# IEC Flickermeter Measurement Results for Distorted Modulating Signal while
Supplied with Distorted Voltage ††thanks: This research was funded in whole or
in part by National Science Centre, Poland – 2021/41/N/ST7/00397, the
Foundation for Polish Science (FNP) – Stipend START 45.2021, the Ministry of
Education and Science – 0212/SBAD/0573. For the purpose of Open Access, the
author has applied a CC–BY public copyright licence to any Author Accepted
Manuscript (AAM) version arising from this submission.

Piotr Kuwałek  Institute of Electrical Engineering and Electronics
Poznan University of Technology
Poznan, Poland
<EMAIL_ADDRESS>
###### Abstract

The paper presents IEC flickermeter measurement results for voltage
fluctuations modelled by amplitude modulation of distorted supply voltage. The
supply voltage distortion caused by electronic and power electronic devices in
the “clipped cosine” form is assumed. This type of supply voltage distortion
is a common disturbance in low voltage networks. Several arbitrary distorted
waveforms of the modulating signal with different modulation depth and
modulating frequency up to approx. 1 kHz are selected to determine the
dependence of severity of voltage fluctuation on their shape. The paper mainly
presents the dependence of voltage fluctuation severity with a frequency
greater than 3$\boldsymbol{f}_{c}$, where $\boldsymbol{f}_{c}$ is the power
frequency. The voltage fluctuation severity and the dependencies associated
with it have been determined on the basis of numerical simulation studies and
experimental laboratory tests.

###### Index Terms:

clipping distortion, flicker, flickermeter, power electronic loads, power
quality, voltage distortion, voltage fluctuation

††publicationid: pubid:

©2022 IEEE. Personal use of this material is permitted. Permission from IEEE
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reprinting/republishing this material for advertising or promotional purposes,
creating new collective works, for resale or redistribution to servers or
lists, or reuse of any copyrighted component of this work in other works. DOI:
10.1109/ICHQP53011.2022.9808552

##  I Introduction

The most common power quality disturbances are voltage fluctuations and
voltage distortions [1], especially considering low voltage (LV) networks. In
LV networks, “clipped cosine” voltage distortion [2, 3, 4] most often occurs,
which is the result of power electronic devices effects, in particular input
stages of switching power supplies. Considering the common use of switching
power supplies, the supply voltage in LV networks is almost always distorted
[1]. In addition, voltage fluctuations occur due to load changes in real power
grids. Taking into account indicated cases, voltage fluctuations and voltage
distortions often occur simultaneously. The effect of simultaneous occurrence
of indicated power quality disturbances, can be an increase in the voltage
fluctuation severity caused by disturbing loads changing their state with a
frequency greater than 3$f_{c}$ [5], where $f_{c}$ is the power frequency. It
is worth noting that so far only voltage fluctuations caused by disturbing
sources with a frequency of up to $3f_{c}$ have been considered [6, 7, 8, 9],
because a sinusoidal supply voltage has been considered. However, such (pure
sine) supply voltage rarely occurs in real power grids, especially in LV
networks. The proven possibility of increase in the voltage fluctuation
severity with a frequency greater than 3$f_{c}$ [5] makes it necessary:

  * •

modification of currently used measuring instruments [10, 11, 12], or use of
other measures of voltage fluctuation severity assessment [13, 14, 15, 16];

  * •

comprehensive analysis and diagnostics of voltage fluctuations, considering
potential voltage fluctuations sources (including power electronic devices
switching with high-frequency, or groups of loads causing voltage fluctuations
with a resultant frequency greater than 3$f_{c}$);

  * •

modification of methods of identification (recognition) and localization of
disturbing sources, including sources of voltage fluctuations with a frequency
greater than 3$f_{c}$ (currently, the most common methods in the literature
are methods of identification and localization disturbing loads changing their
operating state with a frequency to $f_{c}$ [17, 18, 19, 20, 21]; however,
there are also solutions considering loads changing their state with a
frequency of up to 3$f_{c}$ [22, 23, 24]).

Taking the above into account, the paper presents the analysis of voltage
fluctuations with a frequency greater than 3$f_{c}$ focused on investigation
the dependence of severity of voltage fluctuation on their shape in the case
of “clipped cosine” supply voltage distortion (“clipped cosine” distortion is
a good approximation of states occurring in LV networks). On the other hand,
voltage fluctuations in power grids are modelled as amplitude modulation (AM)
without a suppressed carrier wave [25, 26, 27], which is a good approximation
of voltage fluctuations in a stiff power grid (in the case of island
operation, the phase/frequency variation should also be considered, i.e.,
voltage fluctuations should be modelled as amplitude and frequency/phase
modulation (AM-FM/PM) [28, 29]). For the adopted model of voltage
fluctuations, “clipped cosine” supply voltage distortion is adopted as the
carrier signal, and arbitrary signals associated with the influence of
potential voltage fluctuations sources are adopted as the modulating signals.
For the adopted model, numerical simulation studies and experimental
laboratory tests were carried out, on the basis of which the dependence of
severity of voltage fluctuations on their shape are determined.

##  II Adopted model of power quality disturbances

As the model of power quality disturbances used in laboratory tests and in
numerical simulation studies is adopted the signal (AM modulation without
suppressed carrier wave) described by the relationship:

$u_{IN}\left(t\right)=\left(1+\left(\frac{\Delta U}{U}\right)\frac{1}{2\cdot
100}u_{\text{mod}}\left(t\right)\right)\cdot u_{c}\left(t\right),$ (1)

where: $u_{\mathrm{mod}}\left(t\right)$ is the normalized modulating signal
associated with the influence of voltage fluctuations sources, $\left(\Delta
U/U\right)$ is the modulation depth related to the nature of power grids and
disturbing loads expressed in %, $u_{c}\left(t\right)$ is the supply voltage
in the power grid prior to the appearance voltage fluctuations sources
(carrier signal).

The research consider a “clipped cosine” signal with three values of $m_{c}$
as the carrier signal, where $m_{c}$ is the clipping level described by the
relationship:

$m_{c}=\frac{M_{c}}{M},$ (2)

where: $M_{c}$ and $M$ are the signal amplitude after and before clipping,
respectively. The selected $m_{c}$ values are:

  * •

$m_{c}=1$ – rated purely sinusoidal supply voltage;

  * •

$m_{c}=0.8$ – typical distorted voltage in the power grid caused by electronic
devices (by input stages of switching power supplies) for which total harmonic
distortion (THD) ratio is approx. 8% (the limit value for LV networks [30]) –
it is worth noting that usually in real LV networks the value of
$m_{c}\in[0.8;1.0]$;

  * •

$m_{c}=0.1$ – highly distorted voltage in the power grid, typical for example
for island operation supplied with some converters [31, 32, 33], or for UPS
[33], or for resonance interactions in the power grid causing an increase in
the contribution of higher harmonics; for the assumed $m_{c}$ value, the shape
of waveform is similar to a square wave and THD ratio is approx. 43%.

Figure 1: Assumed carrier signals with $m_{c}$ equal to 1.0 (left), 0.8
(centre), 0.1 (right), respectively.

In the research, for every assumed carrier signals, the fundamental frequency
$f_{c}$ was 50$\rm\,Hz$ (rated power frequency [30]) and the rms value $U_{c}$
was 230$\rm\,V$ (rated rms value in LV networks [30]). Assumed carrier signals
(supply voltage before voltage fluctuations) are shown in Fig. 1.

Four types of signals are considered as the modulating signal:

  * •

sinusoidal signal;

  * •

triangular signal;

  * •

trapezoidal signal (duration of the rising edge and falling edge equal to
1/4$T_{m}$, where $T_{m}$ is the fundamental period of modulating signal) –
this shape of voltage fluctuations can be caused by loads equipped with a soft
starter system [35, 36];

  * •

rectangular signal – associated with the influence of loads causing step
voltage changes, which are probably the most common in the power grid nowadays
[36, 37, 38].

It is worth noting that the real (resultant) shapes of voltage fluctuations
probably have more complex waveforms, although assumed arbitrary test signals
make it possible to observe some tendencies [7] and are a good starting point
for further research. The studies were carried out in two stages. In both
stages, for the generated test signal described by (1), the short-term flicker
indicator $P_{st}$ [39] was determined and recorded, and which was adopted in
the research as the reference value determining voltage fluctuation severity.
It is worth noting that despite limitations of this measure of voltage
fluctuation severity, it is currently used in many countries and allows a one-
parameter and numerical assessment of one potential effects of voltage
fluctuations. In numerical simulation studies, test signal generation,
bandwidth limitation (using a low-pass FIR filter of order 200 and a cut-off
frequency of 8$\rm\,kHz$), downsampling and recording $P_{st}$ values (using
the IEC flickermeter implementation [40]) for the assumed modulated signal
with a limited bandwidth was performed in MATLAB (see Fig. 2).

Figure 2: The block diagram of signal chain in MATLAB, where LPF is a low-pass
filter.

In experimental laboratory tests, test signal generation and recording
$P_{st}$ values for it were performed in the laboratory setup shown in Fig. 3.
It is worth noting that $P_{st}$ values for individual test signals were
recorded for 20 minutes (two 10-minute intervals) and the first 10-minute
interval related to transient states [41] was rejected. Hence, considering the
time-consuming of experimental studies, they were carried out only for stage I
of the research for $\left(\Delta U/U\right)=5\%$ for all assumed carrier and
modulating signals. It is worth noting that the results of simulation and
experimental tests are consistent, so without losing generality, it can be
assumed that the other simulations (for which experimental studies are not
carried out) are correct. The inconsistency occurs only for $P_{st}<0.1$,
which results from the imperfection of the signal chain in the real
flickermeter related to, e.g., the resolution of instrument, implementation of
particular filters.

Figure 3: The laboratory setup, where: AWG is a arbitrary waveform generator
[42], PA is power amplifier [43], PQA is class A power quality analyser [44]
with built-in the IEC flickermeter [45, 46].

In the first stage of the research, individual carrier signals were modulated
with selected modulating signals with a modulation depth $\left(\Delta
U/U\right)$ equal to 1%, 5% and 10%, respectively, and with a modulating
frequency $f_{m}\in$ [0.01; 1050]$\rm\,Hz$. In the second stage of the
research, carrier signals were modulated with selected modulating signals with
modulating frequency $f_{m}$ equal to 208.8$\rm\,Hz$ and 1008.8z$\rm\,Hz$ and
modulation depth $\left(\Delta U/U\right)\in$[0.1; 20]%. It is worth noting
that for the modulating frequency $f_{m}$ at the level of$\rm\,kHz$, the
modulation depth $\left(\Delta U/U\right)$ at the level of 10% causes
variations of the rms values determined for the fundamental period every half
of the fundamental period (the fundamental period determined with the use of
zero cross detection) at the level of 0.1%, so in fact for large $f_{m}$
values, the modulation depth $\left(\Delta U/U\right)$ in the voltage
fluctuations model can be much greater than 10%.

##  III Research results and discussion

As a reminder, Fig. 4 shows a comparison of characteristics
$P_{st}=\textbf{f}\left(f_{m},\left(\Delta
U/U\right)=5\%=\mathrm{const}\right)$ for the sinusoidal modulating signal and
for every assumed carrier signals.

