= Solution
A larger-than-predicted rate can be described by adding new fields and symmetry-respecting renormalizable interactions, or, when those fields are much heavier than the energy being probed, by an <effective field theory>. In the latter description, add <gauge-invariant operators> with <Wilson coefficients> divided by appropriate powers of a heavy scale. Their interference with existing amplitudes, or their leading contribution to an otherwise forbidden amplitude, can enhance a decay. The operators should respect the Standard Model gauge group and be consistent with other observables; writing an arbitrary flavour-changing term without its electroweak completion is insufficient.
For charged leptons, the proposed operator has <mass dimension> six: the fermion bilinear contributes three, the scalar doublet one, and $\phi^\dagger\phi$ two. Its hypercharge vanishes by the same calculation as the renormalizable lepton <Yukawa interaction>, and the extra factor $\phi^\dagger\phi$ is a gauge singlet. Thus its coefficient scales as $\Lambda^{-2}$.
In <unitary gauge>, $\phi=(0,(v+h)/\sqrt2)^T$, so the renormalizable and dimension-six terms combine into
$$
\mathcal L\supset-\bar e_L^i
\left[\lambda_{ij}(v+h)
+\frac{\lambda'_{ij}}{2\Lambda^2}(v+h)^3\right]e_R^j+\mathrm{h.c.}
$$
The charged-lepton <fermion mass matrix> and the single-Higgs Yukawa matrix are therefore
$$
M=v\lambda+\frac{v^3}{2\Lambda^2}\lambda',\qquad
Y_h=\lambda+\frac{3v^2}{2\Lambda^2}\lambda'
=\frac Mv+\frac{v^2}{\Lambda^2}\lambda'.
$$
This is the <dimension-six Higgs Yukawa misalignment>. Let $U_L^\dagger MU_R=\operatorname{diag}(m_e,m_\mu,m_\tau)$ and $\widehat\lambda'=U_L^\dagger\lambda'U_R$. In the mass basis,
$$
\boxed{\widehat Y_h
=\frac{\operatorname{diag}(m_e,m_\mu,m_\tau)}v
+\frac{v^2}{\Lambda^2}\widehat\lambda'.}
$$
The two matrices need not be aligned. Choosing a nonzero off-diagonal $\widehat\lambda'_{\mu\tau}$ or $\widehat\lambda'_{\tau\mu}$ produces the required <charged-lepton flavour violation> while keeping the <fermion mass matrix> diagonal.
For example, the amplitude for $h\to\mu^-\tau^+$ is, up to an overall phase,
$$
\mathcal M=-\bar u_\mu\left[
(\widehat Y_h)_{\mu\tau}P_R+
(\widehat Y_h)^*_{\tau\mu}P_L\right]v_\tau.
$$
Neglecting the final lepton masses, summing spins gives $m_h^2(|(\widehat Y_h)_{\mu\tau}|^2+|(\widehat Y_h)_{\tau\mu}|^2)$. The conjugate charge channel has the same rate. The summed width is
$$
\boxed{\Gamma(h\to\mu^-\tau^++\mu^+\tau^-)
=\frac{m_h}{8\pi}\frac{v^4}{\Lambda^4}
\left(|\widehat\lambda'_{\mu\tau}|^2+
|\widehat\lambda'_{\tau\mu}|^2\right).}
$$
This demonstrates an enhancement that is absent in the minimal renormalizable theory. It requires flavour misalignment: an added matrix aligned with the original Yukawa matrix would not generate these decays. The same operator also predicts double-Higgs and triple-Higgs lepton interactions through the remaining powers of $(v+h)^3$.
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