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?