# Looking forward to photon-coupled long-lived particles IV: neutralino-
ALPino/gravitino

Krzysztof Jodłowski<EMAIL_ADDRESS>Particle Theory and Cosmology
Group, Center for Theoretical Physics of the Universe, Institute for Basic
Science (IBS), Daejeon, 34126, Korea

###### Abstract

Various supersymmetric (SUSY) scenarios predict a sub-GeV neutralino decaying
into a single photon and an invisible state. This signature has recently been
studied in a number of intensity frontier experiments, finding constraints
complementary to the usual collider searches. In this work, we study the
prospects of searches for long-lived neutralinos coupled to an ALPino or
gravitino, where each can act as the lightest SUSY particle (LSP). In addition
to the neutralino decays into a LSP and a photon, we also consider three-body
decays into a pair of charged leptons, and signatures related to scattering
with electrons and secondary neutralino production. For both models, we find
that the searches at FASER2 will allow to overcome the current bounds, while
SHIP will extend these limits by more than an order of magnitude in the value
of the coupling constant.

##  I Introduction

Searches for supersymmetric (SUSY) [1, 2, 3, 4, 5] particles with masses close
to the electroweak scale has yielded no conclusive detection thus far [6, 7,
8, 9, 10]. This fact motivates the exploration of alternative possibilities,
including the regime of low-mass neutralinos, possibly connected to other
species by interactions beyond those predicted by the Minimal Supersymmetric
Standard Model (MSSM) [11, 12, 13, 14].

In particular, the neutralino of bino composition is almost unconstrained by
collider searches because its couplings to gauge bosons vanish. Moreover, if
the neutralino decays, and therefore does not constitute dark matter (DM), its
mass could be very light, possibly with masses in the sub-GeV range [15, 16].
Recent studies, e.g., [17, 18, 19, 20, 21] have investigated such light
neutralinos, which decay into SM states due to the R-parity violating
interactions [22, 23, 24]. Consequently, the bino behaves as a long-lived
particle (LLP), making it especially well-suited for intensity frontier
searches [25, 26, 27] looking for light and feebly-interacting particles
beyond the Standard Model (BSM).

Another possibility is that the neutralino is in fact the next-to-lightest
SUSY particle (NLSP), decaying into the LSP and a photon. In this work, we
study two such scenarios, both of them respecting the R-parity.

The first one corresponds to bino decaying into a SUSY partner of axion-like
particle (ALP) called ALPino. This is an attractive BSM scenario that
generalized the axion models invoked as a solution of strong CP problem [28,
29, 30], while the SUSY sector can solve the hierarchy problem [31, 32, 33,
34] and provide a DM candidate [35, 36].

The second scenario assumes that the LPS is composed of the gravitino, which
is the fermionic partner of the spin-2 graviton. It is predicted by
supergravity theories [37, 13, 38, 39], which incorporate local supersymmetry
transformations. For consistency, they necessarily combine SUSY and general
relativity, while at the same time mitigating the hierarchy problem. Moreover,
in certain regime, gravitino can be a DM candidate [40, 41, 42].

We note that the displaced bino decay signature has been investigated in
multiple previous works. In particular, ref. [20] considered bino decays
taking place at various fixed target experiments in the ALPino model,111We
extend their results by considering FASER, MATHUSLA, and NuCal detectors, and
by also investigating the scattering LLP signatures described in Sec. III.2.
while ref. [43], discussing the results of the SLAC beam dump experiment
E-137, considered its decay into gravitino and a photon.222It was an electron
beam dump experiment, so the leading production channel relied on the
t-channel selectron exchange. In light of LEP results on the masses of such
states, the bounds on gravitino mass obtained in [43] are not competitive with
LEP limits [44], while we will show that FASER2 and SHiP may improve them for
neutralinos with masses below $1\,\text{GeV}$. Other relevant works
considering $\sim$sub-GeV neutralinos at the intensity frontier are [45, 46,
47, 48, 49].

Our study is motivated by recent developments of the far-forward LHC detectors
[50], in particular the FASER experiment [51, 52, 53], which began operation
with the LHC Run 3. In particular, a number of papers [54, 55, 21, 56, 57, 58,
59, 60] considered the prospects of multiple BSM scenarios using the signature
of LLP decaying into a single photon and an invisible state. Compared to the
usual signature of LLP decay into a pair of charged leptons or photons, a
single high-energy photon can be present in the detector, e.g., due to the
muon or neutrino induced background. However, dedicated studies performed by
the FASER collaboration [53] have shown that it will be sensitive to them with
the same energy threshold on the visible energy deposited in the calorimeter
by the decaying LLP; see also the extensive discussion in [57].

