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1

When using Kubo formula, you often have to calculate vertex functions. You'd better take care of gauge invariance by imposing Ward-Takahashi identity onto them.

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Just consider the gauge transformation after Fourier transforming everything. A Fourier transform turns derivatives into momenta, such that we get $$\tilde A_\mu \rightarrow \tilde A_\mu - \frac1e k_\mu \tilde\alpha \;.$$ This mean that only the component parallel to $k_\mu$ (the longitudinal one) will change, while the ...

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The Standard Model Yukawa interactions must be $SU(3)\times SU(2) \times U(1)_Y$ gauge invariant. The down-type Yukawa interaction is $$\mathcal{L} \supset -y_d \bar Q \phi d_R + \text{h.c.}.$$ This is indeed gauge invariant. The $\bar Q d_R$ form a colour singlet ($3^* \times 3$), the $\bar Q \phi$ form an $SU(2)$ singlet ($2^*\times2)$, and the whole ...

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The reason is that the $SU(2)$ invariant in $\mathbf{2}\otimes\mathbf{2}$ (or in their complex conjugate $\mathbf{2}^*\otimes \mathbf{2}^*$) is given by contracting the two $\mathbf{2}$ with the anti-symmetric $2\times 2$ matrix $\epsilon_{ab}$, as $i\tau_2$ is. In the case at hand the two $\mathbf{2}^*$ are $\bar{Q}$ and the $\Phi^*$. You could form another ...

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