Handbook of the analytic and expansion formulae for the muon $g-2$ anomaly
Since announcement of the muon $g-2$ anomaly, plenty of papers have devoted to this anomaly. The approximate formulae are always adopted when determining the new physics contributions to $(g-2)_{\mu}$, while clear scope of applications are always absent. This manuscript is dedicated to the comprehensive analytical results and approximations for the canonical interactions at one-loop level, which can be a useful handbook for the model builders. Here, we only collect the analytic and approximate expressions for the scalar mediator case. For the expressions of vector mediator case, they will appear in the future.