Decay rates and electromagnetic transitions of heavy quarkonia

The electromagnetic radiative transition widths for heavy quarkonia, as well as digamma and digluon decay widths, are computed in the framework of the extended harmonic confinement model (ERHM) and Coulomb plus power potential (CPP$_\nu$) with varying potential index $\nu$. The outcome is compared with the values obtained from other theoretical models and experimental results. While the mass spectra, digamma and digluon widths from ERHM as well as CPP$_{\nu=1}$ are in good agreement with experimental data, the electromagnetic transition widths span over a wide range for the potential models considered here making it difficult to prefer a particular model over the others because of the lack of experimental data for most transition widths.


Introduction
Decay properties of mesons are of special experimental and theoretical interest because they provide us with further insight on the dynamics of these system in addition to the knowledge we have gained from the spectra of these families. Large number of experimental facilities world over have provided and continue to provide enormous amount of data which needs to be interpreted using available theoretical approaches [1]. Many phenomenological studies on numerous observables of the cc and bb bound states have established that the non-relativistic nature appears to be an essential ingredient to understand the dynamics of heavy quarkonia [2]. Hence, heavy quarkonium is characterized by the interplay among the several supposedly well-separated scales typical of a nonrelativistic system: the heavy quark mass m, the inverse of the typical size of the quarkonium 1/r ∼ mv and the binding energy E ∼ mv 2 , where v ≪ 1 is the velocity of the heavy quark inside the quarkonium. Two effective field theories, non-relativistic QCD (NRQCD) [3,4] and potential NRQCD (pNRQCD) [5,6], have also been developed. Applications of these two EFTs have led to a plethora of new results for several observables in quarkonium physics [7].
Radiative transitions in heavy quarkonia have been subject of interest as the CLEO-c experiment has measured the magnetic dipole (M1) transitions J/ψ(1S) → γη c (1S) and J/ψ(2S) → γη c (1S) using combination of inclusive and exclusive techniques and reconciling with theoretical calculations of lattice QCD and effective field theory techniques [8,9]. M1 transition rates are normally weaker than E1 rates, but they are of more interest because they may allow access to spin-singlet states that are very difficult to produce otherwise. It is also inter-esting that the known M1 rates show serious disagreement between theory and experiment when it comes to potential models. This is in part due to the fact that M1 transitions between different spatial multiplets, such as J/ψ(1S) → γη c (2S → 1S) are nonzero only due to small relativistic corrections to a vanishing lowest-order M1 matrix element [10].
We use the spectroscopic parameters of extended harmonic confinement model (ERHM) which has been successful in prediction of masses of open flavour mesons from light to heavy flavour sectors [11][12][13]. The mass spectrum of charmonia and bottomonia predicted by this model and a Coulomb plus Power Potential (CPP ν ) with varying potential index ν (from 0.5 to 2.0) employing non-relativistic treatment for heavy quarks [14][15][16][17] have been utilized for the present computations along with other theoretical as well as experimental results.

Theoretical framework
One of the tests for the success of any theoretical model for mesons is the correct prediction of their decay rates. Many phenomenological models predict the masses correctly but overestimate the decay rates [14,15,18]. We have successfully employed phenomenological harmonic potential scheme and Coulomb Plus Power Potential (CPP ν ) with varying potential index for different confinement strengths to compute masses of bound states of heavy quarkonia and the resulting parameters as well as wave functions have been used to study various decay properties [13].
Choice of scalar plus vector potential for the quark confinement has been successful in the predictions of the low lying hadronic properties in the relativistic schemes for the quark confinement [19][20][21] which has been extended to accommodate multiquark states from lighter to heavier flavour sectors with unequal quark masses [11,12]. The coloured quarks are assumed to be confined through a Lorentz scalar plus a vector potential of the form Here A & B are the model parameters and γ 0 is the Dirac matrix.
The wave functions for quarkonia are constructed here by retaining the nature of single particle wave function but with a two particle size parameter Ω N (q i q j ), The coulombic part of the energy is computed using the residual coulomb potential using the colour dielectric "coefficient" which is found to be state dependent [11] so as to get consistent coulombic contribution to the excited states of the hadrons which is a measure of the confinement strength through the non-perturbative contributions to the confinement scale at the respective threshold energy of the quark-antiquark excitations. The spin average (center of weight) masses of the cc and bb ground states are obtained by choosing the model parameters: m c = 1.428 GeV, m b = 4.637 GeV, k = 0.1925 and the confinement parameter A = 0.0685 GeV 3/2 [11,12].
In the other approach using Coulomb plus Power Potential Scheme (CPP ν ) the heavy-heavy bound state systems such as cc and bb, we treat motion of both the quarks and antiquarks nonrelativistically [13]. The Coulomb Plus Power potential (CPP ν ) given by Here, for the study of heavy flavoured mesons, α c = 4α s /3, α s being the strong running coupling constant, A is the potential parameter and ν is a general power, such that the choice, ν = 1 corresponds to the coulomb plus linear potential. We have employed the hydrogenic trial wave function here for the present calculations. For excited states we consider the wave function multiplied by appropriate orthogonal polynomial function such that the generalized variational wave function gets orthonormalized. Thus, the trial wave function for the (n, l) state is assumed to be the form given by Here, µ is the variational parameter and L 2l+1 n−l−1 (µr) is Laguerre polynomial. For a chosen value of ν, the variational parameter, µ is determined for each state using the virial theorem The potential index ν is chosen to vary from 0.5 to 2. Quark mass parameters are fitted to get experimental ground state masses as m c = 1.31 GeV , m b = 4.66 GeV , α c = 0.4 (for cc) and α c = 0.31 (for bb). Potential parameter A also varies with ν [16].
We have done a completely parameter free computation of digamma and digluon decay widths and radiative electric and magnetic dipole transition widths using parameters of these phenomenological models that were fixed to obtain the ground state masses of the quarkonia systems.

