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?

 

[paranormal]

I would suggest approaching this problem by first understanding the basics of Feynman loop integrals and the Feynman rules for constructing them. The Feynman loop integral is a mathematical representation of the probability amplitude for a particle to go from one state to another in a quantum field theory. It involves integrating over all possible paths a particle can take, taking into account the interactions with other particles.

To construct the Feynman loop integral for the given diagram, you would need to follow the Feynman rules for constructing loop integrals. This involves assigning a momentum to each particle and using the propagator for each particle in the loop. Since there is a scalar propagator in the given diagram, you would need to use the Feynman rule for a scalar propagator.

Next, you would need to consider the fact that the neutrinos in the given diagram are left-handed. This means that you would need to include the (1-γ5)/2 factor in the (-igδβα) terms, as you have correctly identified. This factor takes into account the chirality of the neutrinos.

Finally, for a Majorana particle, the propagator would be different from that of a Dirac particle. This is because a Majorana particle is its own antiparticle, so the propagator would not have a spinor structure. Instead, it would involve the mass of the particle and the momentum. You can refer to your textbook or other resources for the specific form of the Majorana propagator.

Overall, constructing a Feynman loop integral requires a good understanding of the Feynman rules and the properties of the particles involved. I would suggest familiarizing yourself with these concepts before attempting to construct the integral for the given diagram.