$B_{u}$ ${\to}$ ${\psi}M$ decays and $S$-$D$ wave mixing effects

The $B_{u}$ ${\to}$ ${\psi}M$ decays are studied with the perturbative QCD approach, where the psion ${\psi}$ $=$ ${\psi}(2S)$, ${\psi}(3770)$, ${\psi}(4040)$ and ${\psi}(4160)$, and the light meson $M$ $=$ ${\pi}$, $K$, ${\rho}$ and $K^{\ast}$. The factorizable and nonfactorizable contributions, and the $S$-$D$ wave mixing effects on the psions are considered in the calculation. With appropriate inputs, the branching ratios for the $B_{u}$ ${\to}$ ${\psi}K$ decays are generally coincident with the experimental data within errors. However, due to the large theoretical and experimental errors, it is impossible for the moment to give a severe constraint on the $S$-$D$ wave mixing angles.


I. INTRODUCTION
The exclusive B meson decays into one psion (ψ) and one light meson (M) are of great interest, and have attracted much attention over the past years. In this paper, unless otherwise specified, the symbol ψ denotes the high excited charmonium states with the quantum number a I G J P Phenomenologically, the nonleptonic B meson weak decays have been studied carefully within the framework of the factorization hypothesis and the low-energy effective Hamiltonian [4]. The naive factorization (NF) assumption [5][6][7] is usually employed in evaluating the nonleptonic B meson decays, where the decay amplitudes in terms of hadronic matrix elements (HME) of the four-quark operators can be expressed as the product of two HME of the diquark currents based on Bjorken's color transparency argument [8]. The diquark HME can be further parameterized by the decay constants or the hadron transition form factors.
The NF hypothesis was verified experimentally to be successful for the class-I nonleptonic B decays, but poor for the class-II ones. It is commonly believed that the characteristic space configuration of psions is compact, with the radius of r ∼ 1/m c . The transverse separation a The symbols of I, J, G, P , C refer to the isospin, angular momentum, G-parity, P -parity, and C-parity of one particle, respectively. b The symbols nL in parentheses are the radial quantum number n and the orbital angular momentum L, with n = 1, 2, · · ·, and L = S, P , D, · · ·. The ψ(2S) particle is thought to be a 2S-wave dominated charmonium state with possible some D-wave components. c The numbers in parentheses indicate the approximate masses of the particles in the unit of MeV. The dominant components of the particles ψ(3770), ψ(4040) and ψ(4160) are usually considered as the 1 3 D 1 , 3 3 S 1 and 2 3 D 1 states, respectively [1 -3]. Here, the spectroscopic notation n 2s+1 L J is used, where s is the total spin of the quark-antiquark pair, and J is the total angular momentum. between the two valence charm quarks should be very small. The massive psions from the B meson decay could be regarded as color singlet states and factorized from the other system, although the velocity of psion might be not very large. The class-II B → J/ψ(1S)M decays have been studied based on the factorization assumption, such as in Refs. [9][10][11][12][13][14][15][16][17][18], where besides the factorizable contributions, the nonfactorizable contributions beyond the NF approximation are also taken into account to accommodate the discrepancies between the experimental data and the theoretical estimations. The B → ψM decays provide a good place to check the factorization postulation and differentiate various theoretical treatments, such as the QCD factorization (QCDF) approach [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] based on the collinear approximation, and the perturbative QCD (pQCD) approach [38][39][40][41][42][43][44][45][46][47] based on the collinear plus k T factorization supposition.
It is well known that according to the quark model assignments, the spin-triplet charmonium states with different orbital angular momentum L can have the same quantum numbers J P C . The conservation of parity and angular momentum implies that the values of L for the mixed states can differ by two units at most. The psions near and above the open-charm threshold can be the admixtures of the S-and D-wave cc states [48][49][50][51][52][53][54][55][56][57][58][59][60]. The wave functions for the S-wave dominant state can receive the D-wave component and vice versa. Additionally, studies of the charmonium spectrum [61][62][63][64] show that the mass of the n 3 S 1 state is close to the mass of the (n − 1) 3 D 1 state. To the first-order approximation, the so-called S-D wave mixing for psions refers mainly to the mixing between the n 3 S 1 and (n − 1) 3 D 1 charmonium states rather than the other states, and this has been used in the previous studies [48][49][50][51][52][53][54][55][56][57][58][59][60]. This S-D wave mixing phenomenon might have certain effects on the production of psions in the B → ψM decays.
In this paper, we will investigate the B u → ψM decays with the pQCD approach. Firstly, the electrically charged final meson M should be easily identified by many specific detectors at the existing and future high energy colliders because of its track curve being saturated with the magnetic field. Secondly, the practicability of the pQCD approach can be checked with the class-II B decays into final states containing the excited psions. Thirdly, the effects of the S-D wave mixing among psions can be examined with the B u → ψM decays, without the disturbances from the mixing between the neutral B mesons and without the pollution from the weak annihilation contributions. This paper is organized as follows. The theoretical framework and the amplitudes for the B u → ψM decays are elaborated in Section II. The numerical results and discussion are presented in Section III. Finally, we give a short summary in Section IV.

