Charting the new physics through one-loop effects in the muon magnetic dipole moment

Since the announcement of the muon $g-2$ measurements, numerous studies have been devoted to exploring the new physics through this precision probe. In specific models, the new physics contributions depend on the couplings and mass scales. Approximate formulae are widely adopted when determining the new physics contributions. This manuscript is dedicated to comprehensive analytical results and approximations for the canonical interactions at one-loop level, which is universal for the spin-half fermions and can serve as a useful reference for the magnetic dipole moment calculations. We present the analytic and approximate expressions for the scalar and vector mediator cases, supplemented by explicit examples in specific models to illustrate their applications. For given muon interactions, we can estimate both the sign and magnitude of the contributions conveniently. Furthermore, we investigate the broader physical implications of these results, offering insights into potential new physics.