# How to calculate the decay rate of Higgs to lepton pair?

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?

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.