In this work, we study such decays to investigate the prospects of sub-GeV
bino-like neutralino at the LHC-based detectors: FASER2 [61, 62, 63],
FASER$\nu$2 [64, 65, 66, 67], and MATHUSLA [68, 69], as well as in the beam
dump experiment CHARM [70], NuCal [71, 72], and SHiP [73, 74]. We also
consider additional LLP signatures - scattering with electrons and secondary
bino production by upscattering of the LSP with nuclei taking place in front
of decay vessel [75]. In particular, the latter process was shown to allow
FASER2 to cover a significant portion of the shorter-lived LLP regime,
characterized by LLP decay length of $d\sim 1\,\text{m}$, for many BSM
scenarios [57, 58, 59, 60].

The paper is organized as follows. In Sec. II, we introduce the SUSY models
for low-energy PQ or SUSY breaking scales. We describe how such a regime
naturally leads to long-lived neutralinos in both scenarios. In Sec. III, we
discuss the production mechanisms of light SUSY species - neutralino and LSP -
in intensity frontier searches. We also introduce signatures involving LLPs,
and we describe details of their simulation. In Sec. IV, we discuss our
results. The main one is the sensitivity reach of each detector considered for
both neutralino decay scenarios. In Sec. V, we summarize our findings.

Figure 1:  Modes of neutralino production by meson decays as a function of
$m_{\tilde{\chi}_{0}}$ for ALPino (left) and gravitino (right). The decays
into neutralino-LSP depending on $f_{a}$ or $F_{\mathrm{SUSY}}$ are denoted by
solid lines, while decays into a pair of neutralinos, which are independent of
such couplings, are indicated by dotted lines; color coding indicates
contributions of each meson according to the legend. In fact, for ALPino, the
$f_{a}$-independent decays dominate for the allowed values of $f_{a}$, while
for gravitino, the decays depending on $F_{\mathrm{SUSY}}$ are the leading
ones.

##  II Models

Since we are interested in the MeV-GeV mass range of a neutralino acting as
the NLSP, we assume that it consists of pure bino, while the LSP is either
ALPino or gravitino.333We note that a combined scenario with both ALPino and
gravitino is possible and has interesting cosmological implications - see,
e.g., [76, 77, 78]. In this scenario, NLSP decay into ALP and LSP is governed
by SUSY breaking coupling, while ALP decays into SM states by PQ-scale
suppressed couplings. Clearly, the LLP phenomenology crucially depends on the
interplay between these two scales, which we leave for future study. In both
of these models, the bino-LSP-photon coupling is the most relevant to our
analysis, while other interactions - or specifics of mass spectrum of other
SUSY states - play only a marginal role; see also the discussion in Sec.
III.1.

Such a coupling is responsible not only for bino decays,
$\tilde{\chi}_{0}\to\mathrm{LSP}+\gamma$, but also for the leading production
mechanism of $\tilde{\chi}_{0}$-$\mathrm{LSP}$ pairs - vector meson decays.
Moreover, it also governs the efficiency of the LSP-electron or LSP-nucleus
upscattering processes that result in NLSP production. When such secondary
production takes place in proximity to the decay vessel, it may allow the
sensitivity reach to be extended to shorter-lived LLPs [75].

The main similarity between our two SUSY scenarios is that the operator
responsible for the $\tilde{\chi}_{0}$-$\mathrm{LSP}$-$\gamma$ coupling is
mass dimension-5 operator, hence it is suppressed by the New Physics energy
scale. In fact, it is the PQ energy breaking scale $f_{a}$ for ALPino, and the
SUSY breaking scale $\sqrt{F_{\mathrm{SUSY}}}$ for gravitino. Both of these
parameters, along with the mass of the neutralino, are among the free
parameters in our analysis.

On the other hand, while the ALPino mass is not necessarily strictly related
to the SUSY or PQ breaking scales, and thus depends on the specifics of the
SUSY scenario considered [79, 78, 80], the gravitino mass is fixed by the SUSY
breaking scale,
$m_{\tilde{G}}=F_{\mathrm{\mathrm{SUSY}}}/(\sqrt{3}\,m_{\mathrm{Pl.}})$, where
$m_{\mathrm{Pl.}}=\sqrt{\hbar c/(8\pi G_{N})}=2.4\times 10^{18}\,\text{GeV}$
is the reduced Planck mass, and $G_{N}$ is the Newton’s gravitational
constant.

###  II.1 Neutralino-ALPino

The relevant part of the Lagrangian is [81, 82, 83]

$\displaystyle\\!\\!\mathcal{L}$
$\displaystyle\supset\frac{\alpha_{\mathrm{EM}}C_{a\gamma\gamma}}{16\pi
f_{a}}\overline{\tilde{a}}\gamma^{5}\left[\gamma^{\mu},\gamma^{\nu}\right]\tilde{\chi}_{0}F_{\mu\nu},$
(1)

where $\tilde{a}$ and $\tilde{\chi}_{0}$ denote the ALPino and neutralino
fields, respectively, $F_{\mu\nu}$ is the electromagnetic (EM) field strength
tensor, $\alpha_{\mathrm{EM}}$ is the fine structure constant,
$C_{a\gamma\gamma}\sim O(1)$ is a mixing constant that depends on the ALP
scenario [35, 36], and $f_{a}$ denotes the PQ breaking scale.

