Medium induced Lorentz symmetry breaking effects in nonlocal Polyakov–Nambu–Jona-Lasinio models

In this paper, we detail the thermodynamics of two-flavor nonlocal Polyakov–Nambu–Jona-Lasinio models for different parametrizations of the quark interaction regulators. The structure of the model is upgraded in order to allow for terms in the quark self-energy that violate Lorentz invariance due to the presence of the medium. We examine the critical properties, the phase diagram, and the equation of state. Furthermore, some aspects of the Mott effect for pions and sigma mesons are discussed explicitly within a nonlocal Polyakov–Nambu–Jona-Lasinio model. In particular, we continued the meson polarization function in the complex energy plane. Under certain approximations we were able to extract the imaginary part as a function of the meson energy. We were not able to calculate the dynamical meson mass and therefore resorted to a technical study of the temperature dependence of the meson width by replacing the meson energy with the temperature-dependent spatial meson mass. Our results show that, while the temperature behavior of the meson widths is qualitatively the same for a wide class of covariant regulators, the special case in which the nonlocal interactions are introduced via the instanton liquid model singles out with a drastically different behavior.

Figure 1

We illustrate the effect of LSB in set B (left) and set C (right) at $\mu =0$. For simplicity, the system has been solved without PL.

Figure 2

Left (right) panel: phase diagrams for set B (set C) for cases with and without PL. With PL and LSB, the results are shown in black, while the results from Ref. [48] are repeated in red. Blue (orange) lines are results without PL and with (without) LSB. The dashed line denotes crossover, and the full line is the first-order transition.

Figure 3

The structure of only real singularities $z$ as functions of the mass gap ${\sigma }_1$ are shown for the Gaussian regulators of a rank-2 model. Setting ${\sigma }_2=0$ leads back to the rank-1 model given by the blue curve. Nonzero values of ${\sigma }_2$ then give a family of red curves, where the green dot gives ${\sigma }_1^c({\sigma }_2)$ (see the text). The full function ${\sigma }_1^c({\sigma }_2)$ is obtained numerically and shown in the inset.

Figure 4

Scaled entropy density as a function of temperature. The left panel is without PL, while the right panel includes the PL. Notice the temperature mismatch in the oscillations for set A vs sets B and C. Since the gap in the ILM model is undercritical, there are no oscillations in the low-$T$ phase even in the case without the PL, as shown in the inset.

Figure 5

Temperature dependence of lowest singularities for set A in the chiral limit and without the Polyakov loop. The full blue and dotted blue lines are the real ($m_R$) and imaginary ($m_I$) parts of the lowest lying singularity, respectively. Beyond a certain temperature given by the condition ${\sigma }_B(T)={\sigma }_B^c$, these poles join on the real axis to form two real poles, $m_L$ given by the red dashed line and $m_H$ given by the green dashed-dotted line. The mass gap ${\sigma }_B$ is also shown to illustrate how, as $T\to T_c$, $m_L$ approaches ${\sigma }_B$.

Figure 6

The panels display screening masses for $\pi$ and $\sigma$ mesons for different sets. The results for sets B and C are only for the LSB case.

Figure 7

The figure displays the approximate pion widths calculated from Eqs. (61) for sets A–C and the ILM.