Magnetic field dependence of the neutral pion longitudinal screening mass in the linear sigma model with quarks

We use the linear sigma model with quarks to study the magnetic-field-induced modifications on the longitudinal screening mass for the neutral pion at one-loop level. The effects of the magnetic field are introduced into the self-energy, which contains the contributions from all the model particles. We find that, to obtain a reasonable description for the behavior with the field strength, we need to account for the magnetic field dependence of the particle masses. We also find that the couplings need to decrease fast enough with the field strength to then reach constant and smaller values as compared to their vacuum ones. The results illustrate the need to treat the magnetic corrections to the particle masses and couplings in a self-consistent manner, accounting for the backreaction of the field effects for the magnetic field dependence of the rest of the particle species and couplings in the model.

Figure 1

Feynman diagram corresponding to the one-loop contribution from the fermion antifermion loop to the neutral pion self-energy in the LSMq.

Figure 2

Feynman diagram corresponding to the tadpole contribution from the fermion loop and a $\sigma$ to the neutral pion self-energy in the LSMq.

Figure 3

Feynman diagram corresponding to the one-loop contribution from charged pions to the neutral pion self-energy in the LSMq.

Figure 4

Feynman diagram corresponding to the tadpole contribution from charged pions and a $\sigma$ to the neutral pion self-energy in the LSMq.

Figure 5

Masses of the quark (blue line), pion (orange line), and $\sigma$ (green line) as a function of the magnetic field following Ref. [72].

Figure 6

Average normalized condensate computed from the model compared to the results from Ref. [86].

Figure 7

Neutral pion longitudinal screening mass as a function of the magnetic field strength normalized to the pion pole mass for $B=0$ computed using $g=2.75$. Solutions to the dispersion relation cease to exist beyond $q_{f}B\sim 0.2{\mathrm{GeV}}^2$.

Figure 8

Neutral pion longitudinal screening mass as a function of the magnetic field strength normalized to the pion pole mass for $B=0$: $g=2.75$ (dotted), $g=1.5$ (dashed), $g=0.75$ (dashed-dot), and $g=0.33$ (solid). Solutions to the dispersion relation cease to exist beyond $q_{f}B\sim 0.2$, 0.35, 0.55, and $0.8{\mathrm{GeV}}^2$, respectively, and the range where solutions exist increases as the coupling decreases.

Figure 9

Neutral pion longitudinal screening mass as a function of the magnetic field strength normalized to the pion pole mass for $B=0$, including all the contributions to the self-energy computed with $g=0.33$ and $\lambda =2.5$ compared to the NJL results from Ref. [82] and to an interpolation of the data for the LQCD results from Ref. [81] for $T=17\mathrm{MeV}$. The green shadow represents the error in the LQCD calculations from Ref. [81]. For comparison we also show the case where only the fermion-antifermion loop is considered, computed with $g=0.33$.

Figure 10

Contour of integration for Eq. (a22).