S, P and D-wave resonance contributions to BðsÞ → ηcð1S;2SÞKπ decays inthe perturbative QCD approach

Ya Li ,1,* Da-Cheng Yan,2,† Zhou Rui,3,‡ and Zhen-Jun Xiao 4,§

1Department of Physics, College of Sciences, Nanjing Agricultural University,Nanjing, Jiangsu 210095, P.R. China

2School of Mathematics and Physics, Changzhou University, Changzhou, Jiangsu 213164, P.R. China3College of Sciences, North China University of Science and Technology, Tangshan 063009, P.R. China

4Department of Physics and Institute of Theoretical Physics, Nanjing Normal University,Nanjing, Jiangsu 210023, P.R. China

(Received 26 November 2019; published 27 January 2020)

In this work, we analyze the three-body BðsÞ → ηcð1S; 2SÞKπ decays within the framework of theperturbative QCD approach (PQCD) under the quasi-two-body approximation, where the kaon-pioninvariant mass spectra are dominated by theK�

0ð1430Þ0,K�0ð1950Þ0,K�ð892Þ0,K�ð1410Þ0,K�ð1680Þ0, and

K�2ð1430Þ0 resonances. The time-like form factors are adopted to parametrize the corresponding S, P,

D-wave kaon-pion distribution amplitudes for the concerned decay modes, which describe the final-stateinteractions between the kaon and pion in the resonant region. The Kπ S-wave component at low Kπ massis described by the LASS line shape, while the time-like form factors of other resonances are modeled bythe relativistic Breit-Wigner function. We find the following main points: (a) the PQCD predictions of thebranching ratios for most considered B → ηcð1SÞðK�0 →ÞKþπ− decays agree well with the currentlyavailable data within errors; (b) for BðB0 → ηcðK�

0ð1430Þ →ÞKþπ−Þ and BðB0 → ηcKþπ−ðNRÞÞ (whereNR means nonresonant), our predictions of the branching ratios are a bit smaller than the measured ones;and (c) the PQCD results for the D-wave contributions considered in this work can be tested once precisedata from the future LHCb and Belle-II experiments become available.

DOI: 10.1103/PhysRevD.101.016015

I. INTRODUCTION

The BðsÞ meson decays into charmonia and a kaon-pionpair are of great interest since only a few color-suppressedmodes in hadronic B decays have been measured so far.Some standard model (SM) parameters can be extractedfrom the b → ccs transitions, while the studies of thesedecay channels can also provide an ideal place to find asignal for physics beyond the SM. The meson ηc and J=ψhave the same quark content but with different spin angularmomentum. As expected, the Bmeson decays involving theηc will garner considerable experimental attention with thedevelopment of experiments. In recent years, significantimprovements in understanding the heavy quarkonium

production mechanism have been achieved [1]. The B0 →ηcðK�ð892Þ0 →ÞKπ decay has been observed by the Belle[2] and BABAR collaborations [3,4]. Very recently, theB0 → ηcKþπ− decay was measured for the first time by theLHCb Collaboration [5], with the ηc meson reconstructedusing thepp decaymode. This decay is expected to proceedthrough K�0 → Kπ intermediate states as well as thenonresonant (NR) S-wave component, where the K�0 refersto various partial-wave resonances, such as K�

0ð1430Þ0,K�

0ð1950Þ0, K�ð892Þ0, K�ð1410Þ0, K�ð1680Þ0, andK�

2ð1430Þ0. TheP-waveK�ð892Þ0 is the largest component,∼50%, while the K�ð1410Þ0, K�ð1680Þ0, and D-waveK�

2ð1430Þ0 states amount to only a few percent.The theoretical study and experimental measurement of

the three-body B meson decays is still in an early stage. Onthe theoretical side, compared with those two-body decaymodes, these three-body B decays are less tractabledue to the entangled resonant and nonresonant contribu-tions, as well as the possible final-state interactions [6–8].An important breakthrough in the theory of three-body Bmeson decays was the confirmation of the validity offactorization. We restrict ourselves to the specific kine-matical configurations in which the three mesons are

*liyakelly@163.com†yandac@126.com‡jindui1127@126.com§xiaozhenjun@njnu.edu.cn

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

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2470-0010=2020=101(1)=016015(13) 016015-1 Published by the American Physical Society

quasialigned in the rest frame of the B meson. Thiscondition is particularly natural in the low-effective-Kπ-mass region of the Dalitz plot, where most of the Kπresonant structures are seen. When the two particles amongthe three final-state mesons move collinearly and generate asmall invariant mass recoiling against the third one, thethree-body interactions are expected to be suppressed.Then, it seems reasonable to assume the validity of thefactorization for these quasi-two-body B decays. In thequasi-two-body mechanism, the two-body scattering andall possible interactions between the two involved particlesare included, but the interactions between the third particleand the pair of mesons are ignored. In recent years, severaldifferent theoretical frameworks based on the factorizationtheorems and symmetry principles have been proposed todeal with the three-body B meson decays. The QCD-improved factorization approach [9–12] has been widelyused in the study of the three-body charmless hadronic Bmeson decays [13–23]. The U-spin and flavor SUð3Þsymmetries were also adopted in Refs. [24–29].It has been known that the collinear factorization of the

charmed and charmless two-body B meson decays sufferfrom end-point singularities. The perturbative QCD(PQCD) factorization approach relying on the kT factori-zation theorem [30,31] was proposed in Refs. [32–34],which has been shown to be infrared finite, gauge invariant,and consistent with the factorization assumption in theheavy-quark limit [35–38]. The operator-level definition ofthe transverse-momentum-dependent hadronic wave func-tions is highly nontrivial in order to avoid the potentiallight-cone divergence and the rapidity singularity [39,40].The Sudakov factors from the kT resummation have beenincluded to suppress the long-distance contributions fromthe large-b region, with b being a variable conjugate to kT .Therefore, the PQCD approach is a self-consistent frame-work and has good predictive power. Based on the PQCDapproach, the quasi-two-body B meson decays have beenstudied in Refs. [41–52] by introducing the two-mesondistribution amplitudes (DAs) [53–59], which catch thedynamics associated with the pair of mesons.In this paper, we continue to study the quasi-two-

body decays B → ηcK�0 → ηcKπ involving the S, P,and D-wave kaon-pion pairs as shown in Fig. 1, withinthe framework of the PQCD factorization approach. Some

studies of B → ηcK� decays have used the two-bodyframework [60–62]. From Refs. [41,43,45], we know thatthe width of the resonant state and the interactions betweenthe final-state meson pair will show their effects on thebranching ratios, especially on the direct CP violations ofthe quasi-two-body decays. Thus, it seemsmore appropriateto treat the K�0 as an intermediate resonance. As addressedbefore, this process is dominated by a series of resonances inS, P, and D waves. The S-wave kaon-pion DAs for theresonanceK�

0ð1430Þ0 have been studied in Ref. [63], andwewill further investigate the dependence of the branchingratios in different scalar scenarios, as proposed in Refs. [64–67]. Besides, we have roughly determined the possiblerange of the first odd Gegenbauer moment B1 for theK�

0ð1950Þ0 resonance by fitting to the existing data, whichmust be tested in the future. We intend to adopt the samefitted parameters as those of the longitudinal kaon-pionDAsin Ref. [52], where the SUð3Þ flavor-symmetry-breakingeffect has been considered and plays an important role in thelongitudinal polarization fractions. The D-wave resonanceK�

2ð1430Þ0 is investigated for the first time in our work. Dueto the limited studies on the tensor resonant states, we treattheD-wave DAs of theK�

2ð1430Þ0 in the same way as thoseof f2ð1270Þ [45].This paper is organized as follows. In Sec. II we give a

brief introduction of the theoretical framework. Thenumerical values, some discussions, and our conclusionswill be given in last two sections. The explicit PQCDfactorization formulas for all of the decay amplitudes arecollected in the Appendix.

II. FRAMEWORK

In the framework of the PQCD factorization approach,the nonperturbative dynamics associated with the pair ofmesons can be absorbed into two-meson DAs; then, therelevant decay amplitude A for the quasi-two-body decaysB → ηcK�0 → ηcKπ can be written in the following form:

A ¼ ΦB ⊗ H ⊗ ΦKπ ⊗ Φηc ; ð1Þ

where ΦB and Φηc are the B meson and charmonium DAs,respectively. The kaon-pion DA ΦKπ absorbs the non-perturbative dynamics in the Kπ hadronization process.

FIG. 1. Typical diagrams for the quasi-two-body decays B → ηcð1S; 2SÞðK�0 →ÞKπ, where the symbol (Bullet) denotes the weakvertex. K�0 represents various partial-wave intermediate states.

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The hard kernel H contains only one hard gluon anddescribes the dynamics of the strong and electroweakinteractions in the three-body hadronic decays, as in theformalism for the corresponding two-body decays.In the light-cone coordinates, we make the kaon-pion

pair and the final-state ηc move along the directions

n ¼ ð1; 0; 0TÞ and v ¼ ð0; 1; 0TÞ, respectively. The Bmeson momentum pB, the total momentum of the kaon-pion pair, p ¼ p1 þ p2, the final-state ηc momentum p3,and the quark momentum ki in each meson are

pB ¼ mBffiffiffi2

p ð1; 1; 0TÞ; kB ¼�0; xB

mBffiffiffi2

p ;kBT

�;

p ¼ mBffiffiffi2

p ð1 − r2; η; 0TÞ; k ¼�zð1 − r2ÞmBffiffiffi

2p ; 0;kT

�;

p3 ¼mBffiffiffi2

p ðr2; 1 − η; 0TÞ; k3 ¼�r2x3

mBffiffiffi2

p ; ð1 − ηÞx3mBffiffiffi2

p ;k3T

�; ð2Þ

where mB is the mass of the B meson, η ¼ ω2

m2Bð1−r2Þ

with

r ¼ mηc=mB, mηc is the mass of charmonia, and theinvariant mass squared ω2 ¼ p2. The momentum fractionsxB, z, and x3 run from zero to unity, respectively.We also define the momenta p1 and p2 in the kaon-pion

pair as

p1 ¼ ðζpþ; ð1 − ζÞηpþ;p1TÞ;p2 ¼ ðð1 − ζÞpþ; ζηpþ;p2TÞ; ð3Þ

with ζ¼pþ1 =P

þ characterizing the distribution of the longi-tudinal momentum of the kaon and p2

1T¼p22T¼ζð1−ζÞω2.

The BðsÞ meson wave function and the charmonium DAsare the same as those widely adopted in the PQCDapproach [42,44,51,68]. Very recently, a new methodwas proposed to calculate the B meson light-cone DAfrom lattice QCD, which can be used as an updatedinput for the B meson DA in the future [69]. Below, webriefly describe the S, P, and D-wave kaon-pion DAsand the corresponding time-like form factors. The S-wavekaon-pion DAs are introduced analogously to the two-pionones [70],

ΦS ¼1ffiffiffiffiffiffiffiffi2Nc

p ½=pϕ0Sðz; ζ;ω2Þ þ ωϕs

Sðz; ζ;ω2Þ

þ ωð=n=v − 1ÞϕtSðz; ζ;ω2Þ�: ð4Þ

In what follows the subscripts S, P, and D are alwaysassociated with the corresponding partial waves.We will use the asymptotic forms for the twist-3 DAs,

but no knowledge on the twist-2 DAs is available at present.We shall adopt similar formulas as those for a scalar meson[66,71], bearing in mind large uncertainties that may beintroduced by this approximation. The detailed expressionsof DAs of various twists are as follows:

ϕ0Sðz; ζ;ω2Þ ¼ 6

2ffiffiffiffiffiffiffiffi2Nc

p FSðω2Þzð1 − zÞ

×

�1

μSþ B1C

3=21 ðtÞ þ B3C

3=23 ðtÞ

�; ð5Þ

ϕsSðz; ζ;ω2Þ ¼ 1

2ffiffiffiffiffiffiffiffi2Nc

p FSðω2Þ; ð6Þ

ϕtSðz; ζ;ω2Þ ¼ 1

2ffiffiffiffiffiffiffiffi2Nc

p FSðω2Þð1 − 2zÞ; ð7Þ

where the Gegenbauer polynomials C3=21 ðtÞ ¼ 3t, C3=2

3 t ¼52tð7t2 − 3Þ with t ¼ 1–2z and μS ¼ ω=ðm2 −m1Þ. The

Gegenbauer moments B1;3 and the related running currentquark masses can be found in Refs. [66,67,72]. It should bestressed that there is less information about the scalarresonanceK�

0ð1950Þ0, and we only test the sensitivity of thebranching ratios on the first odd Gegenbauer moment B1 inour work.If there are overlapping resonances or there is significant

interference with a nonresonant component in the samepartial wave, the relativistic Breit-Wigner (RBW) functionleads to unitarity violation within the isobar model [73].This is the case for the Kπ S-wave at low Kπ mass, wherethe K�

0ð1430Þ0 resonance interferes strongly with a slowlyvarying NR S-wave component. In this work, the time-likescalar form factor FSðω2Þ for the S-wave Kπ system isparametrized by using a modified LASS line shape [74] forthe S-wave resonance K�

0ð1430Þ0, which has been widelyused in experimental analyses [5],

FSðω2Þ¼ ω

jp1j½cotðδBÞ− i�þe2iδBm2

0Γ0=jp0jm2

0−ω2− im20Γ0

ωjp1jjp0j

; ð8Þ

with

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cotðδBÞ ¼1

ajp1jþ rjp1j

2; ð9Þ

where the first term in Eq. (8) is an empirical term from theelastic kaon-pion scattering and the second term is theresonant contribution with a phase factor to retain unitarity.Here m0 and Γ0 are the pole mass and width of theK�

0ð1430Þ state. jp1j is the momentum vector of the reso-nance decay product measured in the resonance rest frame,and jp0j is the value of jp1j when ω ¼ mK� . The para-meters a¼ð3.1�1.0ÞGeV−1 and r ¼ ð7.0� 2.4Þ GeV−1are the scattering length and effective range [5], respec-tively, which are universal in application to the descrip-tion of different processes involving a kaon-pion pair.

The slowly varying part [the first term in the Eq. (8)] isnot well modeled at high masses and it is set to zero formðKπÞ values above 1.7 GeV [5]. For the K�

0ð1950Þ0,we use the relativistic BW line shape to parametrize thetime-like form factors FSðω2Þ, which is adopted in theexperimental data analyses [5].The P-wave kaon-pion DAs related to both longitudinal

and transverse polarizations have been studied in Ref. [52].In the quasi-two-body B → ηcK�ð892Þ0 → ηcKπ decay,the vector meson K�ð892Þ should be completely polarizedin the longitudinal direction due to the angular momentumconservation requirement. The explicit expressions of theP-wave kaon-pion DAs associated with longitudinal polari-zation are listed as follows:

ΦP ¼ 1ffiffiffiffiffiffiffiffi2Nc

p�=pϕ0

Pðz; ζ;ω2Þ þ ωϕsPðz; ζ;ω2Þ þ =p1=p2 − =p2=p1

ωð2ζ − 1Þ ϕtPðz; ζ;ω2Þ

�: ð10Þ

The DAs of various twists in Eq. (10) can be expanded in terms of the Gegenbauer polynomials:

ϕ0Pðz; ζ;ω2Þ ¼ 3Fk

Pðω2Þffiffiffiffiffiffiffiffi2Nc

p zð1 − zÞ�1þ ajj1K�3tþ ajj2K�

3

2ð5t2 − 1Þ

�ð2ζ − 1 − αÞ; ð11Þ

ϕsPðz; ζ;ω2Þ ¼ 3F⊥

P ðω2Þ2

ffiffiffiffiffiffiffiffi2Nc

p ft½1þ a⊥1st� − a⊥1s2zð1 − zÞgP1ð2ζ − 1Þ; ð12Þ

ϕtPðz; ζ;ω2Þ ¼ 3F⊥

P ðω2Þ2

ffiffiffiffiffiffiffiffi2Nc

p t½tþ a⊥1tð3t2 − 1Þ�P1ð2ζ − 1Þ; ð13Þ

where the Legendre polynomial P1ð2ζ − 1Þ ¼ 2ζ − 1 and the factor α ¼ ðm2K −m2

πÞ=ω2 is treated as the SUð3Þ asymmetryfactor.For the Gegenbauer moments ajj1K� , ajj2K� , a⊥1s, and a⊥1t we adopt the same values as those determined in Ref. [52]:

ak1K� ¼ 0.2; ak2K� ¼ 0.5; a⊥1s ¼ −0.2; a⊥1t ¼ 0.2: ð14Þ

The relativistic BW line shape is adopted for the P-wave resonances K�ð892Þ, K�ð1410Þ, and K�ð1680Þ to parametrize

the time-like form factors FkPðω2Þ. The explicit expression is [5]

FkPðω2Þ ¼

c1m2K�ð892Þ

m2K�ð892Þ − ω2 − imK�ð892ÞΓ1ðω2Þ þ

c2m2K�ð1410Þ

m2K�ð1410Þ − ω2 − imK�ð1410ÞΓ2ðω2Þ

þc3m2

K�ð1680Þm2

K�ð1680Þ − ω2 − imK�ð1680ÞΓ3ðω2Þ ; ð15Þ

where the three terms describe the contributions from K�ð892Þ, K�ð1410Þ, and K�ð1680Þ, respectively. The weightcoefficients c1 ¼ 0.72, c2 ¼ 0.135, and c3 ¼ 0.145 are the same as those determined previously [52].The mass-dependent width ΓiðωÞ is defined as

Γiðω2Þ ¼ Γi

�mi

ω

��jp1jjp0j

�ð2LRþ1Þ: ð16Þ

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The mi and Γi are the pole mass and width of thecorresponding resonances, where i ¼ 1, 2, 3 representsthe resonances K�ð892Þ, K�ð1410Þ, and K�ð1680Þ, respec-tively. LR is the orbital angular momentum in the Kþπ−system and LR ¼ 0; 1; 2;… corresponds to the S; P;D;…partial-wave resonances. Following Ref. [41], we alsoassume that

