The Calculation of $f_{\pi }$ and $m_{\pi }$ at Finite Chemical Potential

Based on the previous work in [Y. Jiang, Y.M. Shi, H.T. Feng, W.M. Sun and H.S. Zong, Phys. Rev. C {\bf 78}, 025214 (2008)] on the quark-meson vertex and pion properties at finite quark chemical potential, we provide an analytical analysis of the weak decay constant of the pion ($f_{\pi }[\mu ]$) and the pion mass ($m_{\pi }[\mu ]$) at finite quark chemical potential using the model quark propagator proposed in [R. Alkofer, W. Detmold, C.S. Fischer and P. Maris, Phys. Rev. D {\bf 70}, 014014 (2004)]. It is found that when $\mu$ is below a threshold value ${\mu }_0$ (which equals $0.350\mathrm{G}\mathrm{e}\mathrm{V}$, $0.377 $\mathrm{G}\mathrm{e}\mathrm{V}$$and$0.341 $\mathrm{G}\mathrm{e}\mathrm{V}$$,forthe$$2\mathrm{C}\mathrm{C}$$,$$1\mathrm{R}1\mathrm{C}\mathrm{C}$$and$$3\mathrm{R}$$ parametrizations of the model quark propagator, respectively.), $f_{\pi }[\mu ]$ and $m_{\pi }[\mu ]$ are kept unchanged from their vacuum values. The value of ${\mu }_0$ is intimately connected with the pole distribution of the model quark propagator and is found to coincide with the threshold value below which the quark-number density vanishes identically. Numerical calculations show that when $\mu$ becomes larger than ${\mu }_0$, $f_{\pi }[\mu ]$ exhibits a sharp decrease whereas $m_{\pi }[\mu ]$ exhibits a sharp increase. A comparison is given between the results obtained in this paper and those obtained in previous literatures.