Pion and $\rho$ meson's unpolarized quark distribution functions from $q\bar{q}$ and all Fock-states within Dyson--Schwinger equations

Abstract

We compute the twist-2 unpolarized quark parton distribution functions (PDFs) of the pion and the $\rho$ meson within the Dyson-Schwinger equations (DSEs) framework using the rainbow-ladder (RL) truncation. A new DSE for the spin-1 hadron's quark-quark correlation matrix is derived, from which the PDFs can be extracted. For the $\rho$ meson, we obtain for the first time within RL-DSEs the unpolarized PDFs corresponding to different helicity states. A pronounced difference is observed between the $|\Lambda|=1$ and $\Lambda=0$ cases, where $\Lambda$ denotes the meson helicity, leading to a nonvanishing and numerically sizable tensor-polarized PDF $f_{1LL}(x)$. We further compare these results with those obtained under a leading Fock-state ($|q\bar{q}\rangle$) truncation and find substantial deviations. This comparison demonstrates that the present RL-DSEs framework incorporates higher Fock-state contributions associated with gluonic degrees of freedom, which has a significant impact on the quark distributions.