Let's have interaction between some gauge boson (for example, $W$ boson) and some other field, for example, let assume $\bar{u}\gamma_{\mu}(1 - \gamma_{5})d W^{\mu} + h.c.$. Let's then use gauge $R_{\epsilon}$ gauge, where $\epsilon = 1$. This means that must be $\partial_{\mu}A^{\mu}_{a} = i(t_{a})^{mn}\varphi_{m}u_{n}$, where $u_{n}$ is vacuum averaged goldstone field, and $\varphi_{m}$ is shifted Goldstone field.

The question: how is $\bar{u}\gamma_{\mu}(1 - \gamma_{5})d W^{\mu} + h.c.$ vertex changed under this transformation?

I tried to make transformation $$ A_{\mu}^{a} \to A_{\mu}^{a}{'} = A_{\mu}^{a} + \partial_{\mu}\varepsilon^{a} + A_{\mu}^{b}\varepsilon^{c}c_{abc}, $$ where $W_{\mu} = \frac{1}{\sqrt{2}}(A_{\mu}^{1} - iA_{\mu}^{2})$, and then to solve equation $$ \partial^{\mu}A_{\mu}^{a} = i(t_{a})^{mn}\varphi_{m}u_{n} $$ for getting expression for $\varepsilon$, but I failed.

Maybe someone knows explicit result of changing of vertex for $\epsilon = 0$, this result I'll also glad to see.


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