#  Indirect upper limits on $\boldsymbol{\ell_{i}\to\ell_{j}\gamma\gamma}$
from $\boldsymbol{\ell_{i}\to\ell_{j}\gamma}$

Fabiola Fortuna  Departamento de Física, Centro de Investigación y de Estudios
Avanzados del Instituto Politécnico Nacional, Apdo. Postal 14-740, 07000
Ciudad de México, México.  Alejandro Ibarra  Physik-Department, Technische
Universität München, James-Franck-Straße, 85748 Garching, Germany  Xabier
Marcano  Departamento de Física Teórica and Instituto de Física Teórica
UAM/CSIC,
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain  Marcela
Marín  Departamento de Física, Centro de Investigación y de Estudios Avanzados
del Instituto Politécnico Nacional, Apdo. Postal 14-740, 07000 Ciudad de
México, México.  Pablo Roig  Departamento de Física, Centro de Investigación y
de Estudios Avanzados del Instituto Politécnico Nacional, Apdo. Postal 14-740,
07000 Ciudad de México, México.

###### Abstract

We perform an effective field theory analysis to correlate the charged lepton
flavor violating processes $\ell_{i}\to\ell_{j}\gamma\gamma$ and
$\ell_{i}\to\ell_{j}\gamma$. Using the current upper bounds on the rate for
$\ell_{i}\to\ell_{j}\gamma$, we derive model-independent upper limits on the
rates for $\ell_{i}\to\ell_{j}\gamma\gamma$. Our indirect limits are about
three orders of magnitude stronger than the direct bounds from current
searches for $\mu\to e\gamma\gamma$, and four orders of magnitude better than
current bounds for $\tau\to\ell\gamma\gamma$. We also stress the relevance of
Belle II or a Super Tau Charm Facility to discover the rare decay
$\tau\to\ell\gamma\gamma$.

††preprint: TUM-HEP 1422/22††preprint: IFT-UAM/CSIC-22-124

##  I Introduction

The experimental observation of charged lepton flavor violation (cLFV) would
undoubtedly imply the existence of new physics beyond neutrino oscillations
Calibbi and Signorelli (2018). This has motivated a strong experimental
program over the last 75 years searching for different cLFV processes, see
Fig. 1, each observable providing complementary information about possible
beyond the Standard Model (BSM) scenarios.

Here we consider the cLFV decays of leptons to two photons,
$\ell_{i}\to\ell_{j}\gamma\gamma$ Dreitlein and Primakoff (1962); Bowman  _et
al._ (1978); Davidson  _et al._ (2020), which have been explored in less
detail than other cLFV processes such as the single photon process,
$\ell_{i}\to\ell_{j}\gamma$, specially for the case of
$\tau\to\ell\gamma\gamma$ Gemintern  _et al._ (2003); Cordero-Cid  _et al._
(2005); Aranda  _et al._ (2008, 2009); Bryman  _et al._ (2021).

Experimentally, $\mu\to e\gamma\gamma$ was searched for by several experiments
aiming also for $\mu\to e\gamma$. The latest of these experiments was the
Crystal Box detector, whose result still provides the strongest bound for
$\mu\to e\gamma\gamma$ Grosnick  _et al._ (1986). This limit is however two
orders of magnitude weaker than present $\mu\to e\gamma$ bounds, see Table 1,
since the MEG experiment was optimized for back-to-back topologies and no new
dedicated experiment for $\mu\to e\gamma\gamma$ has been carried out since
Crystal Box. On the other hand, $\tau\to\ell\gamma\gamma$ has rarely been
searched for. To the best of our knowledge, the only existing direct
experimental search was performed by ATLAS, setting an upper limit of
BR$(\tau\to\mu\gamma\gamma)<1.5\times 10^{-4}$ after the LHC run-I Angelozzi
(2017). No direct experimental search exists for $\tau\to e\gamma\gamma$.

