# Interactions of Fermion & Scalar Fields: Exploring the Difference

• jdstokes
In summary, the two interactions described, \mathrm{i} g \bar{\psi}\psi \varphi and \mathrm{i} g \bar{\psi}\gamma_5\psi\varphi, differ in their transformation properties under parity. The first is a scalar coupling, while the second is a pseudoscalar coupling. This distinction is important in understanding the physical differences between these interactions, particularly in the context of the weak force which couples only to left-handed fields. The symmetries of the interactions also play a role in determining their use in modeling different physical phenomena.

#### jdstokes

Suppose I couple a fermion field to a scalar field using $\mathrm{i} g \bar{\psi}\psi \varphi$ and $\mathrm{i} g \bar{\psi}\gamma_5\psi\varphi$.

I'm trying to understand what would be the physical difference between these interactions. I know that $(1/2)(1\pm \gamma_5)$ approximately projects out the left and right handed components of Dirac fields and that this is related to the fact that the weak interaction couples preferentially to left-handed particles and right-handed anti-particles, but other than that I'm pretty clueless.

Last edited:
Your first interaction lagrangian is a scalar coupling, while the second is a pseudoscalar coupling. Check how they both transform under parity.

What is the physical difference in the interactions apart from their transformation properties?

The symmetries of the interaction define what interactions they are used to model.

jdstokes said:
Suppose I couple a fermion field to a scalar field using $\mathrm{i} g \bar{\psi}\psi \varphi$ and $\mathrm{i} g \bar{\psi}\gamma_5\psi\varphi$.

I'm trying to understand what would be the physical difference between these interactions. I know that $(1/2)(1\pm \gamma_5)$ approximately projects out the left and right handed components of Dirac fields and that this is related to the fact that the weak interaction couples preferentially to left-handed particles and right-handed anti-particles, but other than that I'm pretty clueless.

The weak force not only prefers left-handed; at tree level there is no weak coupling to right-handed fields.

The weak force is slightly more complicated than your model because vector bosons are vectors (duh). You can write the weak coupling as a sum of alpha*Vector + beta*Axial Vector couplings. Up to an internal minus sign (always mix it up), the weak coupling is pure V+A, so it couples solely to left-handed fields. The most obvious physical observable used to demonstrate this is the polarized e+/e- beam experiment of SLD...