The ALPino mass is in general a model-dependent quantity [79, 78, 80], so it
essentially acts as a free parameter. However, since the value of the ALPino
mass will not significantly affect our discussion (as long as it is
significantly smaller than the neutralino mass and does not cause large phase
space suppression of the NLSP decay width), we follow [20] and set its value
as follows: $m_{\tilde{a}}=10\,\text{MeV}$.

Ref. [20] considered the prospects of bino decays into ALPino and photon in
various beam dump experiments, in which the NLSP production and decay points
are separated by a large distance, typically $L\sim 100\,\text{m}$. This
length scale largely determines the LLP decay length that can be probed is
such detectors [43, 84]. In the case of a sub-GeV bino, the following
benchmark corresponds to a short-lived NLSP that can be covered in this way:

$\displaystyle d_{\tilde{\chi}_{0}}\simeq$
$\displaystyle\,100\,\text{m}\times\left(\frac{E}{1000\,\text{GeV}}\right)\left(\frac{0.1\,\text{GeV}}{m_{\tilde{\chi}_{0}}}\right)^{4}\left(\frac{f_{a}}{30\,\text{GeV}}\right)^{2},$
(2)

where $d_{\tilde{\chi}_{0}}=c\tau\beta\gamma$, $\tau=1/\Gamma$ is the bino
lifetime, $\gamma=E/m$ is the boost factor in the laboratory reference frame,
and $\beta=\sqrt{1-1/\gamma^{2}}$.

The lifetime of a sub-GeV bino is determined by two-body decays given by Eq.
10, while three-body decays mediated by an off-shell photon typically
contribute less than a percent, see Eq. 11.

###  II.2 Neutralino-gravitino

The Lagrangian is [85, 38, 86]444Feynman rules for this model are given in
ref. [87].

$\displaystyle\\!\\!\mathcal{L}$
$\displaystyle\supset-\frac{i}{8m_{\mathrm{Pl.}}}\bar{\psi}_{\mu}[\gamma^{\rho},\gamma^{\sigma}]\gamma^{\mu}\tilde{\chi}_{0}F_{\rho\sigma},$
(3)

where $\psi_{\mu}$ denotes the gravitino wavefunction, and the Lorentz index
indicates the spin-$\frac{3}{2}$ character of the field.

Compared to the ALPino model, mass of gravitino is not a free parameter.
Instead, the SUSY breaking energy scale determines it by the super-Higgs
mechanism [37, 88, 89, 38, 90]. As a result, the gravitino mass is
$m_{\tilde{G}}=F_{\mathrm{\mathrm{SUSY}}}/(\sqrt{3}\,m_{\mathrm{Pl.}})$.
Moreover, due to the SUSY Equivalence Theorem [91], the gravitino wavefunction
can be approximated at high energies as follows:555In our calculations, we
instead take into the account all gravitino degrees of freedom through the
gravitino polarization tensor given by Eq. 13.

$\displaystyle\psi_{\mu}\simeq
i\sqrt{\frac{2}{3}}\frac{\partial_{\mu}\psi}{m_{\tilde{G}}},$ (4)

where $\psi$ is the spin-$\frac{1}{2}$ goldstino absorbed by the gravitino. As
a result, even though gravitino interactions are suppressed by the Planck mass
(due to its character as a SUSY partner of the graviton), cf. Eq. 3, the
massive gravitino compensates this suppression by the
$\frac{1}{m_{\tilde{G}}}$ factor. Effectively, the bino-gravitino-photon
coupling is therefore proportional to the inverse of the SUSY breaking scale,
$1/F_{\mathrm{SUSY}}$, instead of $1/m_{\mathrm{Pl.}}$.

For $\sim$sub-GeV neutralinos, the long-lived regime corresponds to low-energy
SUSY breaking scales,

$\displaystyle d_{\tilde{\chi}_{0}}\simeq$
$\displaystyle\,100\,\text{m}\times\left(\frac{E}{1000\,\text{GeV}}\right)\left(\frac{0.1\,\text{GeV}}{m_{\tilde{\chi}_{0}}}\right)^{5}\left(\frac{F_{\mathrm{\mathrm{SUSY}}}}{(60\,\text{GeV})^{2}}\right)^{2},$
(5)

where $d_{\tilde{\chi}_{0}}$ is the bino decay length in the laboratory
reference frame. Its lifetime is determined by decays into gravitino and
photon, while decays into gravitino and $e^{+}e^{-}$ pair are suppressed, cf.
Eqs. 12 and 15; see the bottom panels of Fig. 3.

##  III Neutralino at intensity frontier searches

###  III.1 Neutralino-LSP production modes

Light neutralinos with masses below the mass of a proton can be efficiently
produced in rare meson decays. These mesons are generated in large numbers in
both proton-proton (p-p) collisions at the LHC and proton-target collisions in
beam dump experiments.

