aainstitutetext: Department of Physics and McDonnell Center for the Space
Sciences, Washington University, St. Louis, MO 63130, USAbbinstitutetext:
Mitchell Institute for Fundamental Physics and Astronomy, Department of
Physics and Astronomy, Texas A&M University, College Station, TX 77845,
USAccinstitutetext: Theoretical Physics Department, Fermi National Accelerator
Laboratory, P. O. Box 500, Batavia, IL 60510, USA ddinstitutetext: Maryland
Center for Fundamental Physics, Department of Physics, University of Maryland,
College Park, MD 20742, USAeeinstitutetext: School of Physics, Southeast
University, Nanjing 211189, China

# Light, Long-Lived $B-L$ Gauge and Higgs Bosons at the DUNE Near Detector

P. S. Bhupal Dev b Bhaskar Dutta c Kevin J. Kelly d Rabindra N. Mohapatra e,a
Yongchao Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

The low-energy $U(1)_{B-L}$ gauge symmetry is well-motivated as part of beyond
Standard Model physics related to neutrino mass generation. We show that a
light $B-L$ gauge boson $Z{{}^{\prime}}$ and the associated
$U(1)_{B-L}$-breaking scalar $\varphi$ can both be effectively searched for at
high-intensity facilities such as the near detector complex of the Deep
Underground Neutrino Experiment (DUNE). Without the scalar $\varphi$, the
$Z{{}^{\prime}}$ can be probed at DUNE up to mass of 1 GeV, with the
corresponding gauge coupling $g_{BL}$ as low as $10^{-9}$. In the presence of
the scalar $\varphi$ with gauge coupling to $Z{{}^{\prime}}$, the DUNE
capability of discovering the gauge boson $Z{{}^{\prime}}$ can be
significantly improved, even by one order of magnitude in $g_{BL}$, due to
additional production from the decay $\varphi\to
Z{{}^{\prime}}Z{{}^{\prime}}$. The DUNE sensitivity is largely complementary
to other long-lived $Z{{}^{\prime}}$ searches at beam-dump facilities such as
FASER and SHiP, as well as astrophysical and cosmological probes. On the other
hand, the prospects of detecting $\varphi$ itself at DUNE are to some extent
weakened in presence of $Z{{}^{\prime}}$, compared to the case without the
gauge interaction.

††preprint: FERMILAB-PUB-21-200-T, MI-TH-218

##  1 Introduction

The $U(1)_{B-L}$ gauge symmetry is a well-motivated extension Davidson:1978pm
; Marshak:1979fm  of the Standard Model (SM), which provides a natural
framework to account for the tiny neutrino masses via the type-I seesaw
mechanism Minkowski:1977sc ; Mohapatra:1979ia ; Yanagida:1979as ;
GellMann:1980vs ; Glashow:1979nm . It may also accommodate a light dark matter
(DM) particle which interacts with the SM particles through the scalar or
gauge portal of $U(1)_{B-L}$ Mohapatra:2019ysk . In this paper, we focus on
the class of $B-L$ extensions of SM, where $B-L$ does not contribute to
electric charge so that its gauge coupling $g_{BL}$ can be arbitrarily small
making the associated gauge boson $Z^{\prime}$ very light with mass in the
sub-GeV range Brehmer:2015cia ; Dev:2016dja ; Chauhan:2018uuy . This is
especially true if the $U(1)_{B-L}$-breaking scale is low, e.g., below the
GeV-scale. In this case, we can expect the corresponding $U(1)_{B-L}$-breaking
scalar (denoted here by $\varphi$) to be also light. Both $Z^{\prime}$ and
$\varphi$ are essential ingredients of the $U(1)_{B-L}$ extension, and play
important role in connecting DM to the SM, in DM phenomenology Okada:2012sg ;
Kaneta:2016vkq ; Klasen:2016qux ; Mohapatra:2019ysk ; Heeba:2019jho ;
Mohapatra:2020bze ; Borah:2020wyc , and in explaining the observed light
neutrino masses Wetterich:1981bx ; Buchmuller:1991ce ; Emam:2007dy ;
Basso:2008iv ; Perez:2009mu ; Basso:2010jm ; Heeck:2014zfa . Since the seesaw
mechanism does not depend directly on the gauge coupling value, such low gauge
coupling models can still accommodate the seesaw mechanism. For
phenomenological exploration of this class of $B-L$ extensions at a high mass
scale and larger gauge coupling range, see, e.g., Refs. Khalil:2006yi ;
Huitu:2008gf ; Accomando:2016sge ; Accomando:2017qcs ; Dev:2017xry ;
Alioli:2017nzr .

If kinematically allowed, both $\varphi$ and $Z^{\prime}$ can be produced and
detected in the high-intensity experiments such as the Deep Underground
Neutrino Experiment (DUNE) Berryman:2019dme . One or both of these particles
may be relatively long-lived and be able to travel from the DUNE target to its
near detector complex where it decays in a striking fashion. It may then be
possible to search for signatures of this class of beyond Standard Model (BSM)
physics in such experiments. Our goal in this work is to determine how these
gauge bosons, scalar bosons, and their interplay can be searched for in the
DUNE experiment at Fermilab. In particular, we will focus on the following
three scenarios: (i) pure $Z^{\prime}$ case with the scalar $\varphi$
decoupled, (ii) the effect of $\varphi\to Z^{\prime}Z^{\prime}$ on the
prospects of $\varphi$ at DUNE, and (iii) improvement of the DUNE sensitivity
of $Z^{\prime}$ boson due to the scalar source $\varphi\to
Z^{\prime}Z^{\prime}$.

The $Z^{\prime}$ boson in the $U(1)_{B-L}$ model couples to all the SM quarks
and leptons and thus can be produced from meson decays such as
$\pi\to\gamma+Z^{\prime}$ and $\eta\to\gamma+Z^{\prime}$ as well as other
hadronic decays where the $\pi^{0}$ and $\eta$ etc are final state particles
in the $pp$ collision Batell:2009di . The proton-proton bremsstrahlung process
is also very important, in particular when the $Z^{\prime}$ boson is heavier
than the $\eta$ meson Blumlein:2013cua ; deNiverville:2016rqh  (cf. Fig. 3).
After being produced, the $Z^{\prime}$ boson may decay via $Z^{\prime}\to
e^{+}e^{-},\,\mu^{+}\mu^{-},\,\nu\bar{\nu}$ and hadrons such as $\pi\pi\pi$
and $KK$ Buschmann:2015awa ; Ilten:2018crw ; Bauer:2018onh , depending on its
mass, and in the small gauge coupling limit can lead to displaced vertices
with decays in the DUNE near detector hundreds of meters away. The observation
of such displaced vertices may provide the key signatures of the light
$Z^{\prime}$ boson. An analogous analysis of a leptonic gauge boson (e.g. one
of $U(1)_{L_{\mu}-L_{\tau}}$) has been performed in Ref. Berryman:2019dme .
Such gauge bosons can lead to anomalous neutrino-electron scattering and
neutrino trident events, which have been explored in Refs. Ballett:2019xoj ;
Altmannshofer:2019zhy . We will perform a thorough study of the $B-L$
$Z^{\prime}$ gauge boson and determine the DUNE near detector sensitivity. It
turns out to be qualitatively different from the leptonic $Z^{\prime}$ boson,
as presented in Fig. 4.

As for the light scalar $\varphi$, if it mixes with the SM Higgs boson, it
will obtain loop-level flavor-changing neutral current (FCNC) couplings to the
SM quarks. As a result, it can be produced from the loop-level FCNC meson
decays such as $K\to\pi+\varphi$ at DUNE Batell:2009jf , and then decay into
the light SM particles $\varphi\to
e^{+}e^{-},\,\mu^{+}\mu^{-},\,\pi\pi,\,\gamma\gamma$, with the decay also
induced by the mixing of $\varphi$ with the SM Higgs. The sensitivity of a
light scalar $\varphi$ without the $Z^{\prime}$ boson at DUNE has been studied
in Ref. Berryman:2019dme .

In the $U(1)_{B-L}$ model, there is the gauge coupling $\varphi
Z^{\prime}Z^{\prime}$, This will induce the extra decay channel $\varphi\to
Z^{\prime}Z^{\prime}$, which could have multiple effects on the prospects of
$\varphi$ and $Z^{\prime}$ at DUNE:

  * •

This new decay channel of $\varphi$ will produce more $Z^{\prime}$ bosons at
DUNE, compared to the pure $Z^{\prime}$ case, even by a factor of $10^{5}$
(see Fig. 6). As a result, the $Z^{\prime}$ can be probed at DUNE with a
smaller gauge coupling $g_{BL}$, with an improvement factor up to 45 (see Fig.
7). This is only possible for a $Z^{\prime}$ mass $m_{Z^{\prime}}\lesssim 200$
MeV. This is because we require $m_{\varphi}>2m_{Z^{\prime}}$, and $\varphi$
heavier than ${\sim}400$ MeV cannot be produced from the meson decay
$K\to\pi+\varphi$ at DUNE.

  * •

The gauge portal scalar decay will not only change the branching fractions
(BRs) of $\varphi$ (cf. Fig. 2), but also shorten its lifetime. This will have
some effects on the search for $\varphi$ at DUNE, depending on the BR of
$\varphi$ decaying into SM particles, i.e. ${\rm BR}(\varphi\to{\rm SM})$. As
presented in Fig. 5, the effect of $\varphi\to Z^{\prime}Z^{\prime}$ is most
significant when $\varphi$ is relatively heavy and the mixing angle
$\sin\vartheta$ is relatively large.

  * •

The simultaneous existence of $\varphi$ and $Z^{\prime}$ will also induce the
associated production of $\varphi$ and $Z^{\prime}$ at DUNE, for instance from
the meson decay $K\to\pi+Z^{\prime}+\varphi$. However, no matter whether the
scalar $\varphi$ is emitted from the $Z^{\prime}$ boson line or from couplings
to mesons, such decays will always be highly suppressed by either $g_{BL}^{4}$
or $\sin^{2}\vartheta g_{BL}^{2}$, and can thus be neglected at DUNE.

