Can leptophilic-ALP be a solution to the muon $(g-2)$ anomaly?

In the light of recent measurement of muon $(g-2)$, we investigate the phenomenological implications of an axion-like particle (ALP) which only couples to the standard model charged leptons. We find that in a narrow mass range of ALP, it can alleviate the tension between the theoretical prediction and experimental observation of $(g-2)_\mu$ once we consider all possible one and two-loop diagrams. In particular, ALP can either explain the muon $(g-2)$ anomaly for $5\,{\rm GeV} \lesssim m_a \lesssim 6 \,{\rm GeV}$ while satisfying the other experimental constraints, or be restricted by it.