Quark mass dependence of the QCD temperature transition in magnetic fields

The vacuum energy of quarks ${ϵ}_{\text{vac}}^{(q)}=\sum _q{m}_q|⟨\overline{q}q⟩|$ participates in the pressure balance at the temperature transition $T_c$ and defines the dependence of $T_c$ on $m_q$. We first check this dependence in the absence of magnetic fields $eB$ vs known lattice data and then take into account the known strong dependence of the quark condensate on $eB$. The resulting function $T_c(eB,m_q)$ is valid for all $eB$, $m_q<\sqrt{\sigma }$ and explains the corresponding lattice data.

Figure 1

Transition temperature $T_c(m_q,eB,|⟨\overline{q}q⟩{|}_{\mathrm{eff}})$ in GeV as a function of $eB$ for different values of the effective quark condensate defined at zero m.f. Lattice data of Ref. [26] are shown by dotted lines.

Figure 2

Transition temperature $T_c(eB)$ in GeV in two cases: $n_f=2+1$ (solid line with points) and $n_f=3$ with $m_q=0.1\mathrm{GeV}$ and $|⟨\overline{q}q⟩|=(0.17\mathrm{GeV}{)}^3$ (horizontal solid line). Lattice data for $n_f=2+1$ from Ref. [26] (the area between the dotted lines) and for $n_f=3$ (the two points at $eB=0$ and $0.8{\mathrm{GeV}}^2$) from Ref. [22].

Figure 3

Transition temperature $T_c(eB)$ in GeV for the case $n_f=3$ and different quark masses, $m_q=0.2$, 0.28, 0.4 GeV.