# Symmetry factor and coupling constant in scalar field theory

I am just now starting my particles "education" so forgive me if this is elementary...

Looking at interaction terms in a scalar field Lagrangian, I get:

$$\mathcal{L}=\frac{1}{2}\left(\partial\varphi\right)^2 +... + g\chi\varphi^2$$

Where both $\chi$ and $\varphi$ are scalar fields.

I've seen somewhere that the $\chi\varphi\varphi$ coupling constant here is actually $2g$, since the proper interaction Lagrangian form for scalar fields is actually: $$\mathcal{L}_{int}= \frac{g}{\prod_{k}n_{k}!}\prod_{k}\phi_k^{n_k}$$

And if that's the proper form, then I have: $$\mathcal{L}_{int}=\frac{1}{2}\left(2g\chi\varphi^2\right)$$

The question is, is this correct? And if so, please provide a detailed reference to a book (i.e. at least which chapter)

For whomever is wondering, I am trying to justify the hZZ vertex factor to be $$hZZ\rightarrow \frac{2im_z^2 g^{\mu\nu}}{v}$$

Got lost trying to read Peskin&Schroeder :(

• Looks like what you need to clarify is the symmetry factors. There's a good discussion in the first part of Srednicki's book (probably around chap.9-10). If you can ever get hold of it, D. Anselmi's book "Renormalization" is even clearer about that.
– fqq
Sep 22, 2014 at 7:46
• If your $Z$ is for the Z boson, it is not a scalar, btw! Sep 22, 2014 at 10:19

For a scalar field $\phi$, the most widely used convention, based on my experience, is to write the Lagrangian with kinetic and potential terms, followed by interactions like so,

$$\mathcal{L}=\frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2}m^2 \phi^2 - \sum_{n \geq 3} \frac{\lambda^n}{n!}\phi^n$$

where $\lambda_n$ are coupling constants. (We could not have a single coupling constant for multiple interactions, as for each the dimensions must be such that the final quantity has $[\dots] = d$.) The reason for the $n!$ is to ensure the vertex rule has a factor of $\lambda_n$, rather than $n!\lambda_n$ which arises from the differentiations with respect to $\phi$ of the interaction term. Of course, we could have a term,

$$\mathcal{L}_{\mathrm{int}}= g\phi^2$$

with vertex rule $2g$; both definitions just differ by a factor of two. The former is usually preferred simply for convenience. In addition, a diagram may pick up symmetry factors (see my answer provided at Formula for Symmetry Factor).