C9)(MLLSS,S + WLL

SS,S)

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960 Page 8 of 14 Eur. Phys. J. C (2019) 79 :960

+(C5 + C7)(MLRSS,S + WLR

SS,S)]}

, (51)

A(B+ → K ∗+0 S(ss))

= GF√2

{V ∗ubVusC2MLL

SS,S

−V ∗tbVts

[(2C4 + 1

2C10

)MLL

SS,S

+(

2C6 + 1

2C8

)MSP

SS,S

]}, (52)

A(B0 → K ∗00 S(nn))

= GF

2

{V ∗ubVusC1MLL

SS,S

−V ∗tbVts

[(a4 − 1

2a10

)(F LL

SS,S + ALLSS,S)

+(a6 − 1

2a8

)(F SP

SS,S + ASPSS,S)

+(C3 − 1

2C9

)(MLL

SS,S + WLLSS,S)

+(C5 − 1

2C7

)(MLR

SS,S + WLRSS,S)

+(

2C4 + 1

2C10

)MLL

SS,S + (2C6 + C8/2)MSPSS,S

]}, (53)

A(B0 → K ∗00 S(ss))

= GF√2

{− V ∗

tbVts[(

a6 − 1

2a8

)(F SP

SS,S + ASPSS,S)

+(a4 − 1

2a10

)ALL

SS,S

+(C3 + C4 − 1

2C9 − 1

2C10

)MLL

SS,S

+(C3 − 1

2C9

)WLL

SS,S

+(C5 − 1

2C7

)(MLR

SS,S + WLRSS,S)

+(C6 − 1

2C8

)MSP

SS,S

]}, (54)

A(B+ → K ∗+2 S(nn))

= GF

2

{V ∗ubVus

[a1(F LL

SS,S + ALLT S,T )

+C1(MLLT S,T + WLL

T S,T )]

−V ∗tbVts

[(a4 + a10)(F LL

T S,T + ALLSS,S)

+(a6 + a8)(F SPT S,T + ASP

T S,T )

+(C3 + C9)(MLLT S,T + WLL

T S,T )

+(C5 + C7)(MLRT S,T + WLR

T S,T )]}

, (55)

A(B+ → K ∗+2 S(ss))

= GF√2

{V ∗ubVusC2MLL

T S,S − V ∗tbVts

[(2C4 + 1

2C10

)MLL

T S,S

+(

2C6 + 1

2C8

)MSP

T S,S

]}, (56)

A(B0 → K ∗00 S(nn))

= GF

2

{V ∗ubVusC1MLL

T S,T

−V ∗tbVts

[(a4 − 1

2a10

)(F LL

T S,T + ALLT S,T )

+(a6 − 1

2a8

)(F SP

T S,T + ASPT S,T )

+(C3 − 1

2C9

)(MLL

T S,T + WLLT S,T )

+(C5 − 1

2C7

)(MLR

T S,T + WLRT S,T )

+(

2C4 + 1

2C10

)MLL

T S,T + (2C6 + C8/2)MSPT S,T

]},

(57)

A(B0 → K ∗00 S(ss))

= GF√2

{− V ∗

tbVts[(

a6 − 1

2a8

)(F SP

T S,S + ASPT S,S)

+(a4 − 1

2a10

)ALL

T S,S

+(C3 + C4 − 1

2C9 − 1

2C10

)MLL

T S,S

+(C3 − 1

2C9

)WLL

T S,S

+(C5 − 1

2C7

)(MLR

T S,S + WLRT S,S)

+(C6 − 1

2C8

)MSP

T S,S

]}. (58)

Then, we can write down the amplitudes of B → K ∗0,2 f0

and B → K ∗0,2σ as

A(B → K ∗0(2) f0) = A(B → K ∗

0(2)S(nn)) sin θ

+A(B → K ∗0(2)S(ss)) cos θ, (59)

A(B → K ∗0(2)σ ) = A(B → K ∗

0(2)S(nn)) cos θ

−A(B → K ∗0(2)S(ss)) sin θ. (60)

Meanwhile, the direct CP asymmetries of these decays canbe defined as

ACP = A(B0 → K ∗0(2) f0) − A(B

0 → K∗0(2) f0)

A(B0 → K ∗0(2) f0) + A(B

0 → K∗0(2) f0)

, (61)

Obviously, both the amplitudes and the direct CP asymme-tries are related to the mixing angle θ .