Figure 4: The $P_{st}=\textbf{f}\left(f_{m},\left(\Delta
U/U\right)=5\%=\mathrm{const}\right)$ characteristics for the sinusoidal
modulating signal, where $m_{c}$ is 1.0 (blue), 0.8 (green), 0.1 (red),
respectively (line – simulation results, markers – laboratory test results).
Figure 5: The $P_{st}=\textbf{f}\left(f_{m},\left(\Delta
U/U\right)=5\%=\mathrm{const}\right)$ characteristics for sinusoidal (blue),
triangular (green), trapezoidal (purple), rectangular (red) modulating signal,
where $m_{c}$ is 1.0 (top), 0.8 (middle), 0.1 (bottom), respectively (line –
simulation results, markers – laboratory test results).

In Fig. 4 it can be seen that for the sinusoidal carrier signal (supply
voltage), values of $P_{st}>0$ occur only for $f_{m}<3f_{c}$. On the other
hand, when the carrier signal is distorted, values of $P_{st}>0$ can occur for
$f_{m}>3f_{c}$. It is worth noting that, depending on the distortion level,
values of $P_{st}>0$ occur for different intervals of $f_{m}$. When $m_{c}\to
0$ (possible situation for island operation or selected emergency states of
power grids), then values of $P_{st}>0$ can occur for ever greater $f_{m}$
values. The dependence of values of $P_{st}>0$ occurrence for individual
intervals of $f_{m}$ is non-linear, non-monotonic and difficult to precisely
define, as presented in [5]. It is worth noting that the voltage distortion in
power grids (carrier signal) does not affect the $P_{st}$ measurement result
for $f_{m}$ in the range [0; $f_{c}$]$\rm\,Hz$. For $f_{m}$ in the range
($f_{c}$; 3$f_{c}$]$\rm\,Hz$, the influence of voltage distortion on the
$P_{st}$ measurement result is noticeable, and in the range for
$f_{m}>3f_{c}$, the voltage distortion can cause possibility occurrence values
of $P_{st}>0$ (experimentally verified visible obnoxious flicker – luminance
variation).

Figure 6: The $P_{st}=\textbf{f}\left(f_{m},\left(\Delta
U/U\right)\in\left\\{1,5,10\right\\}\%=\mathrm{const}\right)$ characteristics
for sinusoidal (first from top), triangular (second from top), trapezoidal
(third from top), rectangular (fourth from top) modulating signal, where
$m_{c}$ for carrier signal is 0.8 (line – simulation results, markers –
laboratory test results).

Fig. 5 shows a comparison of characteristics
$P_{st}=\textbf{f}\left(f_{m},\left(\Delta
U/U\right)=5\%=\mathrm{const}\right)$ for every assumed modulating signals and
carrier signals. On the basis of characteristics, it can be seen that a some
tendency for the dependence of severity of voltage fluctuations on their shape
for $f_{m}>3f_{c}$ is maintained as for the range $f_{m}<3f_{c}$, i.e., with
the same modulation depth, the most obnoxious are rectangular, trapezoidal,
sinusoidal and triangular voltage fluctuations, respectively. Assuming that
for any $f_{m}$ values there is a tendency to increase severity of voltage
fluctuations depending on their shape, it is possible to generalized
properties for any $f_{m}$, based on the results for $f_{m}$ values in the
range [0; 3$f_{c}$]$\rm\,Hz$. However, this is a hypothesis that requires
analytical proof. In the next part, the paper focuses mainly on the voltage
fluctuation with a frequency $f_{m}>3f_{c}$ ($f_{c}$ = 50$\rm\,Hz$), i.e.,
considered characteristics are presented in the range $f_{m}\in$[150;
1050]$\rm\,Hz$. In this range, the effect of changing the modulation depth
$\left(\Delta U/U\right)$ and the modulating frequency $f_{m}$ on $P_{st}$
measurement results is determined for assumed shapes of modulating signals.

Figure 7: The $P_{st}=\textbf{f}\left(f_{m},\left(\Delta
U/U\right)\in\left\\{1,5,10\right\\}\%=\mathrm{const}\right)$ characteristics
for sinusoidal (first from top), triangular (second from top), trapezoidal
(third from top), rectangular (fourth from top) modulating signal, where
$m_{c}$ for carrier signal is 0.1 (line – simulation results, markers –
laboratory test results).

Fig. 6 and Fig. 7 show a comparison of characteristics
$P_{st}=\textbf{f}\left(f_{m},\left(\Delta
U/U\right)\in\left\\{1,5,10\right\\}\%=\mathrm{const}\right)$ for every
assumed modulating signals and for the assumed carrier signals with $m_{c}$
equal to 0.8 (Fig. 6) and 0.1 (Fig. 7) respectively. Based on the
characteristics, it can be seen that for individual values of $f_{m}$,
regardless of the shape of modulating signal, the same tendency to increase
$P_{st}$ value with increasing modulation depth $\left(\Delta U/U\right)$
occurs.

Fig. 8 shows a comparison of characteristics
$P_{st}=\textbf{f}\left(f_{m}\in\left\\{208.8,1008.8\right\\}\mbox{$\rm\,Hz$}=\mathrm{const},\left(\Delta
U/U\right)\right)$ for every assumed modulating signals and for the assumed
carrier signals with $m_{c}$ equal to 0.8 and 0.1, respectively. On the basis
of the characteristics, it can be seen that for the assumed $f_{m}$ values,
the dependence of $P_{st}$ value on modulation depth $\left(\Delta
U/U\right)$, regardless of the shape of modulating signal, is approximately
linear. It is worth noting that the slope of characteristic depends on the
modulating signal shape and the carrier signal shape. Considering the non-
linearity and non-monotonicity of the dependence of $P_{st}$ value on $f_{m}$
value for different $m_{c}$ value [5] (see Fig. 8 – for $m_{c}=0.8$ greater
$P_{st}$ value is achieved for $f_{m}=208.8\mbox{$\rm\,Hz$}$ at the same
modulation depths $\left(\Delta U/U\right)$ than for $m_{c}=0.1$, and for
$f_{m}=1008.8\mbox{$\rm\,Hz$}$ the relationship is inverse), it is not
possible to explicitly define the dependence of slope of characteristic
$P_{st}=\textbf{f}\left(f_{m}=\mathrm{const},\left(\Delta U/U\right)\right)$
on shape of carrier signal (the supply voltage prior to the occurrence of
voltage fluctuations sources) and on shape of modulating signal (specific for
individual voltage fluctuations sources).

Figure 8: The
$P_{st}=\textbf{f}\left(f_{m}\in\left\\{208.8,1008.8\right\\}\mbox{$\rm\,Hz$}=\mathrm{const},\left(\Delta
U/U\right)\right)$ characteristics for sinusoidal (blue), triangular (green),
trapezoidal (purple), rectangular (red) modulating signal, where $m_{c}$ for
the carrier signal is 0.8 (left) and 0.1(right), respectively (line –
simulation results).

##  IV Conclusion

The paper presents the IEC flickermeter measurement results for assumed
distorted modulating signals with simultaneous “clipped cosine” distortion of
the carrier signal for selected clipping levels $m_{c}$. On the basis of
results obtained for individual test signals, it can be seen that if the
supply voltage (carrier signal) is distorted, then for $f_{m}>3f_{c}$,
obnoxious voltage fluctuations ($P_{st}>>0$) can occur. Moreover, along with
the increase in distortion of supply voltage (carrier signal), the voltage
fluctuations severity can increase for ever greater $f_{m}$ values. Also for
$f_{m}>3f_{c}$, analogous tendencies in the dependence of voltage fluctuation
shape are noticeable as for $f_{m}$ values in the range [0; 3$f_{c}$]$\rm\,Hz$
(also in the conventional situation, where a sinusoidal carrier signal is
assumed). It is worth noting that the presented preliminary research results
are a good start for further research aimed at determining the potential
voltage fluctuation severity caused by specific power electronic devices. In
order to precisely determine the voltage fluctuation severity from the level
of distortion of modulating signal and the level of distortion of carrier
signal, it is necessary to prepare an analytical IEC flickermeter model, and
then carry out experimental verification, considering modulating frequency at
the level of$\rm\,kHz$ associated with commonly used power electronics devices
(e.g., inverters), which is a further research target.

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# Lighting Electroweak-Violating ALP-Lepton Interactions at $e^{+}e^{-}$ and
$ep$ Colliders

Chih-Ting Lu<EMAIL_ADDRESS>Department of Physics and Institute of
Theoretical Physics, Nanjing Normal University, Nanjing, 210023, China

###### Abstract

Recently, Altmannshofer, Dror and Gori (2022) claimed there is a four-point
interaction, $W$-$\ell$-$\nu$-$a$, in electroweak-violating scenario of axion-
like particle (ALP) and lepton interactions which plays a critical role in
searching for ALPs from $\pi^{\pm}$, $K^{\pm}$ mesons and $W$ boson decays
because of the novel energy enhancements. Inspired by this interesting
finding, we first propose new t-channel processes,
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$, for electrophilic ALPs ($e$ALPs) which also
involve $W$-$\ell$-$\nu$-$a$ four-point interaction and have obvious energy
enhancement behaviors in their cross sections when the collision energy is
increasing. On the other hand, heavier $e$ALPs mainly decay to a photon pair
induced by the chiral anomaly instead of an electron-positron pair. Therefore,
studies of these t-channel processes with a photon pair plus missing energy at
$e^{+}e^{-}$ and $ep$ colliders open a new door to search for $e$ALPs at high
energy colliders. The proposed search strategies are not only aiming to
generate a larger production rate of ALPs, but also trying to distinguish
electroweak-violating ALP-lepton interactions from electroweak-preserving
ones.

##  I Introduction

Axion-Like-Particles (ALPs) are the generic extension of the Standard Model
(SM) with the type of pseudoscalars behaving like Nambu–Goldstone bosons.
Compared with the traditional QCD axion which aims to explain the strong CP
problem Peccei:1977hh ; Weinberg:1977ma ; Wilczek:1977pj  and can serve as the
dark matter (DM) candidate Preskill:1982cy ; Abbott:1982af ; Dine:1982ah ,
motivations of new physics models with ALPs are diverse. For example, ALPs may
come from different spontaneous symmetry breaking patterns of global
symmetries Preskill:1982cy ; Abbott:1982af ; Dine:1982ah ; Kim:1979if ;
Bagger:1994hh  or compactifications of very high energy string theories
Svrcek:2006yi ; Arvanitaki:2009fg ; Cicoli:2012sz . ALPs may also affect the
structure of the electroweak phase transition Jeong:2018jqe ; Im:2021xoy  and
explain the hierarchy problem of the Higgs boson mass Graham:2015cka . ALPs
can also be wave-like DM candidates like QCD axions Arias:2012az . In
addition, QCD axions are restricted to a very low and narrow mass range,
$10^{-12}\lesssim m_{a}\lesssim 10^{-2}$ eV. However, the mass of ALPs can
extend from almost massless to the electroweak scale or even higher.
Therefore, we have to set up various search strategies to hunt ALPs in
different mass regions.

