$S$ parameter at nonzero temperature and chemical potential

We compute the finite-temperature and matter density corrections to the $S$-parameter at the one-loop level. At nonzero temperature $T$ and matter density $\mu$ Lorentz symmetry breaks and therefore we suggest a suitable generalization of the $S$-parameter. By computing the plasma correction, we discover a reduction of the $S$-parameter in the physically relevant region of small external momenta for any nonzero $\mu$ and $T$. In particular, the $S$-parameter vanishes at small $m/T$, where $m$ is the mass of the fermions, due to the finite extent of the temporal direction. Our results are directly applicable to the determination of the $S$-parameter via first-principle lattice simulations performed with antiperiodic boundary conditions in the temporal direction.

Figure 1

The finite-temperature $S$-parameter (3) (real part: thick solid line, imaginary part: thin solid line) is given by the sum of the zero-temperature part $S_0$ (dashed-dotted lines) and a plasma contribution $S_{+}$ (6) (dashed lines). In this plot $\coth \eta =x=\sqrt{2}$, $\mu =0$ and $\beta m=1$.

Figure 2

The generalized $S$-parameter in Eq. (5) at finite temperature and chemical potential for $\coth \eta =x=\sqrt{2}$.

Figure 3

The generalized $S$-parameter in Eq. (5) (thick lines) at finite temperature and chemical potential as compared to the high-temperature (16) and low-temperature (20) expansions. The three different curves correspond to $\beta \mu =0$ (top, dashed line), $\beta \mu =2.33$ (middle, solid line) and $\beta \mu =4.33$ (bottom, dashed-dotted line). In this plot we use $\coth \eta =\sqrt{2}$.

Figure 4

The plasma contribution to the $S$-parameter (6) (solid line) at finite temperature $\beta m=5$ and finite chemical potential $\beta \mu =3$ compared to the low-temperature expansion (20) (dashed line).

Figure 5

The rapidity dependence of the $S$-parameter in the limit $k^2/m^2\to 0$ for $\mu =0$. The finite-temperature corrections vanish at large rapidity. The curves correspond to different values of the rapidity ranging from $\eta =0$ (lowermost) to $\eta =3$ (uppermost).

Figure 6

Contour plot of $S/S_0$ at $\mu =0$.

Figure 7

Contour plot of $S/S_0$ for $\coth \eta =\sqrt{2}$ corresponding to Fig. 2.

Figure 8

$S$-parameter at finite temperature as a function of the fermions mass for $k^2{\beta }^2=0.1$ (dashed line) and $k^2{\beta }^2=2$ (solid line). In this figure $\coth \eta =\sqrt{2}$.