F⊥P ðω2Þ=Fk

Pðω2Þ ≈ ðfTK�=fK� Þ; ð17Þ

with fK� ¼ 0.217� 0.005 GeV and fTK� ¼ 0.185�0.010 GeV [75]. Due to the limited studies on thedecay constants of K�ð1410Þ and K�ð1680Þ, we use thetwo decay constants of K�ð892Þ to determine the ratio

F⊥P ðω2Þ=Fk

Pðω2Þ.The D-wave kaon-pion DAs are introduced analogously

to the two-pion ones [45],

ΦD ¼ 1ffiffiffiffiffiffiffiffi2Nc

p�=pϕ0

Dðz; ζ;ω2Þ þ ωϕsDðz; ζ;ω2Þ þ =p1=p2 − =p2=p1

ωð2ζ − 1Þ ϕtDðz; ζ;ω2Þ

�: ð18Þ

TheD-wave Kπ system has similar asymptotic DAs as those for a tensor meson [76–78], but with the time-like form factorreplacing the tensor decay constants:

ϕ0Dðz; ζ;ω2Þ ¼ 6Fk

Dðω2Þ2

ffiffiffiffiffiffiffiffi2Nc

p zð1 − zÞ½3a01ð2z − 1Þ�P2ð2ζ − 1Þ; ð19Þ

ϕsDðz; ζ;ω2Þ ¼ −

9F⊥Dðω2Þ

4ffiffiffiffiffiffiffiffi2Nc

p ½a01ð1 − 6zþ 6z2Þ�P2ð2ζ − 1Þ; ð20Þ

ϕtDðz; ζ;ω2Þ ¼ 9F⊥

Dðω2Þ4

ffiffiffiffiffiffiffiffi2Nc

p ½a01ð1 − 6zþ 6z2Þð2z − 1Þ�P2ð2ζ − 1Þ; ð21Þ

where the Legendre polynomial P2ð2ζ − 1Þ ¼ 1–6ζð1 − ζÞ and the Gegenbauer moment a01 ¼ 0.4� 0.1 [45]. The time-like

form factor FkDðω2Þ for the D-wave Kπ resonance is also described by the relativistic BW function as given in Eq. (15).

Besides, the approximate relation F⊥Dðω2Þ=Fk

Dðω2Þ ≈ fTK�2ð1430Þ=fK�

2ð1430Þ is used in the following calculation with fK�

2ð1430Þ ¼

0.118� 0.005 GeV and fTK�2ð1430Þ ¼ 0.077� 0.014 GeV [76].

III. NUMERICAL RESULTS AND DISCUSSIONS

In our numerical calculations, besides the quantities specified before, the following input parameters (where the massesand decay constants are in units of GeV) are used [79]:

mB0 ¼ 5.28; mB0s¼ 5.367; mb ¼ 4.8; mc ¼ 1.275;

mπ� ¼ 0.140; mK� ¼ 0.494; mηcð1SÞ ¼ 2.9834; mηcð2SÞ ¼ 3.6392;

fB0 ¼ 0.19; fBs¼ 0.23; fηc ¼ 0.42; fηcð2SÞ ¼ 0.243;

τB0 ¼ 1.519 ps; τB0s¼ 1.512 ps: ð22Þ

The pole masses and widths of various partial-wave resonances are summarized in Table I, while we adopt the values of theWolfenstein parameters given in Ref. [79]: A ¼ 0.836� 0.015, λ ¼ 0.22453� 0.00044, ρ ¼ 0.122þ0.018

−0.017 , η ¼ 0.355þ0.012−0.011 .

For the decays B → ηcðK�0 →ÞKπ, the differential decay rate is expressed as

dBdω

¼ τBωjp1jjp3j32π3m3

BjAj2; ð23Þ

where the kaon and charmonium three-momenta in the Kπ center-of-mass frame are given by

jp1j ¼1

2ω

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλðω2; m2

K;m2πÞ

q; jp3j ¼

1

2ω

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλðm2

B;m2ηc ;ω

2Þq

; ð24Þ

with the kaon (pion) mass mK (mπ) and the Källen function λða; b; cÞ ¼ a2 þ b2 þ c2 − 2ðabþ acþ bcÞ.

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By using Eqs. (23)–(24), the decay amplitudes from theAppendix, and all of the input quantities, the resultantbranching ratios B and the available experimental resultsfor the considered B0

ðsÞ → ηcKπ decays involving the

S-wave resonances are summarized in Table II, while thosefor P- and D-wave resonances are shown in Tables III andIV. Since the charged B meson decays differ from theneutral ones only in the lifetimes and the isospin factor inour formalism, we can derive the branching ratios for theBþ decay modes by multiplying those for the B0 decaymodes by the ratio τBþ=τB0 .In our calculations for the various partial-wave reso-

nances, the first uncertainty is induced by the Gegenbauermoments in the S, P, and D-wave kaon-pion DAs, asaforementioned. The second error comes from the varia-tions of the shape parameterωBðsÞ of the BðsÞ meson DA.Weadopt the value ωB ¼ 0.40� 0.04 or ωBs

¼ 0.50�0.05 GeV and vary its value within a 10% range, whichis supported by intensive PQCD studies [33,34]. The last

TABLE I. The pole masses and widths of the various partial-wave resonances [5].

Resonance Mass [MeV] Width [MeV] JP Model

K�ð892Þ0 895.55� 0.20 47.3� 0.5 1− RBWK�ð1410Þ0 1414� 15 232� 21 1− RBWK�

0ð1430Þ0 1425� 50 270� 80 0þ LASSK�

2ð1430Þ0 1432.4� 1.3 109� 5 2þ RBWK�ð1680Þ0 1717� 27 322� 110 1− RBWK�

0ð1950Þ0 1945� 22 201� 90 0þ RBW

TABLE II. PQCD results for the branching ratios of the S-wave resonances in the B0ðsÞ → ηcð1S; 2SÞK�π∓ decays in Scenario I and

Scenario II together with experimental data [5]. The theoretical errors are attributed to the variation of the Gegenbauer moments B1 andB3, the shape parameters ωBðsÞ in the wave function of the BðsÞ meson, and the hard scale t, respectively.