The rest of this paper is organized as follows. First, we provide a brief
review of the model in Section 2. Then, we being the discussion of search
strategies in Section 3 with a brief discussion of the experimental setup and
how proton-beam experiments (including the DUNE beamline) are well-suited for
searches of these types of models. We divide the experimental search for this
model by taking the following approach: first, we discuss in Section 3.1 how
the $Z^{\prime}$ gauge boson of $U(1)_{B-L}$ can be searched for in DUNE
making agnostic assumptions about the existence of the associated scalar
$\varphi$. Then, in Section 3.2, we will demonstrate that searches for
$\varphi$ in accelerator beam environments, including DUNE, are weakened in
the presence of $Z^{\prime}$, where the produced $\varphi$ can decay into
$Z^{\prime}$ instead of SM particles. We will also revisit the $Z^{\prime}$
search strategy and discuss how additional fluxes sourced by $\varphi$
(including a brief discussion of associated production of both $\varphi$ and
$Z^{\prime}$) decays can be detected in Section 3.3. This improves the
capability of discovering the gauge boson $Z^{\prime}$. Finally, we conclude
in Section 4. Some details of the $\varphi$ decay calculations are relegated
to Appendix A. Subdominant contributions to $Z^{\prime}$ flux at DUNE are
collected in Appendix B, and more $Z^{\prime}$ limits can be found in Appendix
C.

##  2 The Model

We consider the minimal $U(1)_{B-L}$ model based on the gauge group
$SU(2)_{L}\times U(1)_{Y}\times U(1)_{B-L}$. For the purpose of anomaly
cancellation, three right-handed neutrinos (RHNs) $N_{i}$ (with $i=1,\,2,\,3$)
are naturally introduced. To break the $U(1)_{B-L}$ gauge symmetry, a complex
singlet scalar field $\phi$ is introduced, which carries two units of $B-L$
charge. When the $\phi$-field develops a non-vanishing vacuum expectation
value (VEV) $\langle\phi\rangle=v_{BL}/2\sqrt{2}$, the $U(1)_{B-L}$ gauge
symmetry is spontaneously broken, which generates the $Z^{\prime}$ boson mass
of $m_{Z^{\prime}}=g_{BL}v_{BL}$, where we have defined the covariant
derivative as
$D_{\mu}\phi=\left[\partial_{\mu}-i2g_{BL}Z^{\prime}_{\mu}\right]\phi$. After
the symmetry breaking, expanding the field around its VEV, we obtain

$\displaystyle\phi=\frac{1}{\sqrt{2}}\left(\frac{v_{BL}}{2}+\varphi+i\chi\right)\,.$
(1)

The CP-odd component $\chi$ is “eaten” by the $Z^{\prime}$ boson, while the
CP-even component $\varphi$ is left as a physical scalar field. In presence of
the $(H^{\dagger}H)(\phi^{\dagger}\phi)$ term in the scalar potential ($H$ is
the SM Higgs doublet), the scalar $\varphi$ mixes with the SM Higgs $h$, with
a mixing angle $\sin\vartheta$.

Given the Yukawa coupling

$\displaystyle{\cal L}_{Y}=-y_{N}\phi\overline{N^{C}}N\,,$ (2)

with $C$ standing here for charge conjugate, the RHNs obtain masses
$m_{N}=\frac{1}{\sqrt{2}}y_{N}v_{BL}$, which can be used to generate the tiny
active neutrino masses via type-I seesaw mechanism Minkowski:1977sc ;
Mohapatra:1979ia ; Yanagida:1979as ; GellMann:1980vs ; Glashow:1979nm . For
simplicity, we will assume the RHN masses $m_{N}>m_{Z^{\prime}}/2$ and
$m_{N}>m_{\varphi}/2$ such that the decays $Z^{\prime}\to NN$ and $\varphi\to
NN$ are kinematically forbidden, and we can neglect the contributions of RHNs
to the DUNE prospects of $Z^{\prime}$ and $\varphi$ in this paper. We also
assume for simplicity that there is no $Z-Z^{\prime}$ mixing throughout this
paper, which would otherwise potentially affect the decay branching ratios
(BRs) of the $Z^{\prime}$ boson. Then there are only four free
phenomenological parameters in the minimal $U(1)_{B-L}$ model, i.e.

$\displaystyle m_{Z^{\prime}}\,,\quad m_{\varphi}\,,\quad
g_{BL}\,,\quad\sin\vartheta\,,$ (3)

which we study to investigate the DUNE prospects. To produce the $Z^{\prime}$
boson and/or the $\varphi$ scalar at the DUNE experiment, their masses
$m_{Z^{\prime}}$ and $m_{\varphi}$ have to be at or below the GeV-scale. As we
will show in Figs. 4 and 5 respectively, the gauge coupling $g_{BL}$ and the
mixing $\sin\vartheta$ have to be sufficiently smaller than one to satisfy the
current experimental constraints.

Neglecting the RHN channel, the gauge boson $Z^{\prime}$ decays predominantly
into pairs of SM fermions (and mesons), and we can express the decay widths of
$Z^{\prime}$ into SM fermions as Ilten:2018crw

$\displaystyle\Gamma(Z^{\prime}\to\bar{f}f)=\frac{m_{Z^{\prime}}g_{BL}^{2}N^{C}_{f}S_{f}}{12\pi}\left(1+\frac{2m_{f}^{2}}{m_{Z^{\prime}}^{2}}\right)\sqrt{1-\frac{4m_{f}^{2}}{m_{Z^{\prime}}^{2}}}\,,$
(4)

with $N^{C}_{f}$ being the color factor ($3$ for quarks and $1$ for leptons),
and the symmetry factor $S_{f}=1$ for quarks and charged leptons and $1/2$ for
neutrinos. Decay widths into hadrons from the quark hadronization are more
complicated, but can be expressed using the $R$-ratio (see, e.g., Refs.
Buschmann:2015awa ; Ilten:2018crw ; Bauer:2018onh  for further discussion). In
practice, we use the darkcast code from Ref. Ilten:2018crw  to determine the
$Z^{\prime}$ lifetime and branching ratio into visible final states accessible
in the DUNE near detector. These final states include $e^{+}e^{-}$,
$\mu^{+}\mu^{-}$, and hadronic ones, dominated by $\pi^{+}\pi^{-}\pi^{0}$ and
$K^{+}K^{-}$ states; Fig. 1 presents these branching ratios as a function of
$m_{Z^{\prime}}$.

Figure 1: Decay branching ratios of the $Z^{\prime}$ in $U(1)_{B-L}$ model
into $\nu\bar{\nu}$, $e^{+}e^{-}$, $\mu^{+}\mu^{+}$ and the dominant hadronic
channels. Data obtained using darkcast Ilten:2018crw .

Through mixing with the SM Higgs, the scalar $\varphi$ decays into the SM
leptons, pion pairs and two photons via the SM $W$ boson and charged fermion
loops. The corresponding partial widths are respectively

$\displaystyle\Gamma(\varphi\to\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle\frac{G_{F}m_{\varphi}m_{\ell}^{2}\sin^{2}\vartheta}{4\sqrt{2}\pi}\left(1-\frac{4m_{\ell}^{2}}{m_{\varphi}^{2}}\right)^{3/2}\,,$
(5) $\displaystyle\Gamma(\varphi\to\pi^{+}\pi^{-})$ $\displaystyle=$
$\displaystyle
2\Gamma(\varphi\to\pi^{0}\pi^{0})=\frac{G_{F}m_{\varphi}^{3}\sin^{2}\vartheta}{8\sqrt{2}\pi}\left(1-\frac{4m_{\pi}^{2}}{m_{\varphi}^{2}}\right)^{1/2}\left|G(m_{\varphi}^{2})\right|^{2}\,,$
(6) $\displaystyle\Gamma(\varphi\to\gamma\gamma)$ $\displaystyle=$
$\displaystyle\frac{G_{F}\alpha^{2}m_{\varphi}^{3}\sin^{2}\vartheta}{128\sqrt{2}\pi^{3}}\left|\sum_{f}N_{f}^{C}Q_{f}^{2}A_{1/2}(\tau_{f})+A_{1}(\tau_{W})\right|^{2}\,,$
(7)

wherer $G_{F}$ is the Fermi constant, $\alpha\equiv e^{2}/4\pi$ is the fine-
structure constant, and $Q_{f}$ is the charge of the fermion $f$ in units of
the proton charge $e$. In Eq. (6), $G(m_{\varphi}^{2})$ is a dimensionless
transition amplitude defined in Appendix A. Similarly in Eq. (7),
$A_{1/2}(\tau_{f})$ and $A_{1}(\tau_{W})$ respectively (with
$\tau_{X}=m_{\varphi}^{2}/4m_{X}^{2}$) are standard fermion and $W$ loop
functions for the Higgs decay, also given in Appendix A for completeness.

As a result of the gauge coupling of $\varphi$ to $Z^{\prime}$ bosons, if
kinematically allowed, we have also the decay channel

$\displaystyle\Gamma(\varphi\to Z^{\prime}Z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{g_{BL}^{2}m_{\varphi}^{3}}{32\pi
m_{Z^{\prime}}^{2}}\left(1-\frac{4m_{Z^{\prime}}^{2}}{m_{\varphi}^{2}}\right)^{1/2}\left(1-\frac{4m_{Z^{\prime}}^{2}}{m_{\varphi}^{2}}+\frac{12m_{Z^{\prime}}^{4}}{m_{\varphi}^{4}}\right)\,.$
(8)

As shown in Eqs. (5) - (7), all the decay channels of $\varphi$ into the SM
particles are universally proportional to the mixing angle $\sin\vartheta$,
while the $Z^{\prime}$ boson channel in Eq. (8) is dictated by the gauge
coupling $g_{BL}$. In the limit of $g_{BL}\to 0$ or
$m_{\varphi}<2m_{Z^{\prime}}$, the scalar $\varphi$ only decays into the SM
particles, which is equivalent to the case of singlet extension of the SM. In
presence of the $U(1)_{B-L}$ gauge couplings and the $Z^{\prime}$ boson, the
decay products of $\varphi$ are changed dramatically, depending on the values
of $m_{Z^{\prime}}$, $\sin\vartheta$ and $g_{BL}$. The BRs of $\varphi$ for
the following two Benchmark Points (BP) of $m_{Z^{\prime}}$ and $g_{BL}$ below
are shown in Fig. 2:

$\displaystyle{\rm BP1}:$ $\displaystyle m_{Z^{\prime}}=30\;{\rm MeV}\,,\quad
g_{BL}=5\times 10^{-9}\,,$ (9) $\displaystyle{\rm BP2}:$ $\displaystyle
m_{Z^{\prime}}=140\;{\rm MeV}\,,\quad g_{BL}=5\times 10^{-9}\,,$ (10)

which correspond to the two stars in Fig. 7.

