# Uncovering Secret Neutrino Interactions at Tau Neutrino Experiments

Pouya Bakhti  Meshkat Rajaee  Seodong Shin  Laboratory for Symmetry and
Structure of the Universe, Department of Physics, Jeonbuk National University,
Jeonju, Jeonbuk 54896, Korea

###### Abstract

We investigate the potential of future tau neutrino experiments for
identifying the $\nu_{\tau}$ appearance in probing secret neutrino
interactions, which is very important in a variety of fields such as neutrino
physics, dark matter physics, grand unified theories, astrophysics, and
cosmology. The reference experiments include the DUNE far detector utilizing
the atmospheric data, which is for the first time in probing the secret
interactions, the Forward Liquid Argon Experiment (FLArE100) detector at the
Forward Physics Facility (FPF), and emulsion detector experiments such as
SND@LHC, AdvSND, FASER$\nu$2, and SND@SHiP. For concreteness, we consider a
reference scenario in which the hidden interactions among the neutrinos are
mediated by a single light gauge boson $Z^{\prime}$ with a mass at most below
the sub-GeV scale and an interaction strength $g_{\alpha\beta}$ between the
active neutrinos $\nu_{\alpha}$ and $\nu_{\beta}$. We confirm that these
experiments have the capability to significantly enhance the current
sensitivities on $g_{\alpha\beta}$ for $m_{Z^{\prime}}\lesssim 500$ MeV due to
the production of high energy neutrinos and excellent ability to detect tau
neutrinos. Our analysis highlights the crucial role of downward-going DUNE
atmospheric data in the search for secret neutrino interactions because of the
rejection of backgrounds dominated in the upward-going events. Specifically,
10 years of DUNE atmospheric data can provide the best sensitivities on
$g_{\alpha\beta}$ which is about two orders of magnitude improvement. In
addition, the beam-based experiments such as FLArE100 and FASER$\nu$2 can
improve the current constraint on $g_{e\tau}$ and $g_{\mu\tau}$ by more than
an order of magnitude after the full running of the high luminosity LHC with
the integrated luminosity of 3 ab-1. For $g_{e\mu}$ and $g_{ee}$ the SHiP
experiment can play the most important role in the high energy region of
$E>{\rm few}~{}100$ MeV. Although our analysis is proceeded under our
reference scenario of secret $Z^{\prime}$, our analysis strategies can be
readily applicable to other types of secret interactions such as Majoron
models.

##  I Introduction

Among the particles in the Standard Model (SM), neutrinos are unique in the
sense that they play the key roles not only in determining the weak
interaction structures but also in guiding a new physics beyond the SM (BSM)
due to their oscillation phenomena, not explained in the context of the SM.
Possible new interactions of neutrinos other than the weak interaction,
therefore, can shed light on identifying the symmetrical structure of BSM.

One area of interest is the Secret Neutrino Interaction (SNI), which involves
new boson(s) mediating the interactions among the active and sterile
neutrinos, or involving only the sterile neutrino sector. Secret neutrino
interactions might arise in the BSM theories with the neutrino masses given
from the breakdown of the global symmetries of the SM such as lepton number
($L$) or the difference between the baryon number and the lepton number
($B-L$) symmetries Chikashige et al. (1981); Gelmini and Roncadelli (1981);
Schechter and Valle (1982); Choi and Santamaria (1991); Acker et al. (1992);
Valle and Romao (2015). Other possibilities include gauging an anomaly free
global symmetry Araki et al. (2012); Asai (2020), which is not technically
“secret”, or introducing a new gauge symmetry completely blind to the SM
particles Farzan and Heeck (2016).

The theoretical scenarios providing SNI have been applied to explain the
neutrino oscillation anomalies Asaadi et al. (2018); Smirnov and Valera
(2021); Dentler et al. (2020); Abdallah et al. (2022); Bennett et al. (2006);
Dutta et al. (2022). Interestingly, the SNIs have been also used to resolve
various issues in cosmological and astrophysical observations. The pseudo
Nambu-Goldstone boson arising after the spontaneous break down of a global
lepton number symmetry ($L$ or $B-L$) and the electroweak symmetry, so called
Majoron, can be a dark matter (DM) candidate Rothstein et al. (1993). The
emission of Majorons or vector bosons can also contribute to the supernova
cooling Choi and Santamaria (1990); Akita et al. (2022). Inclusion of SNI can
make the thermal sterile neutrino DM scenario Dodelson and Widrow (1994)
compatible with the astrophysical observations De Gouvêa et al. (2020). A new
gauge boson mediating the SNI can be used to resolve the small scale problems,
albeit with strong cosmological constraints van den Aarssen et al. (2012);
Ahlgren et al. (2013); Chu et al. (2015), or alleviating the Hubble parameter
tension Escudero et al. (2019); Brinckmann et al. (2021); Lyu et al. (2021).

Due to its importance, the investigation of SNI has been rigorously pursued
across multiple domains including cosmological, astrophysical, and laboratory
experiments. Among them, various astrophysical and cosmological observational
results prefer flavor non-universal secret interactions Das and Ghosh (2021);
Brinckmann et al. (2021). Moreover, the laboratory experiments have provided
stronger constraints on the SNIs with $\nu_{e}$ and $\nu_{\mu}$ than those
with $\nu_{\tau}$ Burgess and Cline (1993); Lessa and Peres (2007); Bauer et
al. (2018); Deppisch et al. (2020); Berryman et al. (2023).

Motivated by these observational and experimental preferences, we explore the
scenarios where the SNIs are flavor non-universal and the mediators do not
interact with the charged leptons in this paper. Notably, we focus on the
exciting potential of a variety of future tau neutrino experiments in directly
probing the $\nu_{\tau}$ SNIs with less constraints from the laboratory
experiments so far, compared to the other flavor SNIs.

For concreteness, in this paper, we adopt a light vector SNI scenario where a
vector with the mass below 1 GeV couples exclusively to the SM active
neutrinos, described by the term
$\sum_{\alpha,\beta}g_{\alpha\beta}Z^{\prime}_{\mu}\bar{\nu}_{\alpha}\gamma^{\mu}\nu_{\beta}$,
which provides Non-Standard Interactions (NSIs) involving light mediators
Machado (2016); Farzan and Shoemaker (2016); Farzan and Heeck (2016). A viable
scenario addressing these NSIs is proposed in Ref. Farzan and Heeck (2016),
which is briefly explained in the next section. The light $Z^{\prime}$ can be
produced via three-body rare decays of pseudoscalar mesons, if kinematically
available, which can be important decay channels due to no chiral suppression.
Sensitivities of meson decay experiments to secret couplings of neutrinos to
$Z^{\prime}$ is studied in Ref. Bakhti and Farzan (2017). We note that while
meson decay experiments are sensitive to the sum of the coupling strength
squares involving charged leptons produced in the decay of charged mesons,
i.e., $\sum_{\alpha\in\\{e,\mu,\tau\\}}|g_{e\alpha}|^{2}$ and
$\sum_{\alpha\in\\{e,\mu,\tau\\}}|g_{\mu\alpha}|^{2}$ (by identifying the
produced charged lepton), neutrino detectors can detect produced neutrinos and
are sensitive to each of the couplings $g_{e\alpha}$, $g_{\mu\alpha}$ and
$g_{\tau\alpha}$. In order to obtain conservative sensitivities, we further
assume that both of the production and the decay of $Z^{\prime}$ are
controlled by a single SNI parameter $g_{\alpha\beta}$. In particular, we
discuss the possibility of using the tau neutrino flux measurement to
constrain the coupling of neutrinos with the new light gauge boson. The
upcoming neutrino detectors can benefit from their capability to detect high-
energy neutrinos produced from heavy mesons. Additionally, their abilities of
detecting $\nu_{\tau}$ directly can provide superb sensitivities on the NSI
couplings Bakhti et al. (2022).

As laboratory experiments probing beam-produced neutrinos, we adopt Forward
Liquid Argon Experiment (FLArE100), SND$@$LHC, FASER$\nu 2$, and SHiP for
reference. The FASER Ariga et al. (2019), FASER$\nu$ Abreu et al. (2021), and
SND$@$LHC Ahdida et al. (2020) detectors are currently under operations in the
tunnels located in the beam forward direction nearby ATLAS and have recently
announced their first phase data Abreu et al. (2023); Albanese et al. (2023),
which has opened up an era of intensity frontier searches for BSM at the LHC.
To succeed these experiments, the Forward Physics Facility (FPF) which aims to
host the next generation experiments during the running of the High Luminosity
LHC (HL-LHC) is proposed Feng et al. (2023).

The FPF neutrino experiments are expected to detect high number of neutrino
interactions at the highest energies ever achieved. Thus, their measurements
are crucial to uncover neutrino interactions at energies above
$\mathcal{O}(100\,{\rm GeV})$. The proposed FLArE, a liquid argon time
projection chamber (LArTPC) located at FPF as well as FASER$\nu$2 are designed
to detect millions of neutrino interactions, including tau neutrinos, and to
search for long-lived BSM particles or dark matter. SHiP is an intensity-
frontier beam dump proposed experiment which aims to explore the domain of
weakly interacting hidden light particles with masses in the MeV- GeV range.
Sensitivity of the currently running FASER$\nu$ to secret neutrino interaction
is previously studied in Ref. Bahraminasr et al. (2021).

