Leading-twist parton distribution amplitudes of S-wave heavy-quarkonia

Minghui Ding,Fei Gao,Lei Chang,Yu-Xin Liu,Craig D Roberts

doi:10.1016/j.physletb.2015.11.075

Abstract

The leading-twist parton distribution amplitudes (PDAs) of ground-state S01 and S13cc¯- and bb¯-quarkonia are calculated using a symmetry-preserving continuum treatment of the meson bound-state problem which unifies the properties of these heavy-quark systems with those of light-quark bound-states, including QCD's Goldstone modes. Analysing the evolution of S01 and S13 PDAs with current-quark mass, mˆq, increasing away from the chiral limit, it is found that in all cases there is a value of mˆq for which the PDA matches the asymptotic form appropriate to QCD's conformal limit and hence is insensitive to changes in renormalisation scale, ζ. This mass lies just above that associated with the s-quark. At current-quark masses associated with heavy-quarkonia, on the other hand, the PDAs are piecewise convex–concave–convex. They are much narrower than the asymptotic distribution on a large domain of ζ; but nonetheless deviate noticeably from φQQ¯(x)=δ(x−1/2), which is the result in the static-quark limit. There are also material differences between S01 and S13 PDAs, and between the PDAs for different vector-meson polarisations, which vanish slowly with increasing ζ. An analysis of moments of the root-mean-square relative-velocity, 〈v2m〉, in S01 and S13 systems reveals that 〈v4〉-contributions may be needed in order to obtain a reliable estimate of matrix elements using such an expansion, especially for processes involving heavy pseudoscalar quarkonia.

Highlights

In studying hard exclusive processes within the Standard Model there are many instances in which one may appeal to factorisation theorems so that, at leading-order in a systematic expansion, the amplitude involved can be written as a convolution of a hard-scattering kernel, calculable in perturbation theory, and the so-called leading-twist parton distribution amplitudes (PDAs) of the hadron involved, φ(x), where x is the light-front fraction of the hadron’s total momentum carried by the struck parton
Having computed a sufficient number of the moments for a light-quark system, one could reconstruct the associated PDA using the “Gegenbauer-α” procedure introduced in Refs. [9, 35], which is ideal for representing the broad, concave amplitudes that are characteristic of such systems
To continue with our calculation of S -wave heavy-quarkonia valence-quark PDAs, the dressed-quark propagators and Bethe-Salpeter amplitudes associated with these bound states are needed

Summary

Introduction

In studying hard exclusive processes within the Standard Model there are many instances in which one may appeal to factorisation theorems so that, at leading-order in a systematic expansion, the amplitude involved can be written as a convolution of a hard-scattering kernel, calculable in perturbation theory, and the so-called leading-twist PDA of the hadron involved, φ(x), where x is the light-front fraction of the hadron’s total momentum carried by the struck parton. (1) we have used the fact that there are only two independent vector-meson PDAs at leading-twist [30]: φV (x), φ⊥V (x) describe, respectively, the light-front fraction of the meson’s total momentum carried by the quark in a longitudinally or transversely polarised bound-state.

Results

Conclusion