MI-HET-797 PITT-PACC-2301

# Anomalous Tau Neutrino Appearance from Light Mediators in
Short-Baseline Neutrino Experiments

P. S. Bhupal Dev<EMAIL_ADDRESS>Department of Physics and McDonnell Center
for the Space Sciences, Washington University, St. Louis, MO 63130, USA
Bhaskar Dutta<EMAIL_ADDRESS>Mitchell Institute for Fundamental Physics and
Astronomy, Department of Physics and Astronomy, Texas A&M University, College
Station, TX 77843, USA  Tao Han<EMAIL_ADDRESS>PITT PACC, Department of
Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
Doojin Kim<EMAIL_ADDRESS>Mitchell Institute for Fundamental Physics and
Astronomy, Department of Physics and Astronomy, Texas A&M University, College
Station, TX 77843, USA

###### Abstract

We point out a new mechanism giving rise to anomalous tau neutrino appearance
at the near detectors of beam-focused neutrino experiments, without extending
the neutrino sector. The charged mesons ($\pi^{\pm},K^{\pm}$) produced and
focused in the target-horn system can decay to a (neutrino-philic) light
mediator via the helicity-unsuppressed three-body decays. If such a mediator
carries non-vanishing hadronic couplings, it can also be produced via the
bremsstrahlung of the incident proton beam. The subsequent decay of the
mediator to a tau neutrino pair results in tau neutrino detection at the near
detectors, which is unexpected under the standard three-flavor neutrino
oscillation paradigm. We argue that the signal flux from the charged meson
decays can be significant enough to discover the light mediator signal at the
on-axis liquid-argon near detector of the DUNE experiment, due to the focusing
of charged mesons. In addition, we show that ICARUS-NuMI, an off-axis near
detector of the NuMI beam, as well as DUNE, can observe a handful of tau
neutrino events induced by beam-proton bremsstrahlung.

Introduction. The discovery of neutrino oscillations has proven the non-zero
neutrino masses [1, 2, 3, 4, 5, 6], requiring an extension of the particle
spectrum and/or interactions in the Standard Model (SM). Precise measurements
of the oscillation phenomena allow us to not only determine physics parameters
in the neutrino sector but also obtain hints to beyond-the-Standard-Model
(BSM) physics upon the observation of any deviation from the expectations of
the standard three-flavor oscillation scheme. Upcoming neutrino experiments
are promising along this line as they are expected to measure the neutrino
oscillations more precisely with their high-capability detectors.

Of them, the appearance of tau-flavor neutrinos is receiving growing
attention, especially in the short-baseline neutrino experiments. According to
the neutrino oscillation theory, the $\nu_{\tau}$ appearance probability, in
particular, with regard to the muon neutrino beam, is given by [6]

$\displaystyle
P_{\mu\to\tau}=\sin^{2}(2\theta_{23})\sin^{2}\left[1.27\frac{\left({\Delta
m_{23}^{2}\over{\rm eV}^{2}}\right)\ \left({L\over{\rm km}}\right)}{E/{\rm
GeV}}\right]\,,$ (1)

where $\theta_{23}$ and $\Delta m^{2}_{23}$ are the atmospheric mixing angle
and mass-squared splitting respectively, $L$ is the baseline length and $E$ is
the neutrino energy. For typical baseline ($L\sim$ 0.5 km) and neutrino energy
($E\sim$ 2 GeV) realized in beam-based neutrino experiments, one can see from
Eq. (1) that the chance of observing any $\nu_{\tau}$-induced events in their
near detectors is very small. Heavy meson (such as $D$, $D_{s}$) decays give
$\nu_{\tau}$, but their production rates are negligible in these experiments.
Therefore, the $\nu_{\tau}$ appearance at near detectors is anomalous in
itself and is a ``smoking-gun'' signature of new physics.

In contrast, there are many scenarios for BSM physics that could lead to
$\nu_{\tau}$ in the final state. In the simplest extension, the existence of
sterile neutrinos mixing with $\nu_{\tau}$ would alter the probability shown
in Eq. (1), allowing for non-negligible $\nu_{\tau}$ detection rates at near
detectors. However, non-conventional oscillation phenomena arising in these
scenarios have been constrained by other existing experiments, either using
atmospheric neutrinos such as Super-K [7], IceCube [8], and ANTARES [9], or
using accelerator-produced neutrinos in far detectors of OPERA [10], MINOS
[11], T2K [12], and NO$\nu$A [13]. Therefore, the regions of sterile neutrino
parameter space to be explored by upcoming beam-based neutrino experiments
like DUNE would be limited [14, 15, 16, 17, 18].

In this Letter, we point out the novelty of light dark-sector scenarios
allowing for an anomalous $\nu_{\tau}$ appearance at the near detectors of
beam-based neutrino experiments, without modifying the neutrino sector. In
particular, we discuss the $\nu_{\tau}$ detection prospects at the DUNE near
detector [19] and ICARUS-NuMI [20] experiments in the context of
$\nu_{\tau}$-philic mediators. From an experimental perspective, we address
the need for $\nu_{\tau}$-optimized target-horn configurations, from which any
new physics scenarios allowing for upscattering processes of new particles
($e.g.$, models of inelastic dark matter) may benefit.