In general, we can apply the effective field theory approach to ALPs
Brivio:2017ije ; Bauer:2017ris ; Bauer:2018uxu ; Ebadi:2019gij ; Bauer:2020jbp
; Bauer:2021mvw . In such a framework, different coupling types between ALPs
and SM particles are possible and each of them can be independent of the
others. Hence, we can explore each coupling type at a time. Although searching
for ALPs is a model-dependent issue, the mass of ALPs below about $10$ MeV
scale has already suffered from severe constraints from cosmological and
astrophysical observations Raffelt:1990yz ; Marsh:2015xka  as well as low
energy beam dump experiments Bjorken:1988as ; Dobrich:2015jyk ;
Dobrich:2019dxc . For ALPs heavier than the MeV scale, bounds from various
meson decay channels Izaguirre:2016dfi ; Benson:2018vya ; Gori:2020xvq ;
Altmannshofer:2022izm  and low energy $e^{+}e^{-}$ colliders such as BaBar
BaBar:2011kau  and Belle II Belle-II:2020jti  play important roles. Finally,
we resort to high energy colliders, like LEP OPAL:2002vhf ; Mimasu:2014nea ;
Jaeckel:2015jla  and LHC ATLAS:2014jdv ; ATLAS:2015rsn ; Knapen:2016moh  to
explore ${\cal O}(1-100)$ GeV ALPs. Hunting for ALPs is not yet a monotonous
task, people are still chasing different new proposals.

Recently, the authors in Ref. Altmannshofer:2022izm  proposed a new way to
explore the ALP and lepton interactions. They emphasized the less attention
$W$-$\ell$-$\nu$-$a$ four-point interaction for leptophilic ALPs
($\ell$ALPs)111The previous studies of $\ell$ALPs can be found in
Kirpichnikov:2020lws ; Han:2020dwo ; Chang:2021myh ; Cheung:2021mol ;
Bertuzzo:2022fcm ; Cheung:2022umw ; Lucente:2022esm . can induce novel energy
enhancement effects in processes of $\ell$ALPs from charged mesons and $W$
boson decays. Indeed, they have shown this type of interaction can explore new
parameter space and cannot be overlooked. Inspired by this work, we first
propose how to explore $W$-$\ell$-$\nu$-$a$ four-point interaction for heavier
electrophilic ALPs ($e$ALPs) and concretely display the energy enhancement
behaviors at $e^{+}e^{-}$ and $ep$ colliders. We point out the t-channel
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$ processes have obvious energy enhancements from
momentum transferring of the initial state $e^{\pm}$ to the final states ALP
and $\nu(\overline{\nu})$. This unique feature not only increases the $e$ALP
production rates, but opens a door to solely explore $W$-$\ell$-$\nu$-$a$
four-point interaction such that we can distinguish electroweak-violating ALP-
lepton interactions from electroweak-preserving ones.

On the other hand, the dominant decay mode of the heavier $e$ALP is a pair of
photons mainly from the chiral anomaly instead of a pair of electron-positron.
As a result, the signal final states are a photon pair plus missing energy (a
photon pair plus missing energy and backward jet) at $e^{+}e^{-}$ ($ep$)
colliders. Most interestingly, when the production of $e$ALP is highly
boosted, two photons are too collimated to pass the photon isolation
criterion. In this situation, we group these two collimated, non-isolated
photons inside a novel jet-like object called ”photon-jet” Dobrescu:2000jt ;
Toro:2012sv ; Draper:2012xt ; ATLAS:2012soa ; Ellis:2012sd ; Ellis:2012zp ;
Knapen:2015dap ; Agrawal:2015dbf ; Chang:2015sdy ; Aparicio:2016iwr ;
Dasgupta:2016wxw ; Domingo:2016unq ; Chiang:2016eav ; Dillon:2016tqp ;
Allanach:2017qbs ; Chakraborty:2017mbz ; ATLAS:2018dfo ; Sheff:2020jyw ;
Wang:2021uyb ; Ren:2021prq ; Ai:2022qvs ; CMS:2022wpu . The photon-jet
signature is clean and unique at both $e^{+}e^{-}$ and $ep$ colliders such
that it can help us to extract the signals from SM backgrounds. We find
searching for these two processes at $e^{+}e^{-}$ and $ep$ colliders with
signatures of both two isolated photons and a photon-jet for different mass
ranges of $e$ALP can provide much stronger future bounds than the existing
ones. Furthermore, these possible future bounds are hard to exceed by other
channels when exploring $\ell$ALPs in collider experiments. Therefore, this
could be another motivation to build $e^{+}e^{-}$ and/or $ep$ colliders in the
future.

We organize this paper as follows. In Sec. II, a brief review on ALP-lepton
interactions is provided and decay modes of the $\ell$ALP are discussed. In
Sec. III, we analytically and numerically display energy enhancement behaviors
in $e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$. In Sec. IV, we demonstrate signal-to-background
analysis at $e^{+}e^{-}$ and $ep$ colliders and predict possible future bounds
as well as compare them with existing bounds. Finally, we will summarize our
findings along with some further discussions in Sec. V. Some kinematic
distributions for both signals and SM backgrounds as well as supplemental
materials are collected in Appendix A.

##  II A Brief Review on ALP-lepton interactions

Taking ALPs as pseudo Nambu-Goldstone bosons arising from the global Peccei-
Quinn (PQ) symmetry Peccei:1977hh , $U(1)_{\text{PQ}}$, breaking, the ALPs
receive a shift symmetry, $a(x)\rightarrow a(x)+\text{const}$. Because of this
shift symmetry, the ALP interactions with SM fermion pairs are characterized
by the derivatively-coupled type. Here we focus on the $\ell$ALP and the
Lagrangian of its interactions is in the form ${\cal
L}_{\ell\text{ALP}}=\partial_{\mu}a~{}J^{\mu}_{\text{PQ},\ell}$. The general
lepton current can be written as Altmannshofer:2022izm ; Bertuzzo:2022fcm

$J^{\mu}_{\text{PQ},\ell}=\frac{c^{V}_{\ell}}{2\Lambda}\overline{\ell}\gamma^{\mu}\ell+\frac{c^{A}_{\ell}}{2\Lambda}\overline{\ell}\gamma^{\mu}\gamma_{5}\ell+\frac{c_{\nu}}{2\Lambda}\overline{\nu_{\ell}}\gamma^{\mu}P_{L}\nu_{\ell}\,.$
(1)

where $\Lambda$ is the relevant new physics scale breaking $U(1)_{\text{PQ}}$
symmetry and $c^{V}_{\ell}$, $c^{A}_{\ell}$, $c_{\nu}$ are dimensionless
couplings. Note the electroweak invariance isn’t imposed in Eq. (1), and the
condition $c_{\nu}=c^{V}_{\ell}-c^{A}_{\ell}$ doesn’t need to be held. As
pointed out in Ref. Altmannshofer:2022izm , the first and the third terms in
the general lepton current can be generated independently from two electroweak
invariant high-order operators
$\left(\overline{HL}\right)\gamma^{\mu}\left(HL\right)$ and
$\left(\overline{H^{\dagger}L}\right)\gamma^{\mu}\left(H^{\dagger}L\right)$.
Here $L$ and $H$ are SM lepton and complex scalar $SU(2)$ doublets,
respectively. After integration by parts of
$\partial_{\mu}a~{}J^{\mu}_{\text{PQ},\ell}$, the ${\cal L}_{\ell\text{ALP}}$
can be represented as Altmannshofer:2022izm

$\displaystyle
a~{}\partial_{\mu}J^{\mu}_{\text{PQ},\ell}=ic^{A}_{\ell}\frac{m_{\ell}}{\Lambda}~{}a\overline{\ell}\gamma_{5}\ell$
(2)
$\displaystyle+\frac{\alpha_{\text{em}}}{4\pi\Lambda}\bigg{[}\frac{c^{V}_{\ell}-c^{A}_{\ell}+c_{\nu}}{4s^{2}_{W}}~{}aW^{+}_{\mu\nu}\tilde{W}^{-,\mu\nu}$
$\displaystyle+\frac{c^{V}_{\ell}-c^{A}_{\ell}(1-4s^{2}_{W})}{2s_{W}c_{W}}~{}aF_{\mu\nu}\tilde{Z}^{\mu\nu}-c^{A}_{\ell}~{}aF_{\mu\nu}\tilde{F}^{\mu\nu}+$
$\displaystyle\frac{c^{V}_{\ell}(1-4s^{2}_{W})-c^{A}_{\ell}(1-4s^{2}_{W}+8s^{4}_{W})+c_{\nu}}{8s^{2}_{W}c^{2}_{W}}~{}aZ_{\mu\nu}\tilde{Z}^{\mu\nu}\bigg{]}$
$\displaystyle+\frac{ig_{W}}{2\sqrt{2}\Lambda}(c^{A}_{\ell}-c^{V}_{\ell}+c_{\nu})~{}a(\bar{\ell}\gamma^{\mu}P_{L}\nu)W_{\mu}^{-}~{}+~{}\text{h.c.}\,,$

where $\alpha_{\text{EM}}$ is the fine structure constant, $g_{W}$ is the weak
coupling constant, $s_{W}$ ($c_{W}$) is the sine (cosine) of the weak mixing
angle.

The term in the first line of Eq. (2) has been intensively studied for light
$\ell$ALPs Bjorken:1988as ; Bauer:2021mvw ; Kirpichnikov:2020lws ;
Chang:2021myh ; Bertuzzo:2022fcm ; Cheung:2022umw  and we label it as
”$\boldsymbol{a\ell\ell}$”. The $\ell$ALP can be produced from this
interaction via the radiation of charged leptons. However, this term has a
$m_{\ell}/\Lambda$ suppression, so we have to resort to high intensity
experiments, like SLAC E137 Bjorken:1988as  and Jefferson Lab BDX BDX:2016akw
to search for $\ell$ALPs. Terms on the second to the fourth lines of Eq. (2)
arise from the chiral anomaly and we label them as
”$\boldsymbol{aVV^{\prime}}$”. These terms are not proportional to $m_{\ell}$,
but they are suppressed by the factor $\alpha_{\text{em}}/4\pi$. The $\ell$ALP
can be produced from its connections with gauge bosons via flavor-changing
meson decays Izaguirre:2016dfi ; Gori:2020xvq ; Bauer:2021mvw  as well as
direct gauge boson fusion and associated gauge boson production processes
Mimasu:2014nea ; Jaeckel:2015jla ; Knapen:2016moh . The terms in the final
line of Eq. (2) were first shown in Ref. Raffelt:1987yt  for stellar bounds on
ALPs and recently have been studied for $\ell$ALP productions from charged
mesons and $W$ boson decays in details Altmannshofer:2022izm . We label them
as ”$\boldsymbol{aW\ell\nu}$”. This interaction vanishes when the general
lepton current in Eq. (1) preserves the electroweak symmetry. This four-point
interaction $W$-$\ell$-$\nu$-$a$ is also not proportional to $m_{\ell}$, and
has obvious $\left(\text{energy}/\Lambda\right)$ enhancement for some specific
processes. The main point of this work is to first show this novel enhancement
behavior is not only crucial for the $\ell$ALP productions from charged mesons
and $W$ boson decays to constrain the light $\ell$ALP Altmannshofer:2022izm ,
but also important to search for heavier $\ell$ALPs from the t-channel
processes, $\ell^{+}\ell^{-}\rightarrow\nu_{\ell}a\overline{\nu_{\ell}}$ and
$\ell^{-}p\rightarrow\nu_{\ell}aj$ in the electroweak violating scenario at
$\ell^{+}\ell^{-}$ and $\ell p$ colliders, respectively.