Digamma and Digluon Decay Widths
Using the model parameters and the radial wavefunctions, we compute the digamma (Γ γγ (η Q )) and digluon (Γ gg (χ Q )) decay widths. Digamma decay width of Pwave QQ state χ Q1 is forbidden according to the Landau-Yang theorem. Most of the quark model predictions for the S-wave η Q → γγ width are comparable with the experimental result, while the theoretical predictions for the P-wave (χ Q0,2 → γγ) widths differ largely from the experimental observations [22]. The contribution from QCD corrections takes care of this discrepancy. The oneloop QCD radiative corrections in the digamma decay widths of 1 S 0 (η Q ), 3 P 0 (χ Q0 ) and 3 P 2 (χ Q2 ) are computed using the non relativistic expressions given by [23,24] where, B 0 = π 2 /3−28/9 and B 2 = −16/3 are the next to leading order (NLO) QCD radiative corrections [25]. Similarly, the digluon decay width of η Q , χ Q0 and χ Q2 states are given by [26], Here, the quantities in the brackets are the NLO QCD radiative corrections [25]

Radiative E1 and M1 transitions
In the non-relativistic limit, the M1 transition width between two S-wave states is given by [9] where e Q is the fraction of electrical charge of the heavy quark (e b = −1/3, e c = 2/3), α is the fine structure constant and R nl (r) are the radial Schrödinger wave functions. The photon energy k γ is about the difference between the masses of the two quarkonia, therefore, it is of order mv 2 or smaller. This is in sharp contrast with radiative transitions from a heavy quarkonium to a light meson, such as J/ψ → ηγ, whereas a hard photon is emitted. Since r ∼ 1/(mv), we may expand the spherical Bessel function j 0 (k γ r/2) = 1−(k γ r) 2 /24+. . . . At leading order in the multipole expansion, for n = n ′ , the overlap integral is 1. Such transitions are usually referred to as allowed. At leading order, for n = n ′ , the overlap integral is 0. These transitions are usually referred to as hindered. The widths of hindered transitions are entirely given by higher-order and relativistic corrections.
In the non-relativistic limit, radiative E1 and M1 transition partial widths are given by [9] The CLEO-c experiment has measured the magnetic dipole (M1) transitions J/ψ(1S) → γη c (1S) and ψ(2S) → γη c (1S) using combination of inclusive and exclusive techniques reconciling with theoretical calculations of lattice QCD and effective field theory techniques [8,9]. M1 transition rates are normally weaker than E1 rates, but they are of more interest because they may allow access to spin-singlet states that are very difficult to produce otherwise. Using the spectroscopic parameters of ERHM and CPP ν are utilized for the present computations.

Conclusion
In this paper, we have employed the masses of the pseudoscalar and vector mesons their wave functions and other input parameters from our earlier work [13] for the Table 2. Di-gluon decay width of charmonia (MeV)    [27] 0.496 0.212 0.135 0.099 ---- [28] 0.527 0.263 0.172 -0.037 0.0066 0.037 0.0067 [29] 0.460 0.20 ------ [30] 0.580       The computation of E1 transition widths are done without any relativistic correction terms. This indicates the possible inclusion of the same in the wave function with single center size parameter. The E1 and M1 transitions of the cc and bb mesons are calculated by several groups (See Tables 5-8) but their predictions are not in mutual agreement. The predictions from references [32,33] and CPP ν model (at ν ≃ 1 for cc and at ν ≃ 1.5 for bb mesons) are in fair agreement with experimental values. One of the limitations of the CPP ν model is inability of obtaining the mass spectra of the cc and bb mesons at same potential index ν. The computed magnetic radiative transition rates are tabulated along with other theoretical predictions and available experimental values in Tables 7 and 8. The values in the parentheses are energy of the photon in MeV. The transition widths obtained by potential models are well off with compared to experimental data, however, the values computed using effective mean field theories are found to be around the same i.e. Γ J/ψ→ηcγ = 1.5 ± 1.0 keV and Γ Υ(1S)→η b γ = 3.6 ± 2.9 eV. The photon energies in all the models are found to be nearly same as the mass splitting. The wide variation in predicted hyperfine splittings leads to considerable uncertainty in predicted rates for these transitions. Differences in theoretical assumptions of the potential models make it difficult to draw sharp conclusions about the validity of a particular model because of lack of experimental data.