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Renormalization Group Analysis of ρ-Meson Properties at Finite Density

We calculate the density dependence of the $ρ$-meson mass and coupling constant($g_{ρNN}$) for $ρ$-nucleon-nucleon vertex at one loop using the lagrangian where the $ρ$-meson is included as a dynamical gauge boson of a hidden local symmetry. From the condition that thermodynamic potential should not depend on the arbitrary energy scale, renormalization scale, one can construct a renormalization group equation for the thermodynamic potential and argue that the various renormalization group coefficients are functions of the density or temperature. We calculate the $β$-function for $ρ$-nucleon-nucleon coupling constant ($g_{ρNN}$) and $γ$-function for $ρ$-meson mass ($γ_{m_ρ}$). We found that the $ρ$-meson mass and the coupling constant for $g_{ρNN}$ drop as density increases in the low energy limit.