Higgs mediated CLFV processes $\mu N(eN)\rightarrow\tau X$ via gluon operators

We revisit charged lepton flavor violating (CLFV) scattering processes $\ell_{i} N \to \tau X \, (\ell_{i} \ni e, \mu)$ mediated by Higgs. We point out that a new subprocess $\ell_{i} g \to \tau g$ via the effective interactions of Higgs and gluon gives the dominant contribution to $\ell_{i} N \to \tau X$ for an incident beam energy of $E_{\ell} \lesssim 1\,\text{TeV}$ in fixed target experiments. Furthermore, in the light of quark number conservation, we consider quark pair-production processes $\ell_{i} g \to \tau q \bar{q}$ ($q$ denotes quarks) instead of $\ell_{i} q \to \tau q$. This corrects the threshold energy of each subprocess contributing to $\sigma(\ell_{i} N \to \tau X)$. Reevaluation of $\sigma(\ell_{i} N \to \tau X)$ including all of relevant subprocesses shows that the search for $\ell_{i} N \to \tau X$ could serve a complementary opportunity with other relevant processes to shed light on the Higgs CLFV.