Hui-feng Fu,   Yue Jiang,   C. S. Kim111cskim@yonsei.ac.kr,   Guo-Li Wang222gl_wang@hit.edu.cn
Department of Physics, Harbin Institute of Technology, Harbin, 150001, China
Department of Physics IPAP, Yonsei University, Seoul 120-749, South Korea

# Probing Non-leptonic Two-body Decays of Bc meson

Hui-feng Fu,   Yue Jiang,   C. S. Kim111cskim@yonsei.ac.kr,   Guo-Li Wang222gl_wang@hit.edu.cn
Department of Physics, Harbin Institute of Technology, Harbin, 150001, China
Department of Physics IPAP, Yonsei University, Seoul 120-749, South Korea
###### Abstract

Rates and CP asymmetries of the non-leptonic two-body decay of are calculated based on the low energy effective Hamiltonian. We concentrate on such quark decays of the processes with and S-wave particles and/or and P-wave particles in the final states. The Salpeter method, which is the relativistic instantaneous approximation of the original Bethe-Salpeter equation, is used to derive hadron transition matrix elements. Based on the calculation, it is found that the best decay channels to observe CP violation are , which need about events in experiment. Decays to are also hopeful channels.

## 1 Introduction

The discovery of the meson [1] has provided a new valuable window for studying the heavy quark dynamics and CP violation. Studies on the decays and their CP asymmetries have drawn much attention in accordance with the coming LHC-b experiment. Since the LHC-b is expected to produce around events per year [2] and to provide detailed information about the meson, it becomes more and more strongly relevant to investigate decay and its CP violation in detail.

There are two major reasons which make the meson special. The first is that it is unique to have two heavy-flavored quarks, composed of a charm quark (anti-quark) and a bottom anti-quark (quark). The other heavy quark in the Standard Model, the top quark, cannot form a hadron because of its too short lifetime to be hadronized. The second reason is that it can decay only via weak interactions, since the pure strong and electromagnetic interacting processes conserve flavors, and the meson, as the ground state of system, is below the mesons decay threshold. Due to these properties, the meson has a long lifetime and rich decay channels.

The quark diagrammatic approach has established and well developed for meson decays. In the approach, there are five diagrams contributing to decays: the color-favored tree diagram, the color-suppressed tree diagram, the time-like penguin diagram, the annihilation diagram and the space-like penguin diagram. The direct CP violation requires at least two diagrams with different weak and strong phases contributing to the relevant process. The weak phases come from CKM matrix elements within the Standard Model, and the strong phases arise from final state interactions including penguin effects (hard strong phases), which can be estimated perturbatively, as well as rescattering effects (soft strong phases), which cannot be estimated solidly now. Therefore, we only discuss penguin effects for the generation of strong phases in this paper.

Non-leptonic two-body decays can play an important role for exploring the direct CP violation. So far many works on non-leptonic decays and their CP violations have been investigated [3][16]. But in those works CP violation of the channels with P-wave final states has not been considered. Here we are going to concentrate on such non-leptonic two-body decay channels that may have direct CP asymmetries: We study the quark decays with final states involving not only pseudoscalar () and vector () particles but also and P-wave particles. Since the contributions from the annihilation diagram and space-like penguin diagram are helicity suppressed, these two type diagrams are ignored in our calculation. Furthermore, the electroweak penguin effects are much smaller compared to the QCD penguin effects, so only the QCD penguin effects are considered here. Therefore, only the two tree diagrams and the time-like QCD penguin diagrams are fully considered (see Fig. 1). The CP asymmetries arise from the interference between the penguin diagrams and tree diagrams or/and the penguin diagrams themselves.

In our calculation the factorization approach is assumed and the Salpeter method is used: With the factorization approach, the amplitude can be expressed by the products of form factors and decay constants. The Salpeter method is used to calculate the form factors at finite recoils. In doing so, a relativistic treatment is needed especially for processes, since the mesons are bound states composed of a heavy and a light quark and the relativistic corrections to such particles may noticeable. It is well known that the Bethe-Salpeter (B-S) equation is a relativistic two-body wave equation and the Salpeter method is just the instantaneous approximation of the B-S equation. With the Salpeter equation and well defined wave functions, we can treat the bound states relativistically. Therefore, in our calculations the relativistic corrections are also systematically covered.

The remainder of this paper is organized as follows: In section 2, the factorization approach based on the low energy effective Hamiltonian is introduced to evaluate the decay amplitudes. Section 3 contains a brief review on the Salpeter method and our model calculation. Section 4 is devoted to numerical results and discussions.

