QCD phase diagram in a magnetic background for different values of the pion mass

We investigate the behavior of the pseudocritical temperature of $N_f=2+1$ QCD as a function of a static magnetic background field for different values of the pion mass, going up to $m_{\pi }\simeq 660\mathrm{MeV}$. The study is performed by lattice QCD simulations, adopting a stout staggered discretization of the theory on lattices with $N_t=6$ slices in the Euclidean temporal direction; for each value of the pion mass the temperature is changed moving along a line of constant physics. We find that the decrease of $T_c$ as a function of $B$, which is observed for physical quark masses, persists in the whole explored mass range, even if the relative variation of $T_c$ appears to be a decreasing function of $m_{\pi }$, approaching zero in the quenched limit. The location of $T_c$ is based on the renormalized quark condensate and its susceptibility; determinations based on the Polyakov loop lead to compatible results. On the contrary, inverse magnetic catalysis, i.e., the decrease of the quark condensate as a function of $B$ in some temperature range around $T_c$, is not observed when the pion mass is high enough. That supports the idea that inverse magnetic catalysis might be a secondary phenomenon, while the modifications induced by the magnetic background on the gauge field distribution and on the confining properties of the medium could play a primary role in the whole range of pion masses.

Figure 1

Lines of constant pion mass in the $m_{\ell }-\beta$ plane.

Figure 2

Dependence of the lattice spacing $a$ on $\beta$ along the lines of constant pion mass.

Figure 3

Lines of constant $B$ together with simulation points (blue dots) on $24\times 6$ lattices at $m_{\pi }=664\mathrm{MeV}$.

Figure 4

Light quark condensate at $T=141\mathrm{MeV}$ for $m_{\pi }=343\mathrm{MeV}$, together with a cubic spline.

Figure 5

Renormalized light quark condensate and susceptibility as a function of $T$ at $B=0$ for different values of the pion mass.

Figure 6

Pseudocritical temperature at $B=0$ as a function of $m_{\pi }$. The dashed line represents the result of a best fit according to Eq. (16).

Figure 7

Renormalized condensate for different values of the magnetic field at the different pion masses.

Figure 8

Renormalized chiral susceptibility for different values of the magnetic field at the different pion masses.

Figure 9

Unrenormalized Polyakov loop for different values of the magnetic field at the different pion masses.

Figure 10

Determinations of $T_c(B)$ obtained from different observables at $m_{\pi }=440\mathrm{MeV}$. Data points have been shifted slightly to improve readability.

Figure 11

Determinations of $T_c(B)$ obtained from the renormalized condensate at the different pion masses.

Figure 12

$T_c(eB)/T_c(0)$ as a function of $eB$ for the different pion masses. Data points have been shifted slightly to improve readability. Data at the physical point correspond to the results on $N_t=6$ lattices reported in Ref. [13].

Figure 13

Curvature coefficient ${\upsilon }_B$ defined in Eq. (17) as a function of the pion mass.

Figure 14

Difference between the condensate at $B\ne 0$ and $B=0$ as a function of $T$ at the different pion masses.

Figure 15

Renormalized sea condensate as a function of $|e|B$ at $m_{\pi }=664\mathrm{MeV}$ for various temperatures around the transition. Data points have been shifted slightly to improve readability.