# New Limits on Exotic Spin-Dependent Interactions at Astronomical Distances

L. Y. Wu  Key Laboratory of Neutron Physics, Institute of Nuclear Physics and
Chemistry, CAEP, Mianyang 621900, Sichuan, China  Institute of Nuclear Physics
and Chemistry, CAEP, Mianyang 621900, Sichuan, China  K. Y. Zhang  Key
Laboratory of Neutron Physics, Institute of Nuclear Physics and Chemistry,
CAEP, Mianyang 621900, Sichuan, China  Institute of Nuclear Physics and
Chemistry, CAEP, Mianyang 621900, Sichuan, China  M. Peng  Key Laboratory of
Neutron Physics, Institute of Nuclear Physics and Chemistry, CAEP, Mianyang
621900, Sichuan, China  Institute of Nuclear Physics and Chemistry, CAEP,
Mianyang 621900, Sichuan, China  J. Gong<EMAIL_ADDRESS>Key Laboratory
of Neutron Physics, Institute of Nuclear Physics and Chemistry, CAEP, Mianyang
621900, Sichuan, China  Institute of Nuclear Physics and Chemistry, CAEP,
Mianyang 621900, Sichuan, China  H. Yan<EMAIL_ADDRESS>Key Laboratory of
Neutron Physics, Institute of Nuclear Physics and Chemistry, CAEP, Mianyang
621900, Sichuan, China  Institute of Nuclear Physics and Chemistry, CAEP,
Mianyang 621900, Sichuan, China

###### Abstract

Exotic spin-dependent interactions involving new light particles address key
questions in modern physics. Interactions between polarized neutrons ($n$) and
unpolarized nucleons ($N$) occur in three forms:
$g_{S}^{N}g_{P}^{n}\bm{\sigma}\cdot\bm{r}$,
$g_{V}^{N}g_{A}^{n}\bm{\sigma}\cdot\bm{v}$, and
$g_{A}^{N}g_{A}^{n}\bm{\sigma}\cdot\bm{v}\times\bm{r}$, where $\bm{\sigma}$ is
the spin and $g$s are the corresponding coupling constants for scalar (S),
pseudoscalar (P), vector (V), and axial-vector (A) vertexes. If such
interactions exist, the Sun and Moon could induce sidereal variations of
effective fields in laboratories. By analyzing existing data from laboratory
measurements on Lorentz and CPT violation, we derived new experimental upper
limits on these exotic spin-dependent interactions at astronomical ranges. Our
limits on $g_{S}^{N}g_{P}^{n}$ surpass the previous combined astrophysical-
laboratory limits, setting the most stringent experimental constraints to
date. We also report new constraints on vector-axial-vector and axial-axial-
vector interactions at astronomical scales, with vector-axial-vector limits
improved by $\sim 12$ orders of magnitude. We extend our analysis to Hari-Dass
interactions and obtain new constraints.

Introduction.$-$ Axions, predicted by the PQ (Peccei-Quinn) mechanism [1, 2,
3], can induce spin-dependent interactions [4]. The originally proposed axions
were quickly ruled out due to the broken energy scale of the electroweak
scale; however, new models with higher broken energy scales were subsequently
proposed. Axions can have arbitrarily small mass and weak couplings to
ordinary matter because the scale at which the PQ symmetry is broken can be
arbitrarily large [5]. Thus, axions might mediate interactions in ranges from
nanometers to astronomical distances. Though the PQ mechanism was originally
proposed to solve the strong CP problem, the axions, which are light, weakly
interacting, and pseudoscalar, are also considered as possible candidates for
cold dark matter. New interactions might also be mediated by vector particles
such as the para-photon (dark, hidden, heavy or secluded photon) [6, 7],
$Z^{\prime}$ boson [8], graviphoton [9], etc., or even unparticles [10].
Reference [11] proposed 16 different types of new interactions, 15 of which
are spin-dependent. Non-Yukawa exotic interactions due to the dark or hidden
sector were also proposed recently [12, 13]. As early as 1980, Fayet [14, 15]
pointed out that the new U(1) vector bosons,characterized by small masses and
weak couplings to ordinary matter, could be generated through the spontaneous
breaking of supersymmetric theories. Searching for the new interactions
mediated by the new particles is related to the strong CP problem, dark
matter, dark energy, and finding evidence of supersymmetry, which is among the
most important unsolved problems in modern physics [16].

Particles with similar properties as axions, predicted by various generalized
theories, are usually called ALPs (Axion Like Particles) [8]. If exist, ALPs
($\phi$) can generate a new interaction through the coupling to a fermion
$\psi$, $\mathcal{L}_{\phi}=\bar{\psi}(g_{S}+ig_{P}\gamma_{5})\psi\phi$, where
$g_{S}$ ($g_{P}$) is the scalar (pseudoscalar) coupling constant [4]. The SP
(scalar-pseudoscalar) interaction between a polarized neutron and an
unpolarized nucleon can be expressed as

$V_{\rm{SP}}=\frac{\hbar^{2}g^{N}_{S}g^{n}_{P}}{8\pi m_{n}}(\frac{1}{\lambda
r}+\frac{1}{r^{2}}){\rm{exp}}(-r/\lambda)\bm{\sigma}\cdot\hat{r},$ (1)

where $\lambda=\hbar/m_{\phi}c$ is the interaction range, $m_{\phi}$ is the
mediator mass, $\bm{\sigma}$ is the spin operator of the polarized neutron,
$m_{n}$ is the neutron mass, $N$ ($n$) represents the nucleon (neutron), and
$r$ is the distance between the two interacting particles. This interaction
could also be generated from
$\mathcal{L}{\phi}=g_{S}\bar{\psi}\psi\phi+g_{P}/(2m)\bar{\psi}\gamma_{\mu}\gamma_{5}\psi\partial^{\mu}\phi$
[4, 17, 18], which includes a derivative term. Such a term appears in the
axion models [19]. Notably, the SP interaction generated from the derivative
coupling between axions and fermions is discussed in Ref. [20]. Thus, studying
this interaction can be used to investigate not only ALPs but also axions.
Recently, the SP interaction has begun to attract more attention [21, 22, 23,
24, 25, 26, 27]. For example, Wei _et al_. [28] proposed a laboratory
experiment scheme that could surpass the astrophysical limit for the SP
interaction. Laboratory limits become closer and closer to the limits derived
by combining $g_{S}^{N}$ from the torsion balance experiment and $g_{P}^{n}$
from SN1987A; however, all the existing limits are less stringent than the
combined astrophysical-laboratory ones at astronomical distances.

VA (vector-axial-vector) and AA (axial-axial-vector) interactions can be
derived from a general Lagrangian
$\mathcal{L}_{X}=X_{\mu}\bar{\psi}(g_{V}\gamma^{\mu}+g_{A}\gamma_{5}\gamma^{\mu})\psi$
in the nonrelativistic limit, where $X$ is a new vector particle and $g_{V}$
($g_{A}$) is the vector (axial-vector) coupling constant. The interactions
between a polarized neutron and an unpolarized nucleon can be expressed as

$\displaystyle V_{\rm{VA}}$ $\displaystyle=$ $\displaystyle\frac{\hbar
g^{N}_{V}g^{n}_{A}}{2\pi}\frac{{\rm{exp}}(-r/\lambda)}{r}\bm{\sigma}\cdot\bm{v},$
(2) $\displaystyle V_{\rm{AA}}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}g^{N}_{A}g^{n}_{A}}{16\pi
m_{n}c}(\frac{1}{\lambda
r}+\frac{1}{r^{2}}){\rm{exp}}(-r/\lambda)\bm{\sigma}\cdot(\bm{v}\times\hat{r}),$
(3)

where $\bm{v}$ is their relative velocity. Many studies have been carried out
to look for the new interactions, detecting either the macroscopic forces or
the torques exerted on the polarized probe spins. For example, Leslie et al.
[29] proposed experimental schemes to detect the new spin-dependent
interaction between a polarized source and a mechanical oscillator. For
another example, Ding et al. [30] used a micro-fabricated magnetic structure
as a polarized source, and then tried to detect the AA interaction at the
range of $\sim\mu$m sensed by a gold-sphere-cantilever. Many groups have been
searching for the new interaction through its rotating effects as a effective
magnetic field on the polarized spin. The VA and AA interactions between
different combinations of fermions have been investigated already, such as
electron-nucleon [31, 32, 33, 30], neutron-nucleon [34, 35, 36], electron-
electron [37], and electron-antiproton [38]. Studies on these exotic
interactions involving muons were performed very recently [39].