The branching ratio of vector and pseudoscalar meson decays into a pair of
neutralinos taking place by $t$-channel squark exchange have been calculated
in [92, 48]. In fact, for the ALPino model, ref. [20] \- see Eq. 9 and the
discussion therein - used these results assuming the following mass spectrum:
squark mass were set at $m_{\tilde{q}}=3\,\text{TeV}$, while the masses of the
other SUSY particles were fixed at $10\,\text{TeV}$.666We adopt this SUSY mass
spectrum in the following discussion. Reducing the value of any of these
masses will not have a significant effect on our results, except for squark
masses, which affect bino pair production, which would increase the detectors
sensitivity to the ALPino model. However, it would typically not affect the
gravitino scenario. As discussed in Sec. IV.1, this production channel allows
for coverage of sizable part of the available parameter space.

On the other hand, in a number of BSM scenarios with a higher-dimensional LLP
coupling to a photon, the leading LLP production channels are vector meson
decays mediated by an off-shell photon, see, e.g., [93, 94, 59]. Therefore, we
also calculated the branching ratios of such decays, which are described by
Eq. 16 for the two SUSY scenarios considered. We also considered the phase-
space suppressed decays of pseudoscalar mesons - the corresponding
differential branching ratios are given by Eq. 17.

As shown in Fig. 1, the leading decay channel among the photon-mediated decays
is $J/\psi$ (gold solid line). It is the heaviest meson among those produced
in sufficiently large quantities - hence our result is consistent with the
discussion in Sec. III in [94], which states that the vector meson branching
ratio is approximately proportional to the mass squared of the meson. The same
relation holds for ALPino, while for gravitino, this branching ratio actually
depends on the fourth power of the meson mass, cf. top and bottom lines of Eq.
16.

For the ALPino model, we found that indeed the dominant neutral meson decays
are those mediated by the exchange of heavy squarks, which are thus
independent of $f_{a}$, and result in the production of a neutralino pair.
These contributions are denoted by dotted lines in Fig. 1, and for the allowed
region of the parameter space, $f_{a}\gtrsim 200\,\text{GeV}$, they clearly
overtake the photon-mediated decays (indicated by solid lines).

On the other hand, for the gravitino model, in the allowed SUSY breaking scale
that is relevant for the intensity frontier searches,
$200\,\text{GeV}\lesssim\sqrt{F_{\mathrm{SUSY}}}\lesssim 3\,\text{TeV}$, the
photon-mediated decays dominate - see the right panel in Fig. 1. Such a
dependence can be explained by comparing the Lagrangians of the two models,
given by Eqs. 1 and 3, respectively - the photon-coupling in the ALPino
scenario has an additional factor of $\alpha_{\mathrm{EM}}/(2\pi)\simeq
10^{-3}$ with respect to the gravitino scenario.

We also checked that the pair production of neutralinos occuring in p-p
collisions - either at the LHC or at beam dumps - does not improve the FASER
or SHiP sensitivity towards the larger mass regime, $m_{\tilde{\chi}_{0}}\gg
1\,\text{GeV}$. The first detector has too small angular coverage, while the
energy beam of the latter is too small to produce heavy neutralinos with
sufficient abundance. On the other hand, this production channel was found to
allow MATHUSLA to cover long-lived very heavy neutralinos,
$m_{\tilde{\chi}_{0}}\gtrsim 100\,\text{GeV}$ \- see discussion in MATHUSLA
physics case paper [69]. The difference between MATHUSLA and FASER is that the
first of these LHC-based detectors has much larger decay vessel volume, is
closer to the beamline, and is placed highly off-axis. As a result, it covers
a much larger part of the solid angle, and covers the LLPs produced with large
transverse momentum $p_{T}$, while FASER is positioned in the far-forward
direction.

Finally, we implemented the equations describing the production of bino-NLSP,
Eq. 16 and Eq. 17, and also the production of bino pairs, Eq. 9 from [20], in
a modified $\tt FORESEE$ package [95]. We use it to generate the spectrum of
bino-NLSP, which is then used to simulate bino decays taking place in each
detector considered and other signatures described in Sec. III.2. We note that
an opposite mass hierarchy between bino and ALPino or gravitino is also
possible, and our simulation is adapted to such a case as well.

###  III.2 Experiments and LLP signatures

In this section, we briefly describe the intensity frontier detectors
sensitive to single-photon LLP decays. We also introduce the main signatures
of LLPs that we use to study our two SUSY scenarios.

The characteristics of both of these topics have been discussed at length in
Sec. III in [59] for the dark axion portal, which is characterized by similar
phenomenology. In particular, in that scenario, the dark sector (DS) states
are connected to the SM by a dimension-5 coupling to a photon. Therefore, the
following presentation is brief, while details can be found in the
aforementioned work.

#### Experiments

We consider a number of intensity frontier experiments in order to take
advantage of their different features that may allow complementary coverage of
the parameter space. The specifics of each detector is given in Tab. 1 in
[59]. We use these parameters in our simulation.

