Quark-meson vertices and pion properties at finite chemical potential

Abstract

Based on the rainbow-ladder approximation of the Dyson-Schwinger equations and the assumption of the analyticity of the quark-meson vertex in the neighborhood of zero chemical potential ($\ensuremath{\mu}=0$) and neglecting the \ensuremath{\mu}-dependence of the dressed gluon propagator, we use the method of studying the dressed quark propagator at finite chemical potential given in [H. S. Zong, L. Chang, F.Y. Hou, W. M. Sun, and Y. X. Liu, Phys. Rev. C 71, 015205 (2005)] to show that the axial-vector quark-meson vertex at finite \ensuremath{\mu} can be obtained from the corresponding one at $\ensuremath{\mu}=0$ by a shift of variable: ${\ensuremath{\Gamma}}_{5\ensuremath{\nu}}^{j}[\ensuremath{\mu}](k, p)={\ensuremath{\Gamma}}_{5\ensuremath{\nu}}^{j}(\stackrel{~}{k}, p)$, where $k$ and $p$ are the relative and total momentum of the quark-antiquark pair, respectively, and $\stackrel{~}{k}=(\stackrel{\ensuremath{\rightarrow}}{k}, {k}_{4}+i\ensuremath{\mu})$. Similar relations hold for any other type of quark-meson vertex. This feature would facilitate the numerical calculations of the quark-meson vertex function at finite \ensuremath{\mu} considerably. Based on these results and using the dressed quark propagator at $\ensuremath{\mu}=0$ proposed in [R. Alkofer, W. Detmold, C. S. Fischer, and P. Maris, Phys. Rev. D 70, 014014 (2004)], we calculate the pion decay constant ${f}_{\ensuremath{\pi}}$ and the pion mass ${m}_{\ensuremath{\pi}}$ at finite \ensuremath{\mu} and a comparison of our results with those in the literature is made.