An alternative for exploring the $\ell_{i}\to\ell_{j}\gamma\gamma$ channels is
to recast the searches for $\ell_{i}\to\ell_{j}\gamma$, as some of the events
of the former would fall into the signal region defined for the latter Bowman
_et al._ (1978). This idea has been recently applied to recast the BABAR
search for $\tau\to\ell\gamma$ Aubert  _et al._ (2010), finding that at 90%CL
BR$(\tau\to\mu\gamma\gamma)<5.8\times 10^{-4}$ and BR$(\tau\to
e\gamma\gamma)<2.5\times 10^{-4}$ Bryman  _et al._ (2021). These limits are
however several orders of magnitude weaker than the associated ones on
$\tau\to\ell\gamma$ due to the low acceptance of these searches for
$\tau\to\ell\gamma\gamma$ events.

Figure 1: Historical evolution for the 90%CL upper limits on several cLFV
leptonic decays. Solid markers correspond to published direct experimental
searches, while empty ones indicate future expected sensitivities at MEG II
Baldini  _et al._ (2018), Mu3e Blondel  _et al._ (2013) and Belle II
Altmannshofer  _et al._ (2019) (arbitrary year). Lighter markers for
$\ell_{i}\to\ell_{j}\gamma\gamma$ were obtained by recasting the available
searches for $\ell_{i}\to\ell_{j}\gamma$ Bowman  _et al._ (1978); Bryman  _et
al._ (2021) or $\mu e\to\gamma\gamma$ Dreitlein and Primakoff (1962). The
evolution for $\tau\to e$ sector is similar to the $\tau\to\mu$ one, with the
exception that no direct search for $\tau\to e\gamma\gamma$ exists.

In this letter, we consider the theoretical correlation between the
$\ell_{i}\to\ell_{j}\gamma\gamma$ and the $\ell_{i}\to\ell_{j}\gamma$ decays.
Clearly, any scenario generating $\ell_{i}\to\ell_{j}\gamma$ would
automatically generate a (model-independent) contribution to
$\ell_{i}\to\ell_{j}\gamma\gamma$, from the radiation of an additional photon
in the final state. Further, any scenario generating
$\ell_{i}\to\ell_{j}\gamma\gamma$ will generate a (model-dependent)
contribution to $\ell_{i}\to\ell_{j}\gamma$ at the quantum level. Barring
cancellations, the quantum-induced contribution should not exceed the
experimental upper limits on $\ell_{i}\to\ell_{j}\gamma$, which in turn allows
to set indirect limits on the rates of $\ell_{i}\to\ell_{j}\gamma\gamma$.

In this work, we will pursue an effective field theory (EFT) approach to study
these correlations, in order to ensure the generality of our conclusions.
Notably, our indirect limits on the process $\ell_{i}\to\ell_{j}\gamma\gamma$
will turn out to be more stringent than the current direct bounds.
Furthermore, these limits do not preclude the possibility of observing the
rare decays $\tau\to\ell\gamma\gamma$ at Belle II or at a Super Tau Charm
Factory, which then represents a competitive probe of cLFV along with the more
studied channels $\tau\to\ell\gamma$ or
$\tau\to\ell_{j}\ell_{k}\bar{\ell}_{k}$.

Decay Mode | Current upper limit on BR (90%CL)
---|---
$\mu\to e\gamma$ | $4.2\times 10^{-13}$ | MEG (2016) Baldini  _et al._ (2016)
$\mu\to e\gamma\gamma$ | $7.2\times 10^{-11}$ | Crystal Box (1986) Grosnick  _et al._ (1986)
$\tau\to e\gamma$ | $3.3\times 10^{-8}$ | BaBar (2010) Aubert  _et al._ (2010)
$\tau\to\mu\gamma$ | $4.2\times 10^{-8}$ | Belle (2021) Abdesselam  _et al._ (2021)
$\tau\to\mu\gamma\gamma$ | $1.5\times 10^{-4}$ | ATLAS (2017) Angelozzi (2017)
Table 1: Experimental upper bounds on the rates of the
$\ell_{i}\to\ell_{j}\gamma(\gamma)$ decays.