## 2 Nonleptonic two-body decay and its CP asymmetry of Bc

In weak decay analysis, the basic starting point is the effective weak Hamiltonian [17], which in the case of decay is

 Heff(ΔB=1)=GF√2{VubV∗uq′(C1Qu1+C2Qu2)−αs(mb)8π(c,t∑i=uVibV∗iq′Ii)(−Q3Nc+Q4−Q5Nc+Q6)}, (1)

where are the tree operators in decay, which would be replaced by in decay. and are the QCD penguin operators. All these local operators are

 Qu1=(¯q′αuβ)V−A(¯uβbα)V−A,Qu2=(¯q′αuα)V−A(¯uβbβ)V−A,Qc1=(¯q′αcβ)V−A(¯cβbα)V−A,Qc2=(¯q′αcα)V−A(¯cβbβ)V−A,Q3=(¯q′αbα)V−A∑qx(¯qxβqxβ)V−A,Q4=(¯q′αbβ)V−A∑qx(¯qxβqxα)V−A,Q5=(¯q′αbα)V−A∑qx(¯qxβqxβ)V+A,Q6=(¯q′αbβ)V−A∑qx(¯qxβqxα)V+A, (2)

where , and the subscript are color indices. ranges from to . The operator , and the operators with represent for the right-handed currents. In Eq. (1), in front of the tree operators are Wilson coefficients. is the number of colors and are the CKM matrix elements. are the QCD loop integrals [4, 18]:

 Iu,c=−4∫10x(1−x)lnmu,c−k2x(1−x)m2Wdx, (3) It=−19+16∫10(1−x)[(2+m2t/m2W)(1−x)(2+x)+12x]m2t/m2W+(1−m2t/m2W)xdx, (4)

where are the current quark masses; is the momentum of the gluon in penguin diagram, see Fig 1 c. Usually one takes a certain value of in the range or  [19]. As argued by the authors in Ref. [6], it is not a good choice to pick up a fixed value of for all decay modes. In this work, we follow the simple kinematic picture presented in Ref. [6] for the value of . One can see from Fig. 1 c, as c quark being a spectator, the relation of the momenta among the quarks and gluon are hold. Since the quark and the anti-quark form a meson, noted as , the momentum of satisfies . With these relations one can get , where is the angle between the 3-momenta of quark and quark in the rest frame of the meson. Since the angle is unknown, we use the averaged value to evaluate the loop-integral functions. After all, one get

 ¯k2m2b=12(1+(m2¯qx−m2q′)(1−m2¯qxm2b)/m2X+(m2q′+2m2¯qx−m2X)/m2b). (5)

Now we turn to evaluate the decay amplitudes in factorization approach [20, 21] and take the channel as an example. The decay amplitude of this process is . First, consider the color-favored tree diagram (see Fig. 1 a), where the tree operators and contribute. Using the Fierz rearrangement

 (¯Ψ1Ψ2)V−A(¯Ψ3Ψ4)V−A=(¯Ψ1Ψ4)V−A(¯Ψ3Ψ2)V−A, (6) (¯Ψ1αΨ2β)V−A(¯Ψ3βΨ4α)V−A=1Nc(¯Ψ1αΨ2α)V−A(¯Ψ3βΨ4β)V−A+Octet, (7) (¯Ψ1Ψ2)V−A(¯Ψ3Ψ4)V+A=−2(¯Ψ1Ψ4)S+P(¯Ψ3Ψ2)S−P, (8)

where the “Octet” is the color-octet term which does not contribute in the factorization approach. One can get the amplitude of the color-favored tree diagram

 GF√2VcbV∗cda1⟨ηc|(¯cb)V−A|B−c⟩⟨D−|(¯dc)V−A|0⟩,

where . The other two amplitudes (corresponding to Fig. 1 b and Fig. 1 c) can be obtained in the same way. In the penguin diagram, we will encounter the term , where . To evaluate these terms, we use the equation of motion, which gives

 ⟨P′|¯q1q2|P⟩=Pμ−P′μm1−m2⟨P′|¯q1γμq2|P⟩, (9) ⟨P′|¯q1γ5q2|P⟩=Pμ−P′μm1+m2⟨P′|¯q1γμγ5q2|P⟩, (10)

where and are the momenta of initial and final states respectively and are the current quark masses. Now we can write the decay amplitude of the process

 M(B−c→ηc+D−)= GF√2{[VcbV∗cda1−αs(mb)8π(VubV∗udIut+VcbV∗cdIct)(1−1N2c)× (11) +VcbV∗cda2⟨D−|(¯db)V−A|B−c⟩⟨ηc|(¯dc)V−A|0⟩},

where and . The unitary condition, with , has been used to achieve the expression. The decay width is , where is the 3-momentum of one of the final state particles in the rest frame of . Generally the amplitude can be written as

 M=VcbV∗cq′T1+VubV∗uq′T2. (12)

The amplitude for CP conjugated process can be obtained by conjugating the CKM matrix elements but not and , .

