Constructing a Feynman loop integral

[ryanwilk]

Homework Statement

I need to construct the Feynman loop integral for the following diagram:

(*)

where $\nu$_L is the left-handed neutrino, $\phi$ is a scalar particle and N is a heavy neutrino with a Majorana mass.

Homework Equations

N/A

The Attempt at a Solution

I'm trying to determine it by comparing it to the self-energy of the electron:

which gives

.

1) Since there's a scalar propagator ($\phi$) in (*), do I need to use these Feynman rules?:

where m = m_$\phi$ in this case. (Taken from http://bolvan.ph.utexas.edu/~vadim/Classes/2011f/QED.pdf).

2) How do I deal with the fact that the neutrinos are left-handed? Do I just add factors of (1-γ₅)/2 in the (-igδ_βα) terms?

3) For Dirac particles, the propagator is:

But how does this change if the particle (in this case, N) is Majorana?Any help would be appreciated.

Thanks!

 

[fzero]

Have you been given a Lagrangian? It would be much better to work out the Feynman rules from that rather than try to guess at bits and pieces that might not quite fit what you want.

 

[ryanwilk]

The Lagrangian is:

L = L_$\varphi$ + L_SM where

L_$\varphi$ = $\frac{1}{2}$∂_μ$\phi$⁺∂^μ$\phi$ + $\frac{m^2}{2}$$\phi$⁺$\phi$ + $\frac{λ}{4}$($\phi$⁺$\phi$)² + g$\phi$$\overline{N_R}$$\nu$_L + $\frac{m_N}{2}$N_R^TCN_R + h.c.

(g = coupling constant, N = Majorana neutrino, $\phi$ = Neutral scalar, $\nu$_L = LH neutrino),

However, I was told just to construct it from Feynman rules. Also, I have no idea how to go from this complicated Lagrangian to the integral =/.

 

[ryanwilk]

Anyone?