Quasi-two-body B (in 10−5)

Modes Scenario I Scenario II Exp. [5]

B0 → ηcKþπ−ðNRÞ 0.85þ0.43þ0.32þ0.10−0.41−0.26−0.15 1.85þ0.94þ0.50þ0.56

−0.59−0.24−0.31 5.90þ1.23−1.29

B0 → ηcðK�0ð1430Þ0 →ÞKþπ− 1.69þ0.71þ0.22þ0.42

−0.69−0.17−0.24 4.75þ2.10þ1.17þ1.05−1.95−1.10−0.68 14.50þ3.36

−3.14

B0s → ηcK−πþðNRÞ 0.03þ0.01þ0.01þ0.00

−0.01−0.01−0.00 0.09þ0.04þ0.03þ0.02−0.04−0.02−0.02 � � �

B0s → ηcðK�

0ð1430Þ0 →ÞK−πþ 0.08þ0.03þ0.02þ0.01−0.02−0.01−0.01 0.21þ0.11þ0.07þ0.05

−0.08−0.05−0.03 � � �B0 → ηcð2SÞKþπ−ðNRÞ 0.17þ0.07þ0.03þ0.04

−0.06−0.02−0.05 0.41þ0.22þ0.12þ0.11−0.11−0.09−0.06 � � �

B0 → ηcð2SÞðK�0ð1430Þ0 →ÞKþπ− 0.56þ0.28þ0.17þ0.12

−0.25−0.24−0.15 0.79þ0.41þ0.24þ0.17−0.19−0.13−0.12 � � �

B0s → ηcð2SÞK−πþðNRÞ 0.007þ0.004þ0.003þ0.001

−0.002−0.001−0.002 0.02þ0.01þ0.01þ0.00−0.01−0.01−0.00 � � �

B0s → ηcð2SÞðK�

0ð1430Þ0 →ÞK−πþ 0.02þ0.01þ0.01þ0.00−0.01−0.00−0.00 0.04þ0.02þ0.01þ0.01

−0.02−0.01−0.01 � � �

TABLE III. PQCD results for the branching ratios of the P-wave resonances in the B0ðsÞ → ηcð1S; 2SÞK�π∓ decays together with

experimental data [5,79]. The theoretical errors are attributed to the variation of the Gegenbauer moments (ajj1K� , ajj2K� and a⊥1s, a⊥1t), theshape parameters ωBðsÞ in the wave function of the BðsÞ meson, and the hard scale t, respectively.

Modes Quasi-two-body B (in 10−5) Exp. (in 10−5)

B0 → ηcðK�ð892Þ0 →ÞKþπ− 46.49þ18.07þ12.63þ12.13−14.80−9.67−8.63 35� 5 [79]a

B0 → ηcðK�ð1410Þ0 →ÞKþπ− 1.35þ0.50þ0.24þ0.43−0.40−0.23−0.26 1.20� 0.90 [5]

B0 → ηcðK�ð1680Þ0 →ÞKþπ− 1.44þ0.63þ0.34þ0.55−0.49−0.26−0.30 1.26þ1.44

−1.51 [5]

B0s → ηcðK�ð892Þ0 →ÞK−πþ 2.13þ0.89þ0.66þ0.54

−0.73−0.46−0.39 � � �B0s → ηcðK�ð1410Þ0 →ÞK−πþ 0.07þ0.02þ0.02þ0.02

−0.02−0.02−0.01 � � �B0s → ηcðK�ð1680Þ0 →ÞK−πþ 0.08þ0.02þ0.02þ0.03

−0.02−0.02−0.02 � � �B0 → ηcð2SÞðK�ð892Þ0 →ÞKþπ− 13.33þ4.72þ3.70þ3.74

−3.51−2.96−2.08 < 26 [79]a

B0 → ηcð2SÞðK�ð1410Þ0 →ÞKþπ− 0.24þ0.11þ0.06þ0.10−0.06−0.04−0.05 � � �

B0 → ηcð2SÞðK�ð1680Þ0 →ÞKþπ− 0.11þ0.04þ0.02þ0.04−0.03−0.02−0.03 � � �

B0s → ηcð2SÞðK�ð892Þ0 →ÞK−πþ 0.60þ0.20þ0.21þ0.16

−0.19−0.16−0.11 � � �B0s → ηcð2SÞðK�ð1410Þ0 →ÞK−πþ 0.01þ0.01þ0.01þ0.01

−0.00−0.00−0.00 � � �B0s → ηcð2SÞðK�ð1680Þ0 →ÞK−πþ 0.008þ0.002þ0.002þ0.002

−0.002−0.002−0.002 � � �aThe experimental results are obtained by multiplying the relevant measured two-body branching ratios according to Eq. (27).

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one is caused by the variation of the hard scale t from 0.75tto 1.25t (without changing 1=bi), which characterizesthe next-to-leading-order (NLO) effects in the PQCDapproach. In Tables II, III, and IV, it is shown that themain uncertainties in our approach come from theGegenbauer moments, which can reach a total magnitudeof about 60%. The scale-dependent uncertainty is less than25% due to the inclusion of the NLO vertex corrections.The other possible errors from the uncertainties of mc andthe Cabibbo-Kobayashi-Maskawa matrix elements areactually very small and can be safely neglected.Combined with the Clebsch-Gordan Coefficients, we

can write the relation

����Kπ; I ¼ 1

2

�¼

ffiffiffi1

3

rjK0π0i −

ffiffiffi2

3

rjKþπ−i: ð25Þ

Isospin conservation is assumed for the strong decaysof an I ¼ 1=2 resonance K�0 to Kπ when we computethe branching fraction of the quasi-two-body processB → ηcK�0 → ηcKþπ−, namely,

ΓðK�0→Kþπ−ÞΓðK�0→KπÞ ¼2=3;

ΓðK�0→K0π0ÞΓðK�0→KπÞ ¼1=3: ð26Þ

Therefore, the corresponding two-body branching fractionBðB → ηcK�0Þ can be extracted directly from the quasi-two-body decay modes in Table III under the narrow-widthapproximation relation:

BðB→ηcK�0→ηcKþπ−Þ¼BðB→ηcK�0Þ ·BðK�0→KπÞ ·23:

ð27Þ

There already exist some results for B0ðsÞ → ηcK�ð892Þ0 in

the two-body framework [60–62]. One can see that thebranching ratios of the quasi-two-body decay modes matchwell with the two-body analyses presented in Ref. [61]using the PQCD approach. These results suggest that thePQCD factorization approach is suitable for describing thequasi-two-body Bmeson decays through analyzing variousresonances by reconstructing Kπ final states and repro-ducing the invariant mass spectra of Dalitz plots.