The CP asymmetry is defined as

 Acp=Γ(B+c→¯f)−Γ(B−c→f)Γ(B+c→¯f)+Γ(B−c→f). (13)

Inserting the expression of the amplitude, one can get

 Acp=∑[2iIm(T1T∗2)(VubV∗uq′VcbV∗cq′−(VubV∗uq′VcbV∗cq′)∗)]∑[2|T1|2+2|VubV∗uq′VcbV∗cq′|2|T2|2+2Re(T1T∗2)(VubV∗uq′VcbV∗cq′+(VubV∗uq′VcbV∗cq′)∗)].. (14)

In the Wolfenstein parameterization of CKM matrix, up to the order, only has weak phase, so we take Then the CP asymmetry drops to a simple form:

 Acp =ϵi2∑Im(T1T∗2)sinγ∑|T1|2/Bi+Bi∑|T2|2+ϵi2∑Re(T1T∗2)cosγ (15) ≡D1sinγ1+D2cosγ,

where corresponding to .

In our calculation, we take numerical values of CKM elements as [22]

 |Vud|=0.97425, |Vus|=0.2252, |Vub|=3.89×10−3, |Vcd|=0.230, |Vcb|=0.0406, |Vcs|=0.97345. (16)

For current quark masses and QCD coupling constant, we take [4] (GeV) and .

## 3 The Salpeter method and the model calculation

To estimate the decay rates and CP asymmetries, the hadron matrix elements need to be calculated. In our work, we use the Salpeter method [23], which is the relativistic instantaneous approximation of the Bethe-Salpeter (B-S) equation, with well defined wave functions to deal with the hadron matrix elements.

The B-S equation [24] is written as

 (⧸p1−m1)χp(q)(⧸p2+m2)=i∫d4k(2π)4V(P,k,q)χp(k), (17)

where is B-S wave function of the relevant bound state. is the four momentum of the state and , , , are the momenta and constituent masses of the quark and anti-quark, respectively. From the definition

 p1=α1P+q, α1≡m1m1+m2,
 p2=α2P−q, α2≡m2m1+m2,

one can deduce the expression of relative momentum between quark and anti-quark . is the interaction kernel which can be treated as a potential after doing instantaneous approximation, the kernel takes the simple form (in the rest frame)

 V(P,k,q)⇒V(|→k−→q|).

For convenience, we divide the relative momentum into two parts,

 qμ=qμ∥+qμ⊥,  qμ∥≡P⋅q/M2Pμ,  qμ⊥≡qμ−qμ∥,

where is the mass of the meson. Correspondingly, we have two Lorentz invariant variables:

 qP≡P⋅q/M,  qT≡√−q2⊥.

With the definitions

 φP(qμ⊥)≡i∫dqP2πχP(qμ∥,qμ⊥),  η(qμ⊥)≡∫dk3⊥(2π)3V(k⊥,q⊥)φP(qμ⊥),

and after performing the integration over in Eq. (17), the B-S equation can be written as

 φP(q⊥)=Λ+1(q⊥)η(q⊥)Λ+2(q⊥)M−ω1−ω2−Λ−1(q⊥)η(q⊥)Λ−2(q⊥)M+ω1+ω2, (18)

where , , and are the generalized projection operators,

 Λ±1(q⊥)≡12ω1[⧸PMω1±(m1+⧸q⊥)],    Λ±2(q⊥)≡12ω2[⧸PMω2∓(m2+⧸q⊥)].

Now we introduce the notations

 φ±±P(q⊥)≡Λ±1(q⊥)⧸PMφP(q⊥)⧸PMΛ±2(q⊥).

With these notations the full Salpeter equation can be written as

 (M−ω1−ω2)φP(q⊥)++=Λ+1(q⊥)η(q⊥)Λ+2(q⊥), (M+ω1+ω2)φP(q⊥)−−=−Λ−1(q⊥)η(q⊥)Λ−2(q⊥), φP(q⊥)+−=0,  φP(q⊥)−+=0. (19)

In our model, the Cornell potential, which is a linear scalar interaction plus a vector interaction, is chosen as the instantaneous interaction kernel .

In solving the equations, the constituent quark masses are taken as

 mu=0.305 GeV,md=0.311 GeV,ms=0.5 GeV,mc=1.62 GeV,mb=4.96 GeV.

