Radiative decays of heavy-light quarkonia through $M1$, and $E1$ transitions in the framework of Bethe-Salpeter equation

In this work we study the radiative decays of heavy-light quarkonia through M1 and E1 transitions, that involve quark-triangle diagrams with two hadron vertices, and are difficult to evaluate in BSE-CIA. We have given a generalized structure of transition amplitude, $M_{fi}$ as a linear superposition of terms involving all possible combinations of $++$, and $--$ components of Salpeter wave functions of final and initial hadron, with coefficients being related to results of pole integrals over complex $\sigma$- plane. We evaluate the decay widths for $M1$ transitions ($^3S_1 \rightarrow ^1S_0 +\gamma$), and $E1$ transitions ($^3S_1 \rightarrow ^1P_0 +\gamma$ and $^1P_0 \rightarrow ^3S_1 +\gamma$). We have used algebraic forms of Salpeter wave functions obtained through analytic solutions of mass spectral equations for ground and excited states of $0^{++},1^{--}$, and $0^{-+}$ heavy-light quarkonia in approximate harmonic oscillator basis, to calculate their decay widths. The input parameters used by us were obtained by fitting to their mass spectra. We have compared our results with experimental data and other models, and found reasonable agreements. We also studied the transition form factors for some M1 transitions.


Introduction
The most important goals of hadronic physics is to bridge the gap between Quantum Chromodynamics (QCD), and the observed hadronic properties. There has a been a renewed interest in recent years in spectroscopy of heavy hadrons in charm and beauty sectors, which was primarily due to experimental facilities the world over such as BABAR, Belle, CLEO, DELPHI, BES etc. [1][2][3][4][5], which have been providing accurate data on cc, and bb hadrons with respect to their masses and decays. In the process many new states have been discovered such as χ b0 (3P ), χ c0 (2P ), X(3915), X(4260), X(4360), X(4430), X(4660) [5], some of which are exotic states, which can not be readily explained through the predictions of the quark model.
The physics of heavy meson decays has provided a useful testing ground for providing precise determination of Standard Model parameters, and providing a deeper understanding of QCD dynamics. Now, single photon decays of quarkonia have been considered as a valuable testing ground for various phenomenological models of hadrons. These radiative transitions of heavy quarkonia are of considerable experimental and theoretical interest, and provide an insight into the dynamics of quarkonium. The radiative transitions between 0 −+ (pseudoscalar), and 1 −− (vector) mesons (for instance, J/Ψ(nS) → η c (n S) + γ), which proceeds through the emission of photon is characterized by ∆L = 0, there is change in C-parity between the initial and final hadron states, though the total C-parity is conserved.
In this work, we first give a generalized framework for handling quark triangle diagrams with two hadron-quark vertices in the framework of 4 × 4 BSE, by expressing the transition amplitude, M f i as a linear superposition of terms involving all possible combinations of ++, and −− components of Salpeter wave functions of final and initial hadron, which should be a feature of relativistic frameworks. Using this generalized expression for M f i in Eq. (21)(22), we study the radiative decays of the heavy-light charmed and bottom quarkonia through the processes, V → P γ, V → Sγ, and S → V γ, where, V, P, S refer to vector, pseudoscalar and scalar quarkonia, and calculate the radiative decay widths of B * , and D * mesons for the above mentioned processes in the framework of 4 × 4 Bethe-Salpeter equation. We had earlier studied M1 transitions, V → P γ between light mesons (e.g. ρ ±,0 → π ±,0 γ, and ω → πγ in [16] in the framework of 16 × 1 BSE, and between equal mass quarkonia in [17]. In our recent works [18,19], we had studied the mass spectrum of ground and excited states of heavy-light scalar In the present work on radiative decays, we use these same input parameters to calculate the single photon decay widths for the above processes, However, due to the highly involved algebra, we take only the leading Dirac structures (γ 5 , I, and γ. respectively) in the hadronic Bethe-Salpeter wave functions of these hadrons, which as we have earlier shown using our naive power counting rule [20,21], contributes maximum to the calculation of any meson observable.
However the price we have to pay for this simplification is that we are unable to get good agreement of decay widths for M1 transitions (nS− > n S + γ, with n = n), with data, which would imply that the incorporation of sub-leading Dirac structures in wave functions of V and P quarkonia are important for studying transitions between different spatial multiplets. Now, as mentioned in our previous works [18,19,22], we are not only interested in studying the mass spectrum of hadrons, which no doubt is an important element to study dynamics of hadrons, but also the hadronic wave functions that play an important role in the calculation of decay constants, form factors, structure functions etc. for QQ, and Qq hadrons. These hadronic Bethe-Salpeter wave functions were calculated algebraically by us in [18,19,22]. The plots of these wave functions [19] show that they can provide information not only about the long distance non-perturbative physics, but also act as a bridge between the long distance, and short distance physics, and are provide us information about the contribution of the short ranged coulomb interactions in the mass spectral calculation of heavy-light quarkonia.