Figure 2: Branching ratios of the scalar $\varphi$ in the $U(1)_{B-L}$ model
into $ee$, $\mu\mu$, $\gamma\gamma$, $\pi\pi$ (including both $\pi^{0}\pi^{0}$
and $\pi^{+}\pi^{-}$) and $Z^{\prime}Z^{\prime}$, as a function of the scalar
mass. In both panels, we take $g_{BL}=5\times 10^{-9}$ and
$\sin^{2}\vartheta=10^{-4}$. The left (right) panel assumes $m_{Z^{\prime}}=$
30 MeV ($140$ MeV).

##  3 Searches for Scalars and Gauge Bosons at DUNE

The Fermilab DUNE facility uses a 120 GeV proton beam striking a graphite
target to generate the high-intensity neutrino beam. Of interest for this
work, the proton collisions produce various hadronic final states such as
pions, Kaons, $\eta$ mesons, etc. Specifically, we are focused on neutral and
charged pions, Kaons and $\eta$ mesons. In their decays, these can produce the
new BSM particles $Z^{\prime}$ and $\varphi$. The proton beam is expected to
deliver at least $1.47\times 10^{21}$ protons on target (POT) per year, with
potential upgrades throughout the lifecycle of the experiment. We will assume
a nominal exposure of $1.47\times 10^{22}$ POT for our analyses,
conservatively corresponding to at most ten years of data collection.

The DUNE near detector complex AbedAbud:2021hpb , including several detector
components, is located 574 meters away from the target and therefore when the
decays of the new particles $Z^{\prime}$ and $\varphi$ occur far away from the
production vertex, the final states can give a signal in the DUNE near
detector. This setup is ideal in probing the parameter space in which $g_{BL}$
and the new particle masses are small, and therefore they are long-lived.

As detailed in Section 3, in our $U(1)_{B-L}$ model,
$\pi^{0},\eta\to\gamma+Z^{\prime}$ provides the source for the new BSM
particle $Z^{\prime}$. This decay will take place promptly, effectively in the
DUNE target, and will be our source for $Z^{\prime}$. There can also be direct
bremsstrahlung production of $Z^{\prime}$ from $pp\to pp+Z^{\prime}$ since
protons have nonzero $B-L$ quantum number. The $\varphi$ production arises
mostly from the decays such $K^{+}\to\pi^{+}+\varphi$ and
$K_{L}\to\pi^{0}+\varphi$. Whether we are looking for $\varphi$ or
$Z^{\prime}$ signatures in the near detector, we will be interested in final
states including opposite-sign pairs of charged particles. This includes
$e^{+}e^{-}$, $\mu^{+}\mu^{-}$, and pionic final states. The DUNE capability,
specifically using its gaseous argon near detector (immediately downstream of
the liquid argon near detector) to identify final-states of particles decaying
in flight has been detailed in Refs. Ballett:2019bgd ; Berryman:2019dme ;
Coloma:2020lgy ; Kelly:2020dda ; Brdar:2020dpr ; Dev:2021ofc ;
Breitbach:2021gvv . We will use the results of these works and consider that a
background-free search for these final states is possible.

As discussed in Section 1, our search strategy naturally divides into three
different categories based on the (non)existence of either the scalar
$\varphi$ and gauge boson $Z^{\prime}$. Phenomenologically speaking, we are
interested in the four free parameters in Eq. (3). We divide the search
strategies by considering the following cases:

  * •

Pure $Z^{\prime}$ case: $\sin\vartheta=0$ – if a scalar boson exists at all,
it is decoupled and does not have any impact on the phenomenology of
$Z^{\prime}$. This is explored in Section 3.1.

  * •

Pure $\varphi$ case: $g_{BL}=0$ – if a gauge boson exists at all, it is
decoupled and does not impact the phenomenology of $\varphi$. This has been
studied in Ref. Berryman:2019dme , and will not be detailed in this paper any
more.

  * •

Combined $Z^{\prime}$ and $\varphi$ case (effects of $\varphi\to
Z^{\prime}Z^{\prime}$): If both particles are relevant at DUNE and
$m_{\varphi}>2m_{Z^{\prime}}$ then the decay channel $\varphi\to
Z^{\prime}Z^{\prime}$ can impact the prospects of the DUNE search for
$\varphi$ – we investigate this effect for DUNE and other $\varphi$ searches
in Section 3.2.2. Furthermore, this can also provide additional flux of
$Z^{\prime}$ at the DUNE near detector. We can also have associated production
of both $Z^{\prime}$ and $\varphi$ in some decay channels in this combined
case. This additional $Z^{\prime}$ flux allows for increased sensitivity as a
function of $m_{Z^{\prime}}$ and $g_{BL}$, subject to current constraints on
$m_{\varphi}$ and $\sin\vartheta$. This is discussed in Section 3.3 and is the
main emphasis of this paper.

###  3.1 Pure Gauge Boson Search

Here we discuss the case in which the only new physics particle is the gauge
boson $Z^{\prime}$ with a mass $m_{Z^{\prime}}$ and gauge coupling $g_{BL}$.
In the context of a search in a proton beam-dump environment, this scenario is
very similar to the study performed for a $U(1)$ dark photon in Ref.
Berryman:2019dme , with a mapping of the kinetic mixing parameter
$\varepsilon$ onto this gauge coupling. This remapping is discussed for proton
beam-dump experiments, as well as other experimental contexts, in Ref.
Bauer:2018onh . In the following subsections, we discuss $Z^{\prime}$
production in Section 3.1.1 and DUNE sensitivity in Section 3.1.2; expressions
for the decay of $Z^{\prime}$ can be found in Section 2.

####  3.1.1 Production of Gauge Bosons

The new gauge boson $Z^{\prime}$ can be produced in the same ways as the
$U(1)$ dark photon – decays of neutral pseudoscalar mesons
$\mathfrak{m}\to\gamma Z^{\prime}$, and proton bremsstrahlung $pp\to
ppZ^{\prime}$. We give expressions for the flux of these $Z^{\prime}$ at the
DUNE near detector here.

The neutral pseudoscalar meson production flux is determined using

$\displaystyle\Phi_{\mathfrak{m}Z^{\prime}}$
$\displaystyle=\frac{c_{\mathfrak{m}}N_{\rm POT}}{A_{\rm
Det.}}\epsilon(m_{Z^{\prime}})\mathrm{BR}\left(\mathfrak{m}\to\gamma
Z^{\prime}\right),$ $\displaystyle=\frac{c_{\mathfrak{m}}N_{\rm POT}}{A_{\rm
Det.}}\epsilon(m_{Z^{\prime}})\left[2\left(\frac{g_{BL}^{2}}{e^{2}}\right)\mathrm{BR}\left(\mathfrak{m}\to\gamma\gamma\right)\left(1-\frac{m_{Z^{\prime}}^{2}}{m_{\mathfrak{m}}^{2}}\right)^{3}\right].$
(11)

The quantities in Eq. (11) are as follows: $c_{\mathfrak{m}}$ is the average
number of mesons $\mathfrak{m}$ produced in a given proton-on-target (POT)
collision; $N_{\rm POT}$ is the POT number considered in the experimental
exposure; $A_{\rm Det.}$ is the detector area as viewed by an incoming
particle; $\epsilon(m_{Z^{\prime}})$ is the geometrical acceptance factor of
the $Z^{\prime}$ particles emerging from the decay $\mathfrak{m}\to\gamma
Z^{\prime}$, determined using Monte Carlo simulations; $e$ is the electric
charge, $m_{\mathfrak{m}}$ the mass of $\mathfrak{m}$, and
$\mathrm{BR}\left(\mathfrak{m}\to\gamma\gamma\right)$ is the SM branching
ratio of the meson $\mathfrak{m}$ into two photons. We find that the dominant
contribution to $\Phi_{\mathfrak{m}Z^{\prime}}$ comes from $\pi^{0}$ and
$\eta$ mesons.

For proton-proton bremsstrahlung, we use the calculations of Refs.
Blumlein:2013cua ; deNiverville:2016rqh  with specifics for DUNE discussed in
Ref. Berryman:2019dme . The total flux from this bremsstrahlung process is

$\Phi_{\mathrm{Brem}Z^{\prime}}=\frac{N_{\rm POT}}{A_{\rm
Det.}\sigma_{pN}(s)}\left\lvert
F_{1,N}(m_{Z^{\prime}}^{2})\right\rvert^{2}\int\int_{\rm
Det.}\mathrm{d}z\>\mathrm{d}p_{T}^{2}\>\sigma_{pN}\left(2m_{p}(E_{p}-E_{Z^{\prime}})\right)w_{ba}(z,p_{T}^{2})\,.$
(12)

Here, $\sigma_{pN}$ is the total proton-target cross section evaluated at
$s=2E_{p}m_{p}$ with $E_{p}=120$ GeV being the DUNE beam energy. $F_{1,N}$ is
a form factor allowing for mixing between $Z^{\prime}$ and the SM vector
mesons, specifically when $m_{Z^{\prime}}$ approaches $m_{\rho}$. The two
integrals are performed over the variables $p_{T}^{2}$ (the transverse
momentum squared of the outgoing $Z^{\prime}$) and $z$, the fraction of the
incoming proton’s initial momentum that is transferred to the longitudinal
momentum of the outgoing $Z^{\prime}$, where we label “Det.” to indicate that
the range of integration of these two is such that the outgoing $Z^{\prime}$
is pointing from its production point to the front face of the near detector.
The photon splitting function $w_{ba}$ is given in Refs. Blumlein:2013cua ;
deNiverville:2016rqh ; Berryman:2019dme  and, for the case of a $U(1)_{B-L}$
gauge boson, is proportional to $g_{BL}^{2}/4\pi$.