Since the signal induced by new interactions is tiny, using a large mass
source and modulating the signal to a high frequency are crucial for the
detection. A mass source with vast constituents could make the signal
measurable. By modulating the signal, on the one hand, the SNR (Signal-to-
Noise Ratio) can be increased with the decrease of the noise bandwidth. On the
other hand, the $1/f$ noise can be significantly reduced at high frequencies.
In this Letter, treating the Sun and the Moon as mass sources and using the
Earth’s rotation as a modulation, we obtain new limits on exotic spin-
dependent SP (1), VA (2), and AA (3) interactions at astronomical distances.

The basic idea.$-$ All three types of spin-dependent interactions are in the
form of $\bm{s}\cdot\bm{B}^{\prime}$, where $\bm{B}^{\prime}$ can be viewed as
a kind of effective magnetic field. Searching for these interactions becomes a
problem probing the effective field acting on polarized spins.

Figure 1: (a) The Sun-centered frame. The relative size of the Sun and the
Earth is not to scale. (b) The Earth-based frame. We take $\hat{z}$ along with
the Earth’s rotation axis. The angle between the ecliptic plane and the
Earth’s equatorial plane is $\eta=23.4^{\circ}$. The red arrows represent the
directions of effective fields of three types of spin-dependent interactions
generated by the Sun’s nucleons.

We first illustrate the basic idea using the Sun as the source mass. In the
Sun-centered frame shown in Fig. 1 (a), the aforementioned new interactions
can generate effective magnetic fields at the Earth center as [40]

$\begin{split}\bm{B}^{\prime}_{\rm{SP}}&=\frac{\hbar
g^{N}_{S}g^{n}_{P}N_{\odot}}{4\pi m_{n}\gamma_{n}}(\frac{1}{\lambda
R}+\frac{1}{R^{2}}){\rm{exp}}(-R/\lambda)[\cos{(\Omega_{\oplus}t)}\hat{X}+\sin{(\Omega_{\oplus}t)}\hat{Y}],\\\
\bm{B}^{\prime}_{\rm{VA}}&=\frac{g^{N}_{V}g^{n}_{A}N_{\odot}}{\pi\gamma_{n}}\frac{{\rm{exp}}(-R/\lambda)}{R}[-\Omega_{\oplus}R\sin{(\Omega_{\oplus}t)}\hat{X}+\Omega_{\oplus}R\cos{(\Omega_{\oplus}t)}\hat{Y}],\\\
\bm{B}^{\prime}_{\rm{AA}}&=-\frac{\hbar g^{N}_{A}g^{n}_{A}N_{\odot}}{8\pi
m_{n}c\gamma_{n}}(\frac{1}{\lambda
R}+\frac{1}{R^{2}}){\rm{exp}}(-R/\lambda)\Omega_{\oplus}R\hat{Z},\end{split}$
(4)

where $R$ is the distance from the Earth to the Sun, $\gamma_{n}$ is the
gyromagnetic ratio of the neutron, $\Omega_{\oplus}$ is the Earth’s orbital
angular frequency, and $N_{\odot}$ is the total nucleon number of the Sun. For
a laboratory frame on the Earth as shown in Fig. 1 (b), the effects of the
Earth’s rotation can be taken into account via the Euler rotations. We first
rotate the frame by the angle $\omega_{\oplus}t$ about the $\hat{Z}$ axis
where $\omega_{\oplus}$ is the Earth’s rotation frequency, and then rotate
about the $\hat{Y}$ axis by an angle $\eta$ which is the Earth’s obliquity. In
the laboratory frame, we will observe effective time-varying fields as

$\displaystyle\bm{b}_{\rm{SP}}=\frac{\hbar g^{N}_{S}g^{n}_{P}N_{\odot}}{4\pi
m_{n}\gamma_{n}}(\frac{1}{\lambda
R}+\frac{1}{R^{2}}){\rm{exp}}(-R/\lambda)\Bigg{[}\begin{array}[]{c}\cos{\eta}\cos{(\Omega_{\oplus}t)}\cos{(\omega_{\oplus}t)}+\sin{(\Omega_{\oplus}t)}\sin{(\omega_{\oplus}t)}\\\
-\cos{\eta}\cos{(\Omega_{\oplus}t)}\sin{(\omega_{\oplus}t)}+\sin{(\Omega_{\oplus}t)}\cos{(\omega_{\oplus}t)}\\\
\sin{\eta}\cos{(\Omega_{\oplus}t)}\end{array}\Bigg{]},$ (8)
$\displaystyle\bm{b}_{\rm{VA}}=\frac{g^{N}_{V}g^{n}_{A}N_{\odot}}{\pi\gamma_{n}}\frac{{\rm{exp}}(-R/\lambda)}{R}v_{\oplus}\Bigg{[}\begin{array}[]{c}-\cos{\eta}\cos{(\omega_{\oplus}t)}\sin{(\Omega_{\oplus}t)}+\sin{(\omega_{\oplus}t)}\cos{(\Omega_{\oplus}t)}\\\
\cos{\eta}\sin{(\omega_{\oplus}t)}\sin{(\Omega_{\oplus}t)}+\cos{(\omega_{\oplus}t)}\cos{(\Omega_{\oplus}t)}\\\
-\sin{\eta}\sin{(\Omega_{\oplus}t)}\end{array}\Bigg{]},$ (12)
$\displaystyle\bm{b}_{\rm{AA}}=\frac{\hbar g^{N}_{A}g^{n}_{A}N_{\odot}}{8\pi
m_{n}c\gamma_{n}}(\frac{1}{\lambda
R}+\frac{1}{R^{2}}){\rm{exp}}(-R/\lambda)v_{\oplus}\Bigg{[}\begin{array}[]{c}\sin{\eta}\cos{(\omega_{\oplus}t)}\\\
-\sin{\eta}\sin{(\omega_{\oplus}t)}\\\ -\cos{\eta}\end{array}\Bigg{]},$ (16)

where $v_{\oplus}=\Omega_{\oplus}R$ is the orbital speed of the Earth. As the
most straightforward case, $\bm{b}_{\rm{AA}}$ clearly shows effective magnetic
fields rotating in the laboratory frame at the Earth’s rotation frequency.
Although $\bm{b}_{\rm{SP}}$ and $\bm{b}_{\rm{VA}}$ appear more complicated due
to their mixture with the Earth’s orbital rotation, the situation can be
greatly simplified by considering the fact that
$\omega_{\oplus}\gg\Omega_{\oplus}$.

In summary, if the exotic spin-dependent interactions exist, the perpendicular
components of the effective fields induced by the Sun are modulated by the
Earth’s rotations; thus, we could observe its signal in the laboratory.
Although the frequency of the Earth’s rotation is not high, its frequency
modulation effect on the nuclear precession in the comagnetometer makes
precision measurements on these new interactions possible.

Constraining the exotic spin-dependent interactions at astronomical
distances.$-$Dual-species comagnetometers are convenient for detecting the
tiny signals caused by new spin-dependent interactions, since the two
components occupy the same space and common-mode background field noise can be
mostly canceled. In principle, we can separate the sidereal modulated signal
from the noisy background during precise measurements. The ultrahigh
sensitivity of the comagnetometer to the magnetic field changes has an
extensive implementation in new physics detection, including electric dipole
moments, CPT and Lorentz violation, spin-gravity interaction, and so on [41,
42, 43, 44]. This comagnetometer method has been used to search for the
constant cosmic background field due to Lorentz violation by detecting the
sidereal variants of the field observed in the laboratory frame on the Earth
in Refs. [45, 46], where a 129Xe+3He comagnetometer and a K+3He one were
respectively adopted. Stringent constraints on the components of the constant
field perpendicular to the Earth’s rotation axis at a similar level were
obtained.

We find that the limits on exotic spin-dependent interactions induced by the
Sun can be obtained by using the Lorentz violation searching results. For
example, for the experiment described in Ref. [45], the $\Omega_{\oplus}t$ in
Eqs. (8), (12) and (16) approximately equals to $\pi/2$, given that the
experiment was performed for $\sim$10 days when the Earth was around the
vernal equinox. The sidereal oscillating effective field $\bm{b}_{\perp}$
perpendicular to the Earth’s rotation axis can be detected as [40]

$\displaystyle\bm{b}_{\rm{SP}\perp}=$ $\displaystyle\frac{\hbar
g^{N}_{S}g^{n}_{P}N_{\odot}}{4\pi m_{n}\gamma_{n}}(\frac{1}{\lambda
R}+\frac{1}{R^{2}})\exp{(-R/\lambda)}$ (17)
$\displaystyle[\sin{(\omega_{\oplus}t)}\hat{x}+\cos{(\omega_{\oplus}t)}\hat{y}],$
$\displaystyle\bm{b}_{\rm{VA\perp}}=$
$\displaystyle-\frac{g^{N}_{V}g^{n}_{A}N_{\odot}}{\pi\gamma_{n}}\frac{{\rm{exp}}(-R/\lambda)}{R}v_{\oplus}\cos{\eta}$
$\displaystyle[-\cos{(\omega_{\oplus}t)}\hat{x}+\sin{(\omega_{\oplus}t)}\hat{y}],$
$\displaystyle\bm{b}_{\rm{AA\perp}}=$ $\displaystyle\frac{\hbar
g^{N}_{A}g^{n}_{A}N_{\odot}}{8\pi m_{n}c\gamma_{n}}(\frac{1}{\lambda
R}+\frac{1}{R^{2}}){\rm{exp}}(-R/\lambda)v_{\oplus}\sin{\eta}$
$\displaystyle[\cos{(\omega_{\oplus}t)}\hat{x}-\sin{(\omega_{\oplus}t)}\hat{y}].$

Using the result of Ref. [45], at the $95\%$ confidence level ($95\%$ C.L.),
we could derive

$|\bm{b}_{\perp}|<0.023~{}\rm{fT}.$ (18)

Plugging in all the known parameters such as $\eta=23.4^{\circ}$,
$N_{\odot}\approx 1.2\times 10^{57}$, $R=1.5\times 10^{11}$ m, and
$v_{\oplus}\approx 3.0\times 10^{4}$ m/s, we can obtain the constraints on the
SP (1), VA (2), and AA (3) interactions between polarized neutrons and
unpolarized nucleons.

