Mass and force relations for extremal Einstein-Maxwell-dilaton-axion black holes

We investigate various properties of extremal dyonic static black holes in Einstein-Maxwell-Dilaton-Axion theory. We obtain a simple first-order ordinary differential equation for the black hole mass in terms of its electric and magnetic charges, which we can solve explicitly for certain special values of the scalar couplings. For one such case we also construct new dyonic black hole solutions, making use of the presence of an enhanced $ SL(2,\mathbb{R})$ symmetry. Finally, we investigate the structure of long range forces and binding energies between nonequivalent extremal black holes. For certain special cases, we can identify regions of parameter space where the force is always attractive or repulsive. Unlike in the case without an axion, the force and binding energies between distinct black holes are not always correlated with each other. Our work is motivated in part by the question of whether long range forces between nonidentical states can potentially encode information about UV constraints on low-energy physics.