Other production mechanisms for $Z^{\prime}$ are possible, however we find
that all of them are subdominant to the neutral pseudoscalar meson decays and
proton-proton bremsstrahlung. These include $K_{L}\to\gamma Z^{\prime}$,
three-body decays of charged mesons $\pi^{+},\,K^{+}\to\ell^{+}\nu Z^{\prime}$
(with $\ell=e,\,\mu$), loop-level flavor-changing decays such as $K\to\pi
Z^{\prime}$ (including both neutral and charged Kaon decays), and heavy baryon
decay $\Delta\to NZ^{\prime}$. For completeness, we give the expressions of
Branching Ratios in these channels in Appendix B.

Figure 3: Expected flux of $Z^{\prime}$ gauge bosons at the DUNE near
detector, assuming the particles are all infinitely long-lived (and have not
decayed en route). This flux scales with $g_{BL}^{2}$ and has been determined
assuming $1.47\times 10^{22}$ protons striking the DUNE target, a conservative
estimate for ten years of operation of the experiment.

As discussed earlier in this Section, we will assume ten years of data
collection.111Planned upgrades to the PIP-II beam will increase this rate. Our
sensitivity can be viewed then as what can be done with at most ten years of
data collection. Assuming only $Z^{\prime}$ exists (either $\varphi$ does not
exist, $\sin\vartheta\to 0$, or $m_{\varphi}<2m_{Z^{\prime}}$), we can
determine the flux of $Z^{\prime}$ at the DUNE Near Detector complex using the
above. This is shown, divided by different contributions, in Fig. 3.

Given the flux of $Z^{\prime}$, we may determine the number of decay events in
the DUNE Near Detector volume using

$N_{\rm Sig.}=\int_{E_{\rm min.}}^{E_{\rm max.}}\int_{A_{\rm
Det.}}\int_{D_{\rm Det.}}^{D_{\rm Det.}+L_{\rm
Det.}}\mathrm{d}E_{Z^{\prime}}\>\mathrm{d}A\>\mathrm{d}x\>\left[\frac{d\Phi_{Z^{\prime}}}{dE_{Z^{\prime}}}P_{\rm
Decay}(E_{Z^{\prime}},x)\right].$ (13)

The outermost integral is over the energy of the $Z^{\prime}$ bosons and can
account for detector thresholds, efficiencies, etc. (with $E_{\rm min}$ and
$E_{\rm max}$ respectively the minimal and maximal energy of $Z^{\prime}$
bosons) – in our simulation we allow for the full range of $E_{Z^{\prime}}$
and assume 100% signal identification efficiency. The second integral is over
the surface area of the detector, $2.5$ m in radius for the gaseous argon time
projection chamber. The innermost integral is over the depth of the detector
$L_{\rm Det.}=5$ m with $D_{\rm Det.}$ being the distance to the detector,
$579$ m222We consider only decays in the gaseous argon detector, situated
directly behind the liquid argon one with a depth of $5$ m.. The probability
of a decay occurring at position $x$ is

$P_{\rm
Decay}(E_{Z^{\prime}},x)=\frac{1}{\gamma_{Z^{\prime}}c\tau_{Z^{\prime}}}e^{-\frac{x}{\gamma_{Z^{\prime}}c\tau_{Z^{\prime}}}}.$
(14)

In this decay expression, $\gamma_{Z^{\prime}}=E_{Z^{\prime}}/m_{Z^{\prime}}$
is the Lorentz boost factor and $\tau_{Z^{\prime}}$ is the proper lifetime of
$Z^{\prime}$, which can be obtained using the expressions for its width given
in Section 2.

The form of Eq. (13) assumes that the entire $Z^{\prime}$ flux
$\mathrm{d}\Phi_{Z^{\prime}}/\mathrm{d}E_{Z^{\prime}}$ originates at a common
location, assumed to be at the proton/target interaction location. If the
$Z^{\prime}$ flux is produced continuously along the beam pipe, Eq. (13) will
require an additional integral over the $Z^{\prime}$ production distribution.
This will be relevant when we consider $Z^{\prime}$ coming from $\varphi\to
Z^{\prime}Z^{\prime}$ decay in Section 3.3. With Eq. (13), we now can obtain
the sensitivity of DUNE to the pure-$Z^{\prime}$ scenario.

####  3.1.2 Sensitivity to Pure Gauge Boson Scenario

Here, we detail the DUNE sensitivity to the pure-$Z^{\prime}$ scenario, in
which the scalar $\varphi$ either does not exist or its decay into
$Z^{\prime}$ pairs is rare. The sensitivity is depicted in Fig. 4, where the
solid black line demonstrates the DUNE discovery capability: in the region
within the black line, at least ten signal events of
$Z^{\prime}\to\ell^{+}\ell^{-}$ are expected in ten years of data collection.
As argued in Ref. Berryman:2019dme , this constitutes a statistically
significant signal, especially since the invariant mass of $m_{Z^{\prime}}$
can be reconstructed by the fully visible decay. The DUNE sensitivity in the
visible channel dies off for $m_{Z^{\prime}}<2m_{e}$, because the $Z^{\prime}$
in that case dominantly decays to invisible final states of light neutrino
pairs.333The additional neutrino flux from $Z^{\prime}\to\nu\bar{\nu}$ decay
can in principle be used to extend the light $Z^{\prime}$ reach at DUNE below
the electron threshold, studied in the context of neutrinophilic vector bosons
in Ref. Bakhti:2018avv . Other complementary probes include the neutrino
trident production and neutrino-electron elastic scattering Ballett:2019xoj ;
Altmannshofer:2019zhy

Figure 4: Prospects of the Pure $Z^{\prime}$ search using the DUNE Near
Detector (black) where the black lines encompass parameter space for which at
least ten signal events (zero background assumed) are expected. This assumes
either that the scalar boson $\varphi$ does not exist, or that its decays into
$Z^{\prime}$ pairs are very rare. We compare against a number of existing and
future experimental studies, as discussed in the text.

In the pure-$Z^{\prime}$ scenario, we see that DUNE can extend on existing
beam-dump searches (shown as a purple region, collected from Ref.
Bauer:2018onh ). This includes the E141 Riordan:1987aw ; Bjorken:1988as ;
Bjorken:2009mm ; Andreas:2012mt , Orsay Davier:1989wz , NuCal Blumlein:2011mv
; Blumlein:2013cua ; Tsai:2019mtm , E137 Bjorken:1988as , and LSND
Athanassopoulos:1997er  experiments. In contrast, DUNE can reach heavier
$m_{Z^{\prime}}$ (due to large $\eta$ meson production and bremmstrahlung
contributions) and lower $g_{BL}$ (due to the large target-detector distance).
We also show the projected sensitivity of the FASER experiment near the LHC in
its first run (dashed purple) and proposed FASER2 run (dot-dashed purple)
Ariga:2018uku . FASER is sensitive to similar $m_{Z^{\prime}}$ with larger
$g_{BL}$, since the $Z^{\prime}$ produced near the LHC collision point have
larger boosts (and therefore longer lab-frame lifetime) than in the DUNE
environment. In this sense, DUNE as a detector searching for decaying new-
physics particles is highly complementary to FASER. In the time between now
and DUNE data collection, other Fermilab-based neutrino experiments could
reach into this parameter space as well, including the Short-Baseline Neutrino
facility detectors. See, e.g., Ref. Batell:2019nwo  for a study on searches
for dark scalars in this setup.

For relatively larger gauge coupling and $Z^{\prime}$ masses, there are limits
on $m_{Z^{\prime}}$ and $g_{BL}$ from the proton beam-dump experiment CHARM
Bergsma:1985qz , the searches of $Z^{\prime}$ in the channel
$e^{+}e^{-}\to\gamma Z^{\prime}\to\gamma\ell^{+}\ell^{-}$ (with
$\ell=e,\;\mu$) at BaBar Lees:2014xha , and the neutrino scattering experiment
TEXONO Deniz:2009mu ; Bilmis:2015lja , which are shown respectively as the
green, cyan and red shaded regions in Fig. 4. Belle II can improve on the
BaBar limits Abe:2010gxa , extending to ever lower $g_{BL}$ as it collects
data Kou:2018nap . More limits can be found in Refs. Aubert:2009cp ;
Bauer:2018onh . For completeness, we also present our calculated limits on
$K^{+}\to\pi^{+}Z^{\prime}\to\pi\nu\bar{\nu}$ from E949 Artamonov:2009sz  and
NA62 Ruggiero:2019 , which are however weaker than the limits from TEXONO,
CHARM and NA62, as indicated by the purple and orange lines in Fig. 4. The
future NA62 data can improve the measurement of ${\rm
BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ by roughly one order of magnitude
Anelli:2005ju , thus extend on the existing limits from CHARM and BaBar, as
denoted by the dashed orange line in Fig. 4. More details of the E949 and NA62
limits and the prospects at future NA62 can be found in Appendix C.

