$\alpha'$ corrections of Reissner-Nordstr\"om black holes

We study the first-order in $\alpha'$ corrections to non-extremal 4-dimensional dyonic Reissner-Nordstr\"om (RN) black holes with equal electric and magnetic charges in the context of Heterotic Superstring effective field theory (HST) compactified on a $T^{6}$. The particular embedding of the dyonic RN black hole in HST considered here is not supersymmetric in the extremal limit. We show that, at first order in $\alpha'$, consistency with the equations of motion of the HST demands additional scalar and vector fields become active, and we provide explicit expressions for all of them. We determine analytically the position of the event horizon of the black hole, as well as the corrections to the extremality bound, to the temperature and to the entropy, checking that they are related by the first law of black-hole thermodynamics, so that $\partial S/\partial M=1/T$. We discuss the implications of our results in the context of the Weak Gravity Conjecture, clarifying that entropy corrections for fixed mass and charge at extremality do not necessarily imply corrections to the extremal charge-to-mass ratio.