Apart from the production of $\ell$ALPs, decay modes of the $\ell$ALP are also
important to search for $\ell$ALPs in collider experiments. For mass of
$\ell$ALP below the electroweak scale ($m_{a}\lesssim M_{W}$), the $\ell$ALP
mainly decays to $\ell^{+}\ell^{-}$ and $\gamma\gamma$ in the form
Bauer:2017ris ; Bauer:2018uxu ; Chang:2021myh

$\Gamma_{a\rightarrow\ell^{+}\ell^{-}}=\frac{(c^{A}_{\ell})^{2}m^{2}_{\ell}m_{a}}{8\pi\Lambda^{2}}\sqrt{1-\frac{4m^{2}_{\ell}}{m^{2}_{a}}}~{},~{}~{}\Gamma_{a\rightarrow\gamma\gamma}=\frac{g^{2}_{a\gamma\gamma}m^{3}_{a}}{64\pi}~{},$
(3)

where

$g_{a\gamma\gamma}=\frac{\alpha_{\text{em}}}{\pi}\frac{c^{A}_{\ell}}{\Lambda}\lvert
1-{\cal F}(\frac{m^{2}_{a}}{4m^{2}_{\ell}})\rvert$ (4)

and the loop function ${\cal
F}(z>1)=\frac{1}{z}\text{arctan}^{2}\left(\frac{1}{\sqrt{1/z-1}}\right)$. Note
the $\ell$ALP interaction with a photon pair comes from both chiral anomaly as
shown in Eq. (2) and one-loop triangle Feynman diagrams222Here only charged
leptons are considered in the loop, and the contribution from $W$ boson is
much suppressed and can be safely ignored..

Figure 1: The decay branching ratios of $e$ALP below the electroweak scale
($m_{a}\lesssim M_{W}$).

Taking $e$ALP as an example, we show the branching ratios for $a\rightarrow
e^{+}e^{-}$ and $a\rightarrow\gamma\gamma$ in Fig. 1. The crossing point of
${\cal B}(a\rightarrow e^{+}e^{-})$ and ${\cal B}(a\rightarrow\gamma\gamma)$
is around $m_{a}=0.6$ GeV. When $m_{a}\gg m_{e}$ as the case in Fig. 1, ${\cal
F}(z)\ll 1$ and the $e$ALP interaction with a photon pair is dominated from
chiral anomaly. Since this chiral anomaly driven $\boldsymbol{a\gamma\gamma}$
interaction is independent of the charged lepton mass, we can expect the decay
of heavier $\ell$ALP becomes photophilic Chang:2021myh .

##  III Energy enhancement behaviors in
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$

Figure 2: Feynman diagrams for
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$. Here the color markers
indicate red for $\boldsymbol{aW\ell\nu}$ interaction, green for
$\boldsymbol{aVV^{\prime}}$ interaction and blue for $\boldsymbol{a\ell\ell}$
interaction.  Figure 3: Feynman diagrams for $e^{-}p\rightarrow\nu_{e}aj$. The
color markers are the same as Fig. 2.

In this section, we concretely study the energy enhancement behaviors in
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$ via the $\boldsymbol{aW\ell\nu}$ interaction.
Using $e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ as an example, the
relevant Feynman diagrams for this process are shown in Fig. 2. We will see
numerical results later that the contributions from
$\boldsymbol{aVV^{\prime}}$ and $\boldsymbol{a\ell\ell}$ interactions in this
process are much smaller than the one from $\boldsymbol{aW\ell\nu}$
interaction. In order to simplify the analytical form for discussions and
highlight the energy enhancement behaviors here, we will only show the
amplitude square with the average (sum) over initial (final) polarization from
the first diagram in Fig. 2 for the process
$e^{-}(p_{1})e^{+}(p_{2})\rightarrow\nu_{e}(q_{1})a(q_{2})\overline{\nu_{e}}(q_{3})$,

$\displaystyle\overline{\lvert{\cal M}\rvert^{2}}=$ (5)
$\displaystyle\frac{g^{4}_{W}\left(c^{A}_{\ell}-c^{V}_{\ell}+c_{\nu}\right)^{2}}{32\Lambda^{2}}\left(\frac{1}{k^{2}-M^{2}_{W}}+\frac{1}{k^{\prime
2}-M^{2}_{W}}\right)^{2}$
$\displaystyle\times\left(s-2m^{2}_{e}\right)\left[s-m^{2}_{a}-2q_{2}\cdot(q_{1}+q_{3})\right]\,,$

where $s=\left(p_{1}+p_{2}\right)^{2}=\left(q_{1}+q_{2}+q_{3}\right)^{2}$,
$k=p_{2}-q_{3}$ and $k^{\prime}=p_{1}-q_{1}$. It’s clear to see this amplitude
square can be enhanced when the momentum transferring in this $t$-channel
process is large enough and it is estimated to be smaller than

$\overline{\lvert{\cal
M}\rvert^{2}}~{}<~{}\frac{g^{4}_{W}\left(c^{A}_{\ell}-c^{V}_{\ell}+c_{\nu}\right)^{2}}{16\Lambda^{2}\left(s-M^{2}_{W}\right)^{2}}\left(s-2m^{2}_{e}\right)\left(\sqrt{s}-m_{a}\right)^{2}.$
(6)

On the other hand, the related Feynman diagrams for the process
$e^{-}p\rightarrow\nu_{e}aj$ are shown in Fig. 3. Again, except for the
$\boldsymbol{aW\ell\nu}$ interaction, contributions from other two types of
$e$ALP interactions can be safely ignored and the analytical form for the
amplitude square from the first diagram in Fig. 3 is similar to Eq. (5) and we
will not show it here.

Since $\boldsymbol{aW\ell\nu}$ interaction can only appear in an electroweak-
violating scenario, we follow the strategy in Ref. Altmannshofer:2022izm  to
define two different scenarios :

$\displaystyle\text{Electroweak
Violating}~{}(\textbf{EWV}):c^{V}_{\ell}=c_{\nu}=0,c^{A}_{\ell}\neq 0,$
$\displaystyle\text{Electroweak
Preserving}~{}(\textbf{EWP}):c_{\nu}=0,c^{V}_{\ell}=c^{A}_{\ell}\neq 0\,.$ (7)

Note the EWV (EWP) scenario is pure axial-vector (right-handed coupling)
current in Eq. (1) and the EWV scenario is usually studied ALP and SM fermion
pair interactions in the literature.

Figure 4: The energy enhancement behaviors of cross sections in
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ with
$c^{A}_{e}/\Lambda=0.01$ GeV-1, $c^{V}_{e}=c_{\nu}=0$ (EWV : solid-blue line)
and $c^{A}_{e}/\Lambda=c^{V}_{e}/\Lambda=0.01$ GeV-1, $c_{\nu}=0$ (EWP :
solid-black line). The dashed-orange, dashed-red and dashed-green lines are
contributions from $\boldsymbol{aW\ell\nu}$, $\boldsymbol{aVV^{\prime}}$ and
$\boldsymbol{aVV^{\prime}}$ interactions in EWV scenario, respectively.
Figure 5: The same as Fig. 4 except for the process
$e^{-}p\rightarrow\nu_{e}aj$ with $E_{e}=50$ GeV.

Even though the energy enhancement behaviors are helpful to explore $e$ALPs
from $e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$ processes at $e^{+}e^{-}$ and $ep$ colliders,
these two processes are still suffered from three-body phase space
suppression. Hence, to know the size of cross sections numerically for varying
the center-of-mass energy is important. We use FeynRules Alloul:2013bka  to
implement ${\cal L}_{\ell\text{ALP}}$ in Eq. (2) and apply Madgraph5 aMC@NLO
Alwall:2014hca  to calculate cross sections for these two processes. As
benchmark points, we set $c^{A}_{e}/\Lambda=0.01$ GeV-1, $c^{V}_{e}=c_{\nu}=0$
($c^{A}_{e}/\Lambda=c^{V}_{e}/\Lambda=0.01$ GeV-1, $c_{\nu}=0$) for the EWV
(EWP) scenario. We vary the center-of-mass energy $\sqrt{s}=100-500$ GeV at
$e^{+}e^{-}$ colliders and $E_{p}=1-15$ TeV with fixed $E_{e}=50$ GeV at $ep$
colliders and show their energy enhancement behaviors in Fig. 4 and Fig. 5,
respectively.

First of all, the cross sections in EWV scenario are more than four (nine)
orders of magnitudes larger than the ones in EWP scenario of Fig. 4 (Fig. 5).
Therefore, these two processes are powerful to distinguish $e$ALPs in EWV
scenario from the EWP scenario. The main reason is that there is
$\boldsymbol{aW\ell\nu}$ interaction in EWV scenario, but not in EWP scenario
and the $\boldsymbol{aW\ell\nu}$ interaction contributes to almost the whole
amount in cross sections of EWV scenario. In order to know quantitative
contributions to cross sections from $\boldsymbol{aW\ell\nu}$,
$\boldsymbol{aVV^{\prime}}$ and $\boldsymbol{a\ell\ell}$ interactions of
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$ processes in EWV scenario, we label them in
dashed lines of both Fig. 4 and Fig. 5. The subleading contribution in EWV
scenario comes from $\boldsymbol{aVV^{\prime}}$ interaction, but it’s about
five (six) orders of magnitude smaller than the one from
$\boldsymbol{aW\ell\nu}$ interaction in Fig. 4 (Fig. 5). Contributions from
$\boldsymbol{a\ell\ell}$ interaction are very tiny in the EWV scenario for
both of these two processes and can be totally ignored. Note for EWP scenario,
both $\boldsymbol{aW\ell\nu}$ and $\boldsymbol{aWW}$ interactions vanish in
$e^{-}p\rightarrow\nu_{e}aj$ such that it overlaps with the dashed line of
$\boldsymbol{a\ell\ell}$ interaction. Therefore,
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$ processes at $e^{+}e^{-}$ and $ep$ colliders are
ideal to explore the novel $\boldsymbol{aW\ell\nu}$ interaction and
distinguish $e$ALPs in EWV scenario from EWP scenario. We will focus on
searching for $eALPs$ in EWV scenario at $e^{+}e^{-}$ and $ep$ colliders in
the next section.

##  IV The study at $e^{+}e^{-}$ and $ep$ colliders

In this section, we further study the signal-to-background analysis at
$e^{+}e^{-}$ colliders in Sec. IV.1 and $ep$ collider in Sec. IV.2. Our goal
is to predict the future bounds for $e$ALPs at $e^{+}e^{-}$ and $ep$ colliders
and compare them with existing constraints which are shown in Sec. IV.3.

###  IV.1 Exploring $e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ at CEPC

Figure 6: Distributions of the opening angle between two photons, $\Delta
R_{\gamma_{1}\gamma_{2}}$, from
$e^{+}e^{-}\rightarrow\nu_{e}(a\rightarrow\gamma\gamma)\overline{\nu_{e}}$ at
CEPC with $\sqrt{s}=240$ GeV (left panel) and
$e^{-}p\rightarrow\nu_{e}(a\rightarrow\gamma\gamma)j$ at LHeC with $E_{e}=50$
GeV and $E_{p}=7$ TeV (right panel) in parton-level. Four benchmark $e$ALP
mass values, $m_{a}=50$, $30$, $20$, $5$ GeV, are shown for comparison.