These wave functions and can also lead to studies on a number of processes involving QQ, and Qq states, and provide a guide for future experiments. This paper is organized as follows: In section 2, we introduce the formulation of the 4 × 4 Bethe-Salpeter equation under the covariant instantaneous ansatz, and derive the hadron-quark vertex. In sections 3, 4, and 5, we calculate the single photon decay widths for the processes, V → P γ, V → Sγ, and S → V γ, where, P, S, and V are the pseudoscalar, scalar and vector heavy-light quarkonium states. In section 6, we provide the numerical results and discussion.

Formulation of the BSE under CIA
Our work is based on QCD motivated BSE in ladder approximation, which is an approximate description, with an effective four-fermion interaction mediated by a gluonic propagator that serves as the kernel of BSE in the lowest order.
The precise form of our kernel is taken in analogy with potential models, which includes a confining term along with a one-gluon exchange term. Such effective forms of the BS kernel in ladder BSE have recently been used in [23][24][25][26][27], and can predict bound states having a purely relativistic origin (as shown recently in [23]). As mentioned above, the BSE is quite general, and provides an effective description of bound quark-antiquark systems through a suitable choice of input kernel for confinement.
The Bethe-Salpeter equation that describes the bound state of two quarks (QQ or Qq) of momenta p 1 and p 2 , relative momentum q, and meson momentum P is where K(q, q ) is the interaction kernel, and S −1 F (±p 1,2 ) = ±i p 1,2 + m 1,2 are the usual quark and antiquark propagators. The 4D B.S. wave function can be written as where the hadron-quark vertex is The 4D B.S. wave function in Eq.(2) can be expressed in terms of the projected wave functions as where and the projection operators with the relation 3. Radiative decays of heavy-light quarkonia through V → P γ The single photon decay of vector (1 −− ) quarkonia (V→ P+γ) is described by the direct and exchange Feynman diagrams as in Figure 1.

Figure 1: Radiative decays of heavy-light quarkonia
To apply the framework of BSE to study radiative decays, V − > P γ, we have to remember that there are two Lorents frames, one the rest frame of the initial meson, and the other, the rest frame of final meson. To calculate further, we first write relationship between the momentum variables of the initial and final meson. Let P , and q be the total momentum and the internal momentum of initial hadron, while P , and q be the corresponding variables of the final hadron, and let k, and λ be momentum and polarization vectors of emitted photon, while λ be the polarization vector of initial meson. Thus if p 1,2 , and p 1,2 are the momenta of the two quarks in initial and final hadron respectively, then,we have, the momentum relations: for initial and final hadrons respectively. From the Feynman diagrams we see that conservation of momentum demands that, P = P + k, while from the first diagram, p 1 = p 1 + k, and −p 2 = −p 2 , where k = P − P is the momentum of the emitted photon. Making use of the above equations, we can express, the relationship between the internal momenta of the two hadrons as, [19] acting like momentum partitioning functions for the two quarks in a hadron. We now decompose the internal momentum q of the initial hadron into two components, q = (q, iM σ), whereq µ = q µ −σP µ is the component of internal momentum transverse to P such thatq.P = 0, while σ = q.P P 2 is the longitudinal component in the direction of P . Similarly for final meson, we decompose its internal momentum, q into two components q = (q , iM σ ).
We now first try to find the relationship between the transverse components of internal momenta of the two hadrons,q, andq . For this, we resolve all momenta in Eq.(9) along the direction transverse to the momentum of the initial meson, P . Thus we can express Eq.(9) as q =q +m 2 (P −P ), where, it can be easily checked thatP .P = 0, and thusP is orthogonal to P . Now, the kinematics gets simplified in the rest frame of the initial meson, where we have P = (0, iM ), while for emitted meson, P = ( − → P , iE ), where E = − → P 2 + M 2 , and since the photon momentum can be decomposed as, final meson and photon would be emitted in opposite directions. Hence we get, . Thus the energy of the emitted meson can be expressed as, Further the dot products of momenta of the initial and the emitted meson can be expressed as, . Thus, we can write the relation between the transverse components of internal momenta of the two hadrons, and their squares as,q From above equations, it can be easily checked that P.q = 0, and P .q = −m 2 Now, we try to find relationship between the time components, σ and σ of the two hadrons. Taking dot product of Eq.(9) with P , the momentum of the initial hadron, we obtain, Making use of the above decomposition of internal momenta, we obtain the relation between the longitudinal components of internal momenta of the two hadrons as, Thus, up to Eq.(13), the kinematics is the same for all the three processes (V → P γ, V → Sγ, and S → V γ) studied in this work.
