Magnetic field dependence of the neutral pion mass in the linear sigma model coupled to quarks: The weak field case

We compute the neutral pion mass dependence on a magnetic field in the weak field approximation at one-loop order. The calculation is carried out within the linear sigma model coupled to quarks and using Schwinger’s proper-time representation for the charged particle propagators. We find that the neutral pion mass decreases with the field strength provided the boson self-coupling magnetic field corrections are also included. The calculation should be regarded as the setting of the trend for the neutral pion mass as the magnetic field is turned on.

Figure 1

One-loop Feynman diagram representing the quark-antiquark contribution to the ${\pi }^0$ self-energy.

Figure 2

One-loop Feynman diagram representing the charged pion contribution to the ${\pi }^0$ self-energy.

Figure 3

One-loop Feynman diagrams contributing to the magnetic field correction to the self-coupling $\lambda$. The double line denotes the charged pion, the solid line is the neutral pion and the dashed line represents the sigma.

Figure 4

Magnetic field-dependent effective potential for different values of the field strength near the minimum $v_0^B$. Shown are three cases $|eB|/m_{\pi }^2=0$, 0.4 and 0.6, with $m_{\pi }=140\mathrm{MeV}$. Notice that the three curves are one on top of the others. Therefore, the minimum does not differ from $v_0^{\prime }$.

Figure 5

Magnetic field dependence of the neutral pion mass (a) for a fixed vacuum $m_{\sigma }=450\mathrm{MeV}$, varying the vacuum pion mass between $m_{\pi }=100–160\mathrm{MeV}$, every 20 MeV and (b) for a fixed vacuum $m_{\pi }=140\mathrm{MeV}$ varying the vacuum sigma mass between $m_{\sigma }=400–550\mathrm{MeV}$, every 50 MeV.

Figure 6

Magnetic field dependence of the ratio of the neutral pion mass, when working with the numerical solution of Eq. (24) (${\tilde{M}}_{\pi }^2$), to the case where we use the approximate solution given by Eq. (25) ($M_{\pi }^2$). (a) is the case for a fixed vacuum $m_{\sigma }=450\mathrm{MeV}$, for two vacuum pion massess $m_{\pi }=100$, 160 MeV and (b) is the case for a fixed vacuum $m_{\pi }=140\mathrm{MeV}$, for two vacuum sigma masses $m_{\sigma }=400$, 550 MeV.