In addition to the laboratory experiments, we consider Deep Underground
Neutrino Experiment (DUNE) far detector to analyze the atmospheric data in
probing SNI for the first time. We place special emphasis on the sensitivity
of DUNE atmospheric data to SNIs by probing the appearance of downward-going
$\nu_{\tau}$ and discuss how this strategy is strong, as well as providing
valuable insights on the flavor structure of $Z^{\prime}$. Note that DUNE will
have both far detector, probing high energy atmospheric neutrinos, and near
detector for high intensity lower energy neutrinos; we can expect the
interplay between those in probing new interactions of neutrinos. Moreover,
the two detectors’ excellent event reconstruction, angular resolution, and
abilities in identifying $\nu_{\tau}$ directly with track reconstruction can
provide key information on SNI. The potential of DUNE near detector (ND) to
constrain the new interaction is studied in Ref. Bakhti et al. (2019). Similar
study on $\nu_{\tau}$ appearance by a neutrino-philic mediator including those
for the SNI in short-baseline laboratory experiments is recently proceeded in
Ref. Dev et al. (2023).

This paper is organized as follows. In Sec. II, we explain our simplified set-
up from which useful analytic formulas are obtained. In Sec. III, we summarize
the experimental details of our reference tau neutrino experiments and discuss
the analysis strategies. In Sec. IV, we then show our analysis results for
each coupling $g_{\alpha\beta}$ while turning off the others, Finally, we
conclude our results and leave discussion in Sec. V.

##  II Theoretical set-up

In this section, we explain our simplified set-up of vector SNI with a new
sub-GeV mass vector boson $Z^{\prime}$. The relevant effective Lagrangian
includes:

$\displaystyle\mathcal{L}\supset\sum_{\alpha,\beta}g_{\alpha\beta}Z^{\prime}_{\mu}\bar{\nu}_{\alpha}\gamma^{\mu}\nu_{\beta}\,,$
(1)

where $g_{\alpha\beta}$ represents the coupling between the new light boson
$Z^{\prime}$ and neutrinos of flavor $\alpha$ and $\beta$, which does not have
to be flavor-diagonal . This interaction can lead to a new decay mode of meson
to lepton, neutrino, and $Z^{\prime}$, which is followed by a subsequent
$Z^{\prime}$ decay. 111 Note that the off-shell $Z^{\prime}$ production leads
to a four-body decay process for $Z^{\prime}$ heavier than the mother meson.
Due to the extra phase space suppression, we do not consider this contribution
here.

Note that this interaction can arise from gauging different combinations of
baryon number and lepton flavor/number He et al. (1991); Allanach et al.
(2019). However, the coupling of the electron to the new gauge boson
$Z^{\prime}$ is subject to stringent constraints across a wide range of
$Z^{\prime}$ mass and hence our models of interest should suppress the sizable
couplings to the SM charged leptons. A possible scenario giving rise to this
interaction can be obtained from adopting a new gauge symmetry U(1)′ along
with a SM singlet heavy fermion $\Psi$ and a scalar particle both of which are
charged under U(1)′ Farzan and Heeck (2016); Farzan and Tortola (2018). Then
the active neutrinos can couple to $Z^{\prime}$ by mixing with $\Psi$ when the
new scalar particle is either a SM singlet or doublet; the active neutrinos of
flavor $\nu_{\alpha}$ can be written as a linear combination of mass
eigenstates $\nu_{i}$, ($i=1,2,3,4$):

$\displaystyle\nu_{\alpha}=\sum_{i=1}^{4}U_{\alpha i}\nu_{i}\,,$ (2)

where $\nu_{4}$ is the heaviest mass eigenstate that gives the main
contribution to $\Psi$. Integrating out the heavy fourth state, the light
active neutrinos receive a coupling of the form
$g_{\beta\alpha}Z_{\mu}^{\prime}\bar{\nu}_{\beta}\gamma^{\mu}\nu_{\alpha}$
where $g_{\beta\alpha}=g_{\Psi}U_{\alpha 4}U_{\beta 4}^{*}$ and $g_{\Psi}$
being the U(1)′ gauge couping of $\Psi$. Note that a kinetic mixing between
U(1)′ and U(1)Y can generically arise. Hence additional theoretical set-up
should be assumed in such a way that the tree level mixing is turned off and
the loop level mixing is induced by very heavy particles.

Figure 1: Three body meson decay $M\rightarrow l\nu Z^{\prime}$ and subsequent
$Z^{\prime}$ decays into a pair of a neutrino and an anti-neutrino.

Our process of interest is depicted in Fig. 1. In order to focus on the SNI
from our reference scenario, we do not consider the baryonic couplings of
$Z^{\prime}$ throughout the whole analysis. The flux of neutrinos in the lab
frame coming from meson decay, $M\rightarrow Z^{\prime}\nu\ell$ with
subsequent decay of $Z^{\prime}\rightarrow\nu\bar{\nu}$, is given by the
equation:

$\displaystyle\Phi(E_{\nu})=\frac{1}{4\pi
L^{2}}\int_{E_{M}^{min}}^{E_{M}^{max}}dE_{M}P_{M}(E_{M})\left(\frac{dN_{\nu}}{dE_{\nu}}\right)_{lab}\frac{d\Omega_{r.M}}{d\Omega_{lab}}\,.$
(3)

Here, $L$ represents the distance from the source to the detector and
$P_{M}(E_{M})$ is the rate of the meson injection in the lab frame.

The spectrum of the neutrino in the lab frame from the decay of a meson with
an energy of $E_{M}$ is expressed as
$\left(\frac{dN_{\nu}}{dE_{\nu}}\right)_{lab}$, which is related to the
spectrum of neutrinos in the rest frame of meson, $dN_{\nu}/dE_{\nu}|_{r.M}$,
as

$\displaystyle\left(\frac{dN_{\nu}}{dE_{\nu}}\right)_{lab}=\left.\left(\frac{dN_{\nu}}{dE_{\nu}}\right)\right|_{r.M}\frac{\partial
E_{\nu}|_{r.M}}{\partial E_{\nu}|_{lab}}\,.$ (4)

The value of $\left.E_{\nu}\right|_{lab}$ can be simply obtained by
kinematics. If the direction of a neutrino reaching the detector coincides
with that of the spatial momentum of the meson beam, for instance, we can
write $E_{\nu}|_{lab}=E_{\nu}|_{r.M}(1+v_{M})\gamma_{M}$, in which $v_{M}$ is
the meson velocity in the lab frame and $\gamma_{M}=(1-v_{M}^{2})^{-1/2}$,
therefore $\frac{\partial E_{\nu}|_{r.M}}{\partial
E_{\nu}|_{lab}}=\gamma_{M}(1-v_{M}).$ Note that $d\Omega_{r.M}/d\Omega_{lab}$
takes care of focusing of the beam in the direction of the detector and is
given by $(1+v_{M})/(4(1-v_{M}))\simeq\gamma_{M}^{2}$ and
$\left.\left(\frac{dN_{\nu}}{dE_{\nu}}\right)\right|_{r.M}$ is the total
spectrum of the (anti-)neutrino produced from both meson and $Z^{\prime}$
decay:

$\displaystyle\left.\left(\frac{dN_{\nu}}{dE_{\nu}}\right)\right|_{r.M}=\left.\left(\frac{dN_{\nu}}{dE_{\nu}}\right)\right|_{r.M}^{Z^{\prime}decay}+\frac{N_{0}}{\Gamma(M\longrightarrow
l\nu Z^{\prime})}\frac{d\Gamma(M\longrightarrow l\nu
Z^{\prime})}{dE_{\nu}}\,,$ (5)

where $N_{0}$ is the total number of the neutrinos produced from the $M^{+}$
($M^{-}$) decay, which is $\nu_{\alpha}$ ($\bar{\nu}_{\alpha}$) in the left
panel of Fig. 1. For the electron decay mode $M\to e\nu Z^{\prime}$, we
neglect the mass of electron and obtain the decay rate analytically, while the
values for the heavier leptons, $M\to\mu\nu Z^{\prime}$ and $M\to\tau\nu
Z^{\prime}$ are obtained numerically.

As previously stated, the presence of the new light gauge boson $Z^{\prime}$
leads to an enhanced three-body decay rate of the pseudo scalar meson without
chiral suppression compared to the two-body decay cases due to the
longitudinal component of the new massive gauge boson. This enhancement arises
from the new interaction between the new massive gauge boson and neutrinos,
and the decay rate scales as $g_{\alpha\beta}^{2}/m^{2}_{Z^{\prime}}$, which
results from the summation over the new gauge boson polarizations, i.e.,
$\Sigma_{i}\epsilon^{\mu}_{i}(k){\epsilon^{\nu}}^{\star}_{i}(k)=-g^{\mu\nu}+k^{\mu}k^{\nu}/m^{2}_{Z^{\prime}}$.
This phenomenon is analogous to the $W$ boson emission in top quark decay
($t\rightarrow bW$), where the decay rate is proportional to $1/m^{2}_{W}$.
The enhancement in both cases arises from the polarization sum resulting from
the spontaneously broken gauge symmetry. For the decay modes into $e^{\pm}$,
the differential decay rate with polarization perpendicular to the
$Z^{\prime}$ momentum $(\epsilon_{1},\epsilon_{2})$ and parallel to the
$Z^{\prime}$ momentum $(\epsilon_{3})$ can be expressed as:

