Decay rate of a scalar particle under scalar/pseudoscalar lagrangian

[torus]

Hi,
I'm trying to solve problem 48.4 of Srednickis QFT-Book. It goes something like this:

Homework Statement

We have a scalar field with mass M and a Dirac field with mass m (M>2m). The interaction part of the lagrangian is
$$L_a = g \varphi \bar{\Psi}\Psi$$
$$L_b = g \varphi \bar{\Psi}i\gamma_5 \Psi$$.
Now the decay rates $$\Gamma_{a/b}$$ of the process $$\varphi \rightarrow e^+ e^-$$ are to be calculated and compared. It turns out $$\Gamma_b > \Gamma_a$$, which should now be explained in light of parity/angular momentum conservation.

Homework Equations

The Attempt at a Solution

I did all the calculations but I am having a hard time with the explanation. I know that $$L_a/L_b$$ is scalar/pseudoscalar under parity, but I don't see why this should affect the decay rate.

Any help is welcome.

Regards,
torus

 

[BigJDubs]

_sThe key here is that the different forms of the interaction represent different angular momentum couplings between the scalar and Dirac fields. The decay process of a scalar field can only occur if the angular momentum of the initial state is conserved, which means that the scalar field must couple to the Dirac field in a way that preserves this angular momentum. In the case of L_a, the coupling includes a vector current, which means that the total angular momentum of the system must be zero because vector currents always couple to opposite spins. This limits the possible decay channels of the scalar field and results in a lower decay rate.On the other hand, L_b includes a pseudoscalar coupling, which means that the total angular momentum of the system need not be zero. This opens up more decay channels for the scalar field, resulting in a higher decay rate.