Proposal Outline. The main idea is based upon the recent realization that the
exotic three-body decays of charged mesons, such as $\pi^{\pm}$ and $K^{\pm}$,
can be great sources of dark-sector particles in the beam-based neutrino
experiments [21]. We envision a situation where a new mediator, which can be
either a (pseudo)scalar or a (massive) vector, is produced from the three-body
decay of charged mesons and decays to a $\nu_{\tau}$ pair, i.e.,

$\pi^{\pm}/K^{\pm}\to\ell^{\pm}\overset{\scalebox{0.4}{{\hbox{(\mkern-1.0mu-\mkern-1.0mu)}}}}{\nu}_{\ell}\
V\quad{\rm with}\ \ V\to\nu_{\tau}\bar{\nu}_{\tau}.$ (2)

Fig. 1 depicts various decay dynamics, with $V$ representing a massive vector
boson. Such three-body decays of charged mesons are free from the helicity
suppression from which their corresponding two-body decays severely suffer,
and thus their resultant branching ratios (BRs) can be significantly enhanced
despite the extra phase-space suppression in the three-body decays [22, 23,
24, 25, 26]. The BR enhancement can be even larger with a massive (light)
vector mediator, owing to the existence of the longitudinal polarization mode
giving rise to the terms proportional to $m_{\pi,K}^{2}/m_{V}^{2}$ in the
decay width [23]. If $V$ has hadronic couplings, the enhancement can be even
more significant; see Supplement for details.

Figure 1: Various three-body decay dynamics of a charged pion (or kaon) to a
massive vector boson $V$: $(a)$ emission from the charged lepton, $(b)$
emission from the neutrino, $(c)$ emission from the charged pion, and $(d)$
contact interaction and QCD-originating pion-structure-dependent
contributions.

Furthermore, it is important to note that in a beam-focused neutrino
experiment, the charged mesons created by the proton collision on a target
enter the magnetic horn system and get focused before they decay. Therefore,
the focusing feature in combination with the above-discussed BR enhancement
can lead to a substantially enhanced flux of $\nu_{\tau}$'s at the near
detectors, resulting in the surprising emergence of $\nu_{\tau}$-induced
charged-current events.

Example scenarios. The anomalous $\nu_{\tau}$ production arises due to the
couplings of a new mediator to the SM lepton sector. While such mediators can
be either (pseudo)scalar or vector bosons, and our discussion is generically
applicable irrespective of underlying model details, we shall consider models
of a massive vector boson $V$ for concreteness. Focusing on the flavor-
diagonal interactions, we write the relevant interactions as

$\mathcal{L}_{\rm int}\supset\sum_{f}g_{V}x_{f}V_{\mu}\bar{f}\gamma^{\mu}f\,,$
(3)

where $g_{V}$ is the new coupling parameter associated with the vector
mediator $V$ and $x_{f}$ denotes the gauge charge of SM fermion species $f$.
Two benchmark scenarios are considered as follows, depending on whether the
mediator couples to only leptons or to both quarks and leptons:

Case (i): Neutrino-philic model. Here the mediator only couples to neutrinos
at tree level. In this case, $x_{f}=1$ for $f=\nu_{e}$, $\nu_{\mu}$,
$\nu_{\tau}$ and $x_{f}=0$ otherwise. Examples of such models include
$\nu$-philic $U(1)$ [27, 28, 29] and $\nu_{R}$-philic $Z^{\prime}$ [30, 31].
Another popular class of models with the $\nu$-philic nature is that the
mediator couples to both neutrinos and charged leptons at tree level, but not
to quarks. Typical examples are $U(1)_{L_{\alpha}-L_{\beta}}$ models [32, 33,
34]. In this case, the $\nu_{\tau}$ production rate is enhanced, because the
mediator boson can be produced from either the neutrino or the charged-lepton
leg in charged-meson decay. However, this scenario does not give better
sensitivity, compared to the other cases, because of relatively low-energy
$\nu_{\tau}$'s from electron bremsstrahlung. Therefore, we will not show the
results separately for the $U(1)_{L_{\alpha}-L_{\beta}}$ scenario.

Case (ii): $B-L$ model. The mediator can couple to both quarks and leptons,
such as in various $U(1)_{X}$ models [35, 36], $X=B-L$ being the most popular
choice [37, 38]. Although mesons carry no baryon number, they can couple to
the mediator via kinetic mixing (see Supplement). Therefore, the production
rate in this scenario is maximal because $V$ can also be emitted from the
charged-meson leg, in addition to the final-state leptons, as shown in Fig. 1.
Although the constraints on such a light-mediator coupling to quarks are
rather severe (coming from beam-dump experiments), we can still get a sizable
rate at ICARUS-NuMI and/or DUNE with the allowed couplings. Moreover, in some
versions of $U(1)_{X}$ models like in $U(1)_{I_{3R}}$ [39, 40], most of the
low-energy constraints can be weakened or even evaded altogether. Besides the
$B-L$ model with a flavor-universal coupling to both quarks and leptons in Eq.
(3), we also consider another interesting incarnation, i.e. the $B-3L_{\tau}$
model [41], with $x_{f}=1$ for $f={\rm quarks}$, $x_{f}=-3$ for
$f=\tau,\nu_{\tau}$, which is best probed via $\nu_{\tau}$ detection.