We note in passing that although a light $Z^{\prime}$ boson of $U(1)_{B-L}$
generates a positive contribution to the muon anomalous magnetic moment
Lindner:2016bgg , the $2\sigma$ parameter space preferred by the $(g-2)_{\mu}$
anomaly Bennett:2006fi ; Abi:2021gix  is deep inside the exclusion region in
Fig. 4, as shown by the gray band. Thus, if the $(g-2)_{\mu}$ anomaly turns
out to be a true signal of BSM physics444This is subject to further scrutiny,
given that a recent lattice result Borsanyi:2020mff  for the SM prediction is
much closer to the experimental value, as compared to the lattice world
average evaluated in Ref. Aoyama:2020ynm ., the minimal $U(1)_{B-L}$ model
with a flavor-universal $Z^{\prime}$ coupling needs to be extended to explain
this result; see, e.g., Refs. Ma:2001md ; Altmannshofer:2016oaq ;
Altmannshofer:2016brv ; CarcamoHernandez:2019ydc ; Dev:2020drf ;
Abdallah:2020biq ; Bodas:2021fsy ; Amaral:2021rzw ; Athron:2021iuf .

Finally, at low masses and small couplings, constraints from Big Bang
Nucleosynthesis (BBN) and the observation of Supernova 1987A (SN1987A) apply,
which are shown respectively as the blue and orange shaded regions in Fig. 4.
The supernova limits are from Ref. Knapen:2017xzo , which combines the limits
from dark photon Chang:2016ntp  and $U(1)_{B}$ models Rrapaj:2015wgs . In the
early Universe, the $Z^{\prime}$ boson may decay into neutrinos and electrons
(if kinematically allowed), and thus contribute to the evolution of neutrinos
and photons at the MeV scale. This could spoil the success of BBN and thus is
constrained by the precise measurement of the light degrees of freedom $N_{\rm
eff}$ by Planck Aghanim:2018eyx . However, the $N_{\rm eff}$ limits on the
$B-L$ $Z^{\prime}$ boson depend largely on the $Z^{\prime}$ mass and the
coupling $g_{BL}$, which dictates whether the $Z^{\prime}$ boson could reach
equilibrium with the SM particles at the MeV scale. Moreover, neutrino
oscillations are also important for the $N_{\rm eff}$ limits Escudero:2018mvt
; Escudero:2020dfa . The dedicated calculations of the cosmological limits are
beyond the main scope of this paper. For simplicity, we take the BBN limit
from Ref. Knapen:2017xzo . Other relevant studies can be found, e.g., in Refs.
Heeck:2014zfa ; Kamada:2018zxi ; Escudero:2019gzq ; Dutta:2020jsy . We
conclude this discussion by noting that both the BBN and SN1987A limits are
model dependent – for example, the chameleon effect (due to the environmental
matter density) can weaken astrophysical limits Nelson:2008tn , and late
reheating can open up parameter space with respect to BBN constraints
Depta:2020wmr . This model-dependence will be more relevant when we revisit
this parameter space in Section 3.2 where we perform a combined search for
$Z^{\prime}$ and the scalar boson $\varphi$ – the interplay of these two new
particles and their associated mass scales complicates the calculations of the
references mentioned above. We leave a detailed analysis of astrophysical
constraints on the combined $Z^{\prime}$, $\varphi$ scenario to future work.

###  3.2 Effects of Gauge Coupling on Searches for Scalar $\varphi$

Now we shift focus to the prospects of detecting the scalar boson $\varphi$
responsible for breaking the $U(1)_{B-L}$ symmetry. Its couplings to SM
particles are controlled by the mixing angle which we refer to as
$\sin\vartheta$. Section 2 detailed the different decay channels and widths of
$\varphi$. In Section 3.2.1 we discuss the production of $\varphi$ in the DUNE
environment. Section 3.2.2 then explores how the existence of both $\varphi$
and $Z^{\prime}$ (and the decay channel $\varphi\to Z^{\prime}Z^{\prime}$) can
hinder current and future searches for the scalar boson $\varphi$. The
improvement of $Z^{\prime}$ prospects at DUNE as a result of the $\varphi\to
Z^{\prime}Z^{\prime}$ source will be investigated in Section 3.3.

####  3.2.1 Production of Scalar Boson

From mixing with the SM Higgs, the light scalar $\varphi$ can be produced
in/near the DUNE target in the following channels:

  * •

Loop-level flavor-changing meson decays $K^{\pm}\to\pi^{\pm}\varphi$ and
$K_{L}\to\pi^{0}\varphi$. This is identical to the production in Fig. 6.1 of
Ref. Berryman:2019dme , with the widths Dev:2019hho

$\displaystyle\Gamma(K^{\pm}\to\pi^{\pm}\varphi)$ $\displaystyle\ \simeq\ $
$\displaystyle\frac{m_{K^{\pm}}\left|y_{sd}\right|^{2}\sin^{2}\vartheta}{64\pi}\frac{m_{K^{\pm}}^{2}}{m_{\varphi}^{2}}\lambda(m_{K^{\pm}},m_{\pi^{\pm}},m_{\varphi})\,,$
(15) $\displaystyle\Gamma(K_{L}\to\pi^{0}\varphi)$ $\displaystyle\ \simeq\ $
$\displaystyle\frac{m_{K_{L}}\left({\rm
Re}\,y_{sd}\right)^{2}\sin^{2}\vartheta}{64\pi}\frac{m_{K^{0}}^{2}}{m_{\varphi}^{2}}\lambda(m_{K_{L}},m_{\pi^{0}},m_{\varphi})\,,$
(16)

where the loop-level FCNC coupling Batell:2009jf

$\displaystyle y_{sd}\ =\
\frac{3\sqrt{2}G_{F}m_{t}^{2}V_{ts}^{\ast}V_{td}}{16\pi^{2}}\,\frac{m_{\varphi}}{\sqrt{2}v_{\rm
EW}}\,,$ (17)

which is induced by the $W-$top loop, and $v_{\rm
EW}=(\sqrt{2}G_{F})^{-1/2}/\sqrt{2}\simeq 174$ GeV is the electroweak VEV.
Note that the partial decay widths in Eqs. (15) and (16) are almost identical,
except for the crucial difference that the decay $K_{L}\to\pi^{0}\varphi$
depends only on the real part of the coupling $y_{sd}$.

  * •

The bremsstrahlung process $pp\to pp\varphi$, analogous to the $pp\to
ppZ^{\prime}$ process discussed in Eq. (12). Ref. Foroughi-Abari:2020gju
found that this production process is dominant for such a search at LSND
(proton beam energy of 800 MeV). In contrast to the $Z^{\prime}$
bremsstrahlung process, the scalar coupling is suppressed by the effective
proton Yukawa coupling. Ref. Boiarska:2019jym  found that this coupling is
small, $\mathcal{O}(10^{-3})$, and Ref. Batell:2019nwo  used this result to
find that the proton bremsstrahlung process is highly suppressed relative to
the Kaon decay processes discussed above for scalar production in the NuMI
beam environment (proton beam energy of 120 GeV). We have verified that it is
also highly suppressed in the DUNE environment.

In Eq. (13) we provided the expected number of decay signal events using some
flux of (potentially) decaying particles. The same formalism applies here with
the $\varphi$ flux instead of $Z^{\prime}$, however, as alluded to previously,
an extra integral must be applied. Because the Kaons $K^{\pm}$ and $K_{L}$ are
long-lived, the production of $\varphi$ is not all in/near the DUNE target –
the distribution of $\varphi$ production points spans hundreds of meters555The
DUNE decay pipe ends roughly 230 meters from the target, so the $\varphi$ can
be produced over this entire distance, including some component from Kaon
decay-at-rest at the absorber at the end of the decay pipe..

####  3.2.2 Effects on $\varphi$ prospects at DUNE

Figure 5: Limits on the existence of a new scalar boson $\varphi$ with mass
$m_{\varphi}$ and Higgs mixing parameterized by $\sin^{2}\vartheta$. For each
panel, we make a different assumption about the branching ratio of $\varphi$
into SM particles. The left panel assumes 100%, the center 10%, and the right
1%. We include constraints from CHARM (blue), KOTO (green), NA62 (red), E949
(purple), and LHCb (gray) and projections from DUNE (black) in all panels, as
well as constraints from LSND (orange) and projections from SBND and ICARUS
(dashed blue) in the left panel. The supernova constraints are shown by the
purple shaded region, with the solid and dot-dashed contours corresponding to
two different values for the luminosity limits.

The decay $\varphi\to Z^{\prime}Z^{\prime}$ not only opens up a new decay
channel, but also contributes to the total width of $\varphi$. Therefore, this
extra channel will affect any existing search for $\varphi$ by (a) modifying
its branching fraction into visible final-state particles and (b) shortening
its lifetime. In general, the constraints on $\varphi$ from searches for
decays like $K^{+}\to\pi^{+}\nu\bar{\nu}$ (which could be mimicked by
$K^{+}\to\pi^{+}\varphi$ and $\varphi$ decaying invisibly/outside a detector)
and beam-dump searches like that of DUNE can be understood by two pieces. At
the bottom of these exclusion regions, the reach of an experiment is limited
by exposure (and/or backgrounds) – for lower values of $\sin^{2}\vartheta$,
not enough $\varphi$ can be produced to create a signal. At the top of these
exclusion regions, the lifetime of $\varphi$ is so short that it decays away
before leaving an imprint in a detector. This is particularly relevant for
beam-dump searches like CHARM, where the target (production point of
$\varphi$) and detector are well separated.

We reproduce existing limits on $m_{\varphi}$ and $\sin^{2}\vartheta$ coming
from a variety of experiments – CHARM Bergsma:1985qz ; Winkler:2018qyg , KOTO
Ahn:2018mvc , NA62 Ruggiero:2019 , and E949 Artamonov:2009sz  – in light of
this modification. We also reproduce the DUNE sensitivity from Ref.
Berryman:2019dme , including this new decay channel. We do so by allowing the
branching fraction of $\varphi$ into SM particles, denoted as BR($\varphi\to\
\mathrm{SM}$), to vary as a free parameter. In actuality, for a given
($m_{\varphi},\leavevmode\nobreak\ \sin^{2}\vartheta$), this branching ratio
is dependent on those parameters as well as $m_{Z^{\prime}}$ and $g_{BL}$. The
limits from these experiments, including the modifications, are shown in Fig.
5 for BR($\varphi\to\ \mathrm{SM}$) = 100% (left), BR($\varphi\to\
\mathrm{SM}$) = 10% (center), and BR($\varphi\to\ \mathrm{SM}$) = 1% (right).
Changes to the no-$Z^{\prime}$ case (effectively, the left panel), are most
apparent at higher $m_{\varphi}$ and larger $\sin^{2}\vartheta$, e.g. the
“lobe” in the CHARM exclusion closes up.

