Vector–axial-vector mixing from a chiral effective field theory at finite temperature

We study the vector–axial-vector mixing in a hot medium and its evolution toward the chiral phase transition using different symmetry restoration scenarios based on the generalized hidden local symmetry framework. We show that the presence of the $a_1$ meson reduces the vector spectral function around the $\rho$ meson mass and enhances it around the $a_1$ meson mass. The coupling strength of $a_1$ to $\rho$ and $\pi$ vanishes at the critical temperature due to the degenerate $\rho$-$a_1$ masses. This feature holds rigorously in the chiral limit and still stays intact to good approximation for the physical pion mass.

Figure 1

The vector spectral function for option A at temperature $T/T_c=0.6$ (upper) and at $T/T_c=0.8$ (lower) with the critical temperature $T_c=200\mathrm{MeV}$, calculated in the $\rho$-meson rest frame. The left side figures are calculated for $m_{\pi }=0$ and the right side for $m_{\pi }=140\mathrm{MeV}$. The solid curve is obtained in the full calculation. The dashed line is calculated eliminating the axial-vector meson and hence V-A mixing from the theory.

Figure 2

The vector spectral function at temperature $T/T_c=0.8$ for $m_{\pi }=0$ (left) and for $m_{\pi }=140\mathrm{MeV}$ (right). The solid lines are obtained for option A where the $a_1$ mass has a temperature dependence given in Eq. (3.3). The dashed lines are calculated for a constant $a_1$ mass.

Figure 3

The vector spectral function (option A) for $m_{\pi }=0$ (left) and for $m_{\pi }=140\mathrm{MeV}$ (right) at several temperatures $T/T_c=0.6–1.0$.

Figure 4

The vector spectral function (option B) for $m_{\pi }=0$ (left) and for $m_{\pi }=140\mathrm{MeV}$ (right) in type (I) at temperature $T/T_c=0.8$ with the critical temperature $T_c=200\mathrm{MeV}$, calculated in the $\rho$-meson rest frame.

Figure 5

The vector spectral function (option B) for $m_{\pi }=0$ (left) and for $m_{\pi }=140\mathrm{MeV}$ (right) at temperature $T/T_c=0.8$ with the critical temperature $T_c=200\mathrm{MeV}$, calculated in the $\rho$-meson rest frame. The solid curve is for type (I). The dashed line is for type (II).

Figure 6

Diagrams contributing to the V-A mixing at one loop. The circle $(\circ )$ denotes the momentum-independent vertex and the dot $(•)$ denotes the momentum-dependent vertex. Vector and axial-vector fields are denoted by $V$ and $A$ and pion by ${\varphi }_{\pi }$. The $A$ is the transverse components of the $a_1$ meson, while the ${\varphi }_q$ the longitudinal one.