We note that a similar idea appeared in Ref. [42] in the pure leptophilic
$Z^{\prime}$ context. As delineated above, our study, however, highlights the
generic BR enhancement feature extended to the mediators carrying hadronic
couplings and hence improved $\nu_{\tau}$ discovery potentials. Furthermore,
we shall show the dependence of the $\nu_{\tau}$ appearance on the target-horn
configuration, motivating the $\nu_{\tau}$-optimized mode of DUNE to exploit
the physics opportunity of this sort.

Benchmark study. To investigate the detection prospects of
$\nu_{\tau}$-induced events, we consider two beam-focused neutrino
experiments, DUNE and ICARUS-NuMI, as concrete examples. In both experiments,
a 120-GeV proton beam strikes a graphite target, and the charged mesons
produced by proton collisions are focused in their magnetic horn systems. The
focused charged mesons are allowed to decay to neutrinos within 204 meters for
DUNE [43] and 715 meters for ICARUS-NuMI [44], before reaching the shielding
and rock area. The DUNE near detector complex consists of three components out
of which we consider the liquid-argon near detector (ND-LAr) that adopts the
liquid argon time projection chamber technology which is also adopted for the
ICARUS-NuMI detector. The DUNE detector is located 575 meters away from the
DUNE target on the beam axis [43], while the ICARUS-NuMI detector is located
$800$ meters away from the NuMI target at an angle of $\sim 6^{\circ}$ from
the NuMI beam line. We summarize some key specifications of both experiments
and their detectors in Table 1.

| DUNE [43] | ICARUS-NuMI [45, 44]
---|---|---
Beam energy | 120 GeV | 120 GeV
Dist. to dump | 204 m | 715 m
Dist. to detector | 575 m | 800 m
Detector angle | On axis | $\sim 6^{\circ}$
Active volume | $3\times 4\times 5$ | $2.96\times 3.2\times 18$
$(w\times h\times l)~{}[{\rm m}^{3}]$ | ($\times$ 2 modules)
Table 1: Key specifications of the DUNE (ND-LAr) and ICARUS-NuMI experiments
and their detectors. The two distances are measured from the beam target, and
the detector angle is measured from the beam axis.

Figure 2: Sensitivities of DUNE-standard-mode (red),
DUNE-$\nu_{\tau}$-optimized-mode (green), and ICARUS-NuMI (blue) to
$\nu_{\tau}$ events expected in ($a$) the $\nu$-philic model, ($b$) $B-L$
model with charged meson form factor parameter Choice I, ($c$) $B-L$ model
with charged meson form factor parameter Choice II, and ($d$) $B-3L_{\tau}$
model with charged meson form factor parameter Choice I. See the text and
supplement for our form factor parameter choices. The gray-shaded regions in
the $B-3L_{\tau}$ and $B-L$ models show the existing limits coming from either
beam-dump, or accelerator/reactor-based neutrino scattering experiments,
compiled in e.g., Refs. [46, 47, 48]. Additional limits (brown-shaded) come
from the search for three-body pion decays $\pi^{+}\to\ell^{+}\nu_{\ell}V$
(PIENU) [49] and three-body kaon decays $K^{+}\to\mu^{+}\nu_{\mu}V$ (NA62)
[50] with $V$ decaying invisibly.

Tagging a $\nu_{\tau}$ event requires the incoming $\nu_{\tau}$ to be
energetic enough (with $E_{\nu}\gtrsim 3.5$ GeV) to upscatter to a tau lepton.
The DUNE Collaboration is investigating the potential of
$\nu_{\tau}$-optimized target-horn configuration to maximize the focusing of
energetic charged mesons that possibly source energetic tau neutrinos [43]. In
our analysis, we also consider the $\nu_{\tau}$-optimized mode of DUNE (in
addition to the standard mode) to assess its impact on detection prospects.

In our simulation study, we first estimate the production rates of charged and
neutral mesons ($\pi^{\pm}$, $\pi^{0}$, $K^{\pm}$, and $\eta$) using the
GEANT4 [51] code package with the QGSP_BIC physics list; we set a 120 GeV
proton beam to impinge on a graphite rod whose specification is similar to
that for DUNE. The focusing-horn effects are modeled, based on simple
assumptions that are described below, and the resulting neutrino spectra are
compared against the expected neutrino spectra that the DUNE and ICARUS
Collaborations have obtained with dedicated simulations.

DUNE ND-LAr is an on-axis detector and the expected neutrino flux is affected
by the magnetic horn configuration. Whether the produced charged mesons are
focused for ND-LAr depends on their production angle and energy. In our
simulation, we assume that if the energy $E$ and angle $\theta$ of a given
charged meson are within certain ranges, its momentum direction gets perfectly
aligned to the beam axis and decays before reaching the shielding area
according to its associated decay law.