$\displaystyle\left.\frac{d\Gamma(M\longrightarrow
e\nu_{\alpha}Z^{\prime})}{dE_{Z^{\prime}}}\right|_{1,2}$
$\displaystyle=\frac{f_{M}^{2}g_{e\alpha}^{2}G_{F}^{2}\cos^{2}\left(\theta_{C}\right)}{96\pi^{3}m_{M}}p_{Z^{\prime}}\left(-2E_{Z^{\prime}}m_{M}+m_{M}^{2}+m_{Z^{\prime}}^{2}\right)\,,$
(6) $\displaystyle\left.\frac{d\Gamma(M\longrightarrow
e\nu_{\alpha}Z^{\prime})}{dE_{Z^{\prime}}}\right|_{3}$
$\displaystyle=\frac{f_{M}^{2}g_{e\alpha}^{2}G_{F}^{2}\cos^{2}\left(\theta_{C}\right)}{96\pi^{3}m_{M}m_{Z^{\prime}}^{2}}p_{Z^{\prime}}\left(E_{Z^{\prime}}m_{M}-m_{Z^{\prime}}^{2}\right)^{2}\,,$
(7)

where $f_{M}$ is the meson decay constant. Observing the decay into the
longitudinal mode, we can see that it is proportional to
$g_{e\alpha}^{2}/m_{Z^{\prime}}^{2}$ and will be enhanced for
$m_{Z^{\prime}}\ll m_{M}$. For the case of pion which is dominantly produced
in various accelerators, the total decay rate is given by:

$\displaystyle\begin{split}\Gamma(\pi\longrightarrow
e\nu_{\alpha}Z^{\prime})&=\frac{g_{e\alpha}^{2}G_{F}^{2}\cos^{2}\left(\theta_{C}\right)f_{\pi}^{2}}{6144\pi^{3}m_{\pi}^{3}m_{Z^{\prime}}^{2}}\times\\\
&\quad\left(m_{\pi}^{8}+72m_{\pi}^{4}m_{Z^{\prime}}^{4}-64m_{\pi}^{2}m_{Z^{\prime}}^{6}+24\left(3m_{\pi}^{4}m_{Z^{\prime}}^{4}+4m_{\pi}^{2}m_{Z^{\prime}}^{6}\right)\log\left(\frac{m_{Z^{\prime}}}{m_{\pi}}\right)-9m_{Z^{\prime}}^{8}\right)\,.\end{split}$
(8)

Figure 2: The branching ratio of the meson three-body decay to electron, anti-
neutrino and new gauge boson $\frac{\Gamma(M\rightarrow e\nu
Z^{\prime})}{\Gamma_{\text{total}}^{\text{SM}}}$ as a function of
$m_{Z^{\prime}}$ for different mesons, namely $\pi$, $K$, and $D_{s}$. We have
fixed the value of $g_{ee}=0.1$ $(g_{ee}={10^{-5}})$ in the left (right)
panel.

Figure 2 shows the branching ratio of the meson’s three-body decay to
electron, anti-neutrino, and new gauge boson, $\left(\frac{\Gamma(M\rightarrow
e\nu Z^{\prime})}{\Gamma_{\text{total}}^{\text{SM}}}\right)$, as a function of
$m_{Z^{\prime}}$ for various mesons, including $\pi$, $K$, and $D_{s}$
assuming $g_{ee}=0.1$ (left) and $g_{ee}={10^{-5}}$ (right). For large new
gauge couplings of $\lesssim\mathcal{O}(0.1)$, we can clearly see that the
three-body branching ratios can be dominant over the conventional chiral
suppressed two-body decays and would be easily constrained by various
experiments. As one expects, the branching ratio decreases rapidly as the
gauge boson mass approaches the mass of the charged meson which makes the
process kinematically forbidden.

The $Z^{\prime}$ gauge boson with masses of $\mathcal{O}({\rm MeV}-100\,{\rm
MeV})$ subsequently decays into $\nu\bar{\nu}$ before reaching the detector
producing signals at neutrino detectors over a wide range of
$g_{\alpha\beta}$. The total decay rate of
$Z^{\prime}\longrightarrow\nu_{\alpha}\bar{\nu}_{\beta}$ for all the
polarizations is given by:

$\displaystyle\Gamma(Z^{\prime}\longrightarrow\nu_{\alpha}\bar{\nu}_{\beta})=\frac{g^{2}_{\alpha\beta}m_{Z^{\prime}}}{24\pi}\,.$
(9)

The number of neutrinos originating from the decay of $Z^{\prime}$ particles
before reaching the detector is given by

$\displaystyle N=N_{0}\left(1-e^{-\Gamma L/\gamma}\right)\,,$ (10)

where $N_{0}$ is the number of produced $Z^{\prime}$, $L$ is the distance
between the $Z^{\prime}$ production point and the detector, and
$\gamma=E_{Z^{\prime}}/m_{Z^{\prime}}$ is the boost factor. In Eq. (10), if
$\Gamma L/\gamma\gg 1$, almost all $Z^{\prime}$ particles decay before
reaching the detector.

Before closing this section, let us briefly comment on the cosmological effect
of a light $Z^{\prime}$. For $Z^{\prime}$ with masses
$m_{Z^{\prime}}\lesssim\mathcal{O}({\rm MeV})$, it is inevitable to consider
its contribution to the radiation energy density without Boltzmann suppression
at the time of Big Bang Nucleosynthesis (BBN). In the presence of SNI, the new
gauge boson can be generated via inverse decay $\nu+\bar{\nu}\rightarrow
Z^{\prime}$ and neutrino-antineutrino annihilation $\nu+\bar{\nu}\rightarrow
Z^{\prime}+Z^{\prime}$ Huang et al. (2018). The new gauge boson $Z^{\prime}$
can contribute to the extra radiation species $\Delta N_{\rm eff}$ when it is
in thermal equilibrium with active neutrinos around $T\sim 1~{}{\rm MeV}$
after neutrino decoupling era. 222This is for $Z^{\prime}$ heavier than the
active neutrinos and much lighter neutrino cases are also discussed in Refs.
Forastieri et al. (2015); Berryman et al. (2023). As a conservative limit, we
adopt the constraints on $\Delta N_{\rm eff}$ with 90 % C.L., which is $\Delta
N_{\rm eff}\lesssim 1$, from Ref. Huang et al. (2018) assuming a flavor
universal SNI. The combined constraints with the abundances of the primordial
elements are similar in the reference. We expect the flavor non-universal and
off-diagonal cases involving $\nu_{\tau}$ which is more proper to be applied
in our analysis would provide weaker bounds but we do not pursue this
direction here and leave more detailed study to a future work. Hence we simply
apply the nominal bound on the vector boson for $m_{Z^{\prime}}\lesssim 5$ MeV
from Ref. Huang et al. (2018). Note that possible baryonic interactions of
$Z^{\prime}$ can provide extra constraints on the abundances of the primordial
elements but we do not include those to focus on SNI here, as stated earlier.
Other scenarios such as scalar SNI can have weaker bounds due to the smaller
degrees of freedom.

##  III Analysis strategies

###  III.1 Accelerator based Neutrino Experiments

In this section we explain the details of the reference experiments and the
analysis strategies. The FLArE100, FASER$\nu$2, SND$@$LHC, and SHiP
experiments allow us to probe the relevant parameter space for the relatively
heavy $Z^{\prime}$ up to $\lesssim 1$ GeV since their beam energies are high
enough to produce heavy mesons such as charmed mesons. Moreover, these
detectors have the potential to collect a large number of tau neutrino events
and identify those, providing an opportunity to use tau neutrino flux
measurements in probing possible new interactions of neutrinos. To compute the
number of events, we have taken the neutrino cross section and the energy
spectra of the charged mesons from Ref. Abreu et al. (2020). We further assume
the perfect energy resolutions and $80\%$ efficiencies for $\nu_{e}$ and
$\nu_{\mu}$ in the aforementioned experiments for simplicity.

The FPF is expected to host far-forward experiments such as FASER$\nu$2, a
20-ton emulsion detector; Advanced SND@LHC (AdvSND), a successor to SND@LHC;
and FLArE, a proposed liquid argon time projection chamber with an active
volume of 100 tons Feng et al. (2023). These experiments have potential to
detect millions of TeV-energy neutrinos. The AdvSND features a 5-ton fiducial
mass that represents a substantial increase of 6.25 times compared to the
SND@LHC experiment. Furthermore, it is expected to have the final integrated
luminosity 20 times higher than SND@LHC (150 fb-1), resulting in a total 125
times larger data. We have taken the details of the aforementioned experiments
from Ref. Kling and Nevay (2021).

The Search for Hidden Particles (SHiP) experiment is a proposal of fixed
target facility at the CERN Super Proton Synchrotron (SPS) which aims to
search for light BSM particles with tiny interactions with the SM particles
avoiding the experimental constraints thus far, so called hidden particles
Ahdida et al. (2022). The other main purpose of SHiP is directly observing
$\nu_{\tau}$ and $\bar{\nu}_{\tau}$ events. Benefiting from high statistics it
can perform active neutrino physics. Inside SHiP, a detector called SND$@$SHiP
will be installed for the study of active neutrino cross-sections and angular
distributions. This is expected to be located about 46 m behind the
interaction point and detect about 12000 $\nu_{\tau}$ events within 5 years of
operation, which is quite large compared to FASER$\nu$ experiment detecting
about 11 $\nu_{\tau}$ events by 2023. Interestingly, SHiP hosts a hadron
absorber that light mesons such as pions or Kaons can interact with before
decaying to neutrinos. Hence, the fraction of the charmed meson increases
compared to the lighter mesons. Since the $D_{s}$ meson decays are the main
sources of tau neutrino production at the SPS with the beam energy 400 GeV, it
is possible to have a large tau neutrino flux. Moreover, SHiP will have the
opportunity to observe the tau anti-neutrino for the first time and perform
its cross section measurements. In this respect, SND@SHiP will also be an
excellent experiment searching for the BSM particles interacting with tau
neutrinos.