4 Numerical results and discussions

We start this section by setting constants used in the calcula-tions. The vector decay constants and the scalar decay con-

123

Eur. Phys. J. C (2019) 79 :960 Page 9 of 14 960

Fig. 2 The branching fractions of B → K ∗0 f0 and B → K ∗

2 σ with variant of the mixing angle θ . The black lines are the center values, and thehorizontal (green) band is the experimental value

stants of the f0 and σ can be found in Ref. [38]. Other inputparameters such as the QCD scale, the masses of mesons, theCKM matrix elements, the decay constant of the B mesonand the lifetimes of the B mesons (in ps) are adopted as fol-lows [23]:

�f =4MS

= 0.25 ± 0.05 GeV, mB = 5.279 GeV, mb = 4.8 GeV,

τB±/0 = 1.638/1.519 ps, Vub = 0.00365, Vus = 0.22452,

Vts = 0.04133, Vtb = 0.999105, fB = 0.19 ± 0.02 GeV.

(62)

By setting the mixing angle θ to be a free parameter,we plot the variation of the branching fractions of decaysB → K ∗0

0 f0(σ ) and B → K ∗02 f0(σ ) with the angle θ in

Figs. 2 and 3, respectively. We acknowledge that the uncer-tainty is the inevitable incident in theoretical evaluation. Inthe present work, three kinds of errors are taken into account:the first errors are caused by the nonperturbative parameters,such as the initial and final mesons’ wave functions afore-mentioned in Sect. 2, which are dominant in our calculation.Fortunately, these errors will be reduced with the improve-ment of the experiments and the update of the theoretical

understanding. The second kind of errors come from the vari-ations of the factorization scale “t” and QCD scale �QCD,characterized by 0.8t ∼ t ∼ 1.2t and �QCD = (0.25±0.05)

GeV. In fact, these errors reflect the contributions of thenext-to-leading order radiative corrections and the next-to-leading order power corrections, since the complete next-to-leading order corrections in PQCD approach have notbeen accomplished [71,72] till now. The last errors are fromthe uncertainties of the unitary angle γ . We combine alluncertainties together and give the bounds as shown in thefigures.

In Ref. [30], BaBar collaboration reported the first mea-surements of branching fractions of the B0 → K ∗0

0,2 f0decays:

B(B0 → K ∗00 f0) = (2.7 ± 0.7 ± 0.6) × 10−6, (63)

B(B0 → K ∗02 f0) = (8.6 ± 1.7 ± 1.0) × 10−6, (64)

which can be also found in the Figs. 2 and 3. Combiningour theoretical results of B0 → K ∗0

0,2 f0 and the experimentalmeasurements, we obtain the mixing angle θ in the rangeof [135◦, 155◦], which is consistent with the conclusions ofthe Refs. [25,73,74]. In Ref. [73], the mixing angle is con-

123

960 Page 10 of 14 Eur. Phys. J. C (2019) 79 :960

Fig. 3 The branching fractions of B → K ∗2 f0 and B → K ∗

2 σ with variant of the mixing angle θ . The black lines are the center values, and thehorizontal (green) band is the experimental value

strained in the range [135◦, 158◦] by studying the charmed B

decays B(s) → D0f0. The authors in [74] obtained the mix-

ing angle θ ∼ 146◦ by analyzing the charmonium decaysBs → J/ψ f0(σ ). If the f0 is composed entirely of ss com-ponent, which indicates θ = 0◦, the branching fractions ofB0 → K ∗0

0 f0 and B0 → K ∗02 f0 are about 1.0 × 10−4 and

3.0 × 10−5, respectively, both of which are much larger thanthe data provided by the BaBar collaboration. When the mix-ing is taken in account and assuming the mixing angle lessthan 90◦, we find that the contributions from the componentnn = (uu+dd)/

√2 and ss have the same sign. Furthermore,

due to the constructive interference between the two differ-ent type amplitudes from the two components, the branchingfractions would be enhanced and overshoot the upper limit ofthe experimental data, which implies that the acute angle θ isunfavored. Conversely, if the angle θ > 90◦, the branchingfractions of B0 → K ∗0

0(2) f0 will be suppressed by the can-cellation between these two amplitudes from nn and ss com-ponents, and the theoretical predictions of PQCD approachwill accommodate the experimental data well.

Now, taking the θ = 145◦ as a benchmark, we present ourpredictions of branching fractions as

B(B0 → K ∗00 f0) = (2.8+3.0

−1.7) × 10−6,

B(B+ → K ∗+0 f0) = (2.7+2.9

−1.7) × 10−6,

B(B0 → K ∗00 σ) = (298.0+96.7

−74.3) × 10−6,

B(B+ → K ∗+0 σ) = (299.7+80.9

−66.3) × 10−6,

B(B0 → K ∗02 f0) = (8.8+3.1

−1.7) × 10−6,

B(B+ → K ∗+2 f0) = (8.6+2.7

−1.7) × 10−6,

B(B0 → K ∗02 σ) = (38.9+14.0

−9.4 ) × 10−6,

B(B+ → K ∗+2 σ) = (38.4+13.3

−8.6 ) × 10−6, (65)

which can be measured in the current experiments, such asLHCb and Belle-2.