Among the beam dumps, these are: CHARM [70], NuCal [71, 72], and SHiP [73,
74]. We also study detectors dedicated for LLP searches at the LHC, such as
FASER2 [61, 62, 63], FASER$\nu$ [64, 65], and MATHUSLA [68, 69].

Moreover, as a result of the efforts of the LHC-based LLP community, another
separate facility, called the Forward Physics Facility (FPF) [96, 67, 97], has
been proposed. It would contain not only a much larger version of the FASER2
detector, but it would also house additional ones. The detectors relevant to
our analysis are FASER$\nu$2 [66, 67] and FLArE [66].

All of these detectors use beams of protons that hit a stationary, dense
target in beam dumps or collide with other protons at the LHC. Since the
energy at the LHC is several orders of magnitude greater than that obtainable
in beam dumps, FASER, MATHUSLA and other such detectors probe much more
boosted LLPs than fixed target experiments. On the other hand, the luminosity
of the proton beam in dedicated beam dump experiments is significantly higher
than at the LHC. This usually results in their deeper reach toward smaller LLP
coupling values.

#### Bino decays

Our main LLP signature is bino decays occuring inside a decay vessel that is
separated from the LLP production point by a distance $L\sim 100\,\text{m}$.
Such large separation, together with a dedicated infrastructure, allows to get
rid of the SM background, perhaps except for neutrinos or muons, depending on
the detector design.

Such decays take place with the following probability:

$\displaystyle p(E)=e^{-L/d}(1-e^{-\Delta/d}),$ (6)

where $\Delta$ is the detector length and $d=d(E)$ indicates the LLP decay
length in the laboratory reference frame.

It is well known that the distance $L$ sets the length scale for the minimal
decay length $d$ that can be probed in such a way [43, 98, 84]. When $d\ll L$,
$p(E)\simeq e^{-L/d}$, while in the opposite regime, $d\gg L$,
$p(E)\simeq\Delta/d$ [99, 26].

Consequently, only sufficiently long-lived species can be probed. Since the
detectors introduced in previous paragraph cover a wide range of the distance
$L$, some variation in their sensitivity to LLP decays can be expected.

#### LSP-NLSP upcattering

LLPs connected to the SM by a photon can also be efficiently produced by
upscattering process of the lighter DS species into the heavier, unstable one
- see Fig. 1 in [75] for an illustration, and [75, 57, 59, 60] for dedicated
studies.

Such scattering is particularly efficient in the coherent regime,
characterized by low-momentum exchange of the off-shell photon. Then, it can
take place in a coherent manner, scattering with the whole nucleus.

In fact, assuming a particular common from of the form factor - given in Eq.
B2 in [58] \- which is responsible for the partial screening of the nucleus by
the atomic electrons, one can obtain a closed-form of the upscattering cross
section, $\mathrm{LSP}+N\to\mathrm{NLSP}+N$:

$\displaystyle\sigma_{\tilde{a}N-\tilde{\chi}_{0}N}\simeq$
$\displaystyle\frac{\alpha_{\mathrm{EM}}^{3}\cos^{2}\theta_{W}Z^{2}}{16\pi^{2}f_{a}^{2}}\times$
(7) $\displaystyle\left(\log\left(\frac{d}{1/a^{2}-t_{max}}\right)-2\right),$
$\displaystyle\sigma_{\tilde{G}N-\tilde{\chi}_{0}N}\simeq$
$\displaystyle\frac{\alpha_{\mathrm{EM}}\cos^{2}\theta_{W}Z^{2}}{2F_{\mathrm{SUSY}}^{2}}\times$
$\displaystyle\left(d+m_{\tilde{\chi}_{0}}^{2}\left(\log\left(\frac{d}{1/a^{2}-t_{max}}\right)-2\right)\right),$

where $N$ is the nucleus, $m_{e}$ is the electron mass, $Z$ and $A$ are the
atomic number and mass number of a nucleus, respectively,
$a=111Z^{-1/3}/m_{e}$, $d=0.164\,\text{GeV}^{2}A^{-2/3}$, and
$t_{\mathrm{max}}\simeq-(m_{\mathrm{NLSP}}^{4}+m_{\mathrm{LSP}}^{4})/(4E_{\mathrm{LSP}}^{2})$.

We obtained these formulas by integrating the differential cross section
following the method used in [100] for photophilic ALP. In the calculation of
the squared amplitude, we included only the leading diagrams involving photon
exchange. In particular, we neglected the diagrams with sleptons in the
$t$-channel, which are negligible for slepton masses in the range of a few
hundred GeV.