##  II Total rate for $\boldsymbol{\ell_{i}\to\ell_{j}\gamma\gamma}$ in the
EFT approach

The effective interaction Lagrangian between two charged leptons of different
flavor and one photon has dimension 5 and reads:

$\mathcal{L}_{\text{dim-5}}=D_{R}^{ij}\,\bar{\ell}_{L_{i}}\sigma_{\mu\nu}\ell_{R_{j}}F^{\mu\nu}+D_{L}^{ij}\,\bar{\ell}_{R_{i}}\sigma_{\mu\nu}\ell_{L_{j}}F^{\mu\nu}+h.c.$
(1)

where the subscripts $L(R)$ indicate the chirality of the lepton and $i,j$ are
generation indices.

On the other hand, the lowest dimensional effective interaction between two
charged leptons of different flavor and two photons has mass dimension 7 and
is given by Bowman  _et al._ (1978) 0pt

$\displaystyle\mathcal{L}_{\text{dim-7}}=$
$\displaystyle\left(G_{SR}^{\,ij}\bar{\ell}_{L_{i}}\ell_{R_{j}}+G_{SL}^{\,ij}\bar{\ell}_{R_{i}}\ell_{L_{j}}\right)F_{\mu\nu}F^{\mu\nu}$
$\displaystyle+$
$\displaystyle\left(\tilde{G}_{SR}^{\,ij}\bar{\ell}_{L_{i}}\ell_{R_{j}}+\tilde{G}_{SL}^{\,ij}\bar{\ell}_{R_{i}}\ell_{L_{j}}\right)\tilde{F}_{\mu\nu}F^{\mu\nu}+h.c.$
(2)

where
$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\lambda}F^{\sigma\lambda}$
is the dual tensor. In Ref. Bowman  _et al._ (1978), the Lagrangian also
contains the dimension-8 operators
$\bar{\ell}_{L_{i}}\gamma^{\sigma}\ell_{L_{j}}F^{\mu\nu}\partial_{\nu}F_{\mu\sigma}$
and
$\bar{\ell}_{L_{i}}\gamma^{\sigma}\ell_{L_{j}}F^{\mu\nu}\partial_{\nu}\tilde{F}_{\mu\sigma}$,
as well as the analogous operators for the right-handed fermions. The effect
of these operators is not only suppressed by higher powers of the cut-off
scale of the EFT, but also by the mass of the decaying lepton, due to the
helicity flip, therefore we will neglect them henceforth.

The effective Lagrangians in Eqs. (1) and (II) generate at tree level the
decay $\ell_{i}\to\ell_{j}\gamma\gamma$, through the diagrams shown in Fig. 2.
The expression for the differential decay rate is complicated and is given in
Appendix A. For the specific limits where the rate is dominated by the
dimension-5 operators, the total decay rate is given by:

$\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)=\frac{\alpha\,m_{i}^{3}}{48\pi^{2}}\Big{(}|D^{ij}_{R}|^{2}+|D^{ij}_{L}|^{2}\Big{)}\,\lambda\left(\tfrac{E_{\gamma}^{\rm
cut}}{m_{i}}\right)\,,$ (3)

with $E_{\gamma}^{\rm cut}$ an energy cut-off introduced to regularize the
infrared and collinear divergences in the rate (see App. A), and

$\displaystyle\lambda(x)$ $\displaystyle\simeq
6+2\pi^{2}+6\log^{2}2+21\log(2x)+6\log(x)\,\log(4x)$
$\displaystyle+18x\big{(}2\log(2x)+1\big{)}+6x^{2}\big{(}8\log(2x)-29\big{)}$
$\displaystyle+\mathcal{O}(x^{3})\,.$ (4)

On the other hand, when it is dominated by the dimension-7 operators, we
obtain

$\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)=\frac{|G_{ij}|^{2}}{3840\pi^{3}}\,m_{i}^{7}\,,$
(5)

where we have neglected the mass of the final lepton and
$|G_{ij}|^{2}=|G_{SL}^{\,ij}|^{2}+|G_{SR}^{\,ij}|^{2}+|\tilde{G}_{SL}^{\,ij}|^{2}+|\tilde{G}_{SR}^{\,ij}|^{2}$.

