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Suppose we have an SU(N) non abelian gauge theory coupled with a multiplet of complex scalar fields $\Phi$. The lagrangian would be

$$ L= - \frac 12 \text{Tr } F_{\mu\nu}F^{\mu\nu} + |D_\mu \Phi|^2 - V(\Phi)$$

where $D_\mu$ is the covariant derivative: $D_\mu \Phi = \partial_\mu \Phi +ig A_\mu^at_a \Phi$, $A_\mu$ is the gauge field and $t_a$ are the generators of $SU(N)$. The kinetic term gives two interaction terms between $\Phi$ $$ \partial_\mu \Phi^* \partial^\mu \Phi + ig A^{\mu a} (\partial_\mu \Phi^* t_a \Phi - \Phi^* t_a \partial_\mu \Phi) + g^2 A_\mu^a A^{\mu b} \Phi^* t_a t_b \Phi$$ I would like to know if I extracted the vertices correctly from the lagrangian. Suppose all the momenta are ingoing. For the $\Phi^*\Phi A_\mu$ vertex I have $$ i g (q+q')^\mu (t^a)_{ij}$$ where $q$,$q'$ are the momenta of $\Phi^*$ and $\Phi$, but I'm not sure about the relative sign of the momenta. For the $\Phi^*\Phi A_\mu^a A_\nu^b$ vertex, I have $$-i g^2 g^{\mu\nu}\{t^a,t^b\}_{ij}$$ ($i,j$ are the component indices for the scalar multiplet).

I would like to underline that this is not homework. Even if the question arises from the preparation of an exam, I'm not asking the solution to an exercise. This is just some speculation following a digression of the lecturer about gauge theories with scalar fields.

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1 Answer 1

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Man, I make this question today. And... The momentum $q'^{\mu}$ in the first vertex is negative, defining the momentum going inward the vertex. The second is correct.

You can easily find these therms using the canonical quantization and making the derivative in the field $\phi$. The momentum of the inward($\phi$) and outward($\phi^{*}$) are $q^{\mu}$ and $-q'^{\,u}$ respectively.

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