Lastly, we will discuss the relations between the directCP asymmetries and the mixing angle. As we already known,both strong and weak phases are the necessary conditions fordirect CP asymmetry. These decays concerned in this workare all governed by the b → s transition, and are dominatedby the penguin operators, because the contributions from thetree operators are either forbidden or suppressed by smallCKM matrix elements |VusVub|. In the naive 2-quark modelwith the ideal mixing, the decay B0 → K ∗0

0 f0 and B0 →

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Eur. Phys. J. C (2019) 79 :960 Page 11 of 14 960

Fig. 4 The direct CP asymmetries of B → K ∗0 f0 and B → K ∗

2 σ with variant of the mixing angle θ

K ∗02 f0 are both induced by b → sss transition, which is a

pure penguin process. In the Wolfenstein parameterizationof CKM matrix, there is no weak phase in this transition,so the direct CP asymmetries of these two decays are zero.For B0 → K ∗0

0 σ and B0 → K ∗02 σ that are induced by

b → sqq (q = u, d), their direct CP asymmetries decayare less than 5%, because |VusVub| � |VtsVub|. Since themixing is supported by many experimental measurementsand taken into account in this work, these considered decaysreceive three distinct types of contributions: the first one fromthe diagrams with emitted K ∗0(+)

0(2) , the second one from thef0/σ emission with qq component and the last one from thef0/σ with ss component. Similar to the branching fractions,these CP asymmetries are also related to the mixing angle θ .We plot the CP asymmetries of these decays with the changesof the mixing angle θ , as shown in Figs. 4 and 5. When themixing angle θ is involved, the qq component contributesto all concerned decays within the tree operators, which cancancel the penguin contributions from ss component whenthe mixing angle θ > 90◦. For instance, when the angle θ =145◦, the CP asymmetry of the B0 → K ∗0

0 f0 can be as largeas −68%. As for the B0 → K ∗0

0(2)σ decays, the interferencebetween qq and ss is contrary to corresponding decays withf0. For the isospin asymmetry, we note that the interference

for the considered B+ decays are similar to the correspondingB0 decays respectively, and 20% differences can be attributedto the effects of tree operators in the annihilation diagrams,which can be found form the Figs. 4 and 5. Because thedirect CP asymmetry is a ratio, the theoretical uncertaintiesfrom the nonperturbative parameters will be cancelled, andthe errors of these asymmetries will decrease, as illustratedin two figures. Therefore, if the two-quark structure will beconfirmed, the CP asymmetries can also be used to determinethe mixing angle θ .

5 Summary

In this paper, it is the first time that the B0(+) →K ∗

0(2)(1430)0(+) f0(980)(σ ) decays were studied in the per-turbative QCD approach under the two-quark assumption.Our theoretical results are hoped to shed light on the oldpuzzle about the inner structure of the scalar meson, espe-cially the mixing angle of the σ − f0(980) system. For thesedecays, due to the charge conjugation invariance and thelorentz invariance, the factorizable emission diagrams areforbidden or suppressed heavily by the vector decay con-stants of scalar mesons, and the nonfactorizale diagrams

123

960 Page 12 of 14 Eur. Phys. J. C (2019) 79 :960

Fig. 5 The direct CP asymmetries of B → K ∗2 f0 and B → K ∗

2 σ with variant of the mixing angle θ

and annihilation ones play the dominant roles. Moreover,for these considered penguin dominant decays, the penguincontributions from nn and ss components are at the samelevel. Thus the interferences are remarkable and affect thebranching fractions and CP asymmetries significantly, whichwill provide us good platforms to determine the mixingangle. After the calculations, combining the experimentalresults of branching fractions, we find that, for the mix-ing angle, the range of [135◦, 155◦] is favored. When themixing angle θ = 145◦, the predicted branching ratiosfor B0 → K ∗

0(2)(1430)0 f0(980) decays are in agreementwith the experimental data well. The future measurements ofCP asymmetries in LHCb and Belle-II can further test ourresults. Finally, we note that our calculation are only basedon the two-quark assumption. The four-quark component orKK threshold effect that may be important components inf0(980) were not included, because the reliable nonpertur-bative input parameters are still absent and left for futurestudy.

Acknowledgements This work was supported in part by the NationalNatural Science Foundation of China under the Grants nos. 11705159,11575151, 11975195, 11765012, 11875033 and 11447032, and by theNatural Science Foundation of Shandong province under the Grant nos.ZR2018JL001 and ZR2019JQ04. X. Liu is also supported by by the

Qing Lan Project of Jiangsu Province under Grant no. 9212218405,and by the Research Fund of Jiangsu Normal University under Grantno. HB2016004.

Data Availability Statement This manuscript has associated datain a data repository. [Authors’ comment: All data analysed in thismanuscript are available from the corresponding author on reasonablerequest.]

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

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