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Magnetic field effects on the static quark potential at zero and finite temperature

We investigate the static $Q\bar{Q}$ potential at zero and finite temperature in the presence of a constant and uniform external magnetic field $\vec{B}$, for several values of the lattice spacing and for different orientations with respect to $\vec{B}$. As a byproduct, we provide continuum limit extrapolated results for the string tension, the Coulomb coupling and the Sommer parameter at $T = 0$ and $B = 0$. We confirm the presence in the continuum of a $B$-induced anisotropy, regarding essentially the string tension, for which it is of the order of 15\% at $|e| B \sim 1~{\rm GeV}^2$ and would suggest, if extrapolated to larger fields, a vanishing string tension along the magnetic field for $|e| B \gtrsim 4$ GeV$^2$. The angular dependence for $|e| B \lesssim 1$ GeV$^2$ can be nicely parametrized by the first allowed term in an angular Fourier expansion, corresponding to a quadrupole deformation. Finally, for $T \neq 0$, the main effect of the magnetic field is a general suppression of the string tension, leading to a precocious loss of the confining properties: this happens even before the appearance of inverse magnetic catalysis in the chiral condensate, supporting the idea that the influence of the magnetic field on the confining properties is the leading effect originating the decrease of $T_c$ as a function of $B$.