Parton distribution functions of ground state mesons composed of $c$ or $b$ quarks

TL;DR

This work computes the valence quark PDFs of all ground-state heavy mesons with charm or bottom quarks by combining a QCD-inspired constituent quark model with a light-front boost from the rest frame. The light-front wave functions are constructed via an inverse Lorentz boost and Melosh rotation, yielding PDFs through transverse-momentum integration and Mellin moments, at the hadron scale $\zeta_H$. The results reveal that heavy-quark PDFs are narrower than a scale-free profile and exhibit clear differences between pseudoscalar and vector states, with heavy-light systems like $B_c$ showing asymmetry in momentum sharing and $\bar{c}(x;\zeta_H)=b(1-x;\zeta_H)$ as expected; comparisons to BLFQ and algebraic DSE approaches show reasonable agreement in pattern. Notably, the $B_c^*$ predictions are new, expanding the understanding of heavy-meson structure and informing future experimental and phenomenological studies.

Abstract

The valence quark parton distribution functions (PDFs) of all ground state heavy mesons that composed of $b$ or $c$ quarks, are discussed; namely, the pseudoscalar $η_c(1S)$, $η_b(1S)$ and $B_c$, together with the corresponding vector ones, $J/ψ$, $Υ(1S)$ and $B_c^\ast$. We use a QCD-inspired constituent quark model, which has been applied with success to conventional heavy mesons, so that one advantage here is that all parameters have already been fixed by previous studies. The wave functions of the heavy mesons in the rest frame are obtained by solving the Schrödinger equation, then boosted to its light-front based on Susskind's Lorentz transformation. The PDFs at the hadron scale, are then obtained by integrating out the transverse momenta of the modulus square of the light-front wave function. Our study shows how the valence quark distributions differ between pseudoscalar and vector mesons, as well as among charmonia, bottomonia and bottom-charmed mesons. Comparisons with other theoretical calculations demonstrate that the PDFs obtained herein are in general narrower but align well with the expected patterns. Moreover, each PDF's point-wise behavior is squeezed with respect to the scale-free parton-like PDF.

Figures (7)

• Figure 1: Valence $c$ quark PDFs at the hadron scale, $\zeta_H$, of $\eta_c(1S)$ (black solid) and $J/\psi$ (black dotted) mesons. The blue dashed and dash-dotted lines correspond, respectively, to the PDFs of $\eta_c(1S)$ and $J/\psi$ given in Ref. Lan:2019img. The red double-dot-dashed line is provided by Ref. Albino:2022gzs. For comparison, the scale-free parton-like profile: $q(x)=30x^2(1-x)^2$, is also shown as a red solid line.
• Figure 2: Valence $b$ quark PDFs at the hadron scale, $\zeta_H$, of $\eta_b(1S)$ (black solid) and $\Upsilon(1S)$ (black dotted) mesons. The blue dashed and dash-dotted lines correspond, respectively, to the PDFs of $\eta_b(1S)$ and $\Upsilon(1S)$ given in Ref. Lan:2019img. The red double-dot-dashed line is provided by Ref. Albino:2022gzs. For comparison, the scale-free parton-like profile: $q(x)=30x^2(1-x)^2$, is also shown as a red solid line.
• Figure 3: Valence $b$ quark and $\bar{c}$ anti-quark PDFs at the hadron scale, $\zeta_H$, of $B_c$ (black solid line) and $B_c^\ast$ (black dotted line) mesons. The blue dashed line corresponds to the PDFs reported in Ref. Lan:2019img. The red dotted line is provided by Ref. Almeida-Zamora:2023bqb. For comparison, the scale-free parton-like profile: $q(x)=30x^2(1-x)^2$, is also shown as a red solid line.
• Figure 4: Mellin moments of $\eta_c(1S)$ light-front wave functions at the hadron scale.
• Figure 5: Mellin moments of $B_c(\bar{c})$ light-front wave functions at the hadron scale.