Pion and $ρ$ 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 $ρ$ 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 $ρ$ 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 $|Λ|=1$ and $Λ=0$ cases, where $Λ$ 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.

Figures (4)

• Figure 1: From top to bottom are the rainbow--ladder truncated DSEs for dressed quark propagator $S(k)$, the amputated Bethe--Salpeter bound state vertex $\Gamma_{\chi}(k;P)$, and the amputated quark--quark correlation matrix $\Gamma_\Phi(k;P)$ respectively.
• Figure 2: An intuitive picture of diagrams considered in RL-DSEs calculation of parton distribution function.
• Figure 3: Pion and $\rho$'s unpolarized full PDFs and $q\bar{q}$-PDFs at hadronic scale.
• Figure 4: Pion and $\rho$'s valence PDFs at hadronic and experimental scales.