Figure 2: Constraint to the coupling constant product $|g^{N}_{S}g^{n}_{P}|$
as a function of the interaction range $\lambda$ (the mediator mass). The
solid lines are the result of this work; the left line uses the Moon as the
source, and the right line uses the Sun. The dashed line is the result of
Refs. [47, 18], which was derived by combining $g_{S}^{N}$ of weak equivalence
with $g_{P}^{n}$ from SN1987A. The dark grey area is excluded by the result of
this work and the light gray one is excluded by the result of Refs. [47, 18].

The derived constraint on $|g_{S}^{N}g_{P}^{n}|$ is shown in Fig. 2. For
$\lambda\gtrsim 2\times 10^{10}$ m, it gives the most stringent limit on
$|g_{S}^{N}g_{P}^{n}|$. For $\lambda>10^{12}$ m, our bounds
$|g_{S}^{N}g_{P}^{n}|<1.6\times 10^{-35}$ ($95\%$ C. L.). Previously, the most
stringent constraints on $|g_{S}^{N}g_{P}^{n}|$ were astrophysical-laboratory
limits that combined astrophysical constraints on $g_{P}^{n}$ from SN1987A
with the laboratory ones on $g_{S}^{N}$ from the weak equivalence principle
experiment. This work improves the existing upper bound by as much as $\sim$70
times. In particular, the new limits on the scalar-pseudoscalar coupling
combination exceed the combined astrophysical-laboratory limits for the first
time.

Figure 3: Constraint to the coupling constant product $|g^{N}_{V}g^{n}_{A}|$
as a function of the interaction range $\lambda$ (the mediator mass). The
solid lines are the result of this work; the left line uses the Moon as the
source, and the right line uses the Sun. The dashed line is the result of Ref.
[36]. The dark grey area is excluded by the result of this work and the light
gray one is excluded by the result of Ref. [36].

Our constraint on $|g^{N}_{V}g^{n}_{A}|$ is shown in Fig. 3. For
$\lambda\gtrsim 3\times 10^{7}$ m, it gives the most stringent limit on
$|g^{N}_{V}g^{n}_{A}|$. For $\lambda>10^{12}$ m, our bounds
$|g^{N}_{V}g^{n}_{A}|<7.1\times 10^{-59}$ ($95\%$ C. L.). Previously, the most
stringent constraint on $|g^{N}_{V}g^{n}_{A}|$ near the interaction range
under consideration was given in Ref. [36]. If the previous result can be
extended to the range of $\sim 10^{12}$ m, the present work improves the
existing upper bound by as much as 12 orders of magnitude.

Figure 4: Constraint to the coupling constant product $|g^{N}_{A}g^{n}_{A}|$
as a function of the interaction range $\lambda$ (the mediator mass). The
solid lines are the result of this work; the left line uses the Moon as the
source, and the right line uses the Sun. The dark grey area is excluded by the
result of this work.

The obtained constraint on $|g_{A}^{N}g_{A}^{n}|$ is shown in Fig. 4. For
$\lambda\gtrsim 10^{7}$ m, it gives the first limit on $|g_{A}^{N}g_{A}^{n}|$.
For $\lambda>10^{12}$ m, our bounds $|g_{A}^{N}g_{A}^{n}|<8.1\times
10^{-31}\rm{(95\%~{}{\mathrm{C}.L.})}$. It is the only known constraint on
$|g_{A}^{N}g_{A}^{n}|$ to us at the astronomical ranges.

We can apply the same analyzing method by using the Moon as a source. In this
case, we shall consider errors due to several systematic effects. We show
details of the analysis in the Supplementary Material [40]. We also plotted
the results using the Moon in Figs. 2, 3, and 4.

Furthermore, the spin-gravity interaction proposed by Leitner and Okubo [48]
and later generalized by Hari Dass [49] can also be strictly constrained.
Assuming CPT invariance, two types of discrete symmetry violation spin-gravity
potentials are constructed as

$V(r)=\frac{G_{N}M\hbar}{2}(\alpha_{1}\frac{\bm{\sigma}\cdot\hat{r}}{cr^{2}}+\alpha_{2}\frac{\bm{\sigma}\cdot\bm{v}}{c^{2}r^{2}}),$
(19)

where $G_{N}$ is the Newton constant of gravitation, $M$ is the mass of the
gravity source, and $\alpha_{1}$ ($\alpha_{2}$) is a dimensionless constant.
These potentials are the starting point of many low-energy experiments [50,
51]. They also provide a direct way to test symmetry violation and the
equivalence principle in General Relativity [49, 52]. Using the Sun as the
mass source, we derive the limits on $\alpha$ as

$\begin{split}&|\alpha_{1}|<2.2\times
10^{2}~{}\rm{(95\%~{}{\mathrm{C}.L.})},\\\ &|\alpha_{2}|<2.4\times
10^{6}~{}\rm{(95\%~{}{\mathrm{C}.L.})}.\end{split}$ (20)

When comparing with the results of Ref. [53], our limit on $\alpha_{1}$
improves the existing one by $\sim$11 times, and on $\alpha_{2}$ we get an
improvement of $\sim$4 orders of magnitude.

Conclusion and discussion.$-$By using the Sun and the Moon as sources, the
Earth’s rotation as modulation, and the existing laboratory limits on the
Lorentz and CPT violation at distances of astronomical scales, we have
constrained three types of possible new interactions between polarized
neutrons and unpolarized nucleons. We derived new limits on the SP interaction
with ranges from $\sim 2\times 10^{10}$ m to $\sim 10^{14}$ m. At the distance
of $\sim 10^{12}$ m, the limit is improved by $\sim$70 times compared to the
previous combined astrophysical-laboratory limit. This result is the first
time the limits from a single laboratory experiment exceed the combined
astrophysical-laboratory ones for the SP interaction. We obtained new limits
on the VA interaction with ranges from $\sim 3\times 10^{7}$ m to $\sim
10^{14}$ m. At the distance of $\sim 10^{12}$ m, the limit is improved by
$\sim$12 orders of magnitude in comparison with the previous result of 3He
spin relaxation experiment. We derived the first limits on the AA interaction
with ranges from $\sim 10^{7}$ m to $\sim 10^{14}$ m. We also constrained the
Hari-Dass spin-dependent interactions and obtained new limits on them.

How can we extend the current work to include other particles, such as
electrons and muons? One possibility is to employ the beam method proposed in
Refs. [54, 39], which uses superconducting magnetic shielding to create a
region with zero background field. By directing spin-polarized particle beams
through this region and detecting sidereal variations in polarization along
the direction perpendicular to the Earth’s rotation axis, we can investigate
spin-dependent new interactions induced by the Sun for the probing electrons,
muons, etc.

Note added.$-$After this work was submitted, Ref. [55] reports improved limits
on $g_{S}^{N}g_{P}^{n}$ in the interaction range of $10^{6}\sim
10^{10}~{}\mathrm{m}$, and Ref. [56] reports similar limits on
$g_{A}^{N}g_{A}^{n}$ in the interaction range of $\sim 10^{12}~{}\mathrm{m}$.

###### Acknowledgements.

We acknowledge supports from the National Natural Science Foundation of China
under grants U2230207 and U2030209. This work was also supported by the
National Key Program for Research and Development of China under grants
2020YFA0406001 and 2020YFA0406002. We thank Dr. C. Fu and Y. M. Ma for their
helpful discussions.

