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Cheng and Li's "Gauge Theory of Elementary Particle Physics" section $13.3$, "$ \mu \to e \gamma,$ an example of $R_\xi$-gauge loop calculations" is a popular question to have issues on this site but I haven't seen this particular issue addressed:

In diagram (a), for instance, the loop momentum integral is power-counting nonrenormalizable: after rewriting the neutrino propagator as $$\frac{i}{\gamma \cdot(p+k)-m_i} = \frac{i(\gamma\cdot(p+k)+m_i)}{(p+k)^2-m_i^2}$$ we count $2$ powers of $k$ in the numerator and $6$ in the denominator giving a total of $-4$. Integrating against the $3$ powers in $d^4k$ gives a logarithmic divergence then. Cheng and Li get past this using the GIM Mechanism $$\sum_i U_{ei}^* U_{\mu i} = 0$$ to eliminate one of the $\sim 1/k^2$ factors in favor of $m_i^2 / [(p+k)^2]^2 \sim 1/k^4$, making the overall dimension $-6$ which is perfectly fine to apply Feynman parameters against and integrate convergently etc etc.

What confuses me is (13.84): $$\sum_i \left[ \frac{U_{ei}^* U_{\mu i}}{(p+k)^2 - m_i^2 }\right] = \sum_i U_{ei}^* U_{\mu i} \left[ \underbrace{\frac{1}{(p+k)^2}}_{\text{cancels by GIM}} + \frac{m_i^2}{[(p+k)^2]^2} + \cdots\right] $$ $$ = \sum_i \frac{U_{ei}^* U_{\mu i}m_i^2}{[(p+k)^2]^2} + \cdots$$

It seems like they've just done a simple Taylor Expansion here, using $m_i^2/(p+k)^2 << 1$, but where $k$ is being integrated over this seems completely unreasonable. Have they done some clever abbreviated dimensional regularization here, or is this a mistake? Applying this same (dubious) technique gives agreeing expressions for the rest of the diagrams as well.

I've had trouble finding another reference corroborating the Br$(\mu \to e \gamma) < 10^{-40}$ value either, so if someone has one handy that would be appreciated. Thanks for bearing with me, any help is appreciated!

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  • $\begingroup$ I don't have the book at hand to give a thorough answer. But let me say that a more rigorous way to do it is to subtract from the integrand $U_{ei}^*U_{\mu i}/(p+k)^2$ (since it's zero) and then observe that in the UV the resulting integrand is well behaved. The reason is the same as the Pauli-Villars regularization. Then you may try to compute it exactly. I also looked for the branching ratio you asked and I didn't find the same value (by a lot!). Page 2 pdg.lbl.gov/2018/tables/rpp2018-sum-leptons.pdf. $\endgroup$ – MannyC Dec 22 '18 at 3:35
  • $\begingroup$ The PDG branching ratio refers to the experimental bounds on the process. The one in question here is the theoretical prediction based on the incorporation of neutrino mixing from the PMNS matrix, and the bound comes from experimental bounds on neutrino masses. $\endgroup$ – user241891 Jan 19 '19 at 18:44
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The Taylor expansion is completely reasonable. Shifting $p+k\mapsto k$ the propagator is $$\frac{\gamma_\mu k^\mu}{k^2-m_i^2}$$ (we ignored the $m_i$ in the numerator as its contribution is zero because of the spinor structure). The Taylor expansion is valid for $k>m_i$. Since $m_i$ is expected to be less than $1\operatorname{eV}$, it's reasonable to discard that part of the integral when $k<m_i$.

If I approximate $\gamma_\mu k^\mu$ as $k$ and replace the W-boson propagators with $1/M_W^2$, then $$\int_0^{2m_i}\frac{k^4dk}{k^2-m_i^2} = 1/3 m_i^3 (14 - 3 \coth^{-1}(2))\approx 4m_i^3$$ according to Wolfram alpha.

Thus ignoring the $\frac{1}{M_W^4}\bar{u}P_R (p\cdot\epsilon) u$, comparing the two contributions, Cheng & Li have something like $$\sum_i|U_{ei}^2|m_i^2m_\mu$$ and our estimate for the correction to the Taylor series is $$\sum_i|U_{ei}^2|m_i^3,$$ which is an order-$m_i/m_\mu\sim 10^{-8}$ correction.

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