  A key ingredient in the evaluation of hadronic light-by-light (HLbL)
scattering in the anomalous magnetic moment of the muon $(g-2)_\mu$ concerns
short-distance constraints (SDCs) that follow from QCD by means of the operator
product expansion. Here we concentrate on the most important such constraint,
in the longitudinal amplitudes, and show that it can be implemented efficiently
in terms of a Regge sum over excited pseudoscalar states, constrained by
phenomenological input on masses, two-photon couplings, as well as SDCs on HLbL
scattering and the pseudoscalar transition form factors (TFFs). Our estimate of
the effect of the longitudinal SDCs on the HLbL contribution is: $\Delta
a_\mu^\text{LSDC}=13(6)\times 10^{-11}$. This is significantly smaller than
previous estimates, which mostly relied on an ad-hoc modification of the
pseudoscalar poles and led to up to a $40\%$ increase with respect to the
nominal pseudoscalar-pole contributions, when evaluated with modern input for
the relevant TFFs. We also comment on the status of the transversal SDCs and,
by matching to perturbative QCD, argue that the corresponding correction will
be significantly smaller than its longitudinal counterpart.