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IFT-UAM/CSIC-23-49

Lepton Flavor Violation from diphoton effective interactions

Fabiola<EMAIL_ADDRESS>Xabier
<EMAIL_ADDRESS>Marcela
<EMAIL_ADDRESS>and Pablo
<EMAIL_ADDRESS>
aDepartamento de Física, Centro de Investigación y de Estudios Avanzados del
Instituto Politécnico Nacional, Apdo. Postal 14-740, 07000 Ciudad de México,
México.

bDepartamento de Física Teórica and Instituto de Física Teórica UAM/CSIC,
Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

We consider charged lepton flavor violating transitions mediated by the
diphoton effective interactions $\ell_{i}\ell_{j}\gamma\gamma$ and explore
which processes can probe them better. Our analysis includes single and double
radiative decays, $\ell_{i}\to\ell_{j}\gamma(\gamma)$, as well as
$\ell_{i}\to\ell_{j}$ conversions in nuclei for all possible flavor
combinations, which we compute for the first time for $\ell\to\tau$
conversions in this framework. We find that currently the best limits are
provided by the loop-induced $\ell_{i}\to\ell_{j}\gamma$ processes, while the
best future sensitivities come from $\mu\to e$ conversion in aluminum and from
potential $\tau\to\ell\gamma\gamma$ searches at Belle II or at the Super Tau
Charm Facility. We also motivate the search for $\mu\to e\gamma\gamma$ at the
Mu3e experiment as a complementary probe of these operators.

##  1 Introduction

The discovery of neutrino oscillations [1, 2, 3] implies the non-conservation
of lepton flavor and it calls for an extension of the Standard Model to
accommodate non-vanishing neutrino masses. Although no other lepton flavor
violating process has been observed so far, it is natural to query if the
dynamics behind neutrino oscillations can produce charged lepton flavor
violating processes [4, 5, 6].

Experimental searches for charged lepton flavor violation (cLFV) date back to
the late 1940’s, and the upper limits have improved ever since until the
impressive current bounds of $\mathcal{O}(10^{-13})$ for some $\mu\to e$
transitions, with an even more promising future landscape [7, 8]. cLFV
processes involving taus imply a greater experimental challenge [9, 10, 11,
8], but the strong experimental effort has pushed current upper limits to the
$\mathcal{O}(10^{-8})$ level, which will be further improved in the future
[11] at Belle II [12], the Super Tau Charm Factory (STCF) [13], or the Future
Circular Collider [14, 15, 16]. We summarize in Table 1 the current and future
status of some cLFV processes, the most relevant ones for our analysis.

In the literature, we find plenty of models proposed to describe cLFV
interactions. The most popular cLFV processes, both from the experimental and
theoretical sides, are probably the radiative $\ell_{i}\to\ell_{j}\gamma$
decays. Nevertheless, if cLFV was discovered, experimental input on a
multitude of independent processes would assist in discriminating among these
UV completions, which introduces the need to explore also other cLFV
transitions. In this work, we will focus on $\ell_{i}\to\ell_{j}$ conversion
in nuclei and on the double radiative $\ell_{i}\to\ell_{j}\gamma\gamma$
decays.

The $\ell_{i}\to\ell_{j}$ conversions in nuclei are well-motivated scenarios
to study cLFV interactions. The $\mu\to e$ conversion in nuclei has already
been pursued in the past, with the strongest limit given by Sindrum II [17].
Nuclei transitions involving $\tau$ leptons are a bit different since they
actually refer to the conversion of an electron or a muon into a tau via deep
inelastic scattering (DIS) with a nucleus [18, 19]. The relevant quantity in
this case is given by the ratio of the cross-sections of two inclusive
processes:

$\mathcal{R}_{\tau\ell}=\frac{\sigma(\ell\mathcal{N}\to\tau
X)}{\sigma(\ell\mathcal{N}\to\ell X)}\,,$ (1)

where the denominator is given by the dominant contribution to the inclusive
$\ell+\mathcal{N}$ process as a result of the lepton bremsstrahlung on nuclei
[20]. At present, there are no experimental limits on these transitions,
however, the NA64 experiment at the CERN SPS plans to search for them [20],
with an expected sensitivity of
$\mathcal{R}_{\tau\ell}\sim[10^{-13},10^{-12}]$. In addition to the NA64,
future foreseen experiments such as the muon collider [21], the electron-ion
collider (EIC) [22], the international linear collider (ILC) [23] or circular
colliders as LHeC [24] might search for this conversion. All these searches
for $\ell\to\tau$ transitions will have a relevant impact on different models
or effective approaches [18, 19, 25, 26, 27, 28, 29, 30, 31].

cLFV obs. | Current upper limit (90%CL) | Expected future limits
---|---|---
$\mu\to e\gamma$ | $4.2\times 10^{-13}$ | MEG (2016) [32] | $6\times 10^{-14}$ | MEG-II [33]
$\mu\to e\gamma\gamma$ | $7.2\times 10^{-11}$ | Crystal Box (1986) [34] | — | —
$\mu A\to eA$ | $7\times 10^{-13}$ | Sindrum II (2006) [17] | $6.2\times 10^{-16}$ | Mu2e [35]
$\tau\to e\gamma$ | $3.3\times 10^{-8}$ | BaBar (2010) [36] | $9\times 10^{-9}$ | Belle-II [11]
$\tau\to e\gamma\gamma$ | $2.5\times 10^{-4}$ | Bryman et al. (2021) [37] | — | —
$\tau\to\mu\gamma$ | $4.2\times 10^{-8}$ | Belle (2021) [38] | $6.9\times 10^{-9}$ | Belle-II [11]
$\tau\to\mu\gamma\gamma$ | $1.5\times 10^{-4}$ | ATLAS (2017) [39] | — | —
$\ell\mathcal{N}\to\tau X$ | — | — | $[10^{-13},10^{-12}]$ | NA64 [20]
Table 1: Experimental upper bounds and future expected sensitivities for the
set of cLFV transitions relevant to our analysis. In the process $\mu A\to eA$
conversion, A=Au for the current upper limit, while A=Al for the expected
future limit. Note there are also promising sensitivities at STCF for LFV
$\tau$ decays [13], nevertheless, we will mostly use Belle-II numbers for the
future, as they are expected to be released sooner.

On the other hand, the $\ell_{i}\to\ell_{j}\gamma\gamma$ decays are a good
example of less considered but interesting processes [40, 41, 42, 43, 44, 45,
37, 46]. Compared to $\ell_{i}\to\ell_{j}\gamma$, the double photon emission
has the advantage of not being chirally suppressed and, from an effective
field theory (EFT) point of view, it is sensitive to a different set of
operators. Experimentally, $\mu\to e\gamma\gamma$ has been searched for by
several experiments, and the best upper limits are still provided by the
Crystal Box detector [34]. The $\tau\to\mu\gamma\gamma$ decays have rarely
been searched for, with the only existing direct experimental search being
performed by ATLAS [39]. No direct search for $\tau\to e\gamma\gamma$ has been
carried out. Nevertheless, experimental upper bounds can be obtained by
recasting the searches for $\tau\to e\gamma$ [37]. Moreover, following an EFT
analysis, stronger upper limits can be derived, as we recently shown in Ref.
[46] and further detailed in this work. For the future, we are unaware of any
concrete plans for new $\ell_{i}\to\ell_{j}\gamma\gamma$ searches.
Nonetheless, they could –in principle– be searched for at future facilities
such as Mu3e [47] for $\mu\to e\gamma\gamma$ and Belle II or the STCF for
$\tau\to\ell\gamma\gamma$. Since they are 3-body decays, and given the
experimental similarities between photons and electrons, we could naively
expect similar sensitivities than for $\mu\to eee$ and $\tau\to eee$, which
are expected to reach the $10^{-16}$ and $10^{-9}$ level111Furthermore, the
prospects for STCF with 10 ab-1 reach sensitivities of $\mathcal{O}(10^{-10})$
for LFV $\tau$ decays [13]., respectively.

Motivated by these cLFV processes, we are interested in the less commonly
considered contact interactions involving two leptons of different flavor and
two photons [40]. Such interactions could mediate various cLFV processes, in
particular, the $\ell_{i}\to\ell_{j}\gamma(\gamma)$ decays and
$\ell_{i}\to\ell_{j}$ conversion in nuclei, as has been studied in the
literature. In particular, Davidson et al. [45] have addressed the role of
these interactions in both $\mu\to e\gamma\gamma$ and $\mu\to e$ conversion in
nuclei222See also, e.g. Refs. [48, 49, 50, 51, 52, 53]., showing that the
latter can probe these diphoton operators more efficiently than the former. It
is natural then to wonder if the same is true for $\tau-\ell$ transitions,
although no such study exists. On the other hand, and as already mentioned,
loop-induced $\ell_{i}\to\ell_{j}\gamma$ transitions have been computed in
Ref. [46], showing a much better sensitivity for these diphoton interactions.

Following all this previous discussion, our goal is to perform a systematic
study of the reach of each of these kinds of processes to probe the diphoton
cLFV interactions. More precisely, we would extend previous works by first
computing the $\ell\to\tau$ conversion in nuclei mediated by these diphoton
interactions, to then conclude on the most promising observables both at the
current and future experimental landscape.

