Cite as: doi:10.26081/K6XC9F

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We consider the shift of charge-to-mass ratio for extremal black holes
in the context of effective field theory, motivated by the Weak Gravity
Conjecture. We constrain extremality corrections in different regimes
subject to unitarity and causality constraints. In the asymptotic IR, we
demonstrate that for any supersymmetric theory in flat space, and for
all minimally coupled theories, logarithmic running at one loop pushes
the Wilson coefficient of certain four-derivative operators to be larger
at lower energies, guaranteeing the existence of sufficiently large
black holes with Q > M. We identify two exceptional cases of
nonsupersymmetric theories involving large numbers of light states and
Planck-scale nonminimal couplings, in which the sign of the running is
reversed, leading to black holes with negative corrections to Q/M in the
deep IR, but argue that these do not rule out extremal black holes as
the requisite charged states for the WGC. We separately show that
causality and unitarity imply that the leading threshold corrections to
the effective action from integrating out massive states, in any weakly
coupled theory, can be written as a sum of squares and is manifestly
positive for black hole backgrounds. Quite beautifully, the shift in the
extremal Q/M ratio is directly proportional to the shift in the on-shell
action, guaranteeing that these threshold corrections push Q > M in
compliance with the WGC. Our results apply for black holes with or
without dilatonic coupling and charged under any number of U(1)s.
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