In Fig. 5, we also include several of other results – in gray, we include
constraints from rare $B$ meson decays at LHCb Aaij:2015tna ; Aaij:2016qsm .
We do not perform our own simulation of how these limits change for
BR($\varphi\to\ \mathrm{SM}$) $\neq 1$, however we expect that their behavior
in the mass/angle range of interest will not vary much. In our simulations in
Section 3.3, we do not allow $\sin^{2}\vartheta>10^{-6}$ for any
$m_{\varphi}$, in addition to requiring it to be consistent with all the
experimental limits shown in Fig. 5. Also, in the left panel of Fig. 5, we
include a limit from a reinterpretation of the LSND experiment’s results
Foroughi-Abari:2020gju  and future projections of the Fermilab SBN detectors,
a combination of using the SBND and ICARUS experiments with the BNB and NuMI
beams as proposed in Ref. Batell:2019nwo . Due to the complicated nature of
simulating the LSND and SBN setups, we do not provide these limits/projections
in the center/right panels, where they should in principle exist as well.

As with the light gauge boson providing a positive contribution to the muon
anomalous magnetic moment, so does a light neutral scalar Lindner:2016bgg .
However, it requires $\sin^{2}\vartheta\simeq 1$ mixing with Higgs for our
light scalar $\varphi$ to explain the $(g-2)_{\mu}$ anomaly Abi:2021gix ,
which is certainly excluded in Fig. 5 (well above the range we present). In
general, a light scalar explanation of the $(g-2)_{\mu}$ anomaly must involve
lepton-flavor-violating couplings to be consistent with the existing
constraints; see, e.g. Refs. Dev:2017ftk ; Cherchiglia:2017uwv ; Dev:2018upe ;
Chun:2019oix ; Wang:2021fkn .

BBN can also provide constraints on the existence of $\varphi$ if it is in
thermal equilibrium and has yet to decay before the formation of light
elements. If the lifetime $\tau_{\varphi}\gtrsim 1$ sec, the scalar $\varphi$
will contribute a factor of $4/7$ to the light degrees of freedom $N_{\rm
eff}$ Kainulainen:2015sva ; Fradette:2017sdd . Although the scalar
contribution to $N_{\rm eff}$ is smaller by a factor of three compared to
$Z^{\prime}$, it is still larger than the allowed range of $\Delta N_{\rm
eff}$ by Planck at the time of CMB formation, which is roughly $0.2-0.5$
depending on the data sets adopted Aghanim:2018eyx . However, to implement the
$\Delta N_{\rm eff}$ limits, $\varphi$ has to be in thermal equilibrium with
the SM particles at the BBN temperature of MeV scale Dev:2017dui . The
decaying of $\varphi$ into $e^{+}e^{-}$ and $\gamma\gamma$ are both very
small, therefore there is almost no parameter space of $m_{\varphi}$ and
$\sin\theta$ to satisfy both the conditions above.

The supernova limits on the scalar $\varphi$ have been investigated in Refs.
Ishizuka:1989ts ; Hanhart:2000er ; Arndt:2002yg ; Diener:2013xpa ;
Krnjaic:2015mbs ; Lee:2018lcj ; Dev:2020eam . In the supernova core, the
scalar $\varphi$ can be produced via the bremsstrahlung process $NN\to
NN\varphi$ (with $N$ being nucleons). Taking into account the partial
cancellation between different Feynman diagrams Dev:2020eam , the observed
neutrino luminosity ${\cal L}_{\nu}$ can be used to set limits on
$m_{\varphi}$ and $\sin^{2}\vartheta$. Setting ${\cal L}_{\nu}=3\times
10^{53}$ erg/sec and $5\times 10^{53}$ erg/sec respectively, the regions
inside the dot-dashed light-purple contours and shaded regions in Fig. 5 can
be excluded. The production amplitudes for the diagrams with the scalar
$\varphi$ coupling to nucleons in Ref. Dev:2020eam  have been corrected by a
factor of $1/2$ merger . One should note that in the supernova limits here we
do not include the possible decay channel $\varphi\to Z^{\prime}Z^{\prime}$.
If this channel is kinematically allowed, the supernova limits on $\varphi$
(and also $Z^{\prime}$) could be potentially changed, but this requires a
dedicated calculation beyond the scope of this paper, and is left for future
work.

###  3.3 Improving $Z^{\prime}$ prospects at DUNE

As discussed in Eq. (8), if both $\varphi$ and $Z^{\prime}$ exist and
$m_{\varphi}>2m_{Z^{\prime}}$, then the decay $\varphi\to
Z^{\prime}Z^{\prime}$ may occur. In Section 3.2.2 we discussed how this decay
channel modifies limits on the scalar boson in the
$(m_{\varphi},\leavevmode\nobreak\ \sin^{2}\vartheta)$ plane. In this
subsection, we will discuss how the additional flux of $Z^{\prime}$ from this
channel can improve search capabilities in the
$(m_{Z^{\prime}},\leavevmode\nobreak\ g_{BL})$ plane.

Before analyzing this flux contribution, we briefly note that additional
associated production of both $\varphi$ and $Z^{\prime}$ may occur from some
meson decays. In the presence of the gauge coupling of $\varphi$ to
$Z^{\prime}$ bosons, we can in principle have the associated production of
$Z^{\prime}$ and $\varphi$ from meson decays, for instance

$\displaystyle\pi^{0}\to\gamma+Z^{\prime}+\varphi\,,\quad
K\to\pi+Z^{\prime}+\varphi\,,\quad\rho\to Z^{\prime}+\varphi\,.$ (18)

If the scalar $\varphi$ is emitted from the $Z^{\prime}$ boson line, the
partial widths will be proportional to $g_{BL}^{4}$ Batell:2009di . If the
scalar $\varphi$ is emitted from the quark fermion lines, the widths will be
proportional to $\sin^{2}\vartheta g_{BL}^{2}$. Therefore for small
$\sin\vartheta$ and $g_{BL}$, these associated production channels are highly
suppressed compared to the channels above. Therefore, we do not consider these
channels in this paper.

Since the main source of $\varphi$ arises from decays of long-lived Kaons
(compared to the prompt direct production of $Z^{\prime}$ – neutral, short-
lived meson decays and proton-proton bremsstrahlung), the $Z^{\prime}$ coming
from $\varphi$ decays will originate all along the distance between the beam
target and the end of the decay pipe. We take this into account by both
allowing the Kaons to be long lived and the $\varphi$ to have a lifetime
according to the relevant parameters – $m_{\varphi}$, $\sin^{2}\vartheta$,
$g_{BL}$, and $m_{Z^{\prime}}$. With these ingredients, we can determine the
resulting $Z^{\prime}$ flux, as well as the number of $Z^{\prime}\to
e^{+}e^{-}$ and $Z^{\prime}\to\mu^{+}\mu^{-}$ events in our detector.

Figure 6: Expected number of signal events of $Z^{\prime}$ decays in ten years
of data collection at DUNE, assuming two benchmark parameters as given in Eqs.
(9) (left panel) and (10) (right panel). The different colored contours
represent increasing signal events (from darker to lighter colors) by an order
of magnitude each, purple corresponding to $N_{\rm sig.}<1$. In both panels,
when $m_{\varphi}<2m_{Z^{\prime}}$, the contribution is purely from direct
production (meson decays and proton-proton bremsstrahlung), where $\varphi\to
Z^{\prime}Z^{\prime}$ production increases the signal rate when
$m_{\varphi}>2m_{Z^{\prime}}$. In black solid lines we show a collection of
current constraints from Section 3.2.2 on $m_{\varphi}$ and
$\sin^{2}\vartheta$, and in dashed black we show the expected DUNE sensitivity
to search for the $\varphi$ decays. In the vertical black shaded region, the
production of $\varphi$ from Kaon decay is kinematically forbidden.

Fig. 6 demonstrates how we determine this. For a given combination of
($m_{Z^{\prime}}$, $g_{BL}$) – for example the benchmark points in Eqs. (9)
and (10) – we determine the flux of $Z^{\prime}$ at the DUNE Near Detector
complex from “direct” production, i.e. the available neutral meson decays and
proton-proton bremsstrahlung. That, combined with the $Z^{\prime}$ lifetime
(determined also by $m_{Z^{\prime}}$ and $g_{BL}$) and branching ratio into
visible final states, determines the number of signal events we observe. For
these two combinations, we would expect ${\sim}0.4$ or ${\sim}0.2$ events in
ten years of data collection at DUNE - not enough to warrant a significant
discovery. However, if there exists $\varphi$ with mass greater than
2$m_{Z^{\prime}}$, then an additional $Z^{\prime}$ flux can be generated. The
total flux, and the number of signal events, would depend on the mass of
$\varphi$ and the mixing $\sin^{2}\vartheta$, however, we cannot take
$\sin^{2}\vartheta$ to be arbitrarily large, given existing constraints
discussed in Section 3.2.2 and Fig. 5. In the left panel of Fig. 6, the
colored contours display the number of expected signal events for the
benchmark point in Eq. (9) as we vary $m_{\varphi}$ and $\sin^{2}\vartheta$,
each color change representing an order-of-magnitude increase in $N_{\rm
sig.}$ from bottom to top (the purple regions correspond to $N_{\rm sig.}<1$
and each subsequent color change goes to $1\leq N_{\rm sig.}\leq 10$ and so
on). We scan over this parameter space, determining the maximum $N_{\rm sig.}$
subject to the existing constraints in this space (solid black line) as well
as the future DUNE sensitivity (dashed black line). For this combination of
$m_{Z^{\prime}}$ and $g_{BL}$, we obtain $N_{\rm sig.}$ as large as
${\sim}10^{4}$ in ten years of data collection (this occurs if
$m_{\varphi}\approx 150$ MeV and for $\sin^{2}\vartheta\approx 3\times
10^{-7}$, where the colored contours in Fig. 6 left panel are lightest), which
would be detectable in DUNE. In contrast, we show the result for the benchmark
point in Eq. (10) with a heavier $Z^{\prime}$ in the right panel of Fig. 6,
where the color changing denotes also the one-order-of-magnitude higher from
bottom to top. As a result of the heavier $Z^{\prime}$ mass, there is only a
narrow window for $m_{\varphi}$ with the enhanced $Z^{\prime}$ production rate
at the DUNE, i.e. $2m_{Z^{\prime}}<m_{\varphi}<m_{K}-m_{\pi}$, or equivalently
$280\,{\rm MeV}<m_{\varphi}\lesssim 363$ MeV. Beyond the mass threshold of
$m_{K}-m_{\pi}$, the scalar $\varphi$ can not be produced from the flavor-
changing Kaon decays, which is shown as the vertical black shaded regions in
both panels of Fig. 6.

