Magnetic-field dependence of the QCD electric screening mass parameter through $g^4$ in hot and dense matter

A constant magnetic field modifies the hard electric screening sector of hot and dense QCD and generates a directional splitting at $\mathcal{O}(g^4)$. We determine the strict-hard screening pole through $g^4$ for three massless flavors with physical $u,d,s$ charges, retaining the full $eB\mathrm{/}{T}^2$ dependence without a weak- or strong-field expansion. The shifted-frequency construction used for the two-loop exchange contribution is a background-field Feynman-gauge curvature; we derive and apply its local conversion to the ordinary Feynman-gauge self-energy before assembling the screening pole. The leading magnetic shift is isotropic, while the $g^4$ directional splitting arises from the one-loop anisotropic kinetic normalization; the symmetry-allowed finite-density $A_0{F}_{12}$ mixing is numerically small over the studied range for physical $u,d,s$ charges. For $\mu \mathrm{/}T=0$ and $1$, the magnetic $g^4$ coefficient is positive over $0\le eB\mathrm{/}{T}^2\le 4$ and is strongly reduced by density. Extending the calculation to $0\le \mu \mathrm{/}T\le 2$ at $eB\mathrm{/}{T}^2=1$ and $4$ reveals sign reversals of both the leading magnetic coefficient and the directional splitting at sufficiently large density. The zero-field limit reproduces the Braaten-Nieto result and its finite-density extension by Vuorinen. The running coupling used only for the phenomenological magnetic mass shift is chosen independently of $B$, so that all magnetic-field dependence through $g^4$ resides in the calculated coefficients.