It is to be noted that 4D BS wave functions of the two hadrons (vector and pseudoscalar)involved in the process are: where S F are the quark propagators, while Γ V (q), and Γ P (q ), are the hadron-quark vertex functions for vector and pseudoscalar mesons respectively, and Ψ P (P , q ) = γ 4 Ψ † P (P , q )γ 4 is the adjoint wave function of the emitted pseudoscalar meson.
The EM transition amplitude of the process is where the first term corresponds to the first diagram, where the photon is emitted from the quark (q), while the second term corresponds to the second diagram where the photon is emitted from the antiquark (Q) in vector meson.
In the above expression, Ψ P and Ψ V are the 4D BS wave functions of pseudoscalar and vector quarkonia involved in the process, and are expressed above, while e q , and e Q are the electric charge of quark, and antiquark respectively, and λ µ is the polarization vector of the emitted photon. Using the fact that the contribution of the second term is the same as that of the first term (except that e q = e Q ), we rewrite above equation in terms of the electronic charge, e as, This can be expressed as, To calculate M f i , we express the propagators S F (±p 1,2 ) as, Here we wish to mention that in transitions involving single photon decays, such as V → P + γ, the process requires calculation of triangle quark-loop diagram, which involves two hadron-quark vertices that we attempt in the 4 × 4 representation of BSE. We now put the propagators expressed as Eq. (18) into Eq. (17), and multiplying this equation from the left by the relation, P where the rest of the terms are anticipated to be zero on account of 3D Salpeter equations. The contour integrations over M dσ are performed over each of the four terms taking into account the pole positions in the complex σ plane: where and the projected wave functions, ψ ±± being taken from the 3D Salpeter equations [19] derived earlier, which for initial meson in internal variableq are:  Fig.2). This superposition of all possible terms in Eq. (21)(22) should be a feature of relativistic frameworks. Now, to calculate the process, we need the 4D BS wave functions for vector and pseudoscalar mesons. We again start with the general 4D decomposition of BS wave functions [28]. Using 3D decomposition under Covariant Instantaneous Ansatz, the wave function of vector mesons of dimensionality, M can be written as [18,22]: where λ is the vector meson polarization vector. Similarly for a pseudoscalar meson, the 3D wave function with dimensionality M can be written as, Now, in accordance with the naive power counting rule [20,21] proposed (in which one of us (SB) was involved), the Dirac structures associated with χ 1 , and χ 2 (in case of vector mesons), and φ 1 , and φ 2 (in case of pseudoscalar mesons), are leading, and would contribute maximum to the calculation of any meson observable. And among these two leading Dirac structures (for both V and P mesons), M , and M γ 5 are the most dominant [22], and contribute the maximum in any calculation. Hence to simplify algebra, we make use of the most dominant Dirac structures for both vector and pseudoscalar mesons. Thus, the 4D Bethe-Salpeter wave functions of heavy-light pseudoscalar and vector quarkonia are taken as, where the 4D Bethe-Salpeter normalizers are The 3D wave functions of ground and excited states of pseudoscalar 0 −+ and vector 1 −− quarkonia are [19], where the inverse range parameters are The ++ and −− components of the B.S. wave function for pseudoscalar meson are [22,26]: Substituting the 4D BS wave function of pseudoscalar meson, the ++ and −− components of the 4D BS wave function of pseudoscalar meson can be obtained and their corresponding adjoint wave functions are written as shown in Appendix A1..
Whereas, the positive and negative energy components of the vector meson wave function are Following the same steps, we obtain the ++ and −− components of the 4D BS wave function of vector meson, as well as their adjoint wave functions.