For the S-wave resonance K�0ð1430Þ0, it should be

mentioned that two scenarios have been proposed todescribe the scalar mesons above 1 GeV using the QCDsum rules method [66,67]. In Scenario I, the K�

0ð1430Þ0 istreated as the first excited state, while a0ð980Þ and f0ð980Þare regarded as the lowest-lying states. In Scenario II, weassume that K�

0ð1430Þ0 is the lowest-lying resonance andthe corresponding first excited states lie between (2.0–2.3) GeV. Scenario II corresponds to the case that lightscalar mesons are four-quark bound states. The Gegenbauermoments B1 ¼ 0.58� 0.07 and B3 ¼ −1.20� 0.08 areadopted in Scenario I, while B1 ¼ −0.57� 0.13 and B3 ¼−0.42� 0.22 are adopted in Scenario II [67]. In this work,we consider two scenarios for the S-wave components andlist the relevant results in Table II. One can see that thepredicted B in Scenario I are always smaller than those inScenario II. This phenomenon is mainly caused by thedifferent signs of the Gegenbauer moment B1 in differentscenarios, which indicates that there is a large cancellationin Scenario I.From Table II, one can see that our predictions for the

branching ratios for the K�0ð1430Þ0 resonance and NR

components in Scenario II are BK�0ð1430Þ0 ¼ ð4.75þ2.62

−2.34Þ ×10−5 and BNR ¼ ð1.85þ1.20

−0.71Þ × 10−5, respectively, whichare a bit smaller than the experimental measurementswithin errors. Anyway, as is well known, in contrast tothe vector and tensor mesons the identification of scalarmesons is a long-standing puzzle. It is difficult to deal withscalar resonances since some of them have wide decaywidths, which cause a strong overlap between resonancesand background. Furthermore, the underlying structure ofscalar mesons is not theoretically well established (for areview, see Ref. [79]). We hope that the situation can beimproved using nonperturbative QCD tools includinglattice QCD simulations. Nonetheless, we define thePQCD prediction of the corresponding ratio for a moredirect comparison with the available experimental data,

RPQCD1 ¼ BðB0 → ηcðK�

0ð1430Þ0 →ÞKþπ−ÞBðB0 → ηcKþπ−ðNRÞÞ ¼ 2.56þ1.98

−1.71 ;

ð28Þ

TABLE IV. PQCD results for the branching ratios of the D-wave resonances in the B0ðsÞ → ηcð1S; 2SÞK�π∓

decays together with experimental data [5]. The theoretical errors are attributed to the variation of the Gegenbauermoment a01, the shape parameters ωBðsÞ in the wave function of the BðsÞ meson, and the hard scale t, respectively.

Modes Quasi-two-body B (in 10−5) Exp. (in 10−5)

B0 → ηcðK�2ð1430Þ0 →ÞKþπ− 3.98þ1.24þ0.59þ0.11

−1.74−0.55−0.04 2.35þ1.08−1.29 [5]

B0s → ηcðK�

2ð1430Þ0 →ÞK−πþ 0.23þ0.13þ0.04þ0.01−0.10−0.05−0.01 � � �

B0 → ηcð2SÞðK�2ð1430Þ0 →ÞKþπ− 0.55þ0.31þ0.12þ0.02

−0.24−0.09−0.01 � � �B0s → ηcð2SÞðK�

2ð1430Þ0 →ÞK−πþ 0.04þ0.02þ0.01þ0.01−0.02−0.01−0.01 � � �

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where the branching fraction of the B0→ηcðK�0ð1430Þ0→Þ

Kþπ− decay is measured relative to that of the correspond-ing NR contributions by the LHCb Collaboration [5]:

RLHCb1 ¼ BðB0 → ηcðK�

0ð1430Þ0 →ÞKþπ−ÞBðB0 → ηcKþπ−ðNRÞÞ ¼ 2.45þ0.81

−0.68 :

ð29ÞOur prediction is quite consistent with the LHCb meas-urement. Combined analyses from the LHCb and Belle-IImeasurements for these decays in the near future could helpus to further study the scalar resonances.For the phenomenological study of the scalar meson

K�0ð1950Þ0, we still lack the distribution amplitudes of the

K�0ð1950Þ0 state at present. Fortunately, we are allowed to

single out theK�0ð1950Þ0 component according to the kaon-

pion DAs. In Fig. 2, we plot the variation of the branchingfraction with the first odd Gegenbauer moment B1 for the

B0 → ηcðK�0ð1950Þ0 →ÞKþπ− decay mode, as well as the

experimental data Bexp ¼ ð2.18þ1.32−1.79Þ × 10−5 [5]. One can

see that the theoretical prediction of the branching ratioshows a strong dependence on the variation of B1.Combined with the available data, we can roughly deter-mine that the possible range of the first odd Gegenbauermoment is from −0.15 to 0.05 or 0.10 to 0.25, whichshould be examined both theoretically and experimentallyin the future.In Fig. 3, we show the ω dependence of the differential

decay rates dBðB0 → ηcKþπ−Þ=dω (the solid curve) anddBðB0 → ηcð2SÞKþπ−Þ=dω (short-dotted curve) after theinclusion of the P-wave resonanceK�ð892Þ0, which exhibita peak at the K�ð892Þ0 meson mass. For the considereddecay modes B0

ðsÞ → ηcð1S; 2SÞKπ, the dynamical limit on

the value of the invariant mass ω is ðmK þmπÞ ≤ ω ≤ðmBðsÞ −mηcð1S;2SÞÞ. Although mK�ð1680Þ0 > ðmB −mηcð2SÞÞ,the resonance K�ð1680Þ0 can still contribute to theB0ðsÞ → ηcð2SÞKπ decay due to its large width