###  III.2 Atmospheric Neutrino Experiments

In addition to the beam produced neutrinos, atmospheric neutrinos can be used
to set stringent bounds on the new couplings. Although the number of charmed
mesons produced in the atmosphere is smaller than those in the aforementioned
accelerator experiments, it is possible to effectively probe tau neutrinos by
reducing most of the backgrounds remarkably. We adopt DUNE far detector to
confirm our expectation.

All of the current neutrino oscillation experiments explore $\nu_{\mu}$ and
$\nu_{e}$ disappearance and $\nu_{\mu}\rightarrow\nu_{e}$ appearance channels
(plus anti neutrino channels). Both atmospheric and neutrino beam experiments
have confirmed the $\nu_{\mu}\rightarrow\nu_{\tau}$ oscillation by the
disappearance of $\nu_{\mu}$. This is because the reconstruction and
identification of $\nu_{\tau}$ events pose significant challenges due to the
prompt and semi-visible decay of the $\tau$ leptons. In particular, the
misidentified neutral current (NC) scattering of any flavor neutrinos can
mimic the $\tau$ lepton signal and is hard to be rejected. Moreover, the
energy threshold to detect the charged current (CC) scattering of $\nu_{\tau}$
off the matter producing $\tau$ lepton is as high as $E_{\nu_{\tau}}\gtrsim
3.35$ GeV for the nuclear scattering process and $E_{\nu_{\tau}}\gtrsim 3.1$
GeV for the electron scattering process, which are mostly beyond the reach of
the current beam neutrino experiments.

On the other hand, DUNE is expected to have capabilities of identifying and
reconstructing the $\tau$ lepton signals due to the characteristic of the
Liquid Argon Time Projection Chamber (LArTPC) detector with an excellent
position resolution De Gouvêa et al. (2019); Machado et al. (2020). In
particular, the Long-Baseline Neutrino Facility (LBNF) will be equipped with
the 120 GeV Neutrinos at the Main Injector (NuMI) beam providing the center-
of-mass energies well above 3 GeV to observe the $\nu_{\tau}$ CC processes and
the near detector complex will host a variety of detectors which can reduce
the backgrounds Hewes et al. (2021).

In LBNF, the first oscillation maxima for DUNE occurs around 2.5 GeV which is
below but very close to the tau neutrino detection energy threshold. This will
cause some ambiguity in the measurement of $\Delta m^{2}_{31}$ and
$\sin^{2}\theta_{23}$ De Gouvêa et al. (2019). By comparison, DUNE far
detector covering a wide range of $L/E$ and benefiting from large flux can
provide a promising tool to search for $\nu_{\tau}$. Hence atmospheric data
can provide a clearer first oscillation maxima together with an excellent
angular resolution (zenith angle resolution is $\sim 5^{\circ}$ for
$\nu_{\tau}$ CC and $\sim 7^{\circ}$ for NC) and energy resolution Mammen
Abraham et al. (2022), making the atmospheric data advantageous in more
accurate measurements of oscillation parameters.

Notice that upward-going atmospheric neutrinos travelling through a larger
baseline can effectively oscillate into $\nu_{\tau}$. On the other hand, we do
not expect $\nu_{\tau}$ signals within the standard oscillation model for
downward-going atmospheric neutrinos since their baselines are too short to
oscillate. Therefore, the unexpected downward-going $\nu_{\tau}$ appearance
will be a unique signal of non-standard interaction without suffering from
large background contamination in the atmospheric neutrino experiments such as
DUNE far detector. In addition, the far detector of DUNE has much larger
fiducial volume than those in the accelerator based experiments, increasing
its capabilities in searching for non-standard interactions. Note that similar
expectation of the unexpected $\nu_{\tau}$ appearance in the beam produced
short-baseline neutrino experiments is studied in Ref. Dev et al. (2023). In
this study we will explore how atmospheric neutrino data of DUNE can provide
wonderful sensitivities in probing the SNI. The details of the experiments we
have used are given in table 1.

Detector number of events
---
Detector name |  mass |  $\nu_{e}+\bar{\nu}_{e}$ |  $\nu_{\mu}+\bar{\nu}_{\mu}$ |  $\nu_{\tau}+\bar{\nu}_{\tau}$
SND@LHC |  800 kg |  250 |  1000 |  11
FASER$\nu$2 |  20 tonnes |  $7.5\times 10^{4}$ |  4 $\times 10^{5}$ |  $1.7\times 10^{3}$
FLArE100 |  100 tonnes |  $2.5\times 10^{4}$ |  $1.38\times 10^{5}$ |  $1.3\times 10^{4}$
SHiP |  10 tonnes |  $3.4\times 10^{4}$ |  $2.35\times 10^{5}$ |  $1.2\times 10^{4}$
DUNE |  40 kilo-tonnes |  $1.6\times 10^{4}$ |  $2.4\times 10^{4}$ |  $150$
Table 1:  Estimated numbers of standard model neutrino events assuming a final
integrated luminosity of 150 fb-1 for SND@LHC, while 3000 fb-1 for FASER$\nu
2$ and FLArE100. For SHiP, we assume $2\times 10^{20}$ POT in five years. We
assume a data-taking period of 10 years for DUNE atmospheric neutrinos.

##  IV Results

Figure 3:  The upper bound on $g_{e\tau}$ vs. $m_{Z^{\prime}}$ at 90% C.L..
The red curve corresponds to the sensitivity of DUNE with the ten years of
atmospheric data assuming no background. The dotted red curve corresponds to
the DUNE atmospheric data taking into account the background by NC scatterings
of any flavor neutrinos estimated in Ref. Aurisano (2021), which is 70 events
for 10 years. The cyan and purple curves show the sensitivity of FLAE100 and
FASER$\nu 2$ to constrain $g_{e\tau}$, respectively. The blue and curve
corresponds to the SHiP, i.e., SND@SHiP experiment and the green one
corresponds to the Advanced SND$@$LHC experiments. The dashed black curve
shows the bound from DUNE near detector Bakhti et al. (2019). The gray region
shows the BBN constraint Huang et al. (2018). The dark gray and light green
regions shows the current constraint from $Z$ boson decay and NA62,
respectively Laha et al. (2014); Lazzeroni et al. (2013). The light blue
region indicates the constraint from core collapse supernova Akita et al.
(2022).

In this section, we show our analysis results by displaying the expected
sensitivities of various reference experiments in the plane of
$g_{\alpha\beta}-m_{Z^{\prime}}$ along with the current constraints. In all of
our results, we assume a chosen $g_{\alpha\beta}$ is the only non-zero SNI
coupling to make the analysis conservative and simple. From observing the
$\nu_{\tau}$ events, the SNI couplings $g_{\alpha\tau}$ can be directly
probed.

The sensitivities on $g_{\tau\tau}$ can be, in principle, dominantly obtained
from the process of $D_{s}\to\tau\nu_{\tau}Z^{\prime}\to\tau 3\nu_{\tau}$.
However, we expect those are very weak due to the small flux of $D_{s}$ and
the phase space suppression for a given mass of $Z^{\prime}$. On the other
hand, $g_{e\tau}$ and $g_{\mu\tau}$ can be probed in the processes $D_{s}\to
e\nu_{e}Z^{\prime}$ and $D_{s}\to\mu\nu_{\mu}Z^{\prime}$, respectively, again
without chiral suppression compared to $D_{s}\to\tau\nu_{\tau}Z^{\prime}$,
providing more phase space. In addition, those SNIs can be probed from the
lighter meson decays. We estimated that BR($D_{s}\to\tau\nu_{\tau}Z^{\prime}$)
for $g_{\tau\tau}=0.1$ and $m_{Z^{\prime}}\sim 10$ MeV is about $10^{-4}$
times smaller than BR($D_{s}\to e\nu_{e}Z^{\prime}$) in Fig. 2.

In order to show the effectiveness of our analysis strategies observing the
$\nu_{\tau}$ events, we analyze the other SNI couplings, i.e., with $\nu_{e}$
or $\nu_{\mu}$ but not $\nu_{\tau}$. Note that our $Z^{\prime}$ from the
reference models might be further constrained by its baryonic interactions but
we do not include such a possibility as a conservative approach.

Figure 3 displays the $90\%$ C.L. current constraints and future sensitivities
on $g_{e\tau}$ versus $m_{Z^{\prime}}$, while all the other SNI couplings are
set to zero. The analysis takes into account all meson decays, including
$\pi$, $K$, and $D_{s}$ to leptons and neutrinos ($\pi,K,D_{s}\rightarrow
l,\nu$). The green region indicates the current exclusion limit from NA62
Lazzeroni et al. (2013), while the dark and light gray regions represent the
constraints from $Z$ boson decay and BBN, respectively Laha et al. (2014);
Huang et al. (2018). Note that the late decay of
$Z^{\prime}\to\nu_{\alpha}\nu_{\beta}$ prior to the recombination epoch can
possibly contribute to extra $\Delta N_{\rm eff}$ from the observation of
Cosmic Microwave Background (CMB), which will be stronger than the currently
displayed BBN bound. However, the detailed fitting of the CMB data in the
presence of flavor non-universal and off-diagonal SNI is nontrivial and hence
we leave more dedicated study to a future work without displaying those bounds
throughout this work. We can also apply a bound from the observation of the
power spectrum of CMB due to the late neutrino free streaming Archidiacono and
Hannestad (2014); Das and Ghosh (2021); Berryman et al. (2023) but it is far
weaker than that from NA62 for a simple universal couplings case,
$g_{ee}=g_{\mu\mu}=g_{\tau\tau}$. We hence do not show the CMB power spectrum
constraint here. The light blue region indicates the constraint from core-
collapse supernovae Akita et al. (2022).