In our case, the production process takes place on tungsten (W) layers of the
emulsion detector FASER$\nu 2$ located upstream of FASER2, which is the main
decay vessel. Such upscattering depends on the coupling constant for
$m_{\tilde{\chi}_{0}}=0.1\,\text{GeV}$ in the following way:

$\displaystyle\sigma^{\text{W}}_{\tilde{a}N\to\tilde{\chi}_{0}N}\simeq\frac{1.5\times
10^{-4}}{\mathrm{GeV}^{2}}\times\left(\frac{\mathrm{GeV}}{f_{a}}\right)^{2},$
(8)
$\displaystyle\sigma^{\text{W}}_{\tilde{G}N\to\tilde{\chi}_{0}N}\simeq\frac{3}{\mathrm{GeV}^{2}}\times\left(\frac{1\mathrm{GeV}^{2}}{F_{\mathrm{\mathrm{SUSY}}}}\right)^{2}.$

We note the suppression of the cross section for the ALPino is again caused by
the $\alpha_{\mathrm{EM}}/(2\pi)\simeq 10^{-3}$ factor in the Lagrangian given
by Eq. 1 with respect to the gravitino scenario. Moreover, even in the latter
case, the upscattering cross section is not very efficient. In fact, comparing
Eq. 8 with the result obtained in [100] for photophilic ALP, see Eq. 15
therein, it is smaller by a factor of $\sim 30$.

In the next section, we also describe the FASER2 sensitivity reach obtained by
secondary neutralino production at FASER$\nu$2, followed by its decay inside
either the neutrino detector, or inside the main decay vessel. Then, instead
of Eq. 6, we use Eq. 6 in [59] to obtain the probability of such decay - for
details, see discussion therein.

In addition to the coherent scattering, incoherent scattering with protons or
electrons is also possible. In fact, the latter signature is particularly
useful for sub-GeV BSM species and it has been investigated in many papers,
see, e.g., [101, 102, 66, 103].

Proceeding analogously to the procedure used to derive Eq. 7, we obtained the
following closed forms of the electron upscattering cross sections:

$\displaystyle\sigma_{\tilde{a}e^{-}\to\tilde{\chi}_{0}e^{-}}\simeq$
$\displaystyle\frac{\alpha_{\mathrm{EM}}^{3}\cos^{2}\theta_{W}}{16\pi^{2}f_{a}^{2}}\times\log\left(\frac{E_{R}^{\mathrm{max}}}{E_{R}^{\mathrm{min}}}\right),$
(9) $\displaystyle\sigma_{\tilde{G}e^{-}\to\tilde{\chi}_{0}e^{-}}\simeq$
$\displaystyle\frac{\alpha_{\mathrm{EM}}\cos^{2}\theta_{W}}{2F_{\mathrm{SUSY}}^{2}}\times$
$\displaystyle\left(2m_{e}(E_{R}^{\mathrm{max}}-E_{R}^{\mathrm{min}})+m_{\tilde{\chi}_{0}}^{2}\log\left(\frac{E_{R}^{\mathrm{max}}}{E_{R}^{\mathrm{min}}}\right)\right).$

For this signature, instead of Eq. 6, we used Eq. 8 in [59]. Since scattering
signature is affected by large neutrino background, we adapt angular and
energy cuts found in [66], see also Tab. 1 in [59].

Figure 2:  The sensitivity of FASER2 to neutralino decays into ALPino and
photon for fixed $m_{\tilde{a}}=10\,\text{MeV}$. The FPF version of the
detector will exceed the current bounds set by NuCal and LEP due to its larger
size compared to the baseline version of FASER2.

Figure 3:  The sensitivity of FASER2, MATHUSLA, and SHiP to the neutralino-
gravitino model. Leading two-body decays will allow FASER2 (solid black line)
to partly extend the LEP bound, while FPF FASER2 will even reach the LHC
(model-dependent) bound. We also present results for three-body neutralino
decays at MATHUSLA (brown) and FASER2 (red solid line), which cover high and
low $p_{T}$ regimes of LLPs produced due to the p-p collisions at the LHC,
respectively. Secondary neutralino production extends the sensitivity of
FASER2 (black dashed and dot-dashed lines) into the short-lived, higher mass
regime, while electron scattering at FASER$\nu$2 and FLArE (solid and dotted
gold lines, respectively) covers the lower mass regime, which, however, are
both already excluded by LEP.

##  IV Results

###  IV.1 Sensitivity reach for ALPino

In Fig. 2, we present our results for the scenario when ALPino is the LSP.

For beam dump experiment, we find agreement with results of [20]. We consider
additional detector of this type, NuCal, and we find that it actually improves
over the NOMAD [104] sensitivity shown in that work.

Since the leading channel of NLSP-LSP production is $f_{a}$-independent meson
decay into a pair of binos, there is hardly any flux of ALPinos - see the
discussion in Sec. III.1, in particular the left panel of Fig. 1. As a result,
neither the secondary production given by the top line of Eq. 7, nor
upscattering on electrons, given by the top line of Eq. 9, are efficient.
Consequently, FASER2 will not have sensitivity to such signatures.

On the other hand, bino-pair production can be quite efficient. While the
baseline versions of FASER2 taking data during the High Luminosity era of the
LHC will not improve over NuCal (but its sensitivity is greater than NOMAD),
the FPF FASER2 will extend it in the $m_{\tilde{\chi}_{0}}\simeq
0.1\,\text{GeV}$ mass regime. For LLP decays produced in the primary
production, its main advantage over the baseline version is simply its larger
size.