Figure 2: Feynman diagrams contributing to $\ell_{i}\to\ell_{j}\gamma\gamma$
(top panel) and $\ell_{i}\to\ell_{j}\gamma$ (bottom panel) in the effective
field theory described by the Lagrangians in Eq. (1) (black squares) and Eq.
(II) (black circles). The dots represent higher order contributions.

##  III Correlating $\boldsymbol{\ell_{i}\to\ell_{j}\gamma\gamma}$ and
$\boldsymbol{\ell_{i}\to\ell_{j}\gamma}$

Let us first consider scenarios where the dimension-5 operators are not
suppressed, so the rate for $\ell_{i}\rightarrow\ell_{j}\gamma\gamma$ is
approximately given by Eq. (3). Clearly, the dimension-5 operators also induce
the decay $\ell_{i}\rightarrow\ell_{j}\gamma$ (see lower panel of Fig. 2). The
rate is given by

$\Gamma(\ell_{i}\to\ell_{j}\gamma)=\frac{m_{i}^{3}}{4\pi}\left(\big{|}D_{R}^{ij}\big{|}^{2}+\big{|}D_{L}^{ij}\big{|}^{2}\right)\,,$
(6)

whence one obtains

$\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)=\frac{\alpha}{12\pi}\,\lambda\left(\frac{E_{\gamma}^{\rm
cut}}{m_{i}}\right)\,\Gamma(\ell_{i}\to\ell_{j}\gamma)\,.$ (7)

Using the upper limits on the rates for $\ell_{i}\to\ell_{j}\gamma$ from Table
1 and imposing $E_{\gamma}^{\rm cut}=7\,(50)$ MeV for $\mu\,(\tau)$ decays,
one finds the indirect limits:

$\displaystyle{\rm BR}(\mu\to e\gamma\gamma)\lesssim 2\times 10^{-16}\,,$
$\displaystyle{\rm BR}(\tau\to e\gamma\gamma)\lesssim 8\times 10^{-11}\,,$
$\displaystyle{\rm BR}(\tau\to\mu\gamma\gamma)\lesssim 1\times 10^{-10}\,.$
(8)

Alternatively, there could be scenarios where the dimension-5 operators are
suppressed, while not the dimension-7 operators (see Section IV). In this
case, the decay $\ell_{i}\rightarrow\ell_{j}\gamma$ is induced at the one
loop-level (see lower panel of Fig. 2). In this case, and keeping only the
leading terms, one finds:

$\Gamma(\ell_{i}\to\ell_{j}\gamma)\sim\frac{\alpha\,|G_{ij}|^{2}}{256\,\pi^{4}}\,m_{i}^{7}\,\log^{2}\left(\frac{\Lambda^{2}}{m_{i}^{2}}\right)\,,$
(9)

where $\Lambda$ is the cut-off energy scale of the effective field theory.
Using Eq. (5) one obtains an approximate correlation between rates

$\Gamma(\ell_{i}\to\ell_{j}\gamma)\sim\frac{15\alpha}{\pi}\,\log^{2}\left(\frac{\Lambda^{2}}{m_{i}^{2}}\right)\,\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)\,,$
(10)

from where one can derive indirect upper limits for
$\ell_{i}\to\ell_{j}\gamma\gamma$ from the upper limits on
$\ell_{i}\to\ell_{j}\gamma$: 0pt