We attempt to reproduce the differential $\nu_{\mu}$ fluxes in Ref. [52],
which we denote by $d\Phi_{\nu}^{\rm DUNE}/dE_{\nu}$, using this simplified
simulation scheme. We find that the following combination of two differential
neutrino flux components $d\Phi_{\nu}^{1}/dE_{\nu}$ and
$d\Phi_{\nu}^{2}/dE_{\nu}$ can reproduce $d\Phi_{\nu}^{\rm DUNE}/dE_{\nu}$
fairly well:

$\frac{d\Phi_{\nu}^{\rm DUNE}}{dE_{\nu}}\approx
3\frac{d\Phi_{\nu}^{1}}{dE_{\nu}}+0.2\frac{d\Phi_{\nu}^{2}}{dE_{\nu}}\,,$ (4)

where the prefactors for $d\Phi_{\nu}^{1,2}/dE_{\nu}$ are rescaling factors
and $\Phi_{\nu}^{1,2}$ consist of the neutrinos from the charged mesons
defined by

$\displaystyle\Phi_{\nu}^{1}$ $\displaystyle:$ $\displaystyle
E_{\pi/K}<10~{}{\rm GeV},~{}\theta_{\pi/K}\in[0.01,1]~{}{\rm rad},$ (5)
$\displaystyle\Phi_{\nu}^{2}$ $\displaystyle:$ $\displaystyle
E_{\pi/K}<120~{}{\rm GeV},~{}\theta_{\pi/K}\in[0.01,1]~{}{\rm rad}.$ (6)

In addition, we consider the $\nu_{\tau}$-optimized mode discussed in Ref.
[52], where the $\nu_{\mu}$ fluxes expected at the far detectors (FD) in the
standard and $\nu_{\tau}$-optimized modes are reported. We observe that the
standard-mode $\nu_{\mu}$ fluxes at both ND and FD are very similar to each
other up to a normalization factor. Our simulated differential $\nu_{\mu}$
flux defined by $E_{\pi/K}\in[5,~{}120]~{}{\rm
GeV},~{}\theta_{\pi/K}\in[0.01,~{}1]~{}{\rm rad}$ reproduces the corresponding
flux fairly well when normalized by a rescaling factor of 1.5.

As for ICARUS-NuMI, since the detector is off the beam axis, the neutrino flux
at the detector benefits from the magnetic horn in a limited manner. It turns
out that the assumption of isotropic charged-meson fluxes can describe the
neutrino fluxes reasonably well [53]. We find that $\nu_{\mu}$ spectra can be
described by the $\nu_{\mu}$'s from isotropic charged mesons together with an
overall rescaling factor of 30.

Throughout our study, we assume that these empirical models can describe the
production of our signal $\nu_{\tau}$ entering the DUNE-ND and ICARUS
detectors. In addition to the charged-meson-induced production, we include
contributions from neutral mesons (e.g., $\pi^{0}$ and $\eta$) and the proton
beam if the dark-sector model of interest allows for hadronic interactions. To
describe the mediator emission from an incident proton, we follow the
formalism in Ref. [54].

Once a mediator is produced by the decay of mesons or the proton
bremsstrahlung, we assume it to promptly decay to a $\nu_{\tau}$ pair as per
the BR that the model under consideration predicts. We then check whether the
$\nu_{\tau}$ enters the detector active volume, and estimate the expected
number of $\nu_{\tau}$-induced events for such a $\nu_{\tau}$ by multiplying
the nucleon number density in liquid argon by the charged-current scattering
cross-section for the associated $E_{\nu_{\tau}}$ [55] and the detector
length.

Results. As outlined earlier, we present our results in the context of
$\nu$-philic and $B-L$ interactions. To make the point of $\nu_{\tau}$
appearance, we also adopt a $B-3L_{\tau}$ interaction model. Our results can
straightforwardly be extended to other gauged $U(1)$ models, as well as to
other light (pseudo)scalar states.

In Fig. 2($a$), we show the sensitivities of the DUNE standard mode (red
curves) and ICARUS-NuMI (blue curves) to the coupling $g_{\nu}$ as a function
of mediator mass for the $\nu$-philic case. The detectors are exposed to
$2\times 10^{22}$ POTs for DUNE [56] and $10^{22}$ POTs for ICARUS-
NuMI.111$10^{22}$ POTs correspond to a $\sim 10$ year data collection of DUNE.
A similar level of POTs is achievable for ICARUS-NuMI [57]. To develop our
intuition on the number of $\nu_{\tau}$ events ($N_{\nu_{\tau}}$) allowed by
the model, we show three representative numbers, $N_{\nu_{\tau}}=2.3$ (solid),
$N_{\nu_{\tau}}=10$ (dashed), and $N_{\nu_{\tau}}=50$ (dot-dashed), assuming
that the $\tau$-lepton identification efficiency is 100%. Therefore, one can
assess the realistic number of $\nu_{\tau}$ events by multiplying each of the
numbers by the true identification efficiency. The brown-shaded regions show
the limits from the search for three-body pion decays
$\pi^{+}\to\ell^{+}\nu_{\ell}V$ (PIENU) [49] and three-body kaon decays
$K^{+}\to\mu^{+}\nu_{\mu}V$ (NA62) [50] with $V$ decaying invisibly.