There are several proposed $e^{+}e^{-}$ colliders which can explore
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$, like CEPC
CEPCStudyGroup:2018ghi , ILC TLEPDesignStudyWorkingGroup:2013myl , CLIC
Baer:2013cma , FCC-ee CLICDetector:2013tfe . Here we choose the CEPC with
$\sqrt{s}=240$ GeV as a concrete example to study how to extract the $e$ALP
signal from relevant SM backgrounds. According to Fig. 1, the $e$ALP
dominantly decays to $\gamma\gamma$ when $m_{a}\gtrsim 1$ GeV. Hence, we focus
on $a\rightarrow\gamma\gamma$ decay mode in this work. Most interestingly, the
$e$ALP becomes highly boosted at CEPC when it is light enough, so two photons
in the final state may be too collimated to pass the photon isolation
criterion at detectors. Take a cone size $R=0.5$ as the photon isolation
criterion at CEPC, we can find two photons cannot be isolated to each other at
detectors when $m_{a}\lesssim 20$ GeV (parton-level) in the left panel of Fig.
6. For this kind of collimated, non-isolated photons, we can group them as a
special signature ”photon-jet” ($J_{\gamma}$) which is a non-QCD jet-like
structure and deposits most of its energy in the the electromagnetic
calorimeter (ECAL) instead of hadron calorimeter (HCAL) as well as leaves less
charged tracks compared with the QCD jet. Therefore, we classify the signal
signatures as (1) two isolated photons plus missing energy
(${\>/\\!\\!\\!\\!E}$) for $m_{a}\gtrsim 20$ GeV, (2) a $J_{\gamma}$ plus
${\>/\\!\\!\\!\\!E}$ for $m_{a}\lesssim 20$ GeV.

cut flow in $\sigma$ [fb] |  signal |  $\nu_{\ell}\overline{\nu_{\ell}}\gamma\gamma$ |  $\nu_{\ell}\overline{\nu_{\ell}}(h\rightarrow\gamma\gamma)$
---|---|---|---
Generator | $0.11$ | $263.60$ | $7.67\times 10^{-2}$
cut-(1) | $7.10\times 10^{-2}$ | $32.23$ | $5.47\times 10^{-2}$
cut-(2) | $6.62\times 10^{-2}$ | $21.85$ | $3.84\times 10^{-5}$
cut-(3) | $6.02\times 10^{-2}$ | $12.62$ | $3.84\times 10^{-5}$
cut-(4) | $5.96\times 10^{-2}$ | $1.17$ | $7.67\times 10^{-7}$
cut-(5) | $5.01\times 10^{-2}$ | $0.68$ |  $0$
cut-(6) | $4.99\times 10^{-2}$ | $0.64$ |  $0$
Table 1: The cut-flow table for
$e^{+}e^{-}\rightarrow\nu_{e}(a\rightarrow\gamma\gamma)\overline{\nu_{e}}$ and
relevant SM backgrounds with signature of two isolated photons plus
${\>/\\!\\!\\!\\!E}$. The benchmark point $m_{a}=50$ GeV with
$c^{A}_{e}/\Lambda=1$ TeV-1 for signal is chosen. Each event selection has
been mentioned in the main text. The ”Generator” means the cross sections in
parton-level calculated by Madgraph5 aMC@NLO.

For the first signal signature, the relevant SM backgrounds are
$e^{+}e^{-}\rightarrow\gamma\gamma\nu_{\ell}\overline{\nu_{\ell}}$ and
$e^{+}e^{-}\rightarrow\nu_{\ell}\overline{\nu_{\ell}}h\rightarrow\nu_{\ell}\overline{\nu_{\ell}}(\gamma\gamma)$.
The benchmark point $m_{a}=50$ GeV with $c^{A}_{e}/\Lambda=1$ TeV-1 for signal
is chosen. We use Madgraph5 aMC@NLO to generate Monte Carlo samples for both
signal and background processes and pass them to Pythia8 Sjostrand:2007gs  for
the QED showering effect. The pre-selection cuts ($E_{\gamma}>5$ GeV and
$\lvert\eta_{\gamma}\rvert<3.0$) at parton-level for both the signal and
backgrounds are imposed. The CEPC template in Delphes3 deFavereau:2013fsa  is
applied for the fast detector simulation where the photon isolation criterion
is consistent with Ref. Cobal:2020hmk ; Ahmed:2022ude . We require the
following event selections to identify the signal signature and suppress
background events :

  * •

(1) $N(\gamma)\geqslant 2$ with $30<E_{\gamma_{1}}<90$ GeV,
$E_{\gamma_{2}}>15$ GeV, $\lvert\eta_{\gamma_{1}}\rvert<1.5$ and
$\lvert\eta_{\gamma_{2}}\rvert<2.0$,

  * •

(2) ${\>/\\!\\!\\!\\!E}>120$ GeV and
$\lvert\eta_{{\>/\\!\\!\\!\\!E}}\rvert<2.0$,

  * •

(3) Veto $85<M_{\>/\\!\\!\\!\\!E}<95$ GeV,

  * •

(4) $\lvert M_{\gamma_{1}\gamma_{2}}-m_{a}\rvert<3$ GeV,

  * •

(5) $\Delta\phi_{\gamma_{1},{\>/\\!\\!\\!\\!E}}>2.5$ and
$\Delta\phi_{\gamma_{2},{\>/\\!\\!\\!\\!E}}>1.8$,

  * •

(6) $2.2<{\>/\\!\\!\\!\\!E}/M_{\gamma_{1}\gamma_{2}}<3.6$,

where $E_{\gamma_{1}}$, $E_{\gamma_{2}}$ ($\eta_{\gamma_{1}}$,
$\eta_{\gamma_{2}}$) are the energy (pseudorapidity) of leading and subleading
energetic photons, $M_{\gamma_{1}\gamma_{2}}$ is the invariant mass of a
photon pair, $\Delta\phi_{\gamma_{i},{\>/\\!\\!\\!\\!E}}$ is the azimuthal
angle between the i-th photon and ${\>/\\!\\!\\!\\!E}$. The cut-flow table
including signal and backgrounds for each event selection is listed in Table.
1 and some kinematic distributions are shown in Fig. 8 of Appendix A.

In signal events, two isolated photons and ${\>/\\!\\!\\!\\!E}$ distribute in
the central region and we set $E_{\gamma_{1}}>30$ GeV, $E_{\gamma_{2}}>15$ GeV
and ${\>/\\!\\!\\!\\!E}>120$ GeV as the trigger for our candidate events.
Moreover, we select $E_{\gamma_{1}}<90$ GeV to further reduce the large
background from
$e^{+}e^{-}\rightarrow\gamma\gamma\nu_{\ell}\overline{\nu_{\ell}}$. The
neutrino pair from these two backgrounds are largely produced from the $Z$
boson decay, so we veto the $Z$ boson mass window,
$85<M_{\>/\\!\\!\\!\\!E}<95$ GeV, to suppress these background events. The
$e$ALP mass window selection is also very powerful to keep most signal events
but highly reduced these two backgrounds. The background events from
$e^{+}e^{-}\rightarrow\nu_{\ell}\overline{\nu_{\ell}}h\rightarrow\nu_{\ell}\overline{\nu_{\ell}}(\gamma\gamma)$
are almost killed up to this step. On the other hand, because two isolated
photons are well-separated from ${\>/\\!\\!\\!\\!E}$, we apply this feature to
reduce some events from
$e^{+}e^{-}\rightarrow\gamma\gamma\nu_{\ell}\overline{\nu_{\ell}}$.
Especially, the distribution of $\Delta\phi_{\gamma_{2},{\>/\\!\\!\\!\\!E}}$
is not so large in both
$e^{+}e^{-}\rightarrow\gamma\gamma\nu_{\ell}\overline{\nu_{\ell}}$ and
$e^{+}e^{-}\rightarrow\nu_{\ell}\overline{\nu_{\ell}}h\rightarrow\nu_{\ell}\overline{\nu_{\ell}}(\gamma\gamma)$
compared to the signal which can be observed in Fig. 8. Finally, we apply the
ratio, ${\>/\\!\\!\\!\\!E}/M_{\gamma_{1}\gamma_{2}}$, as a complementary
selection to the $e$ALP mass window, and it can further reduce a few
background events in the final step.

We consider the most optimistic integrated luminosity of CEPC proposal
CEPCStudyGroup:2018ghi , ${\cal L}=5ab^{-1}$, and define the signal
significance $Z$ as Cowan:2010js

$Z=\sqrt{2\cdot\left((N_{s}+N_{b})\cdot ln(1+N_{s}/N_{b})-N_{s}\right)},$ (8)

where $N_{s}$ and $N_{b}$ are the expected signal and background event
numbers. Here the systematic uncertainties are not taken into account in our
simple analysis since the CEPC is still a future collider. After all of these
event selections in Table. 1, we find the signal significance can reach
$Z=4.34$ for our benchmark point of ${\cal L}=5ab^{-1}$ and it’s detectable in
the future.

For the second signal signature, the only important SM background is
$e^{+}e^{-}\rightarrow\nu_{\ell}\overline{\nu_{\ell}}\gamma$333As we can
expect, the single QCD jet plus large ${\>/\\!\\!\\!\\!E}$ is hard to be
produced at the clean $e^{+}e^{-}$ colliders. Besides, the fake photon-jet
candidate from instruments is beyond the scope of this work and will not be
considered here.. Here we take the signal benchmark point as $m_{a}=5$ GeV and
$c^{A}_{e}/\Lambda=1$ TeV-1444The pre-selection cuts in parton-level are
similar as before, except for $E_{\gamma}>1$ GeV ($E_{\gamma}>10$ GeV) for the
signal (background).. The $J_{\gamma}$ candidate is defined as the following.
We apply the Cambridge/Aachen ($C/A$) jet clustering algorithm
Dokshitzer:1997in ; Wobisch:1998wt  with a cone size $R=0.4$ similar to
previous photon-jet works Ellis:2012sd ; Ellis:2012zp ; Allanach:2017qbs ;
Chakraborty:2017mbz ; Sheff:2020jyw ; Wang:2021uyb ; Ren:2021prq  to group
collimated photons. Then we require the hadronic energy fraction,
$\theta_{J}$, of $J_{\gamma}$ to satisfy $log\theta_{J}<-2$ and $\theta_{J}$
of a jet is defined as

$\theta_{J}=\frac{E_{J,\text{HCAL}}}{E_{J}},$ (9)

where $E_{J}$ is the total energy of the target jet and $E_{J,\text{HCAL}}$ is
the recorded energy from this jet in HCAL. Note the crucial difference for the
$J_{\gamma}$ signature at hadron colliders and $e^{+}e^{-}$ colliders is the
pile-up effect. The pile-up effect in hadron colliders makes it more
challenging to distinguish $J_{\gamma}$ from the single photon and the QCD
jet. However, since there is no pile-up collisions and underlying event at
$e^{+}e^{-}$ colliders, the $J_{\gamma}$ signature becomes very unique and
almost no SM background can mimic it after considering some jet substructure
observables. For this reason, we can choose a more stringent $J_{\gamma}$
criterion, $log\theta_{J}<-2$, than those previous works of $J_{\gamma}$ at
LHC Ellis:2012sd ; Ellis:2012zp ; Chakraborty:2017mbz ; Wang:2021uyb ;
Ren:2021prq .

cut flow in $\sigma$ [fb] |  signal |  $\nu_{\ell}\overline{\nu_{\ell}}\gamma$
---|---|---
Generator | $0.16$ | $4266.80$
cut-(1) | $0.10$ | $520.29$
cut-(2) | $0.10$ | $520.29$
cut-(3) | $9.27\times 10^{-2}$ | $387.77$
cut-(4) | $8.99\times 10^{-2}$ | $373.39$
cut-(5) | $8.14\times 10^{-2}$ |  $0$
cut-(6) | $8.02\times 10^{-2}$ |  $0$
Table 2: Similar to Table. 1, but for the signal benchmark point $m_{a}=5$ GeV
and $c^{A}_{e}/\Lambda=1$ TeV-1 as well as the signature of a $J_{\gamma}$
candidate plus ${\>/\\!\\!\\!\\!E}$.

