Magnetic susceptibility of QCD at zero and at finite temperature from the lattice

Abstract

The response of the QCD vacuum to a constant external (electro)magnetic field is studied through the tensor polarization of the chiral condensate and the magnetic susceptibility at zero and at finite temperature. We determine these quantities using lattice configurations generated with the tree-level Symanzik improved gauge action and ${N}_{f}=1+1+1$ flavors of stout smeared staggered quarks with physical masses. We carry out the renormalization of the observables under study and perform the continuum limit both at $T>0$ and at $T=0$, using different lattice spacings. Finite size effects are studied by using various spatial lattice volumes. The magnetic susceptibilities ${\ensuremath{\chi}}_{f}$ reveal a spin-diamagnetic behavior; we obtain at zero temperature ${\ensuremath{\chi}}_{u}=\ensuremath{-}(2.08\ifmmode\pm\else\textpm\fi{}0.08)\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$, ${\ensuremath{\chi}}_{d}=\ensuremath{-}(2.02\ifmmode\pm\else\textpm\fi{}0.09)\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$ and ${\ensuremath{\chi}}_{s}=\ensuremath{-}(3.4\ifmmode\pm\else\textpm\fi{}1.4)\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$ for the up, down and strange quarks, respectively, in the $\overline{\mathrm{MS}}$ scheme at a renormalization scale of 2 GeV. We also find the polarization to change smoothly with the temperature in the confinement phase and then to drastically reduce around the transition region.