A. The effective Hamiltonian
The B u → ψM decays are actually induced by the weak interaction cascade processes b → c + W * − → c +c q at the quark level within the standard model. Hence, some relevant energy scales are introduced theoretically, such as the infrared confinement scale Λ QCD of the strong interactions, the mass m b for the decaying bottom quark, and the mass m W for the virtual gauge boson W * , with the clear size relation Λ QCD ≪ m b ≪ m W . The effective theory is usually used in practice to deal with the realistic multi-scale problems. With the operator product expansion and the renormalization group (RG) method, the effective Hamiltonian in charge of the B u → ψM decays can be written as [4], where the Fermi coupling constant G F ≃ 1.166×10 −5 GeV −2 [1]. V pb V * pq is the product of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, satisfying the unitarity relation With the Wolfenstein parametrization, the CKM factors can be expanded as the power series of the parameter λ ≈ 0.2[1]. Up to O(λ 7 ), these CKM factors can be written as follows.
The numerical values of the Wolfenstein parameters A, λ, ρ and η are listed in Table II.
From the expression for V pb V * pq above, it is clearly seen that the weak phases for the B u → ψM decays are small, and thus result in a small direct CP violation.
The renormalization scale µ divides the physical contributions into the short-and longdistance parts. The physical contributions from the scale larger than µ are summarized in the Wilson coefficients C i . The Wilson coefficients, C i = {C 1 , C 2 , · · ·, C 10 }, are calculable at the scale µ W ∼ O(m W ) with the perturbation theory, and then evolved to the characteristic scale µ b ∼ O(m b ) for the b quark decay with the RG equation [4], where U(µ b , µ W ) is the RG evolution matrix. The Wilson coefficients are independent of any process and have the same role as the universal gauge couplings. The expressions of the Wilson coefficients C i (m W ) and U(µ b , µ W ), including the next-to-leading order (NLO) corrections, can be found in Ref. [4]. The physical contributions from the scale less than µ are incorporated into the HME, ψM|Q i |B u , where the local four-quark operators Q i are sandwiched between the initial and final hadron states. The operators are expressed as follows.
where Q 1,2 are the tree operators originating from the W -boson emission; Q 3,···,6 and Q 7,···,10 are the QCD and electroweak penguin operators, respectively; (q 1 q 2 ) V ±A = q 1 γ µ (1±γ 5 ) q 2 ; α and β are color indices, i.e., the QCD corrections are considered; q ′ denotes all the active quarks at the scale of O(m b ), i.e., q ′ = u, d, c, s, b; and e q ′ is the fractional electric charge of the quark q ′ in the unit of |e|. To obtain the decay amplitudes, the proper calculation of the HME ψM|Q i |B u will be the focus of the current research.

C. Kinematic variables
In the heavy quark limit, the light quark from the bottom quark decay is assumed to fly quickly away from the interaction point at near the speed of light. The light-cone dynamics can be used to describe the relativistic system. The relations between the four-dimensional x, y, z) and the light-cone coordinates (x + , x − , . The planes of x ± = 0 are called the light-cone. The scalar product of any two vectors is given by In the rest frame of the B u meson, the final ψ and M mesons move in the opposite direction. The light-cone kinematic variables are defined as follows. where the subscript i = 1, 2, 3 on variables (including the mass m i , momentum p i and energy E i ) correspond to the B u , ψ and M mesons, respectively. The parameters k i , x i , k iT are the momentum, the longitudinal momentum fraction, and the transverse momentum of the valence antiquark, respectively. p cm is the center-of-mass momentum of the final states.
The notations of these momenta are displayed in Fig.2(a).