This result suggests that DUNE operating in the standard mode for 20 years may
observe a handful of $\nu_{\tau}$ events if $m_{V}\lesssim 50$ MeV. As briefly
discussed earlier, the DUNE target-horn configuration in the standard mode is
not optimized for the observation of tau neutrinos. Therefore, if DUNE allows
for the $\nu_{\tau}$-optimized mode measurement, $\nu_{\tau}$'s preferentially
reaching the detector are energetic enough to upscatter to $\tau$'s, hence
more $\nu_{\tau}$ events would be detected. To see this potential, we perform
the $\nu_{\tau}$-optimized simulation according to the simulation scheme that
we described earlier and report the expected sensitivity reaches by the green
curves in Fig. 2($a$). For purposes of a fair comparison, the same exposure
and $\tau$ identification efficiency are imposed. As expected, more
$\nu_{\tau}$ events would be observed, and quantitatively $N_{\nu_{\tau}}$
would be increased by up to an order of magnitude over the allowed mass range.

When it comes to the $B-L$ model, hadronic production channels are opened,
viz. (i) the proton beam can radiate a $V$ via bremsstrahlung process, (ii)
$\pi^{\pm}$ and $K^{\pm}$ can internally emit a $V$, and (iii) $\pi^{0}$ and
$\eta$ can decay to $V$ in association with a photon. As mentioned earlier,
since mesons do not carry non-trivial baryon numbers, production (ii) through
hadronic interactions [i.e., Fig. 1($c,d$)] can arise via the usual kinetic
mixing with the SM photon, whereas production (iii) arises through anomaly
hence no kinetic mixing suppression [47]. One may argue that the hadronic
contribution in (ii) would be subdominant. However, some of the parameters
involved in the hadronic interactions in the charged-meson three-body decay
are undetermined [58] and the resulting contributions can be sizable,
depending on the parameter choices. We present the parameter definitions and
technical details in the Supplement. We take two reference hadronic form
factor parameter ($c_{i}$'s) choices for illustration purposes:

$\displaystyle{\rm I}:$ $\displaystyle c_{1}=0.1~{}{\rm
GeV},\,c_{2}=c_{4}=10~{}{\rm GeV}^{-1},$ $\displaystyle{\rm II}:$
$\displaystyle c_{1}=10^{3}~{}{\rm GeV},\,c_{2}=c_{4}=10^{4}~{}{\rm
GeV}^{-1}.$

By contrast, the signal fluxes from (i) and (iii) are proportional to
$g_{B-L}^{2}$. Therefore, for $m_{V}\lesssim 500$ MeV, the contribution from
(ii) can exist even for small $g_{B-L}$ values suppressing the contributions
from (i) and (iii). Fig. 2($b$) shows the sensitivity reaches with Choice I.
Existing limits compiled in e.g., Ref. [47] are shown by the gray region and
the brown regions show the PIENU [49] and NA62 [50] limits interpreted with
parameter choice I. We find that mesonic contributions are subdominant and
proton bremsstrahlung contributions govern the sensitivity reaches, allowing
us to explore a certain range of new parameter space without the
$\nu_{\tau}$-optimized mode of DUNE. On the other hand, the sensitivity
reaches in Fig. 2($c$) suggest that the scenario with parameter choice II
would allow the charged-meson contributions to dominate over the other
contributions. Therefore, the $\nu_{\tau}$-optimized mode measurement (green
curves) would benefit from the charged mesons significantly.

Finally, Fig. 2($d$) shows the sensitivity reaches in the context of the
$B-3L_{\tau}$ model with parameter choice I. Since charged mesons, muons, and
electrons are uncharged under $B-3L_{\tau}$, the $V$ emission from the charged
mesons occurs again via kinetic mixing. Therefore, the charged-meson
contributions with choice I are negligible. The gray region shows the existing
limits compiled in Ref. [47]. The constraints from the neutrino non-standard
interactions (NSI) [48] are somewhat stronger than the other constraints shown
in the plot. However, the NSI constraints are model-dependent and can be
potentially avoided, e.g., in the presence of an additional scalar NSI [59,
60] which could cancel the negative contribution from vector NSI induced by
$V$. Therefore, we omit the NSI limits here. In any case, our results suggest
that ICARUS-NuMI would explore a part of the unexplored regions of parameter
space. DUNE, even without the $\nu_{\tau}$-optimized mode, would extend the
sensitivity reach much further, even surpassing the NSI constraints (if
applicable) and probing up to $m_{V}\sim 3.5$ GeV, before the $Z\to\tau\tau$
constraints take over.

Conclusions and discussion. We investigated the physics potential of an
anomalous $\nu_{\tau}$ appearance at the near detectors of two example beam-
focused neutrino experiments, DUNE and ICARUS-NuMI, in the presence of a dark-
sector mediator $(V)$ interacting with the tau neutrinos. We argued that
charged mesons can be the main source of $\nu_{\tau}$'s as not only the BR of
their three-body decay, e.g.,
$K^{\pm}\to\ell^{\pm}\overset{\scalebox{0.4}{$(\mkern-1.0mu-\mkern-1.0mu)$}}{\nu}_{\ell}\
V$ ($V\to\nu_{\tau}\bar{\nu}_{\tau}$), can be significantly enhanced compared
to their corresponding two-body decay but also the charged-meson flux gets
focused by the magnetic horn system, resulting in a focused $\nu_{\tau}$ flux
toward the detector. For the mediators allowing for hadronic interactions, the
bremsstrahlung of the incident proton beam can be an additional great source
of $\nu_{\tau}$'s, especially when the charged-meson-induced production is
kinematically forbidden. It is thus strongly motivated to search for the
anomalous $\nu_{\tau}$ appearance at the near detectors of the beam-based
neutrino experiments, like ICARUS and DUNE.

