Effects of Longitudinal Short-Distance Constraints on the Hadronic Light-by-Light Contribution to the Muon $g-2$

We present a model-independent method to estimate the effects of short-distance constraints (SDCs) on the hadronic light-by-light contribution to the muon anomalous magnetic moment $a_\mu^\text{HLbL}$. The relevant loop integral is evaluated using multi-parameter families of interpolation functions, which satisfy by construction all constraints derived from general principles and smoothly connect the low-energy region with those where either two or all three independent photon virtualities become large. In agreement with other recent model-based analyses, we find that the SDCs and thus the infinite towers of heavy intermediate states that are responsible for saturating them have a rather small effect on $a_\mu^\text{HLbL}$. Taking as input the known ground-state pseudoscalar pole contributions, we obtain that the longitudinal SDCs increase $a_\mu^\text{HLbL}$ by $(9.1\pm 5.0) \times 10^{-11}$, where the isovector channel is responsible for $(2.6\pm 1.5) \times 10^{-11}$. More precise estimates can be obtained with our method as soon as further accurate, model-independent information about important low-energy contributions from hadronic states with masses up to 1-2 GeV become available.