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

Abstract

The leading-twist parton distribution amplitudes (PDAs) of ground-state ${}^1{S}_0$ and ${}^3{S}_1$ $c\bar{c}$- and $b\bar{b}$-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 ${}^1{S}_0$ and ${}^3{S}_1$ PDAs with current-quark mass, ${\hat{m}}_q$, increasing away from the chiral limit, it is found that in all cases there is a value of ${\hat{m}}_q$ for which the PDA matches the asymptotic form appropriate to QCD's conformal limit and hence is insensitive to changes in renormalisation scale, $\zeta$. 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 $\zeta$; but nonetheless deviate noticeably from ${\phi }_{Q\bar{Q}}(x)=\delta (x-1/2)$, which is the result in the static-quark limit. There are also material differences between ${}^1{S}_0$ and ${}^3{S}_1$ PDAs, and between the PDAs for different vector-meson polarisations, which vanish slowly with increasing $\zeta$. An analysis of moments of the root-mean-square relative-velocity, $⟨v^{2m}⟩$, in ${}^1{S}_0$ and ${}^3{S}_1$ systems reveals that $⟨v^4⟩$-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.

Comments: 6 pages, 2 figures, 3 tables