(ΓK�ð1680Þ0 ¼ 322 MeV). It is shown that the main portionof the differential branching ratio lies in the region aroundthe resonance in Fig. 3, as expected. For B0 →ηcð1SÞðK�ð892Þ0 →ÞKþπ− decay, the central values ofthe branching ratio B are 23.36 × 10−5 and 34.46 × 10−5

when the integration over ω is limited to the range ω¼½mK�−0.5ΓK� ;mK�þ0.5ΓK� � or ω¼½mK� −ΓK� ;mK� þΓK� �,respectively, which amounts to 50% and 74% of the totalbranching ratio B ¼ 46.49 × 10−5 as listed in Table III.From Table III, one can see that our PQCD prediction for

the branching ratio of the B0 → ηcð1SÞK�ð892Þ0 →ηcKþπ− decay is B ¼ ð46.49þ25.16

−19.67Þ × 10−5, the centralvalue of which is a little larger than that in PDG2018:ð35� 5Þ × 10−5 [79]. The measurements from the Belle[2], BABAR [3,4], and LHCb [5] collaborations, aswell as the average value from HFLAV [80], are thefollowing:

BðB0 → ηcðK�ð892Þ0 →ÞKþπ−Þ ¼

8>>>>>><>>>>>>:

ð108þ42−46Þ × 10−5 Belle½2�;

ð53þ28−19Þ × 10−5 BABAR½3�;

ð38� 7Þ × 10−5 BABAR½4�;ð29.5þ3.8

−4.7Þ × 10−5 LHCb½5�;ð41.3� 6.6Þ × 10−5 HFLAV½80�:

ð30Þ

The data point ð41.3� 6.6Þ × 10−5 from HFLAV [80] isobtained by multiplying the relevant measured two-body branching ratio according to Eq. (27). One cansee that the central values of the measured branch-ing ratio for the K�ð892Þ0 resonance from different

experiments vary in the wide range ð29–108Þ × 10−5, whilethe HFLAV world average of the measured values fromthe Belle and BABAR collaborations [2–4] leads toð41.3� 6.6Þ × 10−5, which is in good agreement with ourprediction.

FIG. 2. The branching fraction of the B0 → ηc½K�ð1950Þ0 →�Kþπ− decay as a function of the Gegenbauer moment B1.Shaded bands show the experimental uncertainties.

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For the B0 → ηcð2SÞðK�ð892Þ0 →ÞKþπ− decay, theBABAR Collaboration has measured an upper limit on thebranching ratio BðB0 → ηcð2SÞðK�ð892Þ0 →ÞKþπ−Þ <26 × 10−5 at the 90% confidence level [4]. The PQCDprediction of BðB0 → ηcð2SÞðK�ð892Þ0 →ÞKþπ−Þ ¼ð13.33þ7.07

−5.04Þ × 10−5 agrees with the limit. Meanwhile, onecan see in Fig. 3 that the branching fractions of B0 →ηcð2SÞðK�0 →ÞKþπ− decays are always smaller than those

for B0 → ηcð1SÞðK�0 →ÞKþπ− decays, which is mainlyinduced by the difference between the DAs of ηcð2SÞ andηcð1SÞmesons: the tighter phase space and the smaller decayconstant of the ηcð2SÞ meson result in the suppression.From the numerical results given in Table III, we

obtain the relative ratio R2 between the branching ratioof B meson decays involving ηcð2SÞ and ηcð1SÞ and theresonance K�ð892Þ0,

R2ðK�ð892Þ0Þ ¼ BðB0 → ηcð2SÞðK�ð892Þ0 →ÞKþπ−ÞBðB0 → ηcð1SÞðK�ð892Þ0 →ÞKþπ−Þ ¼ 0.29þ0.21

−0.15 ; ð31Þ

which can be tested by the forthcoming LHCb and Belle-IIexperiments.The branching ratios of the considered D-wave reso-

nance are presented in Table IV. We emphasized that ourpredictions of these decay channels are only rough esti-mates. Although there is not enough data at present, thecalculated value BðB0 → ηcð1SÞðK�

2ð1430Þ0 →ÞKþπ−Þ ¼ð3.98þ1.24

−1.74Þ × 10−5 is compatible with the measurementð2.35þ1.08

−1.29Þ × 10−5 [5] within the large errors. Futureexperimental measurements with high precision can pro-vide a better understanding of the properties of the tensorresonances.For all of the B0

s → ηcð1S; 2SÞKπ decays, such decaymodes can be theoretically related to the corresponding B0

decays since they have identical topologies and similarkinematic properties in the limit of SUð3Þ flavor symmetry.At the quark level, the B0 and B0

s decays correspond to theb → ccs and b → ccd transitions, respectively. The relativeratios of the branching fractions for B0

s and B0 decay modesare dominated by the Cabibbo suppression factor of

jVcdj2=jVcsj2 ∼ λ2 under the naive factorization approxi-mation. It is reasonable to see that the branching fractionsof the B0

s decays are smaller than those of the correspond-ing B0 decays. Though the B0

s channels have relativelysmall branching ratios, some of them can be potentiallymeasurable at future experiments.

IV. CONCLUSION

In this work, by introducing the kaon-pion DAs,we studied the quasi-two-body decays B0

ðsÞ →ηcð1S; 2SÞðK�0 →ÞKπ in the PQCD approach, in whichthe kaon-pion invariant mass spectra are dominated by theK�

0ð1430Þ0, K�0ð1950Þ0, K�ð892Þ0, K�ð1410Þ0, K�ð1680Þ0,

and K�2ð1430Þ0 resonances. These six resonances fall into

three partial waves according to their spins, namely, S, P,and D-wave states. The contributions from each partialwave can be parametrized into the corresponding time-likeform factors involved in the kaon-pion DAs. The Kπ S-wave component at low mass is described by the LASS lineshape, while the time-like form factors of other resonancesare modeled by the relativistic BW function.It has been shown that our predictions of the

branching ratios for most of the considered B0 →ηcð1SÞðK�0 →ÞKþπ− decays are in good agreement withthe existing data within the errors. For the B0 →ηcð1SÞðK�

0ð1430Þ0 →ÞKþπ− decay, although there existsa clear difference between the central value of the PQCDcalculation for BK�

0ð1430Þ0, BNR, and the measured ones, they

are still consistent within three standard deviations due tothe large experimental errors, which should be examined byforthcoming experiments. The new ratio R2ðK�ð892Þ0Þamong the branching ratios of the considered decay modeshas been defined and will be confronted with futuremeasurements.