As the extra production channel of $\varphi\to Z^{\prime}Z^{\prime}$ for
$Z^{\prime}$, the improvement of DUNE sensitivities to the parameter space of
$m_{Z^{\prime}}$ and $g_{BL}$ are shown in Fig. 7 as the dashed and dot-dashed
black lines. As in Fig. 4, we use ten signal events as the required amount to
be statistically significant. The dashed line (reaching lower $g_{BL}$)
denotes the extended reach subject to the current constraints on $m_{\varphi}$
and $\sin^{2}\vartheta$ from CHARM, KOTO, NA62, and E949 discussed in Section
3.2.2. The dot-dashed line (extending below the solid black one but not as low
as the dashed one) indicates the improvement subject to DUNE’s direct
$\varphi$ decay search. In effect, if a signal of $Z^{\prime}$ decay is
observed consistent with the parameter space between the dot-dashed and dashed
lines, then a similar $\varphi\to\ell^{+}\ell^{-}$ or $\varphi\to\pi\pi$
signature should also be observable with the same data. Unlike in Fig. 4, here
we do not show the prospects at future FASER and FASER2, because the presence
of $\varphi$ could in principle modify these projections and it requires a
dedicated FASER simulation666An analogous combined scalar/vector boson search
was explored in the context of dark photons at FASER in Ref. Araki:2020wkq .,
which is beyond the scope of our current work. The other labels are the same
as in Fig. 4. In principle all other limits on $Z^{\prime}$ such as those from
the beam-dump experiments could also be affected by the scalar $\varphi$.
However, for simplicity we do not include these corrections and assume naïvely
that they are small.

Figure 7: Improvements of the $Z^{\prime}$ prospects at DUNE near detector
complex as a result of the new channel $\varphi\to Z^{\prime}Z^{\prime}$. The
dashed line indicates the improvement if we only subject $\varphi$ and
$\sin^{2}\vartheta$ to current constraints from CHARM, KOTO, NA62, and E949
(see Section 3.2.2), where the dot-dashed line also includes constraints from
DUNE’s direct search for $\varphi$. Between the dot-dashed and dashed lines,
any $Z^{\prime}$ signal in DUNE should also require a direct $\varphi$ signal
in the same data. All other constraints and labels are the same as in Fig. 4.

The two benchmark points in Eqs. (9) and (10) are indicated by the stars in
Fig. 7. The benchmark point with $m_{Z^{\prime}}=140$ MeV is out of all
current existing limits and even the $Z^{\prime}$ prospects at DUNE in the
pure gauge boson case in Section 3.1. However, it can be probed at DUNE in
presence of the channel $\varphi\to Z^{\prime}Z^{\prime}$. As shown in Fig. 7,
the benchmark point with $m_{Z^{\prime}}=30$ MeV is out of the reach of DUNE
prospects in the pure $Z^{\prime}$ case, but can be probed in presence of the
scalar $\varphi$. It should be noted that this benchmark point has been
precluded by the supernova limits on $Z^{\prime}$. However, as mentioned in
Section 3.2.2, in presence of $\varphi$ the supernova limits on $Z^{\prime}$
could be very different. Furthermore, the supernova limits on $Z^{\prime}$
could also change dramatically and thus be possibly avoided if $Z^{\prime}$
couples to a dark matter particle Zhang:2014wra . In addition, the interplay
of $\varphi$ and $Z^{\prime}$ would also be important for the evolution of
$Z^{\prime}$ in the early universe and the resultant BBN limits in Fig. 7.

It is clear in Fig. 7 that with the extra contribution from the scalar
$\varphi$, DUNE can extend its search for $Z^{\prime}$ to a factor of 7
smaller $g_{BL}$ when subject to DUNE sensitivity to direct $\varphi$ decays
and by a factor of 45 when considering current $\varphi$ constraints. It is
striking that the scalar $\varphi$ can improve the $Z^{\prime}$ prospects at
DUNE far beyond the current beam-dump limits for $m_{Z^{\prime}}\lesssim 100$
MeV, which would be otherwise very challenging for the pure $Z^{\prime}$ boson
case.

##  4 Conclusion

We have explored the possibility of experimentally searching for a class of
BSM scenarios where the SM is extended by a minimal $U(1)_{B-L}$ gauge
symmetry to explain neutrino masses and possibly provide a portal to DM. We
focused on the light $Z^{\prime}$ boson and light associated
$U(1)_{B-L}$-breaking scalar $\varphi$ in the low gauge coupling regime. We
discussed how the presence of $Z^{\prime}$ and $\varphi$ affects the search
range at high-intensity facilities such as the near detector complex at DUNE.
Focusing primarily on the sub-GeV mass region for $Z^{\prime}$ and $\varphi$,
the main results of this paper can be summarized as follows:

  * •

If the scalar $\varphi$ decouples, the DUNE prospects of pure $Z^{\prime}$
boson is presented in Fig. 4. Compared to the FASER sensitivities, the DUNE
near detector complex can probe a smaller coupling $g_{BL}$ as a result of the
lower center-of-mass proton-beam/target energy at DUNE (compared to the LHC).

  * •

As far as the $B-L$ breaking scalar $\varphi$ is concerned, the presence of
$Z^{\prime}$ somewhat weakens the search range of its parameters at DUNE,
compared to the case without the gauge interaction, as shown in Fig. 5.

  * •

With the extra source of $Z^{\prime}$ from scalar decay $\varphi\to
Z^{\prime}Z^{\prime}$, the number of $Z^{\prime}$ signal events can be greatly
increased for $m_{Z^{\prime}}\lesssim 200$ MeV (cf. Fig. 6), and the DUNE
prospects of $g_{BL}$ can be improved by up to a factor of 45, as demonstrated
in Fig. 7.

Our results are largely complementary to the other works on $B-L$ $Z^{\prime}$
and $\varphi$ searches at DUNE and other facilities such as FASER and SHiP
detectors, as well as to the limits from BBN and supernovae.

## Acknowledgement

The work of P. S. B. D. is supported in part by the U.S. Department of Energy
under Grant No. DE-SC0017987, by the Neutrino Theory Network Program, and by a
Fermilab Intensity Frontier Fellowship. The work of B. D. is supported by the
U.S. Department of Energy (DOE) Grant DE-SC0010813. K. J. K. is supported by
Fermi Research Alliance, LLC, under contract DE-AC02-07CH11359 with the U.S.
Department of Energy. The work of R. N. M. is supported by the National
Science Foundation grant no. PHY-1914631. Y. Z. is partially supported by “the
Fundamental Research Funds for the Central Universities”.

##  Appendix A Details of $\varphi$ Decay

The transition amplitude $G(s)$ appearing in Eq. (6) (with
$s=m_{\varphi}^{2}$) consists of three form factors. In the limit of isospin
conservation, $G(s)$ can be written in the form of Donoghue:1990xh

$\displaystyle
G(s)=\frac{2}{9}\theta_{\pi}(s)+\frac{7}{9}\left[\Gamma_{\pi}(s)+\Delta_{\pi}(s)\right]\,,$
(19)

where in chiral perturbation theory the three form factors are respectively

$\displaystyle\theta_{\pi}(s)$ $\displaystyle=$
$\displaystyle\left(1+\frac{2m_{\pi}^{2}}{s}\right)(1+\psi(s))+b_{\theta}s\,,$
(20) $\displaystyle\Gamma_{\pi}(s)$ $\displaystyle=$
$\displaystyle\frac{m_{\pi}^{2}}{s}(1+\psi(s)+b_{\Gamma}s)\,,$ (21)
$\displaystyle\Delta_{\pi}(s)$ $\displaystyle=$ $\displaystyle
d_{F}(1+\psi(s)+b_{\Delta}s)\,,$ (22)

with the coefficients $b_{\theta}=2.7\;{\rm GeV}^{-2}$, $b_{\Gamma}=2.6\;{\rm
GeV}^{-2}$, $b_{\Delta}=3.3\;{\rm GeV}^{-2}$, $d_{F}=0.09$, and the function

$\displaystyle\psi(s)=\frac{2s-m_{\pi}^{2}}{16\pi^{2}f_{\pi}^{2}}\left[\kappa\log\left(\frac{1-\kappa}{1+\kappa}\right)+2+i\pi\kappa\right]+\frac{s}{96\pi^{2}f_{\pi}^{2}}\,,$
(23)

where $f_{\pi}=130$ MeV is the pion decay constant, and
$\kappa=\sqrt{1-4m_{\pi}^{2}/s}$.