We set up event selections to identify the signal signature and suppress the
background events in the following :

  * •

(1) $N(J_{\gamma})\geqslant 1$ with $log\theta_{J}<-2$,
$30<E_{J_{\gamma}}<100$ GeV, and $\lvert\eta_{J_{\gamma}}\rvert<1.5$,

  * •

(2) ${\>/\\!\\!\\!\\!E}>140$ GeV and
$\lvert\eta_{{\>/\\!\\!\\!\\!E}}\rvert<1.5$,

  * •

(3) Veto $75<M_{\>/\\!\\!\\!\\!E}<105$ GeV,

  * •

(4) ${\>/\\!\\!\\!\\!E}/E_{J_{\gamma}}>1.5$,

  * •

(5) $\lvert M_{J_{\gamma}}-m_{a}\rvert<1$ GeV,

  * •

(6) $\tau_{1}(J_{\gamma})>0.03$.

where $\tau_{1}$ is the case of $N=1$ in the general N-subjettiness,
$\tau_{N}$ Thaler:2010tr ; Thaler:2011gf . The N-subjettiness is a set of
simple but powerful jet substructure observables to effectively count how many
subjets inside a candidate jet. The $\tau_{1}$ can be represented as

$\tau_{1}=\frac{\sum_{k}P_{T_{k}}\cdot\Delta R_{1,k}}{\sum_{k}P_{T_{k}}\cdot
R}$ (10)

where $k$ runs over all constituents from both ECAL and HCAL cells of the jet,
$P_{T_{k}}$ is the transverse momentum for the $k$-th constituent, $\Delta
R_{1,k}$ is the opening angle between the first subjet and the $k$-th
constituent of the jet. Hence, $\tau_{1}$ is nothing but a description of the
energy distribution inside a candidate jet. We can expect the single photon
inside $J_{\gamma}$ is almost no structure, but $J_{\gamma}$ for the signal is
a two-prong structure. Therefore, $\tau_{1}$ can be used to distinguish the
signal from the background in this analysis. The signal and background cut-
flow table for each event selection are listed in Table. 2 and some kinematic
distributions are also shown in Fig. 9 of Appendix A.

We take a $J_{\gamma}$ candidate with $E_{J_{\gamma}}>30$ GeV and
${\>/\\!\\!\\!\\!E}>140$ GeV as the trigger and both of them are distributed
in the central region. The event selections $E_{J_{\gamma}}<100$ GeV and $Z$
boson mass window ($75<{\>/\\!\\!\\!\\!E}<105$ GeV) veto as well as
${\>/\\!\\!\\!\\!E}/E_{J_{\gamma}}>1.5$ are used to keep most signal events
but remove large parts of background events. Note $J_{\gamma}$ and the sum of
missing pieces almost share half of the whole center-of-mass energy and
scatter back-to-back, so cut-(2) keeps the same event rate as cut-(1). As
mentioned before, the jet substructure observables are the most powerful
variables to distinguish $J_{\gamma}$ from the single photon in this analysis.
Here we apply two of them in this work : jet mass and N-subjettiness. The jet
mass of a single photon is close to zero because the photon is massless.
However, the jet mass of signal peaks at $m_{a}$. Similarly, we apply
$\tau_{1}(J_{\gamma})$ to show the different feature between the single photon
and the signal. After involving these two jet substructure observables, the
background has been totally killed. The key reason to dramatically separate
the single from the single photon is that $e^{+}e^{-}$ colliders are free from
the pile-up effect and other pollution from QCD events. With ${\cal
L}=5ab^{-1}$, there are $401$ signal events left for this benchmark point
after all event selections.

###  IV.2 Exploring $e^{-}p\rightarrow\nu_{e}aj$ at LHeC

For $ep$ colliders, there are two proposed experiments, LHeC LHeC:2020van  and
FCC-he FCC:2018byv . Here we choose the LHeC with $E_{e}=50$ GeV and $E_{p}=7$
TeV as a practical example to study the $e^{-}p\rightarrow\nu_{e}aj$ process.
Again, we focus on the $a\rightarrow\gamma\gamma$ decay mode and use Madgraph5
aMC@NLO for event generation as well as Pythia8 for QCD, QED showering and
hadronization effects. The pre-selection cuts ($P_{T}^{\gamma}>5$ GeV,
$P_{T}^{j}>20$ GeV, $\lvert\eta_{\gamma}\rvert<4.5$ and
$\lvert\eta_{\gamma}\rvert<5.0$) at parton-level for both the signal and
backgrounds are imposed. A cone size $R=0.4$ for the photon isolation
criterion at LHeC is considered, and two photons cannot be isolated to each
other at detectors when $m_{a}\lesssim 30$ GeV (parton-level) as shown in the
right panel of Fig. 6. Since the produced $e$ALPs at LHeC are more boosted
than the ones at CEPC, we can expect the opening angle between two photons
from the $e$ALP decay at LHeC is smaller than the one at CEPC. On the other
hand, unlike the LHC, the pile-up effect and multiple interactions at LHeC are
tiny and can be safely ignored Hesari:2018ssq ; Antusch:2019eiz ;
Andre:2022xeh . This is a key point that the $J_{\gamma}$ candidate will not
suffer from serious pile-up pollution. It helps us to easily distinguish
$J_{\gamma}$ from the single-photon and the QCD jet rather than the situation
at LHC.

cut flow in $\sigma$ [fb] |  signal |  $\nu_{e}\gamma\gamma j$ |  $\nu_{e}(h\rightarrow\gamma\gamma)j$ |  $\nu_{e}\gamma jj$
---|---|---|---|---
Generator | $4.31$ | $110.89$ | $0.10$ | $1462.10$
cut-(1) | $2.85$ | $13.16$ | $7.91\times 10^{-2}$ | $0.24$
cut-(2) | $2.49$ | $11.82$ | $6.90\times 10^{-2}$ | $0.11$
cut-(3) | $2.28$ | $10.71$ | $5.55\times 10^{-2}$ | $0.10$
cut-(4) | $2.23$ | $0.82$ | $2.08\times 10^{-6}$ | $4.19\times 10^{-3}$
cut-(5) | $1.94$ | $0.44$ | $1.04\times 10^{-6}$ | $3.47\times 10^{-3}$
cut-(6) | $1.83$ | $0.24$ | $1.04\times 10^{-6}$ | $3.00\times 10^{-3}$
cut-(7) | $1.77$ | $0.22$ | $1.04\times 10^{-6}$ | $2.66\times 10^{-3}$
Table 3: The cut-flow table for
$e^{-}p\rightarrow\nu_{e}(a\rightarrow\gamma\gamma)j$ and relevant SM
backgrounds with the signature of two isolated photons, a backward jet plus
${\>/\\!\\!\\!\\!E_{T}}$. The benchmark point $m_{a}=50$ GeV with
$c^{A}_{e}/\Lambda=1$ TeV-1 for signal is chosen. All related event selections
have been mentioned in the main text.

For the signature with two isolated photons, a backward jet plus missing
transverse energy (${\>/\\!\\!\\!\\!E_{T}}$), the relevant SM backgrounds are
$e^{-}p\rightarrow\nu_{e}\gamma\gamma j$,
$e^{-}p\rightarrow\nu_{e}hj\rightarrow\nu_{e}(\gamma\gamma)j$ and
$e^{-}p\rightarrow\nu_{e}\gamma jj$ where one of QCD jets fakes to photon in
the final process. Here we apply the rate of jet faking to photon as
$P_{j\rightarrow\gamma}=5\times 10^{-4}$ ATLAS:2017muo . We still choose the
benchmark point $m_{a}=50$ GeV with $c^{A}_{e}/\Lambda=1$ TeV-1 for signal-to-
background analysis. For the fast detector simulation, the LHeC template in
Delphes3 is used and the photon isolation criterion is consistent with Ref.
Gutierrez-Rodriguez:2020gsi . The following event selections to identify the
signal signature and suppress the background events are required,

  * •

(1) $N(\gamma)\geqslant 2$ with $p_{T}^{\gamma_{1}}>30$ GeV,
$p_{T}^{\gamma_{2}}>10$ GeV, $-3.0<\eta_{\gamma_{1}}<0.0$ and
$-3.5<\eta_{\gamma_{2}}<0.5$,

  * •

(2) $N(j)\geqslant 1$ with $p_{T}^{j_{1}}>20$ GeV and
$-5.0<\eta_{j_{1}}<-1.0$,

  * •

(3) ${\>/\\!\\!\\!\\!E_{T}}>20$ GeV and $2.0<\eta_{\>/\\!\\!\\!\\!E_{T}}<5.0$,

  * •

(4) $\lvert M_{\gamma_{1}\gamma_{1}}-m_{a}\rvert<3$ GeV,

  * •

(5) $\Delta\phi_{\gamma_{1},{\>/\\!\\!\\!\\!E_{T}}}>1.5$ and
$\Delta\phi_{\gamma_{2},{\>/\\!\\!\\!\\!E_{T}}}>1.0$,

  * •

(6) $\left({\>/\\!\\!\\!\\!E_{T}}+p_{T}^{j_{1}}\right)/M_{\gamma\gamma}>1.0$,

  * •

(7)
$\left({\>/\\!\\!\\!\\!E_{T}}+p_{T}^{j_{1}}\right)/\left(p_{T}^{\gamma_{1}}+p_{T}^{\gamma_{2}}\right)<1.2$.

The cut-flow table for signal and backgrounds from the above event selections
is shown in Table. 3 and some kinematic distributions are shown in Fig. 10 of
Appendix A.

