Axial anomaly and hadronic properties in a nuclear medium

We investigate meson and nucleon dynamics at finite baryon density and temperature by coupling the nucleon field and the omega meson to the three-flavor linear sigma model and calculate hadronic properties around the nuclear liquid-gas transition. We apply the functional renormalization group method, and find that mesonic fluctuations increase the strength of the coefficient of the $U_A(1)$ breaking determinant operator as a function of the chiral condensate. As a consequence, we find that the actual value of the anomaly increases discontinuously at the first order nuclear liquid-gas transition. We calculate how mesonic masses and partial restoration of chiral symmetry are modified due to such an effect.

Figure 1

Shape of the effective potential for three different temperatures, as a function of ${\mu }_B$. The plots demonstrate how the critical end point is approached and how the first order transition is gradually turning into second order and a crossover.

Figure 2

Partial restoration of chiral symmetry due to the nuclear liquid-gas transition. We show how the condensates depend on ${\mu }_B$ at $T=0$ (left) and at $T=18\mathrm{MeV}$ (right), which corresponds to the critical end point, i.e., a second order transition.

Figure 3

Plot of the change in the strength of the anomaly in the minimum of the effective potential as a function of ${\mu }_B-{\mu }_{B,c}$, for different temperatures. Note that ${\mu }_{B,c}$ corresponds to the critical chemical potential and it depends on the temperature.

Figure 4

Profile function $A_{k=0}$ at zero temperature, as a function of the chiral invariant $I_1{|}_{v_{\mathrm{ns}},v_{\mathrm{s}}}=(v_{\mathrm{ns}}^2+v_{\mathrm{s}}^2)/2$. In the temperature range that corresponds to a first order transition (i.e., $0\le T\lesssim 18\mathrm{MeV}$), change of the shape is not visible.

Figure 5

Phase boundary in the ${\mu }_B\text{−}T$ plane. Once the end point is set via parametrization, the shape is not really sensitive to the inclusion of mesonic fluctuations.

Figure 6

Mass spectrum at zero temperature. Notice how the increasing anomaly flattens the ${\eta }^{\prime }$ mass.

Figure 7

Mass spectrum at zero temperature, without taking into account the field dependence of the anomaly coefficient (model parameters are the same as in Fig. 6).