The paper is organized as follows: in section 2 we present the diphoton
effective operators that generate cLFV and describe the processes mediated by
them. This includes in particular our new results for $\ell\to\tau$ conversion
in nuclei, although some details are left for Appendix A. Section 3 is devoted
to our numerical analysis and the discussion about the best observable to
study the diphoton cLFV operators. Finally, we conclude in section 4.

##  2 cLFV form diphoton effective interactions

We are interested in the less explored local interactions between two charged
leptons of different flavor and two photons. The lowest-dimension low-energy
effective Lagrangian that describes the local interaction
$\bar{\ell}_{i}\ell_{j}\gamma\gamma$, where $\ell_{i},\ell_{j}=e,\mu,\tau$,
has energy dimension 7 and is given by [40]

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

where the subscripts $L(R)$ stand for the left(right)-handed chirality of the
lepton and
$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\sigma\lambda}F^{\sigma\lambda}$
is the dual electromagnetic field-strength tensor. Notice that Ref. [40] also
considered the dim-8 operator
$\bar{\ell}_{L_{i}}\gamma^{\sigma}\ell_{L_{j}}F^{\mu\nu}\partial_{\nu}F_{\mu\sigma}$,
as well as the analogous operators for the right-handed fermions.
Nevertheless, we do not include their contribution since it is suppressed by
higher powers of the cut-off scale of the EFT.

This kind of interaction leads to cLFV processes such as
$\ell_{i}\to\ell_{j}\gamma(\gamma)$ and $\ell_{i}\to\ell_{j}$ conversion in
nuclei, as we detail in the following. They also lead to other cLFV processes
including the 3-body $\ell_{i}\to\ell_{j}\bar{\ell_{k}}\ell_{k}$ decays or
semileptonic tau decays $\tau^{-}\to\ell^{-}PP$, where $PP$ are a pair of
light pseudoscalar mesons (which could be detected as a vector resonance).
Nevertheless, we do not expect them to be competitive with respect to the
other decays in restricting the couplings of Eq. (2), and we will therefore
not discuss them here.

Notice that these processes involving two photons can also be induced by the
dim-5 interaction involving just one photon, with the second photon being
emitted from either of the charged particles in the process. Nevertheless,
these dim-5 interactions are already severely constrained by
$\ell_{i}\to\ell_{j}\gamma$ and we do not expect to learn anything new by
considering the $\alpha$-suppressed processes with two photons (see for
instance Ref. [46] for a more quantitative discussion). For all these reasons,
in the following, we will focus only on the phenomenology associated with the
dim-7 Lagrangian in Eq. (2).

###  2.1 $\boldsymbol{\ell_{i}\to\ell_{j}\gamma\gamma}$

At tree level, the effective Lagrangian in Eq. (2) generates the decay
$\ell_{i}\to\ell_{j}\gamma\gamma$, as it was first computed by Bowman et al
[40]. In our analysis, we will only need the total decay rate, which is given
by

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

where we have neglected the mass of the final lepton and $G_{ij}$ is defined
as

$|G_{ij}|^{2}=|G_{SL}^{\,ij}|^{2}+|G_{SR}^{\,ij}|^{2}+|\tilde{G}_{SL}^{\,ij}|^{2}+|\tilde{G}_{SR}^{\,ij}|^{2}.$
(4)

Note that the rate in Eq. (3) seems to imply a non-decoupling behavior, as it
appears to be independent of the new physics. The reason is that the effective
couplings in $G_{ij}$ are defined as dimensionful couplings, as can be seen in
the Lagrangian in Eq. (2). Therefore, they have an implicit dependence on the
cut-off scale of $1/\Lambda^{3}$, which ensures the decoupling behavior of
this and the rest of the transition rates explored in this work.

###  2.2 $\boldsymbol{\ell_{i}\to\ell_{j}\gamma}$

Figure 1: One-loop diagrams generating $\ell_{i}\to\ell_{j}\gamma$ from the
effective dim-7 diphoton operators in Eq. (2), represented as black circles.

At one-loop level, the Lagrangian in Eq. (2) also generates the single photon
decay $\ell_{i}\to\ell_{j}\gamma$. The diagrams, obtained after closing the
loop with one of the photons, are shown in Fig. 1 and were computed first in
Ref. [46]. For our numerical analysis, it will be enough to consider only the
leading-log contributions to the decay rate, which are given by

$\Gamma(\ell_{i}\to\ell_{j}\gamma)\sim\frac{\alpha|G_{ij}|^{2}}{256\pi^{4}}m_{i}^{7}\log^{2}\left(\frac{\Lambda^{2}}{m_{i}^{2}}\right)\,,$
(5)

where the mass of the final lepton has been again neglected, $\Lambda$ is the
cut-off energy scale associated with our EFT framework, and $G_{ij}$ is
defined in Eq. (4). Notice again the apparent non-decoupling behavior, this
time log-enhanced, which is again compensated by the $1/\Lambda^{3}$ scaling
of the dimensionful couplings $G_{ij}$.

###  2.3 $\boldsymbol{\ell_{i}\to\ell_{j}}$ conversion in nuclei

As shown by Davidson et al., the effective cLFV interaction with two photons
also leads to relevant contributions to $\mu\to e$ conversion in nuclei. The
computation is explained minutely in Ref. [45], so we only collect the
relevant contributions here. On the other hand, we extend this idea to flavor
transitions with taus, computing the contributions from the Lagrangian in Eq.
(2) to $\ell\to\tau$ conversion in nuclei.

There are two main contributions to the $\mu\to e$ conversion in nuclei [45].
One is the interaction of the leptons with the classical electromagnetic
field, and the other is a short-distance loop interaction of two photons with
individual protons. The former arises from a contact $\mu e\gamma\gamma$
interaction at momentum transfers of order $m_{\mu}$. The latter stems from
the loop mixing of the $\bar{e}\mu F_{\mu\nu}F^{\mu\nu}$ operator into the
scalar proton operator $(\bar{e}P_{X}\mu)(\bar{p}p)$ ($X=L/R$), where overlap
integrals, energy ratios, and numerical factors overcompensate for the naive
expectation of a loop suppression. Furthermore, the operators with
$F_{\mu\nu}\tilde{F}^{\mu\nu}$ in Eq. (2) are proportional to
$\vec{E}\cdot\vec{B}$, which is negligibly small in the nucleus and can be
therefore disregarded. Then, the conversion rate (CR) in a nucleus $A$ is
given by [45]

${\rm CR}(\mu A\to eA)=\frac{4m_{\mu}^{5}}{\Gamma_{\rm cap}}|\hat{G}_{\mu
e}|^{2}\left|m_{\mu}F_{A}+\frac{18\,\alpha\,m_{p}}{\pi}S_{A}^{(p)}\right|^{2}\,,$
(6)

where $\Gamma_{\rm cap}$ is the muon capture rate on nucleus $A$ [54] and
$F_{A}/S_{A}^{(p)}$ are overlap integrals that can be found in Refs. [45] and
[48], respectively. Notice that we defined a new effective coupling
$|\hat{G}_{\mu e}|^{2}\equiv|G_{SL}^{\mu e}|^{2}+|G_{SR}^{\mu e}|^{2}$ to
emphasize that the contribution from $F_{\mu\nu}\tilde{F}^{\mu\nu}$ terms is
negligible.

The $\mu\to e$ conversion experiments are typically low-energy processes where
the muon becomes bounded before decaying in orbit or being captured by the
nucleus [55]. By contrast, the $\ell\to\tau$ conversion experiments are based
on a fixed-target nucleus hit by an incoming electron or muon beam. The
conversion is expected to occur by DIS of the lepton off the nucleus, meaning
that the energy is high enough as to break the nucleons within the nucleus and
interact with its partons, i.e., quarks and gluons [20]. Therefore, we focus
on the inclusive process, whose products of interaction are a $\tau$ lepton
plus any hadrons, i.e., $\ell+\mathcal{N}(A,Z)\to\tau+X$, where we do not have
any information about $X$.

Husek et al. [28] performed an analysis of LFV tau decays and $\ell\to\tau$
conversion in nuclei using SMEFT operators up to dimension 6. Here we follow
the same methodology to compute the total cross-section in the process
$\ell+\mathcal{N}\to\tau+X$ coming from the Lagrangian in Eq. (2).