Since $E_{p}$ is much larger than $E_{e}$ at LHeC, two isolated photons are
distributed in the relatively backward region, ${\>/\\!\\!\\!\\!E_{T}}$ and
jet are in the forward and backward regions. $p_{T}^{\gamma_{1}}>30$ GeV,
$p_{T}^{\gamma_{2}}>10$ GeV, ${\>/\\!\\!\\!\\!E_{T}}>20$ GeV and
$p_{T}^{j_{1}}>20$ GeV are required as the trigger for this signature. The
$e$ALP mass window selection is the most stringent one which kills large parts
of SM backgrounds. According to the geometric shape of this signal signature
and different energy partitions for each object, we choose the final three
event selections to further separate the signal from SM backgrounds. The most
optimistic integrated luminosity of LHeC proposal LHeC:2020van , ${\cal
L}=5ab^{-1}$, is used in this analysis. After involving all event selections,
the signal significance is $Z=71.97$. This means we can explore much smaller
$c^{A}_{e}/\Lambda$ for $m_{a}=50$ GeV at LHeC.

cut flow in $\sigma$ [fb] |  signal |  $\nu_{e}\gamma j$ |  $\nu_{e}jj$
---|---|---|---
Generator | $5.47$ | $3596.30$ | $42514.00$
cut-(1) | $3.86$ | $831.52$ | $93.02$
cut-(2) | $3.80$ | $795.35$ | $84.22$
cut-(3) | $3.62$ | $724.11$ | $79.08$
cut-(4) | $3.42$ | $646.21$ | $56.63$
cut-(5) | $3.17$ | $7.03$ | $14.88$
cut-(6) | $2.53$ | $0.38$ | $0.77$
Table 4: Similar to Table. 3, but for the signal benchmark point $m_{a}=5$ GeV
with $c^{A}_{e}/\Lambda=1$ TeV-1 and the signature of a $J_{\gamma}$
candidate, a backward jet plus ${\>/\\!\\!\\!\\!E_{T}}$.

For the signature with a $J_{\gamma}$, a backward jet plus
${\>/\\!\\!\\!\\!E_{T}}$, the relevant SM backgrounds are
$e^{-}p\rightarrow\nu_{e}\gamma j$ and $e^{-}p\rightarrow\nu_{e}jj$ where the
single photon and the QCD jet can mimic $J_{\gamma}$555Similar to $J_{\gamma}$
analysis at CEPC, the fake photon-jet candidate from instruments is not
considered in our analysis at LHeC.. Again, we choose the benchmark point
$m_{a}=5$ GeV with $c^{A}_{e}/\Lambda=1$ TeV-1 for signal-to-background
analysis666The pre-selection cuts in parton-level are similar as before,
except for $P_{T}^{\gamma}>1$ GeV ($P_{T}^{\gamma}>10$ GeV) for the signal
(background).. We use the same definition of $J_{\gamma}$ as the previous CEPC
analysis. As shown before, the jet mass and N-subjettiness observables for the
$J_{\gamma}$ candidate are useful to pick out the signal from the single
photon and the QCD jet. We set up event selections to identify the signal
signature and suppress the background events in the following :

  * •

(1) $N(J_{\gamma})\geqslant 1$ with $log\theta_{J}<-2$,
$p_{T}^{J_{\gamma}}>30$ GeV and $-3.5<\eta_{\gamma_{1}}<0.5$,

  * •

(2) $N(j)\geqslant 1$ with $p_{T}^{j_{1}}>20$ GeV and
$-5.0<\eta_{j_{1}}<-0.5$,

  * •

(3) ${\>/\\!\\!\\!\\!E_{T}}>20$ GeV and $1.5<\eta_{\>/\\!\\!\\!\\!E_{T}}<5.0$,

  * •

(4)
$\left({\>/\\!\\!\\!\\!E_{T}}+p_{T}^{j_{1}}\right)/p_{T}^{J_{\gamma}}<1.2$,

  * •

(5) $\lvert M_{J_{\gamma}}-m_{a}\rvert<2$ GeV,

  * •

(6) $0.03<\tau_{1}<0.25$, $\tau_{2}/\tau_{1}<0.3$, $\tau_{3}/\tau_{1}<0.2$ and
$\tau_{3}/\tau_{2}<0.4$,

where the definition of the N-subjettiness observables, $\tau_{2}$ and
$\tau_{3}$, can be found in Ref. Thaler:2010tr ; Thaler:2011gf . The signal
and backgrounds cut-flow table for each event selection is listed in Table. 4
and some kinematic distributions are also shown in Fig. 11 of Appendix A.

We first applied $J_{\gamma}$ selection criterion to pick out the $J_{\gamma}$
candidate from the backward QCD jet. A $J_{\gamma}$ with
$p_{T}^{J_{\gamma}}>30$ GeV, a jet with $p_{T}^{j_{1}}>20$ GeV and
${\>/\\!\\!\\!\\!E_{T}}>20$ GeV are required as the trigger for this
signature. The ratio
$\left({\>/\\!\\!\\!\\!E_{T}}+p_{T}^{j_{1}}\right)/p_{T}^{J_{\gamma}}$ is used
to separate the signal from backgrounds. Furthermore, the powerful jet
substructure observables, jet mass and N-subjettiness, are used to further
distinguish the signal $J_{\gamma}$ from the single photon and the QCD jet as
shown in Fig. 11. Note there is still a small possibility for the backward QCD
jet to be identified as a $J_{\gamma}$ candidate. On the other hand, unlike
the environment in $e^{+}e^{-}$ colliders, there is still some QCD
contamination in this process at LHeC, though we don’t need to consider the
pile-up effect. It indicates SM backgrounds are much suppressed but cannot be
ignored. With the help of these event selections, the signal significance can
reach $Z=59.21$.

###  IV.3 Main results and existing bounds

Figure 7: The future bounds on $c^{A}_{e}/\Lambda$ of $e$ALPs from CEPC with
${\cal L}=5ab^{-1}$ and LHeC with ${\cal L}=1ab^{-1}$ within $95\%$ CL (dashed
lines) as well as existing bounds (bulk regions). Here we label ”$2\gamma$”
and ”$J_{\gamma}$” to identify two kinds of signatures at CEPC and LHeC.
${\cal B}_{W}\lesssim 10^{-5}$ represents ${\cal
B}(W^{\pm}\rightarrow\ell^{\pm}\nu a)<10^{-5}$ Altmannshofer:2022izm  (solid-
black line). Some collider bounds are in order : ATLAS $2\gamma$ ATLAS:2014jdv
; Jaeckel:2015jla ; Knapen:2016moh  (brown bulk), ATLAS $3\gamma$
ATLAS:2015rsn ; Knapen:2016moh  (orange bulk), OPAL $3\gamma$ OPAL:2002vhf ;
Knapen:2016moh  (green bulk), OPAL $2\gamma$ OPAL:2002vhf ; Knapen:2016moh ,
ATLAS/CMS (PbPb) dEnterria:2021ljz  (blue bulk) and LHCb Benson:2018vya ;
CidVidal:2018blh  (magenta bulk). For light $e$ALPs, $B^{\pm}\rightarrow
K^{\pm}a\rightarrow K^{\pm}(\gamma\gamma)$ from BaBar BaBar:2021ich  (red
bulk), $K_{L}\rightarrow\pi^{0}a\rightarrow\pi^{0}(e^{+}e^{-})$ from KTeV
KTeV:2003sls  (black bulk) and $B\rightarrow K^{\ast}a\rightarrow
K^{\ast}(e^{+}e^{-})$ from LHCb LHCb:2015ycz ; Altmannshofer:2022izm  (yellow
bulk) are involved. Finally, we also include the bounds from CHARM
CHARM:1985anb ; Altmannshofer:2022izm  and SLAC E137 Bjorken:1988as  (gray
bulk).

Based on the search strategies in Sec. IV.1 and IV.2, we extend the study of
signal benchmark points to a wide range of $m_{a}$ and try to find possible
future bounds within $95\%$ confidence level (CL) ($Z=1.96$). On the other
hand, in order to conservatively show the signal significance of the case
without the survival background event after all event selections, we require
at least $10$ signal events left. Moreover, only signal efficiency larger than
$10\%$ is considered here. We summarize these possible future bounds from CEPC
with ${\cal L}=5ab^{-1}$ and LHeC with ${\cal L}=1ab^{-1}$ in Fig. 7. Note
some of our event selections for two isolated photons plus
${\>/\\!\\!\\!\\!E}$ at CEPC are sensitive to $m_{a}$, so we dynamically
optimize event selections for different $m_{a}$ to suit each case. Some
benchmark examples are listed in Table. 5 of Appendix A. We also restrict
ourselves to $1$ GeV $\leqslant m_{a}\lesssim M_{W}$ for $e$ALPs below the
electroweak scale.

The lower bound comes from technical issues of $J_{\gamma}$ analysis. As we
have mentioned, the jet mass and N-subjettiness observables are efficient to
separate the signal from SM backgrounds. However, when $m_{a}<1$ GeV, the
above two observables are no longer powerful for discrimination. In this
situation, sophisticated jet substructure observables and/or machine learning
techniques are required Ellis:2012zp ; Wang:2021uyb ; Ren:2021prq ; Ai:2022qvs
, but they are beyond the scope of this work. In addition, when the opening
angle between two photons $\Delta R_{\gamma\gamma}\lesssim 0.04$, the trigger
at ECAL is hard to distinguish a $J_{\gamma}$ from a single photon since this
is close to the size of a standard single photon energy cluster at ECAL
ATLAS:2018dfo . We can clearly find the efficiency of two isolated photons
decreases when $m_{a}\lesssim 20$ GeV because two photons become closer to
each other and cannot pass the photon isolation criterion. Similarly, it’s
hard to group two photons inside a $J_{\gamma}$ candidate for $m_{a}\gtrsim
20(30)$ at CEPC (LHeC). Hence, the analysis of signatures with two isolated
photons and a $J_{\gamma}$ are complementary to each other for the middle
$m_{a}$ of $e$ALP searches.

Some existing bounds are also shown in Fig. 7 for comparison. First of all,
the precise $W$ boson width measurements can indirectly test $\ell$ALPs with
$m_{a}<M_{W}$ in the EWV scenario as pointed out in Ref. Altmannshofer:2022izm
and we conservatively choose ${\cal B}(W^{\pm}\rightarrow\ell^{\pm}\nu
a)<10^{-5}$ as a benchmark value and mark it as black-solid line in Fig. 7.
For heavier $e$ALPs ($m_{a}\gtrsim 5$ GeV), the ATLAS $2\gamma$ ATLAS:2014jdv
; Jaeckel:2015jla ; Knapen:2016moh  (brown bulk), ATLAS $3\gamma$
ATLAS:2015rsn ; Knapen:2016moh  (orange bulk), OPAL $3\gamma$ OPAL:2002vhf ;
Knapen:2016moh  (green bulk), ATLAS/CMS (PbPb) dEnterria:2021ljz  (blue bulk)
as well as LHCb Benson:2018vya ; CidVidal:2018blh  (magenta bulk) can already
bite some parameter space in the upper-right corner. Among these constraints,
the one from ATLAS/CMS (PbPb) is the strongest and extends to a wide $m_{a}$
range. The bound form OPAL $2\gamma$ OPAL:2002vhf ; Knapen:2016moh  can also
extend to $0.5\lesssim m_{a}\lesssim 100$ GeV, but it’s weaker than other
constraints. All of the above bounds rely on the $\boldsymbol{aVV^{\prime}}$
interaction. For lighter $e$ALPs ($m_{a}<5$ GeV), $B^{\pm}\rightarrow
K^{\pm}a\rightarrow K^{\pm}(\gamma\gamma)$ from BaBar BaBar:2021ich  (red
bulk), $K_{L}\rightarrow\pi^{0}a\rightarrow\pi^{0}(e^{+}e^{-})$ from KTeV
KTeV:2003sls  (black bulk) as well as $B\rightarrow K^{\ast}a\rightarrow
K^{\ast}(e^{+}e^{-})$ from LHCb LHCb:2015ycz ; Altmannshofer:2022izm  (yellow
bulk) can already exclude large parameter space. These strong constraints come
from FCNC processes with $\boldsymbol{aVV^{\prime}}$ interaction. Finally, we
also show the bounds from CHARM CHARM:1985anb ; Altmannshofer:2022izm  and
SLAC E137 Bjorken:1988as  (gray bulk) which comes from the
$\boldsymbol{a\ell\ell}$ interaction in the left-bottom region of Fig. 7. Note
all the above bounds have been rescaled according to our definition of ALP-
letpon interactions in Eq. (2) and $e$ALP decay branching ratios in Fig. 1.
Some other bounds from Belle II Belle-II:2020jti , $K\rightarrow\pi
a\rightarrow\pi(\gamma\gamma)$ Izaguirre:2016dfi  and etc are so weak that we
don’t involve them here.

