Radiative decays in bottomonium beyond the long wavelength approximation

Abstract

We revisit the nonrelativistic quark model description of electromagnetic radiative decays in bottomonium. We show that even for the simplest spectroscopic quark model the calculated widths can be in good agreement with data once the experimental masses of bottomonium states and the photon energy are properly implemented in the calculation. For transitions involving the lower lying spectral states this implementation can be easily done via the Long Wave Length approximation. For transitions where this approximation does not apply we develop a new method of implementing the experimental energy dependencies.

Radiative decays in charmonium beyond the p/m approximation

“…In a recent paper [3], we have shown that this discrimination may be rather difficult in bottomonium. By using the standard expansion of the electromagnetic transition operator up to p b =M b order, where p b ðM b Þ stands for threemomentum (mass) of the b quark, we have shown that accurate results for the widths can be obtained from different quark potential models reasonably fitting the spectroscopy once the experimental masses of the bottomonium states instead of the calculated ones are properly implemented in the calculation.…”

“…It is important to emphasize that (i) this potential form arises from spin independent quenched lattice QCD calculations in the Born-Oppenheimer approximation [4], (ii) in the spirit of the nonrelativistic quark model calculations σ, ζ, β and the quark mass M c should be considered as effective parameters through which spin dependent and/or spin independent corrections may be implicitly incorporated. Henceforth we shall make use of two different quark models with the same Hamiltonian form (1), Model I and Model II, that have been used for the analysis of radiative decays in bottomonium [3]. As we are dealing with a radial potential we shall denote the spectroscopic states by n 2sþ1 L J where s, L, and J stand for the spin, orbital angular momentum, and total angular momentum quantum numbers respectively.…”

“…Instead, we prefer to maintain such connection and the modest discrepancy reflected in Table II. Moreover, the chosen values of σ II and ζ II have been used in bottomonium for an accurate description of electromagnetic decay widths [3] what adds interest to the…”

“…The p c M c approximation is defined from (17) in the limit E α ¼ M α . A thorough analysis of this approximation has been carried out in [3]. Here we only give the final expressions for the calculation of the amplitude:…”

“…where jΨi ≡ jJ; m; nL; si ð 19Þ stands for the cc spectroscopic n 2sþ1 L J state and O α for an operator that we detail next. Thus, for 3 S 1 → γ 3 P J transitions we have (here we write only the electric part of the operator; the form for the magnetic part can be found in the appendices of [3])…”