The plot demonstrates that FLArE100 (cyan curve) and FASER$\nu$2 (purple
curve) can set comparable and the most stringent constraints on $g_{e\tau}$
among future beam experiments. Note that FLArE100 has the largest fiducial
volume with the background events comparable to the much smaller detectors,
SND@SHiP or AdvSND, as can be seen in Table 1. Also, FASER$\nu 2$ has a
smaller fiducial volume but with much smaller number of backgrounds even
compared to FLArE100. With the difference in the shape of the neutrino flux,
the above advantages make FLArE100 and FASER$\nu$2 promising in probing the
$g_{e\tau}$ coupling. For $m_{Z^{\prime}}\gtrsim{\rm few}$ MeV, they can
improve the current constraint by about one order of magnitude, and by more
than one order of magnitude for $m_{Z^{\prime}}\lesssim$ 60 keV. Additionally,
the plot shows that SHiP is more sensitive to the new coupling above a few MeV
compared to SND$@$LHC. This is due to the fact that the fraction of the
produced charm mesons to lighter mesons at SHiP experiment is higher than
SND$@$LHC due to the presence of hadron absorber. It is important to note that
our analysis focuses solely on the far detector, where we expect a similar
neutrino flux as SND@LHC. However, it is worth mentioning that incorporating
the near detector may potentially lead to significant improvements in the
obtained results. As depicted in the plot, Advanced SND@LHC demonstrates a
comparable sensitivity on the parameter $g_{e\tau}$ when compared to the
FASER$\nu 2$ and FLArE100 experiments.

It is remarkable that the atmospheric data at DUNE with 10 years of running
can provide most stringent sensitivities for $m_{Z^{\prime}}\gtrsim 1$ MeV and
$m_{Z^{\prime}}\lesssim 60$ keV. Note that the direction of $\nu_{\tau}$ is
crucial, i.e., the downward-going events determine the sensitivities. The red
solid line corresponds to the zero-background assumption while the red dotted
line shows the inclusions of the background by the NC scatterings of any
flavor downward-going neutrinos expected in Ref. Aurisano (2021), which is 70
events for 10 years. In performing our analysis, we use the Honda atmospheric
neutrino flux model Honda et al. (2004). We assume $100\%$ efficiency for
$\nu_{\tau}$ event reconstruction for simplicity, following the relevant study
Machado et al. (2020). Notice that the efficiency for reconstructing tau
neutrino tracks in DUNE atmospheric data depends on various factors such as
the energy, direction of the neutrino, the properties of the detector, and the
reconstruction algorithms used. Nevertheless, DUNE is designed to have
excellent spatial and angular resolutions. The detector consists of a large
volume of liquid argon, which allows for precise tracking and energy
measurements of particles produced in neutrino interactions. The detector is
also complemented by a highly sophisticated software system for event
reconstruction, which is continually being improved to increase the efficiency
and accuracy of tau neutrino reconstruction. We assume angular resolution
($\Theta_{zen}$ resolution) of $5^{\circ}$ for $\nu_{\tau}$ CC Aurisano
(2021). More dedicated study in this direction in collaboration with the
experimentalists is also possible in the future. Although limited, the DUNE
near detector (ND) can also play roles in observing the SNI, which is shown as
black dashed line following Ref. Bakhti et al. (2019). Note that these
experiments can provide the searches for SNIs complementary to cosmological
(gray) and astrophysical (blue) probes. In some parameter regions such as
$m_{Z^{\prime}}\gtrsim\mathcal{O}({\rm MeV})$ or
$g_{e\tau}\lesssim\mathcal{O}(10^{-7})$ while
$m_{Z^{\prime}}\lesssim\mathcal{O}(10^{-2}\,{\rm MeV})$, the ground based
experiments exhibit better sensitivities.

Figure 4:  The upper bound on $g_{\mu\tau}$ vs. $m_{Z^{\prime}}$ at 90% C.L..
The yellow region corresponds to the current constraint from
$K\rightarrow\mu\nu\nu\nu$ Artamonov et al. (2016) Figure 5: The upper bound
on $g_{ee}$ vs. $m_{Z^{\prime}}$ at 90% C.L..  Figure 6: The upper bound on
$g_{e\mu}$ vs. $m_{Z^{\prime}}$ at 90% C.L..

Figure 4 displays the upper bound at $90\%$ C.L. on $g_{\mu\tau}$ as a
function of $m_{Z^{\prime}}$. As observed in the plot, FLArE100 (cyan curve)
and FASER$\nu$2 (purple curve) can improve the current constraint on
$g_{\mu\tau}$ by more than one order of magnitude for $M_{Z^{\prime}}>{\rm
few}$ MeV and for $M_{Z^{\prime}}<$ 60 keV. Furthermore, SHiP can slightly
enhance the current constraint on $g_{\mu\tau}$ for masses larger than a few
MeV, whereas SND$@$LHC can slightly improve the current bound for masses lower
than few keV. Note that the NA62 bound is not applied here since $g_{\mu\tau}$
is the coupling between $\nu_{\mu}$ and $\nu_{\tau}$. Instead, we applied the
experimental constraints from $K\to\mu\nu_{\mu}$ since the new gauge boson can
be produced from $\nu_{\mu}$ and produce
$K\to\mu\nu_{\tau}\nu_{\mu}\nu_{\tau}$ Artamonov et al. (2016). The
corresponding bound is expressed as the yellow shaded region. Again, we expect
the DUNE atmospheric data with the 10 years of running can provide the best
sensitivity.

For comparison, we include the analysis for $g_{ee}$ and $g_{e\mu}$ in Fig. 5
and 6, respectively. We can clearly see the sensitivities are weaker by about
an order of magnitude than those in Figs. 3 and 4 due to the background
contamination for the other flavors of neutrino events. We have taken the SM
events as the background for $\nu_{e}$ and $\nu_{\mu}$. The number of
$\nu_{e}+\bar{\nu}_{e}$ is $1.6\times 10^{4}$. Also the number of
$\nu_{\mu}+\bar{\nu}_{\mu}$ is $2.4\times 10^{4}$. Note that the shape of the
spectrum is important for $g_{ee}$ and $g_{e\mu}$ and the direction of the
neutrinos is non-critical.

For $m_{Z^{\prime}}<$ few keV, FLArE100 can improve the current constraint on
$g_{ee}$, while SHiP shows better sensitivities for $m_{Z^{\prime}}\gtrsim
400$ MeV due to its higher sensitivity to neutrinos originating from heavy
meson decays, such as $D_{s}$. These results highlight the importance of
studying heavy meson decays to further constrain the coupling of secret
neutrino interactions in the higher $Z^{\prime}$ mass region.

Interestingly, the atmospheric neutrino data in the 10 years of running of the
DUNE far detector can still provide excellent sensitivities better than the
accelerator experiments in most of the parameter space of
$m_{Z^{\prime}}\lesssim 500$ MeV due to the size of the fiducial volume and
the shape of the flux. As the mass of $Z^{\prime}$ approaches to GeV level,
SHiP can be more sensitive since the flux of heavy mesons in the atmosphere
decreases while the large backgrounds of $\nu_{\mu}$ or $\nu_{e}$ are still
not effectively rejected.

In addition, the SNI with $g_{ee}$ can induce the two-neutrino double beta
decay ($2\nu\beta\beta$) which is also expected in the SM, even without lepton
number violating interactions Deppisch et al. (2020). However, the current
bound (cyan shaded region in Fig. 5) is still weaker than the combined
constraints from BBN and NA62. We expect the sensitivities on $g_{e\mu}$ are
even weaker that those on $g_{ee}$ because the production rate of $\nu_{\mu}$
in both atmospheric and accelerator data is higher.

As can be observed from Fig. 3$-$6 atmospheric data of DUNE can set the most
stringent bound on the new coupling among the reference experiments. As
indicated in Fig. 3, atmospheric data is the most sensitive probe on
$g_{e\tau}$ even after including the NC background. Notice that, in our
analysis for atmospheric neutrinos, we have fixed the flux value and did not
include the uncertainty of the shape of the flux. The obtained sensitivities
on $g_{ee}$ and $g_{e\mu}$ can be modified significantly including this
uncertainty. On the other hand, the expected sensitivities on $g_{e\tau}$ and
$g_{\mu\tau}$ are quiet robust with respect to the flux uncertainty since the
standard interactions do not produce downward-going $\nu_{\tau}$. We emphasize
again that these two couplings are sensitive to the direction of the tau
neutrino and the shape of the background.

Finally, it is fair to leave a comment that our $Z^{\prime}$ can induce lepton
flavor violating rare decays such as $\mu\to e\gamma$ in the two loop level.
However, our naive estimation shows the contribution can be negligible for the
couplings below $g_{\alpha\beta}\lesssim\mathcal{O}(10^{-2})$ compared to the
experimental limits so far. More exact calculation is beyond the scope of this
paper and we leave the related study to a future work.