To summarize our results (cf. Fig. 2), we may reach a sensitivity for the
coupling $g_{\nu}\sim 10^{-4}$ and $g_{B-L}\sim 10^{-6}$ at $m_{V}\sim 10$ MeV
for the $\nu$-philic and $B-L$ models, respectively, and $g_{B-L}\sim 10^{-4}$
at $m_{V}\sim 1$ GeV for the $B-L$ model. For the $B-3L_{\tau}$ model, DUNE
can extend the coupling sensitivity reach by up to three orders of magnitude
from the current constraints.

Our study is based on a few assumptions at the production and detection of
$\nu_{\tau}$ signals. To estimate the expected $\nu_{\tau}$ flux, we relied on
simple empirical modeling of the focusing effects on the charged mesons and
their decays. For a more precise estimate of not only the $\nu_{\tau}$ signal
but also other BSM signals (e.g., light dark matter), it is highly desired to
perform a dedicated simulation where one keeps track of the charged-meson
behavior in the focusing horn system. Furthermore, the $\nu_{\tau}$-optimized
measurement mode and related simulation study are highly encouraged as it not
only improves the sensitivity to the anomalous $\nu_{\tau}$ signal but also
benefits other BSM signals accompanying upscattering processes (e.g.,
inelastic dark matter).

Regarding the $\nu_{\tau}$ signal detection, we assumed an ideal $\tau$
identification efficiency. In practice, $\tau$'s are detected through their
decay products. Their identification and related backgrounds caused by
misidentification are being carefully investigated by the DUNE Collaboration
in the three major $\tau$-decay channels, in combination with machine learning
techniques [52, 61]. We therefore expect that a more precise assessment of the
detection prospects of anomalous $\nu_{\tau}$ events will be available in the
near future.

Acknowledgments. We thank Antoni Aduszkiewicz, Brian Batell, Matheus Hostert,
Yu Seon Jeong, Pedro Machado, and Alexandre Sousa for useful discussions.
PSBD, BD, and DK acknowledge the warm hospitality provided at PITT PACC, where
part of this work was done. The work of PSBD is supported in part by the U.S.
Department of Energy under Grant No. DE-SC0017987 and by a URA VSP fellowship.
The work of BD and DK is supported by the U.S. Department of Energy Grant DE-
SC0010813. The work of TH is supported in part by the U.S. Department of
Energy under grant No. DE-SC0007914, and by the PITT PACC.

## Supplemental Material

Width of a three-body charged meson decay. For a given charged meson
$\mathfrak{m}$, the width of its three-body decay
$\mathfrak{m}\to\ell\nu_{\ell}V$ ($\ell=e,~{}\mu$) is given by

$\Gamma(\mathfrak{m}\to\ell\nu_{\ell}V)=\frac{1}{64\pi^{3}m_{\mathfrak{m}}}\int_{E_{\ell}^{-}}^{E_{\ell}^{+}}dE_{\ell}\int_{E_{\nu}^{-}}^{E_{\nu}^{+}}dE_{\nu}\overline{\left|\mathcal{M}\right|^{2}},$
(7)

for which the integration ranges are

$\displaystyle E_{\ell}^{-}$ $\displaystyle=$ $\displaystyle
m_{\ell},~{}~{}E_{\ell}^{+}=\frac{m_{\mathfrak{m}}^{2}+m_{\ell}^{2}-m_{V}^{2}}{2m_{\mathfrak{m}}},$
(8) $\displaystyle E_{\nu}^{\pm}$ $\displaystyle=$
$\displaystyle\frac{m_{\mathfrak{m}}^{2}+m_{\ell}^{2}-m_{V}^{2}-2m_{\mathfrak{m}}E_{\ell}}{2(m_{\mathfrak{m}}-E_{\ell}\mp
p_{\ell})}.$ (9)

Case (i): $\nu$-philic model. The mediator $V$ can be emitted only from the
neutrino leg at tree level as shown in Fig. 1($b$) and the resulting matrix
element can be written as

$\displaystyle i\mathcal{M}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}g_{V}V_{\mathrm{C}KM}^{\mathfrak{m}}f_{\mathfrak{m}}\varepsilon^{\mu}\left[\bar{u}_{\ell}p_{\mathfrak{m}}^{\rho}\gamma_{\rho}\frac{\cancel{p}_{V}+\cancel{p}_{\nu}}{(p_{V}+p_{\nu})^{2}}\right.$
(10) $\displaystyle\times$
$\displaystyle\left.\gamma_{\mu}(1-\gamma_{5})v_{\nu}\right]\,,$

where $\varepsilon^{\mu}$ in the polarization vector for the outgoing $V,$
$G_{F}$ is the usual Fermi constant, $f_{\mathfrak{m}}$ is the decay constant
of charged meson species $\mathfrak{m}$, $V_{\rm CKM}^{\mathfrak{m}}$ is
$V_{ud}$ and $V_{us}$ for $\mathfrak{m}=\pi^{\pm}$ and $\mathfrak{m}=K^{\pm}$,
respectively, and $p_{i}$ denotes the four-momentum of particle species $i$.
We then find the spin-averaged matrix element squared to be