###  IV.2 Sensitivity reach for gravitino

On the other hand, when gravitino acts as the LSP, the dominant production
modes produce equal fluxes of gravitinos and neutralinos, allowing the
additional upscattering signatures described in Sec. III.2. In fact, contrary
to the ALPino scenario, both neutralino production and decays are controlled
by the NLSP-LSP-photon coupling, which here depends on the SUSY breaking scale
as $1/F_{\mathrm{SUSY}}$.

This allows one to search not only for the displaced $\tilde{\chi}_{0}$
decays, but also for the electron scattering signature and for the decays of
$\tilde{\chi}_{0}$ produced by upscattering occuring at the FASER$\nu$2
detector located before FASER2.

In Fig. 3, we present our main results for this model. The areas shaded in
gray are excluded by NuCal or LEP [44, 105]. We also indicate a model-
dependent bound from the LHC [106, 107] by dashed gray line.

As mentioned earlier, we consider two versions of the FASER2 detector - the
results for the baseline version are in the left panel, while the results for
FPF FASER2 are in the right panel. The sensitivity lines derived for the two-
body bino decays are marked by black lines for FASER and green for SHiP, while
those for three-body decays are indicated by red (FASER2) and brown (MATHUSLA)
lines. The sensitivity lines correspond to the number of bino decays (number
of LLP signatures in general case) given in Tab. 1 in [59] for each detector
considered.

As is clearly seen, FASER2 will be able to significantly extend the LEP limit
for $m_{\tilde{\chi}_{0}}\gtrsim 0.1\,\text{GeV}$ mass range, while searches
for $e^{+}e^{-}$ pairs produced in the three-body decays at MATHUSLA and
FASER2 will be competitive with current LEP and NuCal bounds. Moreover, the
FPF version of FASER2 may even reach the strongest limit on light gravitinos
coming from the LHC.

The upscattering signatures allow to cover the smaller lifetime regime,
$d_{\tilde{\chi}_{0}}\sim 1\,\text{m}$, which, however, is already excluded by
LEP for both locations of the bino decays: FASER2 (black dashed line) and
FASER$\nu$2 (black dot-dashed line). Finally, the electron scattering
signature covers the low mass region of the bino, which, however is also
already excluded.

##  V Conclusions

Neutralino can act as a LLP decaying into a single photon and LSP in various
SUSY scenarios. Constraining them can be challenging in high-energy detectors
designed for much heavier and short-lived BSM species. However, searches at
the intensity frontier are well suited to such a regime.

We found that FASER2 and SHiP, which are particularly predisposed to such a
difficult signature [108, 57, 20, 21], will be able to meaningfully extend the
current constraints on the low-energy SUSY breaking scale in two scenarios
involving sub-GeV neutralinos. In the former, the LSP is composed of ALPino,
while in the latter case, gravitino plays that role.

For the gravitino model, we also investigated additional LLP signatures:
secondary neutralino production due to upscattering taking place at the
FASER$\nu$2 detector in front of the main decay vessel, FASER2, and three-body
decays depositing visible energy into a $e^{+}e^{-}$ pair.

Finally, we considered the extended version of the FASER2 experiment, the
proposed FPF. Due to its larger size, the FPF scenario of FASER2 may reach the
strongest limit on light gravitinos coming from the LHC, while SHiP, due to
its higher luminosity, may improve them further.

###### Acknowledgements.

This work was supported by the Institute for Basic Science under the project
code, IBS-R018-D1.

##  Appendix A Neutralino decays

In these appendices, we give the relevant decay widths and cross sections,
which we used in our simulation.

As in the rest of our paper, we assume the neutralino is composed of pure
bino, which leads to an additional factor of $\cos\theta_{W}$ \- where
$\theta_{W}$ is the Weinberg angle - when reading the Feynman rules described
by the Lagrangians given by Eqs. 1 and 3.

###  A.1 ALPino

The two-body decay width for bino decaying into an ALPino and a photon is [20]

$\displaystyle\Gamma_{\tilde{\chi}_{0}\to\tilde{a}\gamma}=\frac{\alpha_{\mathrm{EM}}^{2}\cos^{2}\theta_{W}}{128\pi^{3}}\frac{m_{\tilde{\chi}_{0}}^{3}}{f_{a}^{2}}\left(1-\frac{m_{\tilde{a}}^{2}}{m_{\tilde{\chi}_{0}}^{2}}\right)^{3}.$
(10)

Below, we give the integrated decay width for the leading three-body decay
into an ALPino and an electron-positron pair in the limit of
$m_{\tilde{\chi}_{0}}\gg m_{\tilde{G}},m_{e^{-}}$,

$\Gamma_{\tilde{\chi}_{0}\to\tilde{a}e^{+}e^{-}}\simeq\frac{\alpha_{\mathrm{EM}}^{3}\cos^{2}\theta_{W}}{1152\pi^{4}f_{a}^{2}m_{\tilde{\chi}_{0}}^{3}}\left(18m_{\tilde{\chi}_{0}}^{4}m_{e^{-}}^{2}-4m_{\tilde{\chi}_{0}}^{6}-32m_{e^{-}}^{6}+3m_{\tilde{\chi}_{0}}^{6}\log\left(\frac{m^{2}_{\tilde{\chi}_{0}}}{4m_{e^{-}}^{2}}\right)\right).$
(11)

The amplitude squared for the general mass scheme and the code evaluating such
decay width, can be found in the auxiliary materials of the paper.

