General Remarks on the One-loop Contributions to the Muon Anomalous Magnetic Moment

The latest measurement of the muon anomalous magnetic moment $a^{}_{\mu} \equiv (g^{}_\mu - 2)/2$ at the Fermi Laboratory has found a $4.2\,\sigma$ discrepancy with the theoretical prediction of the Standard Model (SM). Motivated by this exciting progress, we investigate in the present paper the general one-loop contributions to $a^{}_\mu$ within the SM and beyond. First, different from previous works, the analytical formulae of relevant loop functions after integration are now derived and put into compact forms with the help of the Passarino-Veltman functions. Second, given the interactions of muon with new particles running in the loop, we clarify when the one-loop contribution to $a^{}_\mu$ could take the correct positive sign as desired. Third, possible divergences in the zero- and infinite-mass limits are examined, and the absence of any divergences in the calculations leads to some consistency conditions for the construction of ultraviolet complete models. Applications of our general formulae to specific models, such as the SM, seesaw models, $Z^\prime$ and leptoquark models, are also discussed.