$\displaystyle\overline{\left|\mathcal{M}\right|^{2}}$ $\displaystyle=$
$\displaystyle\frac{2(G_{F}f_{\mathfrak{m}}m_{\mathfrak{m}}g_{V}V_{\rm
CKM}^{\mathfrak{m}})^{2}}{m_{V}^{2}(m_{\mathfrak{m}}^{2}+m_{\ell}^{2}-2m_{\mathfrak{m}}E_{\ell})^{2}}\left[8E_{\ell}^{3}m_{\mathfrak{m}}\left\\{m_{\mathfrak{m}}(m_{\mathfrak{m}}-2E_{\nu})-2m_{V}^{2}\right\\}\right.$
(11) $\displaystyle+$ $\displaystyle
4E_{\ell}^{2}\left\\{2m_{V}^{4}-(7m_{\mathfrak{m}}^{2}-4E_{\nu}m_{\mathfrak{m}}+2m_{\ell}^{2})m_{V}^{2}+m_{\mathfrak{m}}(3m_{\mathfrak{m}}^{3}-6E_{\nu}m_{\mathfrak{m}}^{2}+3m_{\ell}^{2}m_{\mathfrak{m}}-4E_{\nu}m_{\ell}^{2})\right\\}$
$\displaystyle-$ $\displaystyle
2E_{\ell}\left\\{4m_{\mathfrak{m}}m_{V}^{4}+4(m_{\ell}^{2}+2m_{\mathfrak{m}}^{2})(E_{\nu}-m_{\mathfrak{m}})m_{V}^{2}+(m_{\ell}^{2}+m_{\mathfrak{m}}^{2})(3m_{\mathfrak{m}}^{3}-6E_{\nu}m_{\mathfrak{m}}^{2}+3m_{\ell}^{2}m_{\mathfrak{m}}-2E_{\nu}m_{\ell}^{2})\right\\}$
$\displaystyle+$
$\displaystyle\left.2(m_{\mathfrak{m}}^{2}-m_{\ell}^{2})m_{V}^{4}-(m_{\mathfrak{m}}^{2}+m_{\ell}^{2})(3m_{\mathfrak{m}}^{2}-4E_{\nu}m_{\mathfrak{m}}-m_{\ell}^{2})m_{V}^{2}+(m_{\mathfrak{m}}^{2}+m_{\ell}^{2})^{2}(m_{\mathfrak{m}}^{2}-2E_{\nu}m_{\mathfrak{m}}+m_{\ell}^{2})\right].$

Case (ii): $B-L$ model. In this case, it is instructive to understand first
the decay structure relevant to models of $B-3L_{\tau}$, a variation of the
$B-L$ scenario. Since none of the initial and final state particles in the
decay process are charged under the $B-3L_{\tau}$ massive vector field in this
model, the vector boson emission occurs through kinetic mixing with the
ordinary SM photon. As $\mathfrak{m}$ and $\ell$ are electrically charged,
Figs. 1($a,c,d$) are relevant. The kinetic mixing parameter is given by

$\epsilon_{V}=\sum_{f}\frac{2\alpha}{\pi}\int_{0}^{1}dx(x-1)x\log\left(\frac{\Lambda^{2}}{m_{f}^{2}+x(x-1)k^{2}}\right),$
(12)

where $f$ runs over all fermion degrees of freedom charged under $V$ and the
SM photon field, $\alpha$ is the usual fine structure constant, and $k$ is the
momentum of the photon and $V$ propagators (see Fig. 3). We evaluate
$\epsilon_{V}$ at $k^{2}=m_{V}^{2}$ and find that the dependence of
$\epsilon_{V}$ on $m_{V}^{2}$ and $\Lambda^{2}$ is mild. We choose a fixed
value of $\epsilon_{V}$ over the relevant $m_{V}$ values for illustration
purposes:

$\epsilon_{V}\approx-0.01.$ (13)

Figure 3: Mixing between the new gauge boson $V$ and the ordinary SM photon
via a SM fermion loop.

Taking the kinetic mixing parameter into account, we find that the most
general matrix element for the decay process at hand can be written as

$\displaystyle i\mathcal{M}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}\epsilon_{V}g_{V}V_{\rm
CKM}^{\mathfrak{m}}\varepsilon_{\mu}\left[\bar{u}_{\ell}\gamma_{\rho}(1-\gamma_{5})v_{\nu}T^{\mu\rho}\right.$
(14) $\displaystyle-$
$\displaystyle\left.f_{\mathfrak{m}}\bar{u}_{\ell}\gamma^{\mu}\frac{\cancel{p}_{\ell}+\cancel{p}_{V}+m_{\ell}}{(p_{\ell}+p_{V})^{2}-m_{\ell}^{2}}p_{\mathfrak{m}}^{\rho}\gamma_{\rho}(1-\gamma_{5})v_{\nu}\right],$

where $T^{\mu\rho}$ is a hadronic tensor including QCD structure-dependent
form factors. The most generic form of $T^{\mu\rho}$ is given by [58]

