Can the Higgs Still Account for the g-2 Anomaly?

We use an Effective Field Theory (EFT) approach to evaluate the viability of the Higgs to account for the $g-2$ anomaly. Although the SM contribution of the Higgs to the muon's magnetic dipole moment is negligible, using a \textit{bottom-up} EFT, we show that given the current level of experimental limits on the Higgs sector, the Higgs can still yield a viable solution to the $g-2$ anomaly if its couplings to the rest of the SM particles are allowed to deviate from their SM predictions. Further, applying unitarity arguments, we show that such a solution would indicate a scale of New Physics (NP) of $\sim 10 -18$ TeV, which could be lowered to $\sim 5$ TeV if the Higgs couplings to the $W$ and $Z$ are assumed to conform to their SM predictions. We show that such a scenario could yield significant enhancement to the di-Higgs production in muon colliders, thus providing further motivation for its consideration. A key takeaway of this study is to highlight the importance of measuring the $\bar{\mu} \mu hh$ coupling in future experiments.