The form of wave functions with certain quantum numbers and 333 for general particles, for quarkonium the equal mass system. The wave functions satisfy the correct -parity spontaneously when the masses of quark and anti-quark are equal. are written as

 φ0−(+)(q⊥) = M[⧸PMa1(q⊥)+a2(q⊥)+⧸q⊥Ma3(q⊥)+⧸P⧸q⊥M2a4(q⊥)]γ5, φ1−(−)(q⊥) = (q⊥⋅ϵλ⊥)[b1(q⊥)+⧸PMb2(q⊥)+⧸q⊥Mb3(q⊥)+⧸P⧸q⊥M2b4(q⊥)]+M⧸ϵλ⊥b5(q⊥) +⧸ϵλ⊥⧸Pb6(→6)+(⧸q⊥⧸ϵλ⊥−q⊥⋅ϵλ⊥)b7(q⊥)+1M(⧸P⧸ϵλ⊥⧸q⊥−⧸Pq⊥⋅ϵλ⊥)b8(q⊥), φ0+(+)(q⊥) = f1(q⊥)⧸q⊥+f2(q⊥)⧸P⧸q⊥M+f3(q⊥)M+f4(q⊥)⧸P, φ1+(+)(q⊥) = iεμναβPνqα⊥ϵλβ⊥[g1(q⊥)Mγμ+g2(q⊥)⧸Pγμ+g3(q⊥)⧸q⊥γμ +ig4(q⊥)εμρσδPσq⊥ργδγ5/M]/M2, φ1+(−)(q⊥) = q⊥⋅ϵλ⊥[h1(q⊥)+h2(q⊥)⧸PM+h3(q⊥)⧸q⊥+h4(q⊥)⧸P⧸q⊥M2]γ5, (20)

where and are wave functions to ; is the mass of corresponding bound state; is the polarization vector for state. With these wave functions, we solve the Salpeter equation (Eq. (3)) and get

 M¯D0=1.865,MD−=1.869,MD−s=1.968,Mηc=2...980,M¯D∗0=2.006,MD∗−=2.011,MD∗−s=2.112,MJ/Ψ=3.097,M¯D∗00=2.317,MD∗−0=2.323,MD∗−s0=2.318,Mχc0=3.415,Mχc1=3.510,Mhc=3.526,MD−s1(2460)=2.459,MD−s1(2536)=2.535,

and in unit of GeV. In our method, the wave functions are constructed for certain quantum state, such as , which is a state and also a state and which is a or state. This is the case for quarkonium. For the particles composed of a couple of quark and anti-quark with different masses, the two states are just and states and both are states (such states don’t have C-parity), so the mixture between the and states may happen. If one puts the quark masses equal, the two states are spontaneously deduced to and states respectively. The particles and are considered to be mixed of and states. In this work we take the mixing relation as

 |P1/21⟩=−1√3|1P1⟩+√23|3P1⟩,|P3/21⟩=√23|1P1⟩+1√3|3P1⟩,

where corresponds to the and corresponds to the . Interested reader can find details about the Salpeter method and our model in Ref. [25].

With the wave functions of bound states, we can calculate hadron matrix elements, such as . According to Mandelstam formalism [26], at the leading order, the transition matrix element can be written as [27]

 ⟨ηc|(¯cΓμb)|B−c⟩=∫d3q⊥(2π)3Tr[¯φ++ηc(q⊥+α′2P′⊥)Γμφ++B−c(q⊥)⧸PM], (21)

where ; and is the momentum and mass of initial state, the meson; is the momentum of and ; ; and . The is just a decay constant. For and particles, we define decay constants and as

 ⟨P(0±)|(¯q1q2)V−A|0⟩≡if0±Pμ, (22) ⟨P(1±)|(¯q1q2)V−A|0⟩≡if1±Mϵμ. (23)

Accordingly the transition matrix elements can be expressed with form factors:

 ⟨P′(0±)|(¯q1q2)V−A|P(B−c)⟩≡f+(P+P′)μ+f−(P−P′)μ, (24) ⟨P′(1±)|(¯q1q2)V−A|P(B−c)⟩≡f1ϵ⋅PMPμ+f2ϵ⋅PMP′μ+f3ϵμ+if4εμϵPP′, (25)

where and are form factors. After all the hadron matrix can be expressed in the products of decay constants and form factors.

## 4 Numerical results and discussions

We now use the method previously illustrated to estimate the non-leptonic two-body decay widths of meson and their CP asymmetries. In our calculation the decay constants are taken from experimental values or Lattice QCD results, if available. Otherwise, we use the values shown in Table 1.