ACKNOWLEDGMENTS

Many thanks to Hsiang-nan Li and Wen-Fei Wang forvaluable discussions. This work was supported by the

FIG. 3. Differential branching ratio for the B0 → ηcð1S; 2SÞ½K�ð892Þ0 →�Kþπ− decays.

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National Natural Science Foundation of China under theNo. 11947013, No. 11605060, No. 11775117, andNo. 11547020. Y. L. is also supported by the NaturalScience Foundation of Jiangsu Province under GrantNo. BK20190508 and the Research Start-up Funding ofNanjing Agricultural University. Z. R. is supported in part

by the Natural Science Foundation of Hebei Province underGrant No. A2019209449.

APPENDIX: DECAY AMPLITUDES

The total decay amplitudes for the considered decaymodes B0

ðsÞ → ηcKπ in this work are given as follows:

AðB0ðsÞ → ηcKπÞ ¼ GFffiffiffi

2p

V�cbVcsðcdÞ

��C1 þ

1

3C2

�FLL þ C2MLL

�

− V�tbVtsðtdÞ

��C3 þ

1

3C4 þ C9 þ

1

3C10

�FLL þ

�C5 þ

1

3C6 þ C7 þ

1

3C8

�FLR

þ ðC4 þ C10ÞMLL þ ðC6 þ C8ÞMSP

�; ðA1Þ

where GF ¼ 1.16639 × 10−5 GeV−2 is the Fermi coupling constant and the Vij’s are the Cabibbo-Kobayashi-Maskawamatrix elements. The superscripts LL, LR, and SP refer to the contributions from ðV − AÞ ⊗ ðV − AÞ, ðV − AÞ ⊗ ðV þ AÞ,and ðS − PÞ ⊗ ðSþ PÞ operators, respectively. The explicit amplitudes FðMÞ from the factorizable (nonfactorizable)diagrams in Fig. 1 can be obtained straightforwardly by replacing the twist-2 or twist-3 DAs of the ππ andKK systems withthe corresponding twists of the Kπ ones in Eqs. (5)–(7), (11)–(13), and (19)–(21), since the kaon-pion distributionamplitudes considered in this work [Eqs. (4), (10), and (18)] have the same Lorentz structure as the two-pion (kaon) ones inRefs. [42,51].TheWilson coefficientsCi are evaluated at the corresponding scale t. At themW scale, theWilson coefficients at the NLO

level can be written in the following form (as in Ref. [81]):

C1ðmWÞ ¼11

2

αsðmWÞ4π

; C2ðmWÞ ¼ 1 −11

6

αsðmWÞ4π

−35

8

αem4π

;

C3ðmWÞ ¼ −αsðmWÞ24π

�E0ðxÞ −

2

3

�þ αem

6π

1

sin2θW½2B0ðxÞ þ C0ðxÞ�; C4ðmWÞ ¼

αsðmWÞ8π

�E0ðxÞ −

2

3

�;

C5ðmWÞ ¼ −αsðmWÞ24π

�E0ðxÞ −

2

3

�; C6ðmWÞ ¼

αsðmWÞ8π

�E0ðxÞ −

2

3

�;

C7ðmWÞ ¼αem6π

�4C0ðxÞ þD0ðxÞ −

4

9

�;

C9ðmWÞ ¼αem6π

�4C0ðxÞ þD0ðxÞ −

4

9þ 1

sin2θWð10B0ðxÞ − 4C0ðxÞÞ

�; CiðmWÞ ¼ 0; i ¼ 8; 10; ðA2Þ

where the relevant Inami-Lim functions B0ðxÞ; C0ðxÞ; D0ðxÞ, and E0ðxÞ [82] are

B0ðxÞ ¼1

4

�x

1 − x−

x ln xðx − 1Þ2

�; C0ðxÞ ¼

6x − x2

8ð1 − xÞ −ð3x2 þ 2xÞ ln x

8ðx − 1Þ2 ;

D0ðxÞ ¼−25x2 þ 19x3

36ð1 − xÞ3 −ð8 − 32xþ 54x2 − 30x3 þ 3x4Þ ln x

18ðx − 1Þ4 ;

E0ðxÞ ¼18x − 11x2 − x3

12ð1 − xÞ3 −ð4 − 16xþ 9x2Þ ln x

6ðx − 1Þ4 ; ðA3Þ

with x ¼ m2t =m2

W . In the region mb < t < mW , we evaluate the Wilson coefficients at the t scale using the followingrenormalization group equation:

CðtÞ ¼ Uðt; mWÞCðmWÞ; ðA4Þ

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whereCðmWÞ ¼ ðC1ðmWÞ;…; C10ðmWÞÞT andUðt; mWÞ is the renormalization group running matrix at the NLO level (fordetails, see Ref. [81]). The Wilson coefficients CiðtÞ evaluated at the scale t ¼ mb ¼ 4.8 GeV (as given, for example, inRef. [83]) are as follows:

C1 ¼ −0.17474; C2 ¼ 1.07737; C3 ¼ 0.01249; C4 ¼ −0.03304;

C5 ¼ 0.00942; C6 ¼ −0.03929; C7 ¼ −0.00003; C8 ¼ 0.00023;

C9 ¼ −0.00999; C10 ¼ 0.00201: ðA5Þ

If the scale t < mb, we can evaluate the Wilson coefficients at the t scale using the evolution equationCðtÞ ¼ Uðt; mbÞCðmbÞ, where CðmbÞ ¼ ðC1ðmbÞ;…; C10ðmbÞÞT are given in Eq. (A5).

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