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In class, the generic formula we've been given for calculating scattering amplitudes for the weak decay (at low energies so that the interaction is effectively point-like) is

$$ M= <f|L_{eff}|i> $$ where $L_{eff}=-\frac{G_F}{\sqrt(2)}(J^{\alpha \dagger} J_{\alpha} + J^{\alpha_n \dagger} J_{\alpha,n})$, where $ J^\alpha$ is the weak current and $J^\alpha _n$ is the neutral current.

My question is, what do I use as the effective Lagrangian in the case of the weak decay with a Higgs to a lepton-antilepton pair, since the Higgs does not appear in either the weak or neutral current?

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You have to use the Standard model gauge invariant lagrangian of Yukawa interaction of leptons with Higgs doublet, $$ L_{\text{Yuk}} = -\sum_{l}y_{L}\bar{l}_{L}Hl_{R}, $$ where in the unitary gauge $$ H = \begin{pmatrix} 0 \\ h + v\end{pmatrix}, \quad L \equiv \begin{pmatrix} \nu^{l}_{L} \\ l_{L} \end{pmatrix} $$ and $$ y_{L} \equiv \frac{m_{l}}{v} $$ Such term (and coupling constant $y_{l}$) is required theoretically for introducing gauge invariant leptons mass terms in the lagrangian.

In explicit form interaction $hl\bar{l}$ takes the form $$ L_{\text{int}} = -h\sum_{l}y_{l}\bar{l}l $$ Now calculating the decay rate is straightforward.

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