Quark fluctuations and anisotropic confinement in magnetic fields

We study the string tension $\sigma$ of quantum chromodynamics (QCD) in external magnetic fields $B$ by combining the quasi-particle treatments for quarks with the collective treatments for gluons. We utilize the 1/Nc-expansion (Nc: number of colors) and the framework of Wilson's strong coupling expansion at long distance, and compare the resulting string tensions with the lattice data. The effects of magnetic fields are taken into account through the dielectric functions, which are evaluated from the one-loop polarization due to light flavor quarks with $B$-dependent effective masses $M_{\rm f}(B)\ ({\rm f}=u,d,s)$. All Landau levels are adopted in the calculations so that we can explore the string tensions from weak to strong magnetic fields. For the effective quark masses, we use a simple parameterization: $M_{\rm f}(B) = M_{\rm f}(0) + C_B |q_{\rm f}B|^{1/2}$ with $q_{\rm f}$ the electric charge of a given flavor and $C_B$ an adaptive parameter. We find that the parameter $C_B$ needs to be small, $C_B \lesssim 0.05$, to qualitatively reproduce the trends of lattice QCD results for $0 \leq |eB| \lesssim 1.5\, {\rm GeV}^2$. The quantitative agreement is found for $C_B \simeq 0$ and the strong coupling $\alpha_S \simeq 5$ in the infrared.