All in all, compared with these existing bounds, our proposals to search for
$e$ALPs via $e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$ at $e^{+}e^{-}$ and $ep$ colliders are still
attractive and much stronger than some existing bounds. The possible future
bounds of $c^{A}_{e}/\Lambda$ can reach to less than $0.1-1.0$ TeV-1 which
open a new door to explore $e$ALPs in EWV scenario below the electroweak
scale.

##  V Discussions and Conclusion

The ALP is a well-motivated, postulated new particle beyond the SM. Searching
for couplings among ALPs and various SM particles in different ALP mass ranges
is an important mission to explore their particle nature. For ALP-lepton
interactions, the recent work in Ref. Altmannshofer:2022izm  has shown the
less discussed four-point interaction, $W$-$\ell$-$\nu$-$a$, in electroweak-
violating (EWV) scenario plays an important role to explore leptophilic ALPs
for some charged current interaction processes. Especially, for concreteness,
they have applied this kind of interaction to search for electronphilic ALPs
($e$ALPs) from $\pi^{\pm}$, $K^{\pm}$ mesons and $W$ boson decays and find the
novel energy enhancement effect in these decay modes can largely increase the
$e$ALP production rate.

In this work, we further explore heavier $e$ALPs via $W$-$\ell$-$\nu$-$a$
four-point interaction in collider experiments. New t-channel processes,
$e^{+}e^{-}\rightarrow\nu_{e}a\overline{\nu_{e}}$ and
$e^{-}p\rightarrow\nu_{e}aj$ to search for $e$ALPs at $e^{+}e^{-}$ and $ep$
colliders are proposed. In the EWV scenario, there are obvious energy
enhancement behaviors in production cross sections for these two processes as
shown in Fig. 4 and 5 such that they can be used as new channels not only to
search for $e$ALPs but also to distinguish EWV ALP-lepton interactions from
electroweak-preserving ones at high energy colliders.

For $m_{a}\gtrsim 1$ GeV, the $e$ALPs dominantly decays to a photon pair
induced by the chiral anomaly as shown in Fig. 1. It is noteworthy that two
photons from the $e$ALP decay at colliders may be too collimated to pass the
photon isolation criterion when $e$ALP is highly boosted. We apply photon-jet
analysis to this situation. Therefore, we study two kinds of signal
signatures, two isolated photos and a photon-jet, at $e^{+}e^{-}$ and $ep$
colliders. Taking CEPC with ${\cal L}=5ab^{-1}$ and LHeC with ${\cal
L}=1ab^{-1}$ as two examples, we find the possible future bounds of
$c^{A}_{e}/\Lambda$ can be lower than $0.1-1.0$ TeV-1 for $1$ GeV $\leqslant
m_{a}\lesssim M_{W}$ which is much stronger than existing bounds as shown in
Fig. 7.

Before closing, we discussed some possible extensions of this work. First, we
don’t consider the polarized $e^{+}$, $e^{-}$ at $e^{+}e^{-}$ and $ep$
colliders. The signal significance should be enhanced once the specific
$e^{+}$, $e^{-}$ polarization is chosen. Moreover, boosted decision tree (BDT)
Roe:2004na , convolutional neural network (CNN) Ayyar:2020ijy  and other
advanced machine learning techniques can further improve the simple analysis
here, but the main features to explore these two new $e$ALP production
channels have been emphasized in this work. On the other hand, for light
$e$ALPs with $c^{A}_{e}/\Lambda\lesssim 1$ TeV-1, they may become long-lived
particles inside colliders with lab frame decay lengths of ${\cal O}(100)$ mm.
If we can identify this special displaced photon-jet signature in detectors,
almost no SM background can mimic it. Detecting this displaced photon-jet
signature is highly dependent on the design of detectors and it is beyond the
scope of this paper. Finally, the extension to muonphilic ALPs at
$\mu^{+}\mu^{-}$ and $\mu p$ colliders is left for future work.

###### Acknowledgements.

We thank Lei Wu and Shu-Yu Ho for helpful discussions.

##  Appendix A Some kinematic distributions and supplemental information

Figure 8: Some signal and background kinematic distributions for the signature
of two isolated photons plus ${\>/\\!\\!\\!\\!E}$ at CEPC.

Figure 9: Some signal and background kinematic distributions for the signature
of a $J_{\gamma}$ plus ${\>/\\!\\!\\!\\!E}$ at CEPC.

Figure 10: Some signal and background kinematic distributions for the
signature of two isolated photons, a backward jet plus
${\>/\\!\\!\\!\\!E_{T}}$ at LHeC.

Figure 11: Some signal and background kinematic distributions for the
signature of a $J_{\gamma}$, a backward jet plus ${\>/\\!\\!\\!\\!E_{T}}$ at
LHeC.

In this Appendix, we choose some representative kinematic distributions for
both signals and backgrounds at CEPC and LHeC in the following :

  * •

For the signature of two isolated photons plus ${\>/\\!\\!\\!\\!E}$ at CEPC,
$E_{\gamma_{1,2}}$, $\eta_{\gamma_{1,2}}$, $M_{\>/\\!\\!\\!\\!E}$,
$M_{\gamma_{1}\gamma_{2}}$, $\Delta\phi_{\gamma_{2},{\>/\\!\\!\\!\\!E}}$ and
${\>/\\!\\!\\!\\!E}/M_{\gamma_{1}\gamma_{2}}$ distributions are shown in Fig.
8.

  * •

For the signature of a $J_{\gamma}$ plus ${\>/\\!\\!\\!\\!E}$ at CEPC,
$E_{J_{\gamma}}$, $\eta_{J_{\gamma}}$, $M_{\>/\\!\\!\\!\\!E}$,
${\>/\\!\\!\\!\\!E}/E_{J_{\gamma}}$, $M_{J_{\gamma}}$ and
$\tau_{1}(J_{\gamma})$ distributions are shown in Fig. 9.

  * •

For the signature of two isolated photons, a backward jet plus
${\>/\\!\\!\\!\\!E_{T}}$ at LHeC, $M_{\gamma_{1}\gamma_{2}}$,
$({\>/\\!\\!\\!\\!E_{T}}+P^{j_{1}}_{T})/M_{\gamma_{1}\gamma_{2}}$,
$\Delta\phi_{\gamma_{1},{\>/\\!\\!\\!\\!E_{T}}}$ and
$\Delta\phi_{\gamma_{2},{\>/\\!\\!\\!\\!E_{T}}}$ distributions are shown in
Fig. 10.

  * •

For the signature of a $J_{\gamma}$, a backward jet plus
${\>/\\!\\!\\!\\!E_{T}}$ at LHeC, $P^{J_{\gamma}}_{T}$,
${\>/\\!\\!\\!\\!E_{T}}$, $M_{J_{\gamma}}$, $\theta_{J}$, $\tau_{1}$,
$\tau_{2}/\tau_{1}$, $\tau_{3}/\tau_{1}$, $\tau_{3}/\tau_{2}$, distributions
are shown in Fig. 11.

Note that only physical quantities are shown in these figures. The $M_{J}$
distribution for the single photon inside $J_{\gamma}$ suffers from detector
smearing effects and fluctuations which make parts of events with $M_{J}$
slightly less than zero. Besides, there are some events in signals and
backgrounds with $\tau_{1}$, $\tau_{2}$, $\tau_{3}$ are equal or close to
zero, which cause $\tau_{2}/\tau_{1}$, $\tau_{3}/\tau_{1}$ and
$\tau_{3}/\tau_{2}$ are ill-defined. Finally, in the $\theta_{J}$ distribution
of Fig. 11, we enlarge the event shape of $e^{-}p\rightarrow\nu_{e}jj$ by ten
times to make it easily be visualized.

$m_{a}$ [GeV] |  ${\>/\\!\\!\\!\\!E}$ [GeV] |  $\Delta M_{\gamma_{1}\gamma_{2}}$ [GeV] | $\Delta\phi_{\gamma_{1},{\>/\\!\\!\\!\\!E}}$ | $\Delta\phi_{\gamma_{2},{\>/\\!\\!\\!\\!E}}$ | ${\>/\\!\\!\\!\\!E}/M_{\gamma_{1}\gamma_{2}}$
---|---|---|---|---|---
$10$ | $>125$ | $<1.0$ | $>3.0$ | $>2.5$ | $(13,21)$
$20$ | $>125$ | $<1.0$ | $>2.9$ | $>2.4$ | $(6,9)$
$30$ | same | $<1.5$ | $>2.8$ | $>2.2$ | $(4,6.2)$
$40$ | same | $<2.0$ | $>2.7$ | same | $(3,4.5)$
$60$ | same | same | same | same | $(2,2.6)$
$70$ | $>110$ | same | $>2.3$ | $>1.5$ | $(1.6,2.2)$
$80$ | $>110$ | same | $>2.3$ | $>1.5$ | $(1.4,1.8)$
Table 5: The changes of some event selections with some $m_{a}$ benchmark
points for two isolated photons plus ${\>/\\!\\!\\!\\!E}$ at CEPC where
$\Delta M_{\gamma_{1}\gamma_{2}}\equiv\lvert
M_{\gamma_{1}\gamma_{2}}-m_{a}\rvert$ and ”same” means the same event
selection as the benchmark point $m_{a}=50$ GeV in the main text.

On the other hand, we dynamically optimize some event selections for two
isolated photons plus ${\>/\\!\\!\\!\\!E}$ with different $m_{a}$ at CEPC and
these changes for some benchmark examples are listed in Table. 5. Note we also
changed the setting of $e$ALP mass window selection, $\lvert
M_{\gamma_{1}\gamma_{2}}-m_{a}\rvert<2(1.5)$ GeV for $m_{a}\lesssim 40(20)$
GeV to optimize our analysis for two isolated photons, a backward jet plus
${\>/\\!\\!\\!\\!E_{T}}$ at LHeC. Similarly, we changed the settings of $e$ALP
mass window selection, $\lvert M_{J_{\gamma}}-m_{a}\rvert<1$ GeV,
$0.02<\tau_{1}<0.15$ and $\tau_{3}/\tau_{2}<0.6$ for $m_{a}\lesssim 2$ GeV to
optimize our analysis for a $J_{\gamma}$, a backward jet plus
${\>/\\!\\!\\!\\!E_{T}}$ at LHeC.

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