We now calculate the individual terms, P ψ ) are the Wightman-Garding definitions of masses of the quarks, that act as momentum partitioning parameters. Further, ω 2 1,2 = m 2 1,2 +q 2 , while, ω 2 1,2 = m 2 1,2 +q 2 . Thus the relationship betweenq 2 andq 2 ,q The transition amplitude, M f i is expressed as, where the antisymmetric tensor, µναβ ensures its gauge invariance. Here, F V P (k 2 ) with k = P − P being the fourmomentum transfer (andk being the component of k transverse to initial hadron momentum, P ), is the transition form factor for V → P γ, with expression, where the explicit dependence onk 2 is obtained through use of the kinematical relation,q 2 =q 2 +k 2 (m 1 − 1) 2 + 2(m 1 − 1)k.q, connecting transverse components of internal hadron momenta of the two hadrons, given in the above equation. Andq 2 enters into the expression for F V P through, ω 2 1,2 = m 2 1,2 +q 2 given earlier, whileq 2 in Eq.(34) is integrated out. Here, as stated above,k = k − k.P P 2 P is the photon momentum transverse to initial hadron momentum P , and satisfying the relation,k.P = 0. Here, it is to be noted that althoughk is an effective 3D vector, it is indeed a 4-vector, and further,k 2 = k 2 − (k.P ) 2 P 2 ≥ 0 over the entire 4D space. And in the rest frame of initial hadron, Thus, we have plotted the curves for F V P (k 2 ) for the physical region,k 2 > 0, and to see the overall nature of the curves, have chosen, 0 <k 2 < 2GeV 2 . A similar dependence of transition form factors onk 2 (in their notation q 2 , with time component, q 0 = 0) has recently been studied by Karmanov and co-workers [15]. These form factors [15] were evaluated in the Breit frame, in framework of Bethe-Salpeter Equation.
Our plots of transition form factors for some transitions V → P γ are given in Fig.3. Further treatment of F V P is relegated to Discussions section. Our plots are also similar to the plots in [10]. Now, we proceed to calculate the decay widths, for which we need to calculate the spin averaged amplitude square, where we average over the initial polarization states λ of V-meson, and sum over the final polarization λ of photon. We make use of the normalizations, Σ λ λ µ λ ν = 1 3 (δ µ,ν + PµPν M 2 ) for vector meson, and Σ λ λ µ λ ν = δ µ,ν , for the emitted photon, with M f i taken from the previous equation. The spin-averaged amplitude square of the process, obtained after dividing by the total spin states (2j + 1) of the initial vector meson can be obtained as In ). Thus, |M f i | 2 can be expressed as, The decay width of the process (V → P γ) in the rest frame of the initial vector meson is expressed as where we make use of the fact that modulus of the momentum of the emitted pseudoscalar meson can be expressed in terms of masses of particles as, where, ω k is the kinematically allowed energy of the emitted photon. Thus, Γ in turn can be expressed as: We now calculate the radiative decay widths for the process, V → S + γ in the next section.

Radiative decays of heavy-light quarkonia through V → Sγ
After the 3D reduction of the 4D BS wave function of scalar meson under CIA, we express the 3D BS wave function with dimensionality M as Making use of the fact that the most leading Dirac structure in scalar meson BS wave function is M I (I being the unit 4 × 4 unit matrix), and making use of [18], we express the 3D scalar meson BS wave function as, where φ S (q) is the spatial part of this wave function, whose analytic form is obtained by solving the 3D mass spectral equations for scalar mesons, given in [18] are The 4D BS normalizer of scalar meson, N S , can be obtained by solving the current conservation conditions, and is expressed as, We We can now express the transition amplitude, M f i for the process, V → Sγ as, which can be reexpressed as where where To calculate the decay widths, we need to calculate the spin averaged amplitude square, where we average over the initial polarization states λ of V-meson, and sum over the final polarization λ of photon. The spin-averaged amplitude square of the process, obtained after dividing by the total spin states (2j +1) of the initial vector meson can be written as a double integral over d 3q as, The averaged square of the scattering amplitude can be expressed as where andq where we have made use of the relations, (P. λ ) = 0, (P.ε λ ) = 0, and (P.q ) = 0. To obtain spin averaged amplitude square, we sum over the polarizations of the vector meson and the emitted photon by making use of the normalizations, Σ λ λ µ λ ν = 1 3 (δ µ,ν + PµPν M 2 ) for vector meson, and Σ λ λ µ λ ν = δ µ,ν , for the emitted photon. This gives λ λ | λ . λ | 2 = 1. We can write the decay width, where we make use of the fact that modulus of the momentum of the emitted scalar meson can be expressed in terms of masses of particles as,

Radiative decays of heavy-light quarkonia through S → V γ
We proceed to evaluate the process in the same manner as V → Sγ, using Fig.1, where the initial scalar meson decays into a vector meson and a photon. Drawing analogy from V → P γ, and V → Sγ, the effective 3D form of transition amplitude, M f i for S → V γ under Covariant Instantaneous Ansatz can be expressed as, The transition amplitude of the S → V γ process can be obtained as where Evaluating trace over gamma matrices, M f i can be expressed as, where To calculate the decay widths, we again need to calculate the spin averaged amplitude square, where we sum over the final polarization states, λ of photon, and λ of V-meson.