The dynamics of the interacting parton living in the hadronic environment of
the nucleus is influenced by low-energy non-perturbative QCD effects. Given
that the perturbative cross-sections are calculated within the framework of
perturbation theory, this computation includes a characteristic energy scale,
$Q^{2}$, at which perturbative and non-perturbative effects are factorized.
The non-perturbative behavior is encoded in the so-called parton distribution
functions (PDFs). Both the PDFs and the perturbative cross-sections are
functions of $Q^{2}$, typically related to the transferred momentum $q^{2}$ of
the system as $Q^{2}=-q^{2}$. In addition, the PDFs are also characterized
through the fraction of the nucleus momentum carried by the interacting
parton, $\xi$. Therefore, we express the perturbative cross-section as well as
the non-perturbative PDFs as a function of the two discussed invariant
quantities. Using the QCD factorization theorems, we can obtain the total
cross-section of the process by calculating the convolution of the
perturbative cross-section ($\hat{\sigma}$) with the non-perturbative PDFs
($f$):

$\sigma_{\ell-\tau}=\hat{\sigma}(\xi,Q^{2})\otimes f(\xi,Q^{2})\,.$ (7)

The evolution of the PDFs in terms of $Q^{2}$ is achieved by using the DGLAP
evolution equations [56, 57, 58], whose dependence on the momentum fraction
$\xi$ is completely non-perturbative and has to be extracted from the data.
Moreover, given that the scattering happens with nucleons bounded in a
nucleus, the non-perturbative effects relevant to describe the $\ell\to\tau$
conversion in nuclei are better captured by the nuclear parton distribution
functions (nPDFs). For this computation we use the nCTEQ15-np fit of the
nPDFs, provided by the nCTEQ15 project [59], and incorporated within the
ManeParse Mathematica package [60].

The contributions to the perturbative cross-sections come from the dimension
seven operators in Eq. (2). These bring the following contributions in:

  1. (a).

The process $\ell q\rightarrow\tau q$ (see Fig. 2) involves a loop with a
quark and two photons.

  2. (b).

The same process as in (a), but with antiquarks:
$\ell\bar{q}\rightarrow\tau\bar{q}$. Note that the non-perturbative behavior
of anti-quarks inside the nucleons differs from their opposite-charged
partners, and also the perturbative cross sections of the process are
different from those involving quarks.

Figure 2: One-loop contribution to $\ell q\rightarrow\tau q$ ($\ell=e,\mu$),
stemming from effective $\tau\ell\gamma\gamma$ interactions.

The process $\ell g\to\tau g$ also contributes to the perturbative cross-
sections, although the operators in Eq. (2) generate it at two-loop level. The
diagram for this contribution would be similar to the one in Fig. 2, closing
the quarks line in a loop and adding initial and final gluons coupled to the
quarks. Given that this process has an additional loop suppression compared
with the ones mentioned in (a) and (b), we neglect it.

Accounting for the contributions to the perturbative cross-section previously
mentioned, we obtained the following expression for the unpolarized squared
amplitude as a function of $\xi$ and $Q^{2}$

$\displaystyle\overline{|\mathcal{M}_{qq}(\xi,Q^{2})|^{2}}$
$\displaystyle=~{}6e^{4}\Big{(}|G_{SR}^{\tau\ell}|^{2}+|G_{SL}^{\tau\ell}|^{2}\Big{)}\left[\left(m_{\ell}^{2}+m_{\tau}^{2}+Q^{2}\right)\left(\left(m_{i}+\xi
M\right)^{2}+Q^{2}\right)\right]\Gamma_{qq}(\xi,Q^{2})$
$\displaystyle+\frac{3e^{4}}{2}\left(|\tilde{G}_{SR}^{\tau\ell}|^{2}+|\tilde{G}_{SL}^{\tau\ell}|^{2}\right)\left[\left(m_{\ell}^{2}+m_{\tau}^{2}+Q^{2}\right)\left(\left(m_{i}-\xi
M\right)^{2}+Q^{2}\right)\right]\tilde{\Gamma}_{qq}(\xi,Q^{2})\,,$ (8)

where $\Gamma_{qq}(\xi,Q^{2})$ and $\tilde{\Gamma}_{qq}(\xi,Q^{2})$ result
from the evaluation of the loop in Fig. 2 and are given in Appendix A. Notice
that the interference between $F_{\mu\nu}F^{\mu\nu}$ and
$\tilde{F}_{\mu\nu}F^{\mu\nu}$ term vanishes, while the interference between
$L$ and $R$ operators is chirality suppressed and we therefore neglect it. For
the process with antiquarks, we obtain an identical expression as Eq. (2.3)
but with different loop functions (see Appendix A).

The perturbative unpolarized differential cross sections can be computed from
the squared amplitude in Eq. (2.3), leading to

$\displaystyle\frac{d\hat{\sigma}(\ell\,q_{i}(\xi
P)\rightarrow\tau\,q_{i})}{d\xi dQ^{2}}$
$\displaystyle=\frac{1}{16\pi\lambda(s(\xi),m_{\ell}^{2},m_{i}^{2})}\overline{|\mathcal{M}_{qq}(\xi,Q^{2})|^{2}}\,,$
(9) $\displaystyle\frac{d\hat{\sigma}(\ell\,\bar{q}_{i}(\xi
P)\rightarrow\tau\,\bar{q}_{i})}{d\xi dQ^{2}}$
$\displaystyle=\frac{1}{16\pi\lambda(s(\xi),m_{\ell}^{2},m_{i}^{2})}\overline{|\mathcal{M}_{\bar{q}\bar{q}}(\xi,Q^{2})|^{2}}\,,$
(10)

with $p_{i}=\xi P$ the momentum of the interacting parton, $P$ the total
momentum of the nucleus and the subscript $i$ labeling quark flavor. We also
defined $m_{i}^{2}=\xi^{2}M^{2}$, being $M$ the nucleus mass.
$\lambda(s(\xi),m_{\ell}^{2},m_{i}^{2})$ is the usual Källén function.
Finally, working at leading order in QCD, the total cross-section reads

$\displaystyle\sigma(\ell\mathcal{N}(P)\rightarrow\tau
X)=\sum_{i}\int_{\xi_{\text{min}}}^{1}\int_{Q^{2}_{-}(\xi)}^{Q^{2}_{+}(\xi)}d\xi
dQ^{2}$ $\displaystyle\left\\{\frac{d\hat{\sigma}(\ell\,q_{i}(\xi
P)\rightarrow\tau\,q_{i})}{d\xi\,dQ^{2}}f_{q_{i}}(\xi,Q^{2})\right.$
$\displaystyle+\left.\frac{d\hat{\sigma}(\ell\,\bar{q}_{i}(\xi
P)\rightarrow\tau\,\bar{q}_{i})}{d\xi
dQ^{2}}f_{\bar{q}_{i}}(\xi,Q^{2})\right\\}\,,$ (11)

being $f_{q_{i}}(\xi,Q^{2})$ and $f_{\bar{q}_{i}}(\xi,Q^{2})$ the quark and
antiquark PDFs, respectively. The integration limits can be found in Appendix
E of Ref. [28].

##  3 Results and discussion

As detailed in Section 2, the diphoton effective interactions in Eq. (2)
contribute to $\ell_{i}\to\ell_{j}\gamma$, $\ell_{i}\to\ell_{j}\gamma\gamma$
and $\ell_{i}\to\ell_{j}$ conversion in nuclei. Therefore, it is interesting
to compare the sensitivity of each of these processes as a probe of the
effective couplings $G_{ij}$ or, alternatively, of the new physics scale
associated with these effective interactions, which we can naively define as
$\Lambda=1/\sqrt[3]{|G_{ij}|}$.

The transition rates for (double) radiative decays and $\mu\to e$ conversion
depend on the combination of couplings in Eq. (4) and thus it is
straightforward to translate the experimental upper limits in Table 1 into
upper limits on $G_{ij}$. In the case of $\ell\to\tau$ conversions, however,
the dependence on the effective couplings does not factorize out in the same
manner, see Eq. (2.3), so we need to make some assumption on their structure.
For concreteness, we will consider three benchmark scenarios in our numerical
analysis:

$\displaystyle(i)~{}|G_{\tau\ell}|^{2}$
$\displaystyle=|G_{SR}^{\tau\ell}|^{2}+|G_{SL}^{\tau\ell}|^{2}=|\tilde{G}_{SR}^{\tau\ell}|^{2}+|\tilde{G}_{SL}^{\tau\ell}|^{2}\,,$
$\displaystyle(ii)~{}|G_{\tau\ell}|^{2}$
$\displaystyle=|G_{SR}^{\tau\ell}|^{2}+|G_{SL}^{\tau\ell}|^{2};~{}\tilde{G}_{SR}^{\tau\ell}=\tilde{G}_{SL}^{\tau\ell}=0\,,$
(12) $\displaystyle(iii)~{}|G_{\tau\ell}|^{2}$
$\displaystyle=|\tilde{G}_{SR}^{\tau\ell}|^{2}+|\tilde{G}_{SL}^{\tau\ell}|^{2};~{}G_{SR}^{\tau\ell}=G_{SL}^{\tau\ell}=0\,.$

Furthermore, according to the prospects of the NA64 experiment [20], we use
two specific nuclei in our analysis, Fe(56,26) and Pb(208,82), as well as
$E_{e}=100\,$GeV and $E_{\mu}=150\,$GeV for the energies of the incident
lepton beams. We calculated the integral in Eq. (2.3) using either light
quarks only or light plus heavy quarks. The criterion to choose our results
was the following: if the error of the calculation (due to the PDF
uncertainties) using light quarks only, was larger than the contribution of
the heavy quarks, then we neglected the latter contributions. That is the case
for scenarios $(i)$ and $(iii)$. Conversely, when the contribution of the
heavy quarks was larger than the computation uncertainty, we included them, as
in scenario $(ii)$.

