How are the heavy fields integrated out in Type II seesaw?

Question

I have been trying to understand a detail from this paper for quite a long time. In it, a Yukawa Lagrangian

$$L_{Y} = \frac{Y_{\alpha\beta}}{\sqrt{2}}\Delta^+(\overline{l^c_{\alpha}}P_L\nu_{\beta} + \overline{\nu^c_{\alpha}}P_Ll_{\beta}) $$

is introduced with symmetric Yukawa matrix $Y$, and according to the paper by integrating out the heavy triplet Higgs scalar field ($\Delta^+$ here) one would arrive to

$$ L_{NSI} = \frac{Y_{\sigma \beta}Y_{\alpha \rho}^{\dagger}}{m_{\Delta}^2}(\overline{\nu_{\alpha}}\gamma_{\mu}P_L\nu_{\beta})(\overline{l_{\rho}}\gamma^{\mu}P_Ll_{\sigma}). $$

At this point I am very confused. The gamma matrices have appeared between the lepton fields and projection operators and the Yukawa matrix elements indices have changed. If you look how Fermi interaction is derived from standard model, the heavy $W$ field is replaced by leptonic current $J^l_{\mu} = \sum \limits_l \overline{\nu_l}\underbrace{(I_4 -\gamma_5)}_{=2P_L}l$, but an approach like that doesn't appear to work in this case. How should one replace the heavy $\Delta^+$ field in $L_Y$ to arrive at $L_{NSI}$?

Answer 1

First notice that $\Delta^+$ is just one component of the 2x2 representation of the new Higgs triplet $\xi=(\xi^{++},\xi^+,\xi^0)^T$. Sadly, the story is more complicated.

The paper by Chao and Zhang states the methodology. (see https://arxiv.org/pdf/hep-ph/0611323.pdf) Essentially, a more general procedure to obtain the corresponding effective field theory (EFT) is to employ the steepest-descent method on the sector of the path integral one is interested in. The book by Zinn-Justin, Path Integrals in Quantum Mechanics, has some more mathematical details on it. However any standard textbook on QFT has some chapter dedicated to EFT's in this more general setting.

Schematically what one does is to expand the fields of the theory around a specific configuration (minima,. or local minima), in this case around zero for all fields except for the Higgs which is expanded around its non-zero v.e.v., then the action's quadratic dependence can be integrated out as it will be a Gaussian functional integral. To 1-loop, this leads to factors of $\sqrt{\det O}$ where $O$ is inverse effective propagator for the $\xi$ fluctuations thus eliminating the dependence on $\xi$ of the theory as it should for a low energy theory.