In the book “Gauge theory of elementary particles-Cheng and Li, section 13.3, the $\mu\rightarrow e\gamma$ decay amplitude is calculated in the $R_{\xi}$ gauge. Regarding this derivation, I'm stuck with a couple of questions.
(i) I think this calculation would have been easier in the unitary gauge since in that case we will not have loops with unphysical Higgs. I can't understand why is it not calculated in the unitary gauge. I think, this amplitude should give finite and identical result in unitary gauge as well. Isn't it?
Are there any other useful references where diagrams such as $\mu\rightarrow e\gamma$, $\mu\rightarrow e\nu\bar{\nu}$ etc are computed in reasonable detail?
(ii) Although it is explained in one statement, it is not clear to me how the contributions from four diagrams of figure 13.6(e) cancel from similar terms appearing in diagrams 13.6(a)-(d).
(iii)Why did they say that “we need to concentrate on the $p\cdot \epsilon$ term in Eqn. 13.79. What did they left out the $\gamma\cdot\epsilon$ term? My guess is that it is the $\gamma\cdot\epsilon$ term from the diagrams 13.6(a)-(d), is cancelled by contributions from the four diagrams of 13.6(e). But is this cancellation so trivial to see that one need not calculate them at all?
EDIT: The Lorentz invariant transition amplitude is proportional to $T(\mu\rightarrow e\gamma)\sim \bar{u}_e\sigma_{\mu\nu}(A+B\gamma_5)u_\mu$. Then he argues that neglecting the electron mass i.e., $m_e=0$ leads to $A\approx B$. How does that work out? In another account, when we do not throw the electron mass the amplitude is proportional to $T(\mu\rightarrow e\gamma)\sim \bar{u_{e}} i \sigma_{\mu\nu}q^\nu[m_e(1-\gamma_5)+m_\mu(1+\gamma_5)]u_\mu$. With $m_e=0$, though, the Lorentz invariant transition amplitude from Cheng and Li can be reproduced. I couldn't prove the second proportionality either.
I used the fact that the operators $\sigma_{\mu\nu}$ and $\sigma_{\mu\nu}\gamma_5$ connects the left-chiral state with the right-chiral one i.e., $\bar{\psi}\sigma_{\mu\nu}\psi=\overline{\psi_R}\sigma_{\mu\nu}\psi_L+\overline{\psi_L}\sigma_{\mu\nu}\psi_R$ and same for $\sigma_{\mu\nu}\gamma_5$. Using these I proceeded upto the step $$T(\mu\rightarrow e\gamma)\sim A\overline{u_{eL}}\sigma_{\mu\nu}u_{\mu R}+A\overline{u_{eR}}\sigma_{\mu\nu}u_{\mu L}+B\overline{u_{eL}}\sigma_{\mu\nu}\gamma_5 u_{\mu R}+B\overline{u_{eR}}\sigma_{\mu\nu}\gamma_5 u_{\mu L}$$
But it is not clear to me that how this to reduce to an expression of the form $T(\mu\rightarrow e\gamma)\sim i \sigma_{\mu\nu}q^\nu[m_e(1-\gamma_5)+m_\mu(1+\gamma_5)]$ or in the $m_e=0$ limit $T(\mu\rightarrow e\gamma)\sim \bar{u_{e}}i \sigma_{\mu\nu}q^\nu[m_\mu(1+\gamma_5)]u_\mu$?