##  V CONCLUSION

The upcoming beam and atmospheric tau neutrino experiments offer a promising
avenue to explore the hidden interactions between neutrinos, whose
identification is highly crucial in various fields, including neutrino
physics, dark matter physics, grand unified theories, astrophysics, and
cosmology. For concreteness we adopted a scenario with a light gauge boson
$Z^{\prime}$ with coupling $g_{\alpha\beta}$ to $\nu_{\alpha}$ and
$\nu_{\beta}$. Our analysis highlights the importance of DUNE atmospheric data
in obtaining the best sensitivities on $g_{\alpha\beta}$ for the $1~{}{\rm
MeV}\lesssim m_{Z^{\prime}}\lesssim 500~{}{\rm MeV}$ mass range as well as for
$m_{Z^{\prime}}\lesssim\mathcal{O}({\rm keV})$, with the potential to improve
the current constraint by up to two orders of magnitude. Notice that we have
assumed angular resolution of $5^{\circ}$ to perform our analysis for the DUNE
atmospheric data, which is a conservative choice. In particular, the downward-
going $\nu_{\tau}$ events, together with the help of exact identification and
reconstruction of tau leptons, can be highly efficient in proging
$g_{\alpha\tau}$ couplings. We observed that including the NC background does
not change our conclusions significantly. Additionally, FLArE100 and
FASER$\nu$2 have the potential to significantly enhance the current bounds on
$g_{e\tau}$ and $g_{\mu\tau}$, while also slightly improving the constraints
on $g_{ee}$ and $g_{e\mu}$. Notably, in the case of $g_{ee}$ and $g_{e\mu}$,
above a few hundred MeV, SHiP is more sensitive in probing the couplings due
to the larger number of produced $D_{s}$ mesons compared to the atmospheric
case and the background contamination by conventional $\nu_{\mu}$ and
$\nu_{e}$ in the atmosphere.

It is worth noting that in our analysis of DUNE atmospheric neutrinos we have
assumed a fixed flux shape. Inclusion of flux shape uncertainty could
significantly modify the obtained sensitivities on $g_{ee}$ and $g_{e\mu}$. On
the other hand, the sensitivities on $g_{e\tau}$ and $g_{\mu\tau}$ will be
highly reliable on the direction of the tau neutrino and the shape of the
background instead, so our results are quite robust on the flux uncertainties.
Our analysis results here can guide future experimental searches for new
physics beyond the Standard Model. It is also important to note that the
sensitivities obtained in this study are based on the reference scenario with
a sub-GeV level new gauge boson. Other theoretically well-motivated models
that predict different types of secret interactions may result in different
sensitivities as well as the astrophysical and cosmological constraints, which
is worth to be studied in a future work. In conclusion, the currently on-going
and future tau neutrino experiments such as DUNE, FLArE100, FASER$\nu$2,
SND$@$LHC, and SND$@$SHiP, have great potential to search for a hidden
interaction between neutrinos mediated by a new light sub-GeV gauge boson. In
this regard, we emphasize on the importance of increasing the efficiency and
accuracy of tau neutrino reconstruction in searching for BSM interacting with
tau neutrinos and encourage the experimental colleagues in this direction.

### Acknowledgments

Authors are grateful to Osamu Seto for useful remarks. Hospitality at APCTP
during the program “Dark Matter as a Portal to New Physics” is kindly
acknowledged. This work is supported by the National Research Foundation of
Korea (NRF-2020R1I1A3072747 and NRF-2022R1A4A5030362).

## References

  * Chikashige et al. (1981) Y. Chikashige, R. N. Mohapatra, and R. D. Peccei, Phys. Lett. B 98, 265 (1981).
  * Gelmini and Roncadelli (1981) G. B. Gelmini and M. Roncadelli, Phys. Lett. B 99, 411 (1981).
  * Schechter and Valle (1982) J. Schechter and J. W. Valle, Physical Review D 25, 774 (1982).
  * Choi and Santamaria (1991) K. Choi and A. Santamaria, Phys. Lett. B 267, 504 (1991).
  * Acker et al. (1992) A. Acker, A. Joshipura, and S. Pakvasa, Phys. Lett. B 285, 371 (1992).
  * Valle and Romao (2015) J. W. F. Valle and J. Romao, _Neutrinos in high energy and astroparticle physics_ (John Wiley & Sons, 2015).
  * Araki et al. (2012) T. Araki, J. Heeck, and J. Kubo, JHEP 07, 083 (2012), eprint 1203.4951.
  * Asai (2020) K. Asai, Eur. Phys. J. C 80, 76 (2020), eprint 1907.04042.
  * Farzan and Heeck (2016) Y. Farzan and J. Heeck, Phys. Rev. D 94, 053010 (2016), eprint 1607.07616.
  * Asaadi et al. (2018) J. Asaadi, E. Church, R. Guenette, B. J. P. Jones, and A. M. Szelc, Phys. Rev. D 97, 075021 (2018), eprint 1712.08019.
  * Smirnov and Valera (2021) A. Y. Smirnov and V. B. Valera, JHEP 09, 177 (2021), eprint 2106.13829.
  * Dentler et al. (2020) M. Dentler, I. Esteban, J. Kopp, and P. Machado, Phys. Rev. D 101, 115013 (2020), eprint 1911.01427.
  * Abdallah et al. (2022) W. Abdallah, R. Gandhi, and S. Roy, JHEP 06, 160 (2022), eprint 2202.09373.
  * Bennett et al. (2006) G. W. Bennett et al. (Muon g-2), Phys. Rev. D 73, 072003 (2006), eprint hep-ex/0602035.
  * Dutta et al. (2022) B. Dutta, D. Kim, A. Thompson, R. T. Thornton, and R. G. Van de Water, Phys. Rev. Lett. 129, 111803 (2022), eprint 2110.11944.
  * Rothstein et al. (1993) I. Z. Rothstein, K. S. Babu, and D. Seckel, Nucl. Phys. B 403, 725 (1993), eprint hep-ph/9301213.
  * Choi and Santamaria (1990) K. Choi and A. Santamaria, Phys. Rev. D 42, 293 (1990).
  * Akita et al. (2022) K. Akita, S. H. Im, and M. Masud, JHEP 12, 050 (2022), eprint 2206.06852.
  * Dodelson and Widrow (1994) S. Dodelson and L. M. Widrow, Phys. Rev. Lett. 72, 17 (1994), eprint hep-ph/9303287.
  * De Gouvêa et al. (2020) A. De Gouvêa, M. Sen, W. Tangarife, and Y. Zhang, Phys. Rev. Lett. 124, 081802 (2020), eprint 1910.04901.
  * van den Aarssen et al. (2012) L. G. van den Aarssen, T. Bringmann, and C. Pfrommer, Phys. Rev. Lett. 109, 231301 (2012), eprint 1205.5809.
  * Ahlgren et al. (2013) B. Ahlgren, T. Ohlsson, and S. Zhou, Phys. Rev. Lett. 111, 199001 (2013), eprint 1309.0991.
  * Chu et al. (2015) X. Chu, B. Dasgupta, and J. Kopp, JCAP 10, 011 (2015), eprint 1505.02795.
  * Escudero et al. (2019) M. Escudero, D. Hooper, G. Krnjaic, and M. Pierre, JHEP 03, 071 (2019), eprint 1901.02010.
  * Brinckmann et al. (2021) T. Brinckmann, J. H. Chang, and M. LoVerde, Phys. Rev. D 104, 063523 (2021), eprint 2012.11830.
  * Lyu et al. (2021) K.-F. Lyu, E. Stamou, and L.-T. Wang, Phys. Rev. D 103, 015004 (2021), eprint 2004.10868.
  * Das and Ghosh (2021) A. Das and S. Ghosh, JCAP 07, 038 (2021), eprint 2011.12315.
  * Burgess and Cline (1993) C. P. Burgess and J. M. Cline, Phys. Lett. B 298, 141 (1993), eprint hep-ph/9209299.
  * Lessa and Peres (2007) A. P. Lessa and O. L. G. Peres, Phys. Rev. D 75, 094001 (2007), eprint hep-ph/0701068.
  * Bauer et al. (2018) M. Bauer, P. Foldenauer, and J. Jaeckel, JHEP 07, 094 (2018), eprint 1803.05466.
  * Deppisch et al. (2020) F. F. Deppisch, L. Graf, W. Rodejohann, and X.-J. Xu, Phys. Rev. D 102, 051701 (2020), eprint 2004.11919.
  * Berryman et al. (2023) J. M. Berryman et al., Phys. Dark Univ. 42, 101267 (2023), eprint 2203.01955.
  * Machado (2016) P. A. N. Machado, AIP Conf. Proc. 1743, 030005 (2016).
  * Farzan and Shoemaker (2016) Y. Farzan and I. M. Shoemaker, JHEP 07, 033 (2016), eprint 1512.09147.
  * Bakhti and Farzan (2017) P. Bakhti and Y. Farzan, Phys. Rev. D 95, 095008 (2017), eprint 1702.04187.
  * Bakhti et al. (2022) P. Bakhti, M. Rajaee, and S. Shin, Phys. Rev. D 106, 115029 (2022), eprint 2206.02594.
  * Ariga et al. (2019) A. Ariga et al. (FASER) (2019), eprint 1901.04468.
  * Abreu et al. (2021) H. Abreu et al. (FASER), Phys. Rev. D 104, L091101 (2021), eprint 2105.06197.
  * Ahdida et al. (2020) C. Ahdida et al. (SHiP) (2020), eprint 2002.08722.
  * Abreu et al. (2023) H. Abreu et al. (FASER), Phys. Rev. Lett. 131, 031801 (2023), eprint 2303.14185.
  * Albanese et al. (2023) R. Albanese et al. (SND@LHC), Phys. Rev. Lett. 131, 031802 (2023), eprint 2305.09383.
  * Feng et al. (2023) J. L. Feng et al., J. Phys. G 50, 030501 (2023), eprint 2203.05090.
  * Bahraminasr et al. (2021) M. Bahraminasr, P. Bakhti, and M. Rajaee, J. Phys. G 48, 095001 (2021), eprint 2003.09985.
  * Bakhti et al. (2019) P. Bakhti, Y. Farzan, and M. Rajaee, Phys. Rev. D 99, 055019 (2019), eprint 1810.04441.
  * Dev et al. (2023) P. S. B. Dev, B. Dutta, T. Han, and D. Kim (2023), eprint 2304.02031.
  * He et al. (1991) X. G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Phys. Rev. D 43, 22 (1991).
  * Allanach et al. (2019) B. C. Allanach, J. Davighi, and S. Melville, JHEP 02, 082 (2019), [Erratum: JHEP 08, 064 (2019)], eprint 1812.04602.
  * Farzan and Tortola (2018) Y. Farzan and M. Tortola, Front. in Phys. 6, 10 (2018), eprint 1710.09360.
  * Huang et al. (2018) G.-y. Huang, T. Ohlsson, and S. Zhou, Phys. Rev. D 97, 075009 (2018), eprint 1712.04792.
  * Forastieri et al. (2015) F. Forastieri, M. Lattanzi, and P. Natoli, JCAP 07, 014 (2015), eprint 1504.04999.
  * Abreu et al. (2020) H. Abreu et al. (FASER), Eur. Phys. J. C 80, 61 (2020), eprint 1908.02310.
  * Kling and Nevay (2021) F. Kling and L. J. Nevay, Phys. Rev. D 104, 113008 (2021), eprint 2105.08270.
  * Ahdida et al. (2022) C. Ahdida et al. (SHiP), Eur. Phys. J. C 82, 486 (2022), eprint 2112.01487.
  * De Gouvêa et al. (2019) A. De Gouvêa, K. J. Kelly, G. V. Stenico, and P. Pasquini, Phys. Rev. D 100, 016004 (2019), eprint 1904.07265.
  * Machado et al. (2020) P. Machado, H. Schulz, and J. Turner, Phys. Rev. D 102, 053010 (2020), eprint 2007.00015.
  * Hewes et al. (2021) V. Hewes et al. (DUNE), Instruments 5, 31 (2021), eprint 2103.13910.
  * Mammen Abraham et al. (2022) R. Mammen Abraham et al., J. Phys. G 49, 110501 (2022), eprint 2203.05591.
  * Aurisano (2021) A. Aurisano, Workshop on Tau Neutrinos from GeV to EeV (2021), talk presented at the University of Cincinnati.
  * Laha et al. (2014) R. Laha, B. Dasgupta, and J. F. Beacom, Phys. Rev. D 89, 093025 (2014), eprint 1304.3460.
  * Lazzeroni et al. (2013) C. Lazzeroni et al. (NA62), Phys. Lett. B 719, 326 (2013), eprint 1212.4012.
  * Archidiacono and Hannestad (2014) M. Archidiacono and S. Hannestad, JCAP 07, 046 (2014), eprint 1311.3873.
  * Honda et al. (2004) M. Honda, T. Kajita, K. Kasahara, and S. Midorikawa, Phys. Rev. D 70, 043008 (2004), eprint astro-ph/0404457.
  * Artamonov et al. (2016) A. V. Artamonov et al. (E949), Phys. Rev. D 94, 032012 (2016), eprint 1606.09054.