$\displaystyle{\rm BR}(\mu\to e\gamma\gamma)$ $\displaystyle\lesssim 6.4\times
10^{-14}\Big{[}1+0.15\log\tfrac{\Lambda}{100\,{\rm GeV}}\Big{]}^{-2},$
$\displaystyle{\rm BR}(\tau\to e\gamma\gamma)$ $\displaystyle\lesssim
1.5\times 10^{-8}\,\,\Big{[}1+0.25\log\tfrac{\Lambda}{100\,{\rm
GeV}}\Big{]}^{-2},$ $\displaystyle{\rm BR}(\tau\to\mu\gamma\gamma)$
$\displaystyle\lesssim 1.9\times
10^{-8}\,\,\Big{[}1+0.25\log\tfrac{\Lambda}{100\,{\rm GeV}}\Big{]}^{-2},$ (11)

which have a mild sensitivity to the cut-off scale.

Regardless of the underlying physics generating the process
$\ell_{i}\to\ell_{j}\gamma\gamma$, our indirect limits are significantly more
stringent than the current direct limits. Concretely, the limit on $\mu\to
e\gamma\gamma$ is about three orders of magnitude stronger than the direct
search using the Crystal Box detector and the limit on
$\tau\to\mu\gamma\gamma$ is about four orders of magnitude stronger than the
direct search performed at ATLAS. Future foreseeable sensitivities of MEG II
searching for $\mu\to e\gamma$ and of Belle II for $\tau\to\ell\gamma$ will
improve our indirect limits by about one order of magnitude.

Furthermore, the results in Eqs. (11) motivate a dedicated experimental search
for the $\tau\to\ell\gamma\gamma$ decays, since this decay might be at the
reach of future experiments Banerjee  _et al._ (2022). Let us consider the
specific case of the Belle II experiment. Assuming that Belle II could achieve
the same sensitivity for double than for single photon processes, as occurred
in the Crystal Box Detector for muon decays Bolton  _et al._ (1988), Belle II
could probe the $\tau\to\ell\gamma\gamma$ decays with branching ratios as
small as $\mathcal{O}(10^{-9})$ Banerjee  _et al._ (2022). If this sensitivity
is reached, Belle II will probe uncharted parameter space of the dimension-7
operators, and possibly find evidence for cLFV by the observation of the decay
$\tau\rightarrow\ell\gamma\gamma$.

##  IV Models with enhanced $\boldsymbol{\ell_{i}\to\ell_{j}\gamma\gamma}$

There are several scenarios where the dimension-5 operators could be
suppressed with respect to the dimension-7 operators, thereby enhancing the
rate of $\ell_{i}\to\ell_{j}\gamma\gamma$ compared to
$\ell_{i}\to\ell_{j}\gamma$. For instance, it was argued in Bowman  _et al._
(1978) that in models where cLFV was mediated by charged heavy leptons Wilczek
and Zee (1977), the GIM suppression Glashow  _et al._ (1970) could be stronger
for $\ell_{i}\to\ell_{j}\gamma$ than for $\ell_{i}\to\ell_{j}\gamma\gamma$.
Also, in Ref. Davidson  _et al._ (2020) it was argued that in some new physics
models the dimension-7 operators could actually arise at
$\mathcal{O}(1/\Lambda^{2})$, instead of the naive expectation of
$\mathcal{O}(1/\Lambda^{3})$, so their contributions would be less suppressed
than expected.

Another interesting possibility arises in models where the cLFV is mediated by
heavy scalars, such as a two Higgs doublet model (2HDM) with off-diagonal
Yukawa interactions. In this scenario, $\ell_{i}\to\ell_{j}\gamma$ decays are
induced at one-loop level, however they are suppressed by three chiral flips
and therefore the two-loop (Barr-Zee diagrams) contributions are actually the
dominant ones Hisano  _et al._ (2011). On the other hand, the
$\ell_{i}\to\ell_{j}\gamma\gamma$ decays do not suffer from this chirality
suppression, the dominant contributions are at the one-loop and, consequently,
they can have ratios comparable to those of $\ell_{i}\to\ell_{j}\gamma$.