Similarly in Eq. (7), the loop functions are defined as Djouadi:2005gi

$\displaystyle A_{1/2}(\tau)$ $\displaystyle\ =\ $ $\displaystyle
2\left[\tau+(\tau-1)f(\tau)\right]\tau^{-2}\,,$ (24) $\displaystyle
A_{1}(\tau)$ $\displaystyle\ =\ $
$\displaystyle-\left[2\tau^{2}+3\tau+3(2\tau-1)f(\tau)\right]\tau^{-2}\,,$
(25)

with $\tau_{X}=m_{\varphi}^{2}/4m_{X}^{2}$, and the function

$\displaystyle f(\tau)\ =\begin{cases}\arcsin^{2}\sqrt{\tau}\,,&\tau\leq
1\,,\\\
-\frac{1}{4}\left[\log\left(\frac{1+\sqrt{1-1/\tau}}{1-\sqrt{1-1/\tau}}\right)-i\pi\right]^{2}\,,&\tau>1\,.\end{cases}$
(26)

##  Appendix B Additional Subdominant Contributions to the Gauge Boson Flux

For completeness, we list here additional possible production channels of
$Z^{\prime}$ in the $U(1)_{B-L}$ model at DUNE, although they are small with
respect to the dominant channels in Section 3.1.1.

  * •

Two-body neutral meson $K_{L}\to\gamma Z$, with the BRs Batell:2009di

$\displaystyle{\rm BR}(K_{L}\to\gamma Z^{\prime})$ $\displaystyle\ \simeq\ $
$\displaystyle 2{\rm
BR}(K_{L}\to\gamma\gamma)\times\frac{g_{BL}^{2}}{e^{2}}\left(1-\frac{m_{Z^{\prime}}^{2}}{m_{K^{0}}^{2}}\right)^{3}\,,$
(27)

which is highly suppressed by the small SM ${\rm
BR}(K_{L}\to\gamma\gamma)=5.47\times 10^{-4}$ Zyla:2020zbs .

  * •

Three-body charged pion/Kaon decays $\pi^{+}\to\ell+\nu Z^{\prime}$ and
$K^{+}\to\ell^{+}\nu Z^{\prime}$ with $\ell=e,\,\mu$. Because the incoming
mesons have zero baryon or lepton number, this decay width should be identical
to the leptonic $Z^{\prime}$ case derived in Appendix B of Ref.
Berryman:2019dme  (see also Ref. Krnjaic:2019rsv ). This process can be
calculated numerically – we find that the production of $Z^{\prime}$ $B-L$
gauge bosons from charged meson decays is suppressed relative to neutral meson
decay/bremsstrahlung production, which we considered in the main text.

  * •

Loop-level flavor-changing two-body decays $K^{+}\to\pi^{+}Z^{\prime}$ and
$K_{S}\to\pi^{0}Z^{\prime}$. Based on the low-energy chiral theory
Ecker:1987qi ; DAmbrosio:1998gur , the width is Pospelov:2008zw ;
Davoudiasl:2014kua

$\displaystyle\Gamma(K\to\pi Z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{g_{BL}^{2}m_{Z^{\prime}}^{2}W^{2}(m_{Z^{\prime}}^{2})}{2^{12}\pi^{5}m_{K}}\lambda(m_{K},\,m_{\pi},\,m_{Z^{\prime}})$
(28)
$\displaystyle\times\left[\left(1-\frac{m_{\pi}^{2}}{m_{K}^{2}}\right)^{2}+\frac{m_{Z^{\prime}}^{2}}{m_{K}^{2}}\left(2+\frac{2m_{\pi}^{2}}{m_{K}^{2}}-\frac{m_{Z^{\prime}}^{2}}{m_{K}^{2}}\right)\right]\,,$

where $W^{2}(q^{2})$ incorporates both the polorization and $\pi\pi$
contributions DAmbrosio:1998gur , and

$\displaystyle\lambda(a,b,c)\equiv\left[1-\frac{2(b^{2}+c^{2})}{a^{2}}+\frac{(b^{2}-c^{2})^{2}}{a^{4}}\right]^{1/2}\,.$
(29)

Note that in the limit of $m_{Z^{\prime}}\to 0$, the width vanishes, as
required by angular momentum conservation. The decay width in Eq. (28) applies
to both $K^{+}\to\pi^{+}Z^{\prime}$ and $K_{S}\to\pi^{0}Z^{\prime}$, which,
however, are highly suppressed by the loop factor:

$\displaystyle{\rm BR}(K^{\pm}\to\pi^{\pm}Z^{\prime})$ $\displaystyle\ \simeq\
$ $\displaystyle 8\times 10^{-4}\times\left(\frac{m_{Z^{\prime}}}{100\,{\rm
MeV}}\right)^{2}\lambda(m_{K^{\pm}},\,m_{\pi^{\pm}},\,m_{Z^{\prime}})\,,$ (30)
$\displaystyle{\rm BR}(K_{S}\to\pi^{0}Z^{\prime})$ $\displaystyle\ \simeq\ $
$\displaystyle 6\times 10^{-6}\times\left(\frac{m_{Z^{\prime}}}{100\,{\rm
MeV}}\right)^{2}\lambda(m_{K^{0}},\,m_{\pi^{0}},\,m_{Z^{\prime}})\,.$ (31)

On the contrary, the process $K_{L}\to\pi\gamma^{\ast}$ is CP-violating, and
as a result the decay $K_{L}\to\pi^{0}Z^{\prime}$ is highly suppressed
Ecker:1987qi ; DAmbrosio:1998gur . There are also the flavor-violating decays
$B\to\pi Z^{\prime}$ and $B\to KZ^{\prime}$. However, the production rate of
$B$ mesons at the DUNE is highly suppressed Adams:2013qkq , and we will not
consider these $B$ meson decay channels any more.

  * •

Baryon decay $\Delta(1232)\to NZ^{\prime}$ with $N$ here referring to nucleon,
The corresponding BR is Batell:2009di

$\displaystyle{\rm Br}(\Delta\to NZ^{\prime})={\rm BR}(\Delta\to
N\gamma)\times\frac{g_{BL}^{2}}{e^{2}}\left(1-\frac{m_{Z^{\prime}}^{2}}{(m_{\Delta}-m_{N})^{2}}\right)^{3/2}\,,$
(32)

which is also suppressed by the SM ${\rm BR}(\Delta\to N\gamma)\simeq 6\times
10^{-3}$ Zyla:2020zbs .

##  Appendix C Additional $Z^{\prime}$ Boson Constraints

The current limits on the light $Z^{\prime}$ boson of the $U(1)_{B-L}$ model
are summarized in Fig. 4. Here we discuss a couple more limits from rare meson
decays, which are not included in Ref. Bauer:2018onh . The most relevant ones
are those from the loop-level flavor-changing $K^{+}$ decay (see Eq. (28) for
the width)

$\displaystyle K^{+}\to\pi^{+}Z^{\prime}\,,\quad Z^{\prime}\to\nu\bar{\nu}\,.$
(33)

The current most stringent limits are from NA62 Ruggiero:2019  and E949
Artamonov:2009sz . The NA62 experiment has obtained a 95% C.L. upper limit
${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})<2.44\times 10^{-10}$, which can be
translated into the limit on $m_{Z^{\prime}}$ and $g_{BL}$ plane, as shown by
the orange line in Fig. 4. The E949 limit on ${\rm BR}(K^{+}\to\pi^{+}+X)$
(with $X$ a long-lived particle) can reach up to $5.4\times 10^{-11}$, which
is shown as the purple line in Fig. 4. The future NA62 data can improve the
${\rm BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$ constraint down to $2.35\times
10^{-11}$ Anelli:2005ju , and the resulting limit on $g_{BL}$ can be enhanced
by a factor of 3, as indicated by the dashed orange line in Fig. 4.

The gaps in the NA62 and E949 limits are due to the significant background due
to the decays $K^{+}\to\pi^{+}\pi^{0}$ with $\pi^{0}\to\nu\bar{\nu}$. Note
that because the width of $K^{+}\to\pi^{+}Z^{\prime}$ is proportional to
$Z^{\prime}$ mass (cf. Eq. (28)), both the NA62 and E949 limits get weaker
when $Z^{\prime}$ is lighter. There are in principle also the limits for a
long-lived $Z^{\prime}$, with $Z^{\prime}$ decaying outside the detectors.
However, to have a long-lived $Z^{\prime}$ the coupling $g_{B-L}$ needs to be
very small, which will in turn highly suppress the production of $Z^{\prime}$.
It turns out that no parameter space of $m_{Z^{\prime}}$ and $g_{B-L}$ is
excluded due to the long-lived $Z^{\prime}$ at NA62 and E949. The KOTO
experiment has put an upper bound on the decay ${\rm
BR}(K_{L}\to\pi\nu\bar{\nu})<3.0\times 10^{-9}$ Ahn:2018mvc ; Ahn:2020opg .
However, this limit can not be used to set constraints on the decay
$K_{S}\to\pi^{0}Z^{\prime}$.

There are more limits from the meson and baryon decays which can be used to
constrain $m_{Z^{\prime}}$ and $g_{BL}$, which are however much weaker, thus
are not shown in Fig. 4:

  * •

The two-body meson decay $K_{L}\to\gamma Z^{\prime}$ with the subsequent
decays $Z^{\prime}\to e^{+}e^{-},\,\mu^{+}\mu^{-}$ contributes respectively to
the decays $K_{L}\to e^{+}e^{-}\gamma$ Ohl:1990qw ; Barr:1990zh ; Fanti:1999rz
and $K_{L}\to\mu^{+}\mu^{-}\gamma$ AlaviHarati:2001wd . However, as stated in
Appendix B, the meson decay $K_{L}\to\gamma Z^{\prime}$ is highly suppressed
by the corresponding SM ${\rm BR}(K_{L}\to\gamma\gamma)$.

  * •

The baryon decay $\Delta\to pZ^{\prime}$ with $Z^{\prime}\to e^{+}e^{-}$
contributes to $\Delta\to pe^{+}e^{-}$, which is highly suppressed by the
small SM ${\rm BR}(\Delta\to N\gamma)$, as mentioned in Appendix B.

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