# Implementation of a Model of the Cortex Basal Ganglia Loop

Naoya Arakawa1 1 The Whole Brain Architecture Initiative
<EMAIL_ADDRESS>
###### Abstract.

This article presents a simple model of the cortex-basal ganglia-thalamus
loop, which is thought to serve for action selection and executions, and
reports the results of its implementation. The model is based on the
hypothesis that the cerebral cortex predicts actions, while the basal ganglia
use reinforcement learning to decide whether to perform the actions predicted
by the cortex. The implementation is intended to be used as a component of
models of the brain consisting of cortical regions or brain-inspired cognitive
architectures.

###### Key words and phrases:

cognitive neuroscience, whole brain architecture, machine learning,
biologically inspired cognitive architecture

##  1\. Introduction

The cortico-basal ganglia-thalamo-cortical loop is thought to serve for action
selection and execution [1](Chapter 10). Since the loop is common to most
cortical areas, a model of the loop could serve as a component of models of
the entire brain; it could serve not only for computational cognitive
neuroscience, but also for the design of architectural parts for brain-
inspired artificial general intelligence (AGI), as the cerebrum is involved in
a variety of intellectual functions. This article reports the implementation
of a model based on the hypothesis that the cerebral cortex predicts actions,
while the basal ganglia use reinforcement learning to decide whether to
perform the actions predicted by the cortex (details follow).

###  1.1. The Hypothesis

The cerebral cortex selects/predicts actions. The basal ganglia determine
whether the cortex’s selection is suitable for execution and disinhibit the
thalamus to send a Go signal to the cortex at the appropriate timing. The
basal ganglia (BG), as reinforcement learners, receive information on the
state based on the input to the cortex and the action predicted, and decide
whether to perform the action (Go/NoGo). The BG receive rewards and learn the
State/Action $\Rightarrow$ Go/NoGo policy [1]. The cortex learns from executed
State/Action pairs and comes to predict (select) the appropriate State
$\Rightarrow$ Action. The hypothesis reconciles the role of reinforcement
learning and prediction in cerebral action selection.

The following would support the hypothesis.

  * •

The cortex selects/predicts actions:

    * –

The cortex is said to have predictive functions [2].

    * –

The prefrontal cortex is supposed to generate goal representation [3]. Goals
are thought to be generated predictively from inputs in the cortical areas.

  * •

The BG control the timing of the start of an action (Go/NoGo) rather than the
selection of the type of action:

    * –

The BG are supposed to control the timing of motion initiation [4].

    * –

As the matrix part of the thalamus, which is inhibited by the BG, projects to
a relatively wide range of cortical areas [5][6], there may not be enough
“resolution” to distinguish between action options.

    * –

The number of cells in the GPi of the BG, which controls the thalamus, is
smaller than the corresponding number of minicolumns in the cortex (the number
of minicolumns in the human neocortex is in the order of $10^{8}$ ($10^{10}$
neurons in the cortex [7] / $10^{2}$ neurons in a minicolumn [8]), while the
number of neurons in GPi is in the order of $10^{5}$ [9]) (It is arguable
whether mini-columns code action choices).

###  1.2. Things Implemented

The following were implemented.

  * •

Winner-take-all (mechanism that selects from multiple actions)

  * •

Separation of the cortex and BG

  * •

Reinforcement learning in the BG

  * •

Prediction learning in the cortex

##  2\. Implementation

An overview of the implementation for the cortex, thalamus, and BG is
described below.

The implementation code is at https://github.com/rondelion/MinimalCtxBGA.

###  2.1. The Cortex

It consists of the following parts. The cortex receives observation and
outputs action selection.

  1. (1)

Output predictor: Learns to predict actions from observation using the action
‘executed’ by the BG as the teacher signal. A two-layer perceptron was used
for implementation.

  2. (2)

Output moderator: Calculates the output from the output predictor and noise as
input. As the output predictior learns to make correct predictions, the
contribution of noise decreases. Specifically, the mean squared error was used
as the probability for using output prediction.

  3. (3)

Output selector: The largest output from the output moderator is selected
(winner-take-all) as one-hot-vector, and the output is gated with Go/NoGo
signals from the BG.

###  2.2. The Thalamus

The actual thalamus receives inhibitory signals from the BG (NoGo) and sends
Go signals when disinhibited. In the implementation, the Go/NoGo signal from
the BG is directly transmitted to the output selector of the cortex, to omit
the thalamus part (i.e., the thalamus was not implemented).

###  2.3. The Basal Ganglia

The BG were implemented as a monolithic reinforcement learner, where the
concatenation of observation and action selection (one-hot-vector) in the
output selector was used as the state, and reinforcement learning for Go/NoGo
decisions was performed with reward from the environment.

###  2.4. Overall Architecture

Figure 1 shows the outline of the implementation with connections between the
parts.

Figure 1. Overall Architecture

##  3\. Test Environment

A simple delayed reward task in which a reward was given after action
selection was implemented (CBT1Env.py).

  * •

Action selection is accepted while the cue is presented for a certain period
of time. A cue is one of the following: [1,1,0], [0,0,1], [0,1,1], [1,0,0]
Observation outside the presentation period is [0,0.0]

  * •

If the relationship of the cue and action selection satisfies the following,
reward will be given after a delay.

    * –

Cues [1,1,0] or [0,0,1] with action [1,0]

    * –

Cues [0,1,1] or [1,0,0] with action [0,1]

  * •

Penalty: If an output other than [0,0] is produced within the presentation
period and the above reward conditions are not met, a negative reward will be
given as a penalty (-0.7 in the experiment below). If the delay before reward
is greater than 1 unit time, the environment is no longer a Markov Decision
Process and basic reinforcement learning algorithms cannot learn, so the
experiment was conducted with a delay of 1 unit time.