D. Wave functions
The wave functions and/or distribution amplitudes (DAs) are the essential ingredient in the master pQCD formula of Eq. (17). Although nonperturbative, the WFs and DAs are generally considered to be universal for any process. The WFs and DAs determined by nonperturbative methods or extracted from data can be employed here to make predictions.
where f B and f ψ are the decay constants of the B u and ψ mesons, respectively. ǫ (ǫ ⊥ ) is the longitudinal (transverse) polarization vector. n + = (1, 0, 0) and n − = (0, 1, 0) are the positive and negative null light-cone vectors satisfying the conditions of n 2 ± = 0 and n + ·n − = 1. The chiral parameter µ P is given by [68], According to the twist classification in Refs. [66][67][68][69][70], the WFs of Φ a B,P and Φ v,T ψ,V are twist-2, while the WFs of Φ p,t B,P and Φ t,s,V,A ψ,V are twist-3. The WFs for the nS and nD psion states are given in Appendix A. In general, these mesonic WFs are the functions of two variables, the longitudinal momentum fractions x i and the transverse momentum k iT of the valence quarks. It is unanimously assumed with both the QCDF and pQCD approaches that outside the soft regions, the contributions from the transverse momentum can be neglected and the collinear approximation should work well [19][20][21][22][23][24][25][26][38][39][40][41][42][43][44][45][46][47]. One can obtain the corresponding DAs by integrating out the transverse momentum from the WFs. Near the endpoint regions where x i → 0 or 1, the collinear factorization approximation should no longer be valid [21][22][23][24]. The pQCD approach [38][39][40] suggests that the effects of the transverse momentum cannot be overlooked. In addition, the valence quarks have different momentum fractions and velocities near the endpoint. The hadrons cannot be regarded as color transparent. The Sudakov factors should be introduced for the participating WFs in order to suppress the soft and nonperturbative contributions from the small x i and the large k iT regions [38][39][40][41][42][43][44][45][46][47].
A distinguishing feature of the DAs in Eqs. (34)(35)(36)(37)(38)(39)(40)(41)(42)(43) is the exponential functions. These exponential factors are proportional to the ratio of m 2 i /x i , so that the shape lines of DAs  Table I, where the decay constant f ψ is defined by 0|c γ µ c|ψ = f ψ m ψ ǫ µ ψ and can be extracted from the electronic ψ → e + e − decay through the formula including the QCD radiative corrections [48][49][50][51][78][79][80][81][82][83], where the RG evolution equation for the coupling α QED (α s ) of the electromagnetic (strong) and Γ ee denote the branching ratio and partial width for the pure leptonic ψ → e + e − decay; f ψ is the decay constant obtained with Eq. (58); α s (m ψ ) is the QCD coupling at the scale µ = m ψ .  Table I that there exist differences in the dielectric psion decay widths, which is assumed to be accommodated appropriately with the interferences between the S-and D-states [48][49][50][51][52][53][54][55][56][57][58][59][60]. Although with nearly the same shape lines for the 2S-and 1D-wave (and the 3S-and 2D-wave) psion DAs (see Fig.1 where the subscript i of the S-D mixing angle θ i corresponds to the radial quantum number  Within the pQCD framework, the Feynman diagrams for the B u → ψK decay are shown in Fig.2. The spectator quark always interacts with one hard gluon in each subdiagram.

