# Lepton flavor violating process on loop level $\mu \rightarrow e \gamma$ via Higgs, Divergent integral

I am stuck with calculating the proces $$\mu \rightarrow e \gamma$$ as in this diagram: I wrote down the matrix element like this: $$\mathcal{M}=\bar{u}(p-q)\left[\int \frac{d^4k}{(2\pi)^4}A\frac{(k+m_\tau)(k'+m_\tau)}{(k'^2-m^2_\tau)(k^2-m^2_\tau)[(p-k)^2-m^2_\tau)]}(e\gamma^\sigma\epsilon_\sigma)\frac{m\tau}{v}\right]u(p)$$

I'm new to calculating loop diagrams so I have trouble with the k integral. I know I have to integrate about the momentum from $$-\infty$$ to $$\infty$$.

$$I=\int \frac{d^4k}{(2\pi)^4}\frac{(k+m_\tau)(k'+m_\tau)}{(k'^2-m^2_\tau)(k^2-m^2_\tau)[(p-k)^2-m^2_\tau)]}$$

With the denominator I have to go through all the standard stuff. Using Feynman parameters $$x,y,z$$, since I have three propagators. Then shifting the integration variable $$\ell=k-xq-zp$$ In the end I get $$2\int_0^1dx dy dz \frac{\delta(x+y+z-1)}{[\ell^2-\Delta+i\epsilon]^3}$$ Now for $$I$$ there are 3 types of integrals the ones with no $$k$$ in the numerator with $$k$$ in the numerator and with $$k^2$$ in the numerator.
The one with no k is working fine so far. I can do a Wick rotation and then integrate in 4 dimensional spherical coordinates in Euklidean space. With a substitution I can simply calculate the integral here...
And when substituting $$k$$ by $$k=\ell+xq+zp$$ I can drop all the terms linear in $$\ell$$ in the numerator since they cancel out due to symetric integration.
Am I correct so far? The main question is: for the integral of the type: $$\int_{-\infty}^\infty d^4\ell \frac{\ell^2}{[\ell^2-\Delta+i\epsilon]^3}$$ If I do the Wick rotation here and with the spherical coordinates I get an extra $$\ell³$$. This integral becomes divergent for large $$\ell$$. As far as I know there should be no infinities here, as compared to the vacuum polarization in QED. Am I missing something? Or how can I calculate this integral?

• Looks like it's just log divergent, have you tried using dim reg? – InertialObserver Mar 21 '19 at 0:12
• Your integral is schematically the same as the calculation of the electron vertex function in QED (anomalous magnetic moment and all that). This is standard and appears in any QFT textbook (for instance Peskin and Schroeder 6.2, 6.3, and the end of 7.2). The log divergence you are seeing is expected. – octonion Mar 21 '19 at 5:34
• @InertialObserver thanks for your comments. So first I have to say I don't really know anything about regularization and that stuff. The proffesor who gave me that task said the calculation should be fine for me as the integrals are finite. So obviously they aren't. – higgshunter Mar 21 '19 at 7:37
• @octonion the anomalous magnetic moment calculation is the only loop diagram I have ever calculated so far... so I will take a second look at it. But my professor told me there shouldn't be any infinities in this calculation. That's why I was confused. And also... Isn't the problem in QED arising because the photon is massless and one puts an extra term in the calculation with a massive photon to cancel the divergency? But here the Higgs boson is massive... But yes I understand that there is a log divergency... So would I just do the same as in the QED calculation here? – higgshunter Mar 21 '19 at 7:43
• It looks like the powers of k all check out and so the naive divergence is log divergence.. the easiest thing to do would just to impose a momentum cutoff, which just means integrating to some scale $\Lambda$ – InertialObserver Mar 21 '19 at 7:45

Ok, I found out that the Integral might be divergent but this divergency will cancel out later in the calculation, so it's not important to calculate. Actually one should not just start computing this diagram straight away but first find out the form of the amplitude.

A good reference for this is Cheng and Li "Gauge theory of elementary particle physics". In section 13.3 they calculate a similar diagram but with neutrino oscillation. The further calculation might be different but the derivation of the form of the amplitude is the same. Thus one finds out that all terms proportional to $$\ell^2$$ in the numerator cancel.

If one wants to know more about it he can also take a look here:

where I explained the steps. I just need help to reproduce them. But this question should be answered by this...

You can use a program I wrote, Package-X for Mathematica, which allows you to directly calculate diagrams like this. By the way, I don't think you wrote down your amplitude correctly.

Use LoopIntegrate in conjunction with FermionLine to initiate the computation of the integral

Then replace the dot products with the relevant on shell conditions (p.p->mμ^2, q.q->0, and p.q->(mμ^2-me^2)/2), and finally apply LoopRefine to obtain an explicit analytic representation of that integral:

However, the original setup of the amplitude (in OP's post) is likely wrong, and therefore this output is not the value of the diagram.