I am trying to derive Feynman rules from a given Lagrangian and I got stuck on some vertex factors. What for example is the vertex factor that corresponds to the four-scalar interaction that is decribed by the following Lagrangian?

\begin{equation} L = -\frac{1}{4} g_3^2 \phi^\dagger \lambda^a \phi \chi^\dagger \lambda^a \chi + \frac{2}{9} g_1^2 \phi^\dagger \phi \chi^\dagger \chi \,, \end{equation}

where $\phi,\chi$ are complex scalar (color triplet) fields, $\lambda^a$ are the Gell-Mann matrices, and $g_1,g_3$ are the coupling constants corresponding to $\text{U}(1)$ and $\text{SU}(3)$ respectively.

If we would have only had the second term here, say, then the vertex factor would simply be found by "dropping" the fields and multiplying by $i$. But now there are two terms contributing, and in the first term the Gell-Mann matrices even mix the color components of the scalar triplets. So how do I proceed in this case?

And could anyone give me some general strategies on how to derive vertex factors for "complicated" interactions? For example, I also find it tricky to get the sign right if there is a derivative in an interaction.

(If you are interested in the context of this Lagrangian, for $\phi = \tilde{u}_R$ and $\chi = \tilde{d}_R$ this Lagrangian describes the interaction between two up squarks and two down squarks in a supersymmetric theory.)


You can compute the feynman rule for the $\phi$-$\phi$-$\chi$ vertex by taking $$e^{-i \int \mathrm d^4x L_\mathrm{full} }\frac{\delta}{\delta \phi^a} \frac{\delta}{\delta \phi^b} \frac{\delta}{\delta \chi^c} e^{i\int \mathrm d^4x L_\mathrm{full}}$$ where $L_\mathrm{full}$ is the sum of the free and interaction Lagrangeans and afterwards remove any propagator connecting to external degrees of freedom.


If you have two terms then you would have two vertices which contribute to a certain graph.

For your first term, as far as I can tell, you would have a vertex for $\phi_i+\chi_j\to\phi_k+\chi_l$ given by $\propto g_3^2(\lambda^a)_{ik}(\lambda^a)_{jl}$.

The general recipe to derive the Feynman rules is to feed your Lagrangian into the path integral and just see what propagators / vertices come out.

I cannot help you with the signs because I never get them right myself. But the path-integral could tell you that if you follow through with the computation.

Notice that then, if you want to add the total contribution for the $\phi_i+\chi_j\to\phi_k+\chi_l$ then the second term would give you exactly the same type of vertex, but now replace the Gell-Mann matrices with unit matrices.


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