More concretely, one can consider a scenario containing a heavy scalar $S$
with off-diagonal Yukawa couplings to leptons and an effective vertex to two
photons (which matches to the framework in Ref. Hisano  _et al._ (2011) when
integrating out the top and $W$ boson). The double and single photon decays
are then generated by diagrams such as those in Fig. 3, which in the heavy
scalar limit reduce to a local interaction and Fig. 2, respectively. We have
explicitly checked that, in this heavy limit, we recover our EFT result of Eq.
(10) with $\Lambda=m_{S}$.

Figure 3: Example of diagrams generating $\ell_{i}\to\ell_{j}\gamma\gamma$ and
$\ell_{i}\to\ell_{j}\gamma$ mediated by a scalar with off-diagonal Yukawa
couplings and an effective vertex to two photons.

##  V Conclusions

Pursuing an effective field theory approach, we have derived model-independent
upper limits on the rates of $\ell_{i}\to\ell_{j}\gamma\gamma$ from the
current experimental limits on $\ell_{i}\to\ell_{j}\gamma$. Our indirect
limits are, even under the most conservative assumptions, significantly more
stringent than the current direct limits, concretely three orders of magnitude
for $\mu\rightarrow e\gamma\gamma$ and four orders of magnitude for
$\tau\rightarrow\mu\gamma\gamma$. When $\ell_{i}\to\ell_{j}\gamma\gamma$ is
dominantly generated by the same dimension-5 operators generating
$\ell_{i}\to\ell_{j}\gamma$, the stringent limits on the latter preclude the
observation of the former in planned experiments. In contrast, in scenarios
where the dimension-5 operators are suppressed compared to the dimension-7
operators, the rare decays $\ell_{i}\to\ell_{j}\gamma\gamma$ can be enhanced
compared to $\ell_{i}\to\ell_{j}\gamma$, which in our EFT approach is only
generated at the one-loop level. In this class of scenarios, the rare decay
$\tau\rightarrow\mu\gamma\gamma$ could be at the reach of the Belle II
experiment or at a proposed Super Tau Charm Facility, and could constitute an
stringent probe of lepton flavor violation. We also discussed some possible
UV-complete scenarios where $\ell_{i}\to\ell_{j}\gamma\gamma$ is enhanced.

###### Acknowledgements.

F.F., M.M. and P.R. are grateful to Eduard de la Cruz, Iván Heredia and Alex
Miranda for discussions. This work has been supported by the Collaborative
Research Center SFB1258, by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2094 -
390783311, and by the Science and Technology Facilities Council [grant number
ST/T004169/1]. F.F. and M.M. acknowledge Conacyt scholarships and P.R. is
grateful to funding from ‘Cátedras Marcos Moshinsky’ (Fundación Marcos
Moshinsky) and ‘Paradigmas y Controversias de la Ciencia 2022’ (project number
319395, Conacyt), which supported M.M. X.M. acknowledges funding from the
European Union’s Horizon Europe Programme under the Marie Skłodowska-Curie
grant agreement no. 101066105-PheNUmenal.

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##  Appendix A Differential decay rates

In this appendix we present the double differential decay width for the
$\ell_{i}\to\ell_{j}\gamma\gamma$ processes, including the contributions from
effective operators up to dim-7. The differential rate can be cast as:

$\displaystyle\frac{\differential^{2}\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)}{\differential
E_{\gamma}\differential
E_{\gamma^{\prime}}}=\frac{\differential^{2}\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)}{\differential
E_{\gamma}\differential E_{\gamma^{\prime}}}\Big{|}_{\rm
dim-5}+\frac{\differential^{2}\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)}{\differential
E_{\gamma}\differential E_{\gamma^{\prime}}}\Big{|}_{\rm
dim-7}+\frac{\differential^{2}\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)}{\differential
E_{\gamma}\differential E_{\gamma^{\prime}}}\Big{|}_{\rm int}\,,$ (12)

where in an obvious notation, dim-5 denotes the contribution from the
Lagrangian Eq. (1), dim-7 from the Lagrangian Eq. (II), and int is the
interference term. Explicitly, and neglecting the mass of the lepton in the
final state, we obtain:

$\displaystyle\frac{\differential^{2}\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)}{\differential
E_{\gamma}\differential E_{\gamma^{\prime}}}\Big{|}_{\rm dim-7}$
$\displaystyle=\frac{\big{|}G_{ij}\big{|}^{2}}{16\pi^{3}}\,m_{i}^{2}\big{(}m_{i}-E_{\gamma}-E_{\gamma^{\prime}}\big{)}\big{(}m_{i}-2(E_{\gamma}+E_{\gamma^{\prime}})\big{)}^{2}\,,$
(13)
$\displaystyle\frac{\differential^{2}\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)}{\differential
E_{\gamma}\differential E_{\gamma^{\prime}}}\Bigg{|}_{\rm dim-5}$
$\displaystyle=\frac{\alpha\left(|D^{ij}_{R}|^{2}+|D^{ij}_{L}|^{2}\right)}{4E_{\gamma}E_{\gamma^{\prime}}+m_{i}^{2}-2m_{i}(E_{\gamma}+E_{\gamma^{\prime}})}\frac{m_{i}-2(E_{\gamma}+E_{\gamma^{\prime}})}{4\pi^{2}E_{\gamma}^{2}E_{\gamma^{\prime}}^{2}}\bigg{\\{}48E_{\gamma}^{3}E_{\gamma^{\prime}}^{3}-m_{i}^{4}\big{(}E_{\gamma}-E_{\gamma^{\prime}}\big{)}^{2}$
$\displaystyle+2E_{\gamma}E_{\gamma^{\prime}}m_{i}^{2}\left(E_{\gamma}^{2}+6E_{\gamma}E_{\gamma^{\prime}}+E_{\gamma^{\prime}}^{2}\right)-E_{\gamma}E_{\gamma^{\prime}}m_{i}(E_{\gamma}+E_{\gamma^{\prime}})\left(24E_{\gamma}E_{\gamma^{\prime}}+m_{i}^{2}\right)\bigg{\\}}\,,$
(14)
$\displaystyle\frac{\differential^{2}\Gamma(\ell_{i}\to\ell_{j}\gamma\gamma)}{\differential
E_{\gamma}\differential E_{\gamma^{\prime}}}\Bigg{|}_{\rm int}$
$\displaystyle=-\frac{e\,m_{i}\big{(}m_{i}-2(E_{\gamma}+E_{\gamma^{\prime}})\big{)}^{2}}{4\pi^{3}}\real\\{D^{ij*}_{L}\left({G}^{ij}_{SL}+i\tilde{G}^{ij}_{SL}\right)+D^{ij*}_{R}\left(G^{ij}_{SR}-i\tilde{G}^{ij}_{SR}\right)\bigg{\\}}\,,$
(15)

with kinematical ranges for the photon energies $0\leq E_{\gamma}\leq
m_{i}/2$, $m_{i}/2-E_{\gamma}\leq E_{\gamma^{\prime}}\leq m_{i}/2$. Notice
that the dimension-5 contribution suffers from both infrared and collinear
singularities, which we can avoid by introducing a regulator such that
$E_{\gamma}^{\rm cut}\leq E_{\gamma}\leq m_{i}/2-E_{\gamma}^{\rm cut}$,
$m/2-E_{\gamma}\leq E_{\gamma^{\prime}}\leq m_{i}/2-E_{\gamma}^{\rm cut}$.
Strictly, one should introduce different regulators, however we assume them to
be the same for simplicity.