E. Decay amplitudes
The diagrams Fig.2(a,b) are the factorizable emission topologies, where the gluons are exchanged between the initial B u meson and the recoil K meson. It is possible to completely isolate the emission psion particle from the B u K system, and hence the integral of the psion WFs will reduce to the psion decay constant. The diagrams Fig.2(c,d) are the nonfactorizable emission topologies, where the gluons are exchanged between the psion particle and the B u K system, and hence no meson can escape from the interferences of other mesons.
The diagrams Fig.2(c,d) are also called the spectator scattering topologies with the QCDF approach [20][21][22][23][24][25]. The nonfactorizable HME can be written as the convolution integral of all the participating meson WFs. Compared with the factorizable contributions from Fig.2(a,b), the nonfactorizable contributions from Fig.2(c,d) are color-suppressed, which is quite similar to the cases between the external and internal W emission topologies.
After a direct calculation, the amplitudes for the B u → ψM decays are written as follows: where the color factor C F = 4/3 and the color number N c = 3. For the amplitude building block A k i,j , the subscript i corresponds to the subdiagram indices of Fig.2; the subscript j = P , L, N, T denotes the invariant polarization amplitudes, and the superscript k refers to the two possible Dirac structures Γ 1 ⊗Γ 2 of the operators (q 1 q 2 ) Γ 1 (q 3 q 4 ) Γ 2 , namely k = LL for (V − A)⊗(V − A) and k = LR for (V − A)⊗(V + A). The explicit expressions of the building blocks A k i,j are collected in Appendix B. In addition, the amplitudes for the B u → ψV decays are conventionally expressed as the helicity amplitudes. The relation between the helicity amplitudes H 0, ,⊥ and the scalar amplitudes A L,N,T is [85][86][87][88]:
The numerical values of the input parameters are listed in Tables I and II,  (1) It has been shown in Refs. [22][23][24] that the contributions from the spectator scattering topologies to the coefficient a 2 with the QCDF approach are amplified by the large Wilson coefficient C 1 , and the contributions are notable for the B → J/ψM decays [9,17]. Hence, it is initially expected that the nonfactorizable contributions from Fig.2(c,d) should be significant for the B u → ψM decays. However, it is seen from the numbers in Tables III and IV that compared with the factorizable contributions, the nonfactorizable contributions to the branching ratios are important, but not so obvious as expected. One of the reasons might be that the opposite signs of the charm quark propagators of Fig.2(c) and Fig.2(d) results in the destructive interference between their amplitudes. In addition, the amplitudes of Fig.2(c,d) are suppressed by the color factor 1/N c relative to the amplitudes of Fig.2(a,b) (see the expressions listed in Appendix B). It is also shown that the nonfactorizable contributions are positive (negative) to branching ratios for the B u → ψV (ψP ) decays.
(2) The B u → ψ(2S)M decays have been studied with the pQCD approach in Refs. [15,16], by considering part of the NLO factorizable vertex corrections, but without the 2S-1D mixing effects on psions. Our numerical results generally agree with those of Refs. [15,16] within theoretical uncertainties, although with different parameters. In the future, a careful and comprehensive study of the NLO corrections to the B → ψM decays is desperately needed, and will be essential for the forthcoming precision measurements at the LHCb and Belle-II experiments.
(3) The S-D wave mixture has literally altered the branching ratios for the B u → ψM  clear these doubts, it is necessary to check how many shares come from the perturbative domain. The ψ(4160) meson has the largest mass among the psions concerned. To make the analysis more persuasive, we take the B u → ψ(4160)K * decay as an example. The percentage contributions to the branching ratio from different α s /π regions are shown in Fig.4. It is seen that more than 60% of contributions come from the α s /π ≤ 0.4 regions.
Our study also shows that more than 80% of contributions to the B u → ψ(2S)π decay come from the α s /π ≤ 0.4 regions. These facts imply that the perturbative calculation with the pQCD approach might be feasible. Besides the suppression on the soft contributions from both the Sudakov factors and the exponential functions of DAs in Eqs. (34)(35)(36)(37)(38)(39)(40)(41)(42)(43) is also an important factor to further ensure the perturbative calculation with the pQCD approach.  [17,18]. The potential FSIs deserve much attention for the nonleptonic B → ψM decays, but this is beyond the scope of this paper. (6) There are lots of theoretical uncertainties, especially from m b , hadronic parameters and the S-D mixing angles. It is shown in Refs. [12][13][14][15] that the pQCD's results are sensitive to the model of mesonic WFs/DAs and input parameters. Besides, many other factors, such as FSIs, different models for mesonic WFs/DAs, higher order corrections to HME, and so on, are not scrutinized here, in spite of the value of dedicated study. Most of the theoretical uncertainties actually result from our inadequate comprehension of the long-distance and nonperturbative dynamics. Great efforts should be made to improve the reliability of theoretical results.

IV. SUMMARY
The color-suppressed nonleptonic B u → ψM decay provides an important place to explore the S-D wave mixing among psions, and test the QCD-inspired approaches for dealing with the hadronic matrix elements. In this paper, the B u → ψM decays are investigated with the pQCD approach, including the contributions of factorizable and nonfactorizable emission topologies. We also consider the effects of 2S-1D and 3S-2D mixing on psions. It is found that with appropriate inputs, there is generally agreement with the experimental data for the branching ratios for the B u → ψK decays within theoretical uncertainties. However, due to the large experimental and theoretical uncertainties, the angle θ 1 (θ 2 ) for the 2S-1D (3S-2D) wave mixing cannot be determined properly for the moment. corresponds to S, P , D · · · waves, respectively. The wave functions for the nS and nD states associated with the isotropic linear harmonic oscillator potential are written as follows.
where the parameter ω determines the average transverse momentum of the oscillator, i.e., 1S| k 2 T |1S = ω 2 . With the power counting rules of the nonrelativistic QCD effective theory [75][76][77], the characteristic velocity v of the valence quark in heavy quarkonium is about v ∼ α s . The parameter ω ≃ m α s is taken for the psions in our calculation, where α s is the QCD coupling constant. We adopt the light-cone momentum and employ the commonly used substitution [89], where x i , k iT , m q i are the longitudinal momentum fraction, transverse momentum, and mass of the valence quark. These variables satisfy the relations x i = 1 and k iT = 0. After integrating out k iT and combining the results with their asymptotic forms [68][69][70], one can obtain the distribution amplitudes of Eqs. (36)(37)(38)(39)(40)(41)(42)(43) for the charmonium states.
where x i and b i are the longitudinal momentum fraction and the conjugate variable of the transverse momentum k iT , respectively. The subscript i of A k i,j corresponds to the indices of Fig.2; the subscript j = P , L, N, T correspond to the different helicity amplitudes; the superscript k refers to the two possible Dirac structures Γ 1 ⊗Γ 2 of the operators (q 1 q 2 ) Γ 1 (q 3 q 4 ) Γ 2 , namely k = LL for (V − A)⊗(V − A) and k = LR for (V − A)⊗(V + A).