Figure 3: Values of the new physics scale $\Lambda$ that are accessible by
each of the cLFV observables with current bounds (solid bars) and future
sensitivities (lighter bars). Striped bars indicate our estimations for future
sensitivities in $\ell_{i}\to\ell_{j}\gamma\gamma$ decays (see main text for
details). We define $\Lambda\equiv 1/\sqrt[3]{|G_{ij}|}$. For $\ell\to\tau$
conversion in nuclei, we used the benchmark (i) in Eq. (3).

Our results are shown in figure 3, where we present the new physics scale
$\Lambda$ accessible by each of the cLFV observables, with solid (light) bars
covering current (future) experimental sensitivities. We see that at present
the radiative decays $\ell_{i}\to\ell_{j}\gamma$ provide the best
sensitivities for $\ell_{i}\ell_{j}\gamma\gamma$ effective interactions, as
already pointed out in Ref. [46], although $\mu\to e$ conversion is also
competitive. In fact, future reach in the $\mu\to e$ sector will be dominated
by the impressive future sensitivities expected at conversion rate experiments
such as Mu2e, going beyond the expected improvement at MEG-II 333
Additionally, $\mu\to eee$ could also become competitive in the future, since
the improved sensitivity of $\mathcal{O}(10^{-16})$ at Mu3e may overcome the
$\alpha$ suppression with respect to $\mu\to e\gamma$. Nevertheless, we expect
it to still be below the $\mu\to e$ conversion reach.. On the other hand, and
even being a tree-level process in our framework, $\mu\to e\gamma\gamma$
currently provides a much lower sensitivity, since the last search for this
process was done by the Crystal Box detector.

We are not aware of any future plan to search for $\mu\to e\gamma\gamma$.
Nevertheless, Fig. 3 shows how would the future landscape look like assuming
an improvement of one, three and five orders of magnitude in this branching
ratio. We see that an improvement of three orders of magnitude is needed to
reach the current sensitivity levels of $\mu\to e\gamma$ and $\mu\to e$
conversion, while five orders would make this observable quite competitive
with respect to future $\mu\to e$ conversion searches. Interestingly, the
latter corresponds to a sensitivity level of $10^{-16}$, precisely the future
goal for $\mu\to eee$, which motivates a dedicated search for $\mu\to
e\gamma\gamma$ at Mu3e.

The situation in the tau sector is different, especially in the future. So
far, the only direct search for $\tau\to\mu\gamma\gamma$ was performed by
ATLAS [39], although the searches for $\tau\to\ell\gamma$ have been recast
into upper limits on $\tau\to\ell\gamma\gamma$ [37]. Their sensitivity for the
$\tau\ell\gamma\gamma$ operator is, however, much lower than current ones from
$\tau\to\ell\gamma$, which provides, again, the best probe for
$\tau\ell\gamma\gamma$ operators [46]. In the near future, the double photon
channels can be searched for at Belle II or at STCF with improved
sensitivities. In Fig. 3 we display the future landscape assuming that Belle
II can probe $\tau\to\ell\gamma\gamma$ transitions at the
$\mathcal{O}(10^{-9})$ level, as expected for other cLFV tau decays such as
$\tau\to\ell\gamma$ or $\tau\to\ell\ell\ell$. Then, we see that direct
searches for $\tau\to\ell\gamma\gamma$ will improve our current knowledge of
$\tau\ell\gamma\gamma$ operators and that they will be competitive with future
improvements from $\tau\to\ell\gamma$ decays. This is an exciting result that
motivates dedicated searches for $\tau\to\ell\gamma\gamma$ decays at Belle II
and STCF.

Figure 4:  Maximum $\ell\to\tau$ conversion rates assuming the scenarios in
Eq. (3) and imposing the upper limits on $\tau\to\ell\gamma$ decays. The
latter is done following Eq. (2.3) with $\Lambda=100$ GeV. The upper shadowed
region shows the expected future sensitivity at NA64.

On the other hand, $\ell\to\tau$ conversion searches at NA64 will not be
competitive probing $\tau\ell\gamma\gamma$ operators, especially for
$e\to\tau$ conversions. This is manifest in Fig. 3, which was done considering
Pb and scenario $(i)$, and the situation does not improve for other cases we
explored, as shown in Fig. 4. Here, we considered the maximum $G_{\tau\ell}$
allowed by current upper limits on $\tau\to\ell\gamma$ and translated them
into $\ell\to\tau$ conversion rates for both Fe and Pb, and for all the
scenarios in Eq. (3). In other words, this figure shows how large the
conversion rates can be in each scenario without inducing too large rates for
other cLFV transitions. We see that the maximum rates are always far from
future expected sensitivities at NA64. Consequently, and unlike in the $\mu\to
e$ sector, the $\ell\to\tau$ conversion does not provide a competitive probe
for the local $\tau\ell\gamma\gamma$ interaction.

Finally, a word on the validity of our EFT approach is in order. In principle,
we only focused on low-energy processes and, therefore, the choice of the low-
energy EFT Lagrangian in Eq. (2) is well justified. The only exception could
lie in the $\ell\to\tau$ conversion rates, although being a DIS process is not
straightforward to conclude it. The incoming electron (muon) beam has an
energy of 100 (150) GeV, which translates into maximum center-of-mass energy,
$\hat{s}\leq\sqrt{2ME_{\ell}}$, around the electroweak scale. In that sense,
it would be more appropriate to consider a SMEFT approach, where the low-
energy diphoton interactions in Eq. (2) are generated from a $SU(2)_{L}\times
U(1)_{Y}$ invariant operators. Nevertheless, our low-energy analysis showed
that the $\ell\to\tau$ conversions are far from being competitive in probing
the diphoton operators, which will not change if SMEFT diphoton operators are
considered.

Besides, we see in Fig. 3 that the $\Lambda$ scales we are sensitive to are
well above the relevant energy scale of each process 444Again with the only
exception of $e\to\tau$ conversion. However, the sensitivity is so far away
from other $\tau\to e$ transitions, that the discussion about the validity
becomes pointless.. In fact, the accessible $\Lambda$ values are in most cases
higher than the electroweak scale, motivating a dedicated analysis for
diphoton SMEFT operators in cLFV processes, although this goes beyond the
scope of our work.

##  4 Conclusions

Searches for cLFV processes constitute one of our best prospects to discover
new physics, and it is, therefore, important to consider as many cLFV
observables as possible. This will not only enhance our chances of discovering
cLFV transitions, but also provide crucial information about the model behind
a potential positive signal, since each BSM scenario predicts in general
different correlations between these processes.

In this paper, we have focused on the less studied cLFV diphoton interactions
$\ell_{i}\ell_{j}\gamma\gamma$ and performed a phenomenological analysis of
their implications for the cLFV processes $\ell_{i}\to\ell_{j}\gamma$,
$\ell_{i}\to\ell_{j}\gamma\gamma$, and $\ell_{i}\to\ell_{j}$ conversions in
nuclei for all possible lepton flavor combinations. In order to do this, we
presented a computation of the contribution to the $\ell\to\tau$ conversion on
fixed-target nuclei from the interaction $\tau\ell\gamma\gamma$.

By comparing with the current experimental upper limits for these processes,
we showed that at present the strongest limits for these diphoton operators
are provided by the loop-induced $\ell_{i}\to\ell_{j}\gamma$ decays, although
$\mu\to e$ conversion in nuclei also provides competitive bounds. For the
future, the strong improvement at Mu2e experiment for $\mu\to e$ conversion in
nuclei will dominate the sensitivity for $\mu e\gamma\gamma$ interaction. This
is in contrast to the $\tau$ sector, where future NA64 searches for
$\ell\to\tau$ conversion will not be competitive for probing
$\tau\ell\gamma\gamma$ operators, and they will be better explored by
$\tau\to\ell\gamma$ decays.

Nevertheless, we showed that future direct searches for
$\ell_{i}\to\ell_{j}\gamma\gamma$ could be competitive or even provide the
best limits if they could reach the same sensitivity levels as similar
processes such as the 3-body $\ell_{i}\to\ell_{j}\ell_{j}\bar{\ell}_{j}$
decays. This result motivates ongoing or planned experiments to search also
for the diphoton channel, in particular, $\mu\to e\gamma\gamma$ searches at
Mu3e or $\tau\to\ell\gamma\gamma$ at Belle-II, STCF, EIC or FCC.

