Scaling / engineering dimensions for these interacting fields?

[Dixanadu]

Hi guys,

So I have the following Lagrangian:

$\mathcal{L}=\bar{\psi}(i\partial_{\mu}\gamma^{\mu}-m)\psi+\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-\frac{M^{2}}{2}\phi^{2}-\lambda\bar{\psi}\psi\phi$

where m is the is the mass of the spin-half Dirac field and M is the mass of the spin-0 field.

I was wondering: how do I find the scaling/engineering dimensions for the $\lambda$ and both fields?

 

[eljose79]

Hi there,

To find the scaling/engineering dimensions for \lambda and both fields, you can use the following equations:

For the spin-half Dirac field \psi: [m]=L^{-1} (where L is length)

For the spin-0 field \phi: [M]=L^{-1}

For the coupling constant \lambda: [\lambda]=L^{d-4} (where d is the number of spatial dimensions)

These dimensions can also be written in terms of energy by using the conversion factor [E]=L^{-1}. So the dimensions for m and M would be [m]=E and [M]=E, and the dimension for \lambda would be [\lambda]=E^{d-4}.

I hope this helps! Let me know if you have any further questions.