Light-front wave functions of vector mesons in an algebraic model

Abstract

Inspired by the recent development of an algebraic model which provides an adequate and unified description of the internal structure of the lowest-lying pseudoscalar mesons, belonging both to the light quarks sector and to the one of heavy quarks, we perform its first extension to the vector-meson case. The algebraic model describes the meson's structure in terms of the spectral density function that appears in a Nakanishi integral representation of the covariant quark-antiquark bound-state amplitude, i.e., the Bethe-Salpeter amplitude. We compute the leading-twist light-front wave functions of the $\ensuremath{\rho}(770)$, $\ensuremath{\phi}(1020)$, $J/\ensuremath{\psi}$, and $\mathrm{\ensuremath{\Upsilon}}(1S)$ mesons through their connection with the parton distribution amplitudes. Among the results we present, the following are of particular interest: (i) transverse light-front wave functions can be obtained algebraically from the corresponding parton distribution amplitudes, whereas that is not the case for longitudinal light-front wave functions, which requires an intermediate step where a spectral density function must be derived from the particular parton distribution amplitude; (ii) the derived spectral density functions show marked differences between light and heavy vector mesons, the latter being narrower as compared to the former, and these are also nonpositive definite, although the integral over the entire curve is larger than zero as expected; and (iii) the longitudinal and transverse light-front wave functions of vector mesons with light quark content exhibit steep $x$ and ${p}_{\ensuremath{\perp}}^{2}$ dependence, while those of the $J/\ensuremath{\psi}$ and $\mathrm{\ensuremath{\Upsilon}}(1S)$ mesons are characterized by narrow distributions in the $x$ range but, comparatively, much more gradual falloffs with respect to the ${p}_{\ensuremath{\perp}}^{2}$ range depicted.