$\displaystyle T^{\mu\rho}$ $\displaystyle=$ $\displaystyle
c_{1}g^{\mu\rho}+c_{2}(p_{\ell}+p_{\nu})^{\mu}p_{V}^{\rho}+c_{3}(p_{\ell}+p_{\nu})^{\rho}p_{V}^{\mu}$
(15) $\displaystyle+$ $\displaystyle
c_{4}(p_{\ell}+p_{\nu})^{\mu}(p_{\ell}+p_{\nu})^{\rho}+c_{5}p_{V}^{\mu}p_{V}^{\rho}$
$\displaystyle+$ $\displaystyle
F_{V}\epsilon^{\mu\rho\lambda\sigma}(p_{\ell}+p_{\nu})_{\lambda}p_{V,\sigma}\,,$

where $c_{i}$ are form factor parameters or coefficients and $F_{V}$ is the
hadronic form factor of the vector current.

The squared matrix element is not very illuminating, but one can find that the
terms proportional to $c_{3}$ and $c_{5}$ eventually vanish and we end up with
$c_{1}$, $c_{2}$, $c_{4}$, and $F_{V}$. In principle, these dimensionful
parameter values (together with coupling $g_{V}$ that appears as an overall
factor) would be determined by experiments. While they are unknown as of now,
one could attempt to estimate the scales of these parameters. For example, if
$V$ were massless, the Ward identity or the current conservation holds,
resulting in the following relations [58]:

$\displaystyle c_{1}+c_{2}(p_{\ell}+p_{\nu})\cdot p_{V}$ $\displaystyle=$
$\displaystyle f_{\mathfrak{m}},$ (16) $\displaystyle
c_{4}(p_{\ell}+p_{\nu})\cdot p_{V}$ $\displaystyle=$ $\displaystyle
f_{\mathfrak{m}}.$ (17)

A massive $V$ does not necessarily obey the above relations, but one could
still assume $c_{1}\sim f_{\mathfrak{m}}$ and $c_{2}\sim
c_{4}\sim\frac{f_{\mathfrak{m}}}{(p_{\ell}+p_{\nu})\cdot p_{V}}$. In this
context, we take the following reference parameter values (form factor
parameter Choice I):

$c_{1}=0.1~{}{\rm GeV},~{}(c_{2},c_{4},F_{V})=(10,10,0.2)~{}{\rm GeV}^{-1}\,,$
(18)

where the $F_{V}$ value is inferred from the $\pi^{+}\to e^{+}\nu_{e}\gamma$
consideration [62]. The form factor parameter values for $K^{\pm}$ are not
necessarily the same as those for $\pi^{\pm}$, but we set both sets of
parameter values to be the same for purposes of illustration.222We do not use
the photon form factors $F_{V}$ and $F_{A}$ for the dark photon case as in
Ref. [63].

When it comes to the $B-L$ gauge boson case, it can directly couple to the
charged lepton and the neutrino, whereas it can interact with the charged
meson again via kinetic mixing. The resulting matrix element takes the form of

$\displaystyle i\mathcal{M}$ $\displaystyle=$
$\displaystyle-\frac{G_{F}}{\sqrt{2}}g_{V}V_{\rm
CKM}^{\mathfrak{m}}\varepsilon_{\mu}\left[\epsilon_{V}\bar{u}_{\ell}\gamma_{\rho}(1-\gamma_{5})v_{\nu}T^{\mu\rho}\right.$
(19) $\displaystyle-$ $\displaystyle
f_{\mathfrak{m}}\bar{u}_{\ell}\gamma^{\mu}\frac{\cancel{p}_{\ell}+\cancel{p}_{V}+m_{\ell}}{(p_{\ell}+p_{V})^{2}-m_{\ell}^{2}}p_{\mathfrak{m}}^{\rho}\gamma_{\rho}(1-\gamma_{5})v_{\nu}$
$\displaystyle+$
$\displaystyle\left.f_{\mathfrak{m}}\bar{u}_{\ell}p_{\mathfrak{m}}^{\rho}\gamma_{\rho}\frac{\cancel{p}_{\nu}+\cancel{p}_{V}}{(p_{\nu}+p_{V})^{2}}\gamma^{\mu}(1-\gamma_{5})v_{\nu}\right].$

Note that the third term has a sign opposite to the second term because the
charged lepton (anti-charged lepton) comes along with an antineutrino
(neutrino), and therefore, the two terms interfere destructively.

The squared matrix element is not illustrative either, so we again omit the
expression here. When evaluating the partial decay widths, we take the same
form factor parameter values as in Eq. (18). To illustrate the impact of
different parameter choices, we consider a different set of parameter values
(form factor parameter Choice II) as follows:

$c_{1}=10^{3}~{}{\rm GeV},~{}(c_{2},c_{4},F_{V})=(10^{4},10^{4},0.2)~{}{\rm
GeV}^{-1}\,.$ (20)

The two choices (18) and (20) are used to obtain the numerical results shown
in Figs. 2($b,c$), respectively.

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