#### Acknowledgments.

We would like to thank Alejandro Ibarra for his collaboration in the first
stage of the project and for very valuable discussions. This project has
received support through the Grant IFT Centro de Excelencia Severo Ochoa No
CEX2020-001007-S funded by MCIN/AEI/10.13039/501100011033. FF and MM are
indebted to Conacyt for funding. XM acknowledges funding from the European
Union’s Horizon Europe Programme under the Marie Skłodowska-Curie grant
agreement no. 101066105-PheNUmenal. PR is thankful to Cátedras Marcos
Moshinsky (Fundación Marcos Moshinsky) and ‘Paradigmas y Controversias de la
Ciencia 2022’ (project number 319395, Conacyt) for funding.

##  Appendix A Functions from the evaluation of the loops in
$\boldsymbol{\ell\to\tau}$ conversion in nuclei

Here we provide the relevant functions needed to evaluate the perturbative
contribution to $\ell\to\tau$ conversion, as given in Eq. (2.3), which are the
result of computing the loop in Fig. 2. In particular, we define

$\displaystyle\Gamma_{qq}(\xi,Q^{2})$
$\displaystyle=\frac{1}{64\pi^{4}}|\text{F}1(\xi,Q^{2})|^{2}\,,$ (13)
$\displaystyle\tilde{\Gamma}_{qq}(\xi,Q^{2})$
$\displaystyle=\frac{1}{64\pi^{4}}|\text{F}2(\xi,Q^{2})|^{2}\,,$
$\displaystyle\Gamma_{\bar{q}\bar{q}}(\xi,Q^{2})$
$\displaystyle=\frac{1}{64\pi^{4}}|\text{F}3(\xi,Q^{2})|^{2}\,,$
$\displaystyle\tilde{\Gamma}_{\bar{q}\bar{q}}(\xi,Q^{2})$
$\displaystyle=\frac{1}{64\pi^{4}}|\text{F}4(\xi,Q^{2})|^{2}\,,$

with

F1 $\displaystyle=2\big{[}m(Q^{2})+M\xi\big{]}{\bf
B}_{0}(M^{2}\xi^{2};m(Q^{2}),0)+2\big{[}m(Q^{2})+m_{i}\big{]}{\bf
B}_{0}(m_{i}^{2};m(Q^{2}),0)$ $\displaystyle+2\big{[}m(Q^{2})-m_{i}\big{]}{\bf
B}_{0}(-Q^{2};0,0)+2M\xi\,{\bf B}_{1}(M^{2}\xi^{2};m(Q^{2}),0)+2\big{[}M\xi-
m_{i}\big{]}{\bf B}_{1}(-Q^{2};0,0)$ $\displaystyle+2m_{i}\,{\bf
B}_{1}(m_{i}^{2};m(Q^{2}),0)+2\big{[}m^{3}(Q^{2})+Mm_{i}\,m(Q^{2})\xi-M^{2}m_{i}\xi^{2}+m(Q^{2})Q^{2}-Mm_{i}^{2}\xi\big{]}$
$\displaystyle{\bf
C}_{0}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)+2\big{[}m^{2}(Q^{2})-m_{i}^{2}+m_{i}m(Q^{2})-m_{i}M\xi-M^{2}\xi^{2}+Mm(Q^{2})\xi\big{]}$
$\displaystyle\big{(}M\xi\,{\bf
C}_{2}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)+m_{i}\,{\bf
C}_{1}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)\big{)}$
$\displaystyle+m_{i}-4m(Q^{2})+M\xi\,,$ (14) $\displaystyle\text{F}2$
$\displaystyle=-2i\bigg{(}2\big{[}M\xi+m(Q^{2})\big{]}{\bf
B}_{0}(M^{2}\xi^{2};m(Q^{2}),0)+2\big{[}m_{i}+m(Q^{2})\big{]}{\bf
B}_{0}(m_{i}^{2};m(Q^{2}),0)$
$\displaystyle+2\big{[}M\xi+m_{i}-2m(Q^{2})\big{]}{\bf
B}_{0}(-Q^{2};0,0)+2M\xi\,{\bf B}_{1}(M^{2}\xi^{2};m(Q^{2}),0)+2m_{i}\,{\bf
B}_{1}(m_{i}^{2};m(Q^{2}),0)$
$\displaystyle+2\big{[}m_{i}m^{2}(Q^{2})-2m^{3}(Q^{2})+Mm^{2}(Q^{2})\xi-
Mm_{i}^{2}\xi-M^{2}m_{i}\xi^{2}+2Mm_{i}m(Q^{2})+m(Q^{2})Q^{2}\big{]}$
$\displaystyle{\bf
C}_{0}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)+2\big{[}m_{i}^{2}-2m_{i}m(Q^{2})-M^{2}\xi^{2}+2Mm(Q^{2})\xi\big{]}$
$\displaystyle\big{(}m_{i}\,{\bf
C}_{1}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)-M\xi\,{\bf
C}_{2}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)\big{)}$
$\displaystyle-3(m_{i}+M\xi)\bigg{)}\,,$ (15) $\displaystyle\text{F}3$
$\displaystyle=2\big{[}m(Q^{2})-M\xi\big{]}{\bf
B}_{0}(M^{2}\xi^{2};m(Q^{2}),0)+2\big{[}m(Q^{2})-m_{i}\big{]}{\bf
B}_{0}(m_{i}^{2};m(Q^{2}),0)$ $\displaystyle+2\big{[}m(Q^{2})+m_{i}\big{]}{\bf
B}_{0}(-Q^{2};0,0)-2M\xi\,{\bf B}_{1}(M^{2}\xi^{2};m(Q^{2}),0)-2m_{i}\,{\bf
B}_{1}(m_{i}^{2};m(Q^{2}),0)$ $\displaystyle+2\big{[}m_{i}-M\xi\big{]}{\bf
B}_{1}(-Q^{2};0,0)+2\big{[}M^{2}\xi^{2}+m_{i}\big{(}M\xi+m(Q^{2})\big{)}+Mm(Q^{2})\xi+m_{i}^{2}-m^{2}(Q^{2})\big{]}$
$\displaystyle\big{(}M\xi\,{\bf
C}_{2}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)+m_{i}\,{\bf
C}_{1}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)\big{)}$
$\displaystyle+2\big{[}m^{3}(Q^{2})+Mm_{i}m(Q^{2})\xi+M^{2}m_{i}\xi^{2}+m(Q^{2})Q^{2}+Mm_{i}^{2}\xi\big{]}{\bf
C}_{0}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)$
$\displaystyle-\big{(}m_{i}+4m(Q^{2})+M\xi\big{)}\,,$ (16)
$\displaystyle\text{F}4$
$\displaystyle=2i\,\bigg{(}2\big{[}M\xi-m(Q^{2})\big{]}{\bf
B}_{0}(M^{2}\xi^{2};m(Q^{2}),0)+2\big{[}m_{i}-m(Q^{2})\big{]}{\bf
B}_{0}(m_{i}^{2};m(Q^{2}),0)$
$\displaystyle+2\big{[}m_{i}+2m(Q^{2})+M\xi\big{]}{\bf
B}_{0}(-Q^{2};0,0)+2m_{i}\,{\bf B}_{1}(m_{i}^{2};m(Q^{2}),0)+2M\xi\,{\bf
B}_{1}(M^{2}\xi^{2};m(Q^{2}),0)$
$\displaystyle+2\big{[}2m^{3}(Q^{2})+m_{i}m^{2}(Q^{2})+M\xi m^{2}(Q^{2})-M\xi
m_{i}^{2}-M^{2}\xi^{2}m_{i}-2M\xi m_{i}m(Q^{2})-m(Q^{2})Q^{2}\big{]}$
$\displaystyle{\bf
C}_{0}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)+2\big{[}2m_{i}m(Q^{2})-M^{2}\xi^{2}-2M\xi\,m(Q^{2})+m_{i}^{2}\big{]}$
$\displaystyle\big{(}m_{i}\,{\bf
C}_{1}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)-M\xi\,{\bf
C}_{2}(m_{i}^{2},-Q^{2},M^{2}\xi^{2};m(Q^{2}),0,0)\big{)}$
$\displaystyle-3(m_{i}+M\xi)\bigg{)}\,.$ (17)

The function $m(Q^{2})$ represents the running of the quark mass in the loop.
For the computation of the quark masses at different energy scales, we use
RunDec [61]. The notation employed for the Passarino-Veltman loop functions is
standard. We use Package-X [62] to analytically evaluate the loop integrals.
CollierLink extends Package-X so that the Passarino-Veltman functions can be
directly evaluated, using the COLLIER library [63]. Then we use CollierLink to
numerically evaluate our expressions.

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