The spin averaged amplitude modulus square gives, where |T R| 2 = λ λ |T R| 2 . Thus, The decay widths Γ for the process, S → V γ, are given by Eq.(50), with P , now the momentum of the emitted vector meson.

Results and Discussion
We have studied radiative decays of conventional heavy-light quarkonia through M1 and E1 transitions in the framework of Bethe-Salpeter equation. Such processes involve quark-triangle diagrams, and involve two hadron-quark vertices, and are difficult to calculate in BSE under CIA [14,16]. In this work, we have given a generalized method for handling quark triangle diagrams with two hadron-quark vertices in the framework of 4×4 BSE, by expressing the transition amplitude, that were obtained by fitting to their mass spectra [19]. We have compared our results with experimental data and other models, and found reasonable agreements.
To simplify the algebra, we take only the leading Dirac structures (γ 5 , I, and γ. respectively) in the hadronic Bethe-Salpeter wave functions of P, S and V quarkonia, which as we have earlier shown using our naive power counting rule [20,21], contribute maximum to the calculation of any meson observable. However the price we have to pay for this simplification is that though we get reasonable agreements of our decay widths for M1 transitions, nS → n S + γ (with n = n), with data, but for n = n, the agreement is not good. This can be seen from Table 1, that for J/Ψ(n S) → η c (nS) + γ, though we get reasonable results for 1S → 1S, and 2S → 2S, but for 2S → 1S, our calculated decay widths are half of central values of data, which would imply that the incorporation of sub-leading Dirac structures in wave functions of V and P quarkonia might be important for studying transitions between different spatial multiplets.
Incorporation of all Dirac structures would lead to better agreements with data. We wish to study this further.
We also calculated the transition form factors, F V P (k 2 ), which are obtained from transition matrix elements, M f i through use of the kinematical relation connecting the transverse components of internal momenta of the two hadrons, q 2 =q 2 +k 2 (m 1 − 1) 2 + 2(m 1 − 1)k.q, whereq 2 enters into the expression for F V P through, ω 1,2 = (m 2 1 +q 2 ), whilê q 2 in Eq.(34) is integrated out. Here,k = k − k.P P 2 P is the photon momentum transverse to initial hadron momentum P , and as mentioned in Section 3,k 2 = k 2 − (k.P ) 2 P 2 ≥ 0 over the entire 4D space. Thus, we have plotted the form factor curves for F V P (k 2 ) for the physical region,k 2 > 0, and to see the overall nature of the curves, have chosen, We thus give the plots of transition form factors, F V P (k 2 ) versusk 2 for the transitions, J/Ψ → η c γ, D * → Dγ, and D s * → D s γ in Fig.2. Our plots of form factors for M1 transitions are very similar to the corresponding plots in [10]. And the coupling constant g V P γ for a real photon is obtained from F V P (k 2 ) in the limitk 2 → 0. It is seen that our coupling constant, |g J/Ψηcγ | = 0.668GeV −1 (Expt. = 0.570 ± 0.110)GeV −1 , while the coupling constant, |g D * Dγ | = 1.536GeV −1 , which can be compared with results of other models [10,29,30] that obtain |g D * Dγ | in the range (1.783 − 1.940)GeV −1 . Our |g Ds * Dsγ | = 0.351GeV −1 , which is higher than the corresponding value 0.161GeV −1 [11].
Similarly we again see a wide range of variations in decay widths of different models for M 1 transitions, particularly for decays of J/Ψ, and Ψ(2S). Our decay widths for J/Ψ → η c γ, and Ψ(2S) → η c (2S)γ are reasonable, though Ψ(2S) → η c (1S) is nearly half of central value of data. However our nS → nS transitions show a marked decrease as we go from ground to higher excited states, which is in conformity with data and other models. We have also given our predictions for radiative decays of B c * (1S), B c * (2S), B s * (2S), B * (2S), D * (2S), for which data is not yet available, and for D s * (1S), where PDG [40] gives only the upper limit of the decay width. As regards E1 transitions, our decay width result for Ψ(2S) is in good agreement with data, but for χ c0 is higher than data, though again there is a lot of variation in results of other models. However, as mentioned above, incorporation of all Dirac structures is expected to give better agreements with data.