##  4\. Implementation Details

###  4.1. Frameworks used

####  4.1.1. Cognitive architecture description framework

The model was implemented as a module of BriCA (Brain-inspired Computing
Architecture) [10], a computational platform for brain-inspired software
development. As mentioned in Introduction, the cortico-BG-thalamo-cortical
loop is considered to be a generic component, so the model (module) would be
incorporated as part of a larger cognitive architecture. In the experiment, a
cognitive architecture only consisting of the model was constructed and
evaluated.

####  4.1.2. Environment description framework

OpenAI’s Gym (the current version is Gymnasium maintained by the Farama
Foundation) was used. Gym is widely used as a framework for agent learning
environments.

####  4.1.3. Machine learning framework

PyTorch, a widely used machine learning framework, was used. The perceptron
was used for the cortical model, and reinforcement learning was used for the
BG model. For reinforcement learning, the DQN (Deep Q-Network) code from the
PyTorch tutorial was used. Though TensorForce, a reinforcement learning
framework, was also tried, due to problems with its instability and not being
maintained, the final version used the PyTorch DQN.

###  4.2. Learning in the BG model

The state value is the observation + output selector’s selected output
concatenated. The latter is the one-hot-vector before gating. Its action is
binary Go/NoGo to decide whether the entire loop is to take an action based on
the state value. The reward is from the external environment. Action choice
and its Go were carried out only once for each episode. If no penalty was
given, learning would not be successful, as Go would be opted for due to the
positive reward. Figure 2 shows the result of learning (average of 5 trials).

Figure 2. Learning Results
Horizontal axis: the number of episodes (×100)
Red line: average reward when Go is selected
Blue line: average reward
Yellow line: mean squared error of the cortical prediction

Comparison with Reinforcement Learning

Figure 3 shows the results in which DQN (without the architecture being
reported) was used to learn the above task (average of 5 trials). As with the
experiment above, cues were presented in one step and reward delay was one
step.

Figure 3. RL Results
Horizontal axis: the number of episodes (×100)
Vertical axis: average reward

###  4.3. Cortical Learning

With external environmental input, the perceptron learned to predict action
selection when the BG issued a Go. The mean squared error was used as the
probability that the prediction is used as the output candidate for which the
BG part determines Go/NoGo (otherwise the output candidate is randomly
determined). Binary cross entropy (BCE – which is said to be suitable for
binary classification tasks) was used as the loss function of the perceptron.
Stochastic gradient descent (SGD) was used as the optimization algorithm, and
its learning coefficient was set to 0.5 (through some trial and error).

The mean squared error is plotted with the yellow line in Figure 2. The loss
was decreasing as the BG learned. Separate epoch loops (100 times per main
episode) were run inside the cortex to augment the data (as the loss decreases
quickly enough, it may need less augmentary loops).

##  5\. Discussion

###  5.1. Biological Plausibility

The interior structure of the BG and thalamus was not implemented, and thus
not subject to the plausibility evaluation. While the implementation for the
cortical model consists of three parts (output predictor, output moderator,
and output selector), there is no evidence regarding the biological
plausibility for this division. Yet, the output of the cortex is at least
related to Layer 5 of the cortex. It is also known that cortical output is
gated by Go/NoGo signals via the BG and thalamus [4][1]. In the
implementation, random selections of output values in the output moderator and
the use of predictions in the output predictor were used as engineering
hypotheses, to leave them as homework for neuroscience to search for
functionally similar mechanisms.

If an action option corresponds to a cortical (mini-)column, lateral
inhibition should be found between (mini-)columns to select the maximum value
in a winner-takes-all way. It may be realized as inhibition around columns
[11].

As for connections between the cortex and the BG, the implemented cortical
model passes the observation (or cortical state) and the output selected to
the BG model. The former corresponds to the connection from Layers III-Va to
the striatal striosome, the latter to the connection from Layers Vb & VI to
the matrix in Fig.8 of [12].

In the implementation, the cortex learns to predict executed action, and it
could cause fixation on non-optimal solutions. Possible countermeasures
include always making random predictions with a certain probability and using
rewards in learning. The latter could be biologically plausible as dopamine
signals are provided to the cortex [13]. While a reinforcement learner was
used in the BG model, more biologically plausible implementations could be
tried. In particular, the DQN may not be biologically plausible as it uses
tricks such as replay buffers.

###  5.2. Related Topics

Accumulation of evidence is claimed to occur in action decisions made around
the BG [14][15]. The current implementation assumed that the input was
reliable and it does not have an accumulation mechanism. When dealing with
issues involving uncertain input or multiple types of evidence, an
accumulation mechanism should be introduced. The Drift-Diffusion Model [4][16]
to deal with the trade-off between action time pressure and situational
understanding would also have to be implemented.

###  5.3. Future Tasks

The implementation was tested with a minimal delayed reward task. Addressing
more complex tasks such as working memory tasks and tasks with non-Markov
Decision Processes would require the introduction of short-term memory models
in cortical regions, models of broader brain regions including hippocampal
regions, and cortical control over those regions. Wherever the cortical
control is involved, the proposed model could be embedded to be evaluated.

Acknowledgement. This work was partially financed by the Kakenhi project
“Brain information dynamics underlying multi-area interconnectivity and
parallel processing” in FY2021.

## References

  * [1] O’Reilly , R.C. , Munakata , Y. , Frank , M.J. , Hazy , T.E.: Computational Cognitive Neuroscience. Wiki Book, 4th Edition (2020). https://CompCogNeuro.org
  * [2] Millidge, B., et al: Predictive Coding: a Theoretical and Experimental Review, (2021). https://doi.org/10.48550/arXiv.2107.12979
  * [3] Passingham, R. and Wise, S.: The Neurobiology of the Prefrontal Cortex: Anatomy, Evolution, And The Origin Of Insight, Oxford Psychology (2012) https://doi.org/10.1093/acprof:osobl/9780199552917.001.0001
  * [4] Dunovan, K. et al.: Believer-Skeptic Meets Actor-Critic: Rethinking the Role of Basal Ganglia Pathways during Decision-Making and Reinforcement Learning, Front. Neurosci., 10 (2016). https://doi.org/10.3389/fnins.2016.00106
  * [5] Benarroch E.E.: The midline and intralaminar thalamic nuclei: anatomic and functional specificity and implications in neurologic disease. Neurology, 71(12):944-9. (2008) https://doi.org/10.1212/01.wnl.0000326066.57313.13
  * [6] Granger, R.H. and Hearn, R.A.: Models of thalamocortical system. Scholarpedia, 2(11):1796 (2007). http://www.scholarpedia.org/article/Models_of_thalamocortical_system
  * [7] Pakkenberg, B., et al.: Neocortical neuron number in humans: Effect of sex and age, The Journal of Comparative Neurology, 384(2) (1997).
https://doi.org/10.1002/(SICI)1096-9861(19970728)384:2<312::AID-
CNE10>3.0.CO;2-K

  * [8] Sporn, O., et al.: The Human Connectome: A Structural Description of the Human Brain, PLoS Computational BiologyT, 1(4) (2005). http://dx.doi.org/10.1371/journal.pcbi.0010042
  * [9] Thörner G, Lange H, Hopf A.: Morphometrisch-statistische Strukturanalysen des Striatum, Pallidum und Nucleus Subthalamicus beim Menschen. II. Pallidum [Morphometrical-statistical structure analysis of human striatum, pallidus and subthalamic nucleus. II. Globus pallidus]. J Hirnforsch. 16(5):401-13 (1975).
  * [10] Takahashi, K., et al.: A Generic Software Platform for Brain-Inspired Cognitive Computing, Procedia Computer Science, 71:31-37 (2015). https://doi.org/10.1016/j.procs.2015.12.185
  * [11] Helmstaedter, M., Sakmann, B., and Feldmeyer, D.: Neuronal Correlates of Local, Lateral, and Translaminar Inhibition with Reference to Cortical Columns, Cerebral Cortex, 19 (4) (2009). https://doi.org/10.1093/cercor/bhn141
  * [12] Fujiyama, F., et al.: Exclusive and common targets of neostriatofugal projections of rat striosome neurons: a single neuron-tracing study using a viral vector, European Journal of Neuroscience, 33 (4) (2011) https://doi.org/10.1111/j.1460-9568.2010.07564.x
  * [13] Goldman-Rakic, P.S.: The Cortical Dopamine System: Role in Memory and Cognition, Advances in Pharmacology, 42 (1997). https://doi.org/10.1016/S1054-3589(08)60846-7
  * [14] Agarwal, A. et al.: Better Safe than Sorry: Evidence Accumulation Allows for Safe Reinforcement Learning. arXiv:1809.09147 [cs.LG] (2018). https://arxiv.org/abs/1809.09147
  * [15] Bogacz, R. et al. The Basal Ganglia and Cortex Implement Optimal Decision Making Between Alternative Actions, Neural computation, 19. 442-77 (2007). https://doi.org/10.1162/neco.2007.19.2.442
  * [16] Ratcliff, R. and McKoon, G.: The Diffusion Decision Model: Theory and Data for Two-Choice Decision Tasks, Neural Computation, 20 (4) (2008). https://doi.org/10.1162/neco.2008.12-06-420