###  A.2 Gravitino

The two-body decay width for bino decaying into an gravitino and a photon is

$\displaystyle\Gamma_{\tilde{\chi}_{0}\to\tilde{G}\gamma}=$
$\displaystyle\frac{\cos^{2}\theta_{W}m^{5}_{\tilde{\chi}_{0}}}{16\pi
F_{\mathrm{SUSY}}^{2}}\left(1-\frac{m^{2}_{\tilde{G}}}{m^{2}_{\tilde{\chi}_{0}}}\right)^{3}\left(1+\frac{m^{2}_{\tilde{G}}}{m^{2}_{\tilde{\chi}_{0}}}\right).$
(12)

We used the Feynman rules described in [87], where in particular we used the
full form of the gravitino polarization tensor. It is defined as the sum of
the gravitino field with momentum $p$ over its spin degrees of freedom,

$\displaystyle\Pi^{\pm}_{\mu\nu}(k)\equiv\sum_{s=\pm\frac{1}{2},\pm\frac{3}{2}}\psi^{\pm,s}_{\mu}(k)\overline{\psi}^{\pm,s}_{\nu}(k).$
(13)

In the high-energy limit, where
${\Pi^{\pm}_{\mu\nu}(k)\simeq-\not{k}(g_{\mu\nu}-\frac{2p_{\mu}p_{\nu}}{3m^{2}_{\tilde{G}}})}$,
we obtain the well-known result [109, 110, 111],

$\displaystyle\Gamma_{\tilde{\chi}_{0}\to\tilde{G}\gamma}=\frac{\cos^{2}\theta_{W}m^{5}_{\tilde{\chi}_{0}}}{16\pi
F_{\mathrm{SUSY}}^{2}}\left(1-\frac{m^{2}_{\tilde{G}}}{m^{2}_{\tilde{\chi}_{0}}}\right)^{3}\left(1+3\frac{m^{2}_{\tilde{G}}}{m^{2}_{\tilde{\chi}_{0}}}\right).$
(14)

Since the MATHUSLA detector may not be sensitive to single-photon decays, we
also considered phase-space suppressed decays into a gravitino and an
electron-positron pair. In the limit of $m_{\tilde{\chi}_{0}}\gg
m_{\tilde{G}},m_{e^{-}}$, the following formula describes it:

$\displaystyle\Gamma_{\tilde{\chi}_{0}\to\tilde{G}e^{+}e^{-}}\simeq$
$\displaystyle\frac{\alpha_{\mathrm{EM}}\cos^{2}\theta_{W}m_{\tilde{\chi}_{0}}^{5}}{576\pi^{2}F_{\mathrm{SUSY}}^{2}}\times$
(15)
$\displaystyle\left(24\log\left(\frac{m_{\tilde{\chi}_{0}}}{m_{e^{-}}}\right)-25-12\log(4)\right),$

while the general formula for the amplitude squared can be found in the
Mathematica notebook linked to the paper.

##  Appendix B Pseudoscalar and vector meson decays

###  B.1 Vector meson decays

The following are formulas for vector meson decays mediated by an off-shell
photon that result in the production of LSP-NLSP pair,
$V(p_{0})\\!\to\\!\gamma^{*}(p_{1}+p_{2})\\!\to\\!\mathrm{LSP}(p_{1})+\mathrm{NLSP}(p_{2})$:

$\displaystyle\frac{{\rm BR}_{V\rightarrow\tilde{a}\tilde{\chi}_{0}}}{{\rm
BR}_{V\rightarrow e^{+}e^{-}}}\\!=\\!\cos^{2}\theta_{W}$
$\displaystyle\frac{\alpha_{\text{EM}}\left(m_{V}^{2}+2(m_{\tilde{a}}-m_{\tilde{\chi}_{0}})^{2}\right)(m_{V}^{2}-(m_{\tilde{a}}+m_{\tilde{\chi}_{0}})^{2})\sqrt{\left(-m_{V}^{2}+m_{\tilde{a}}^{2}+m_{\tilde{\chi}_{0}}^{2}\right)^{2}-4m_{\tilde{a}}^{2}m_{\tilde{\chi}_{0}}^{2}}}{128\pi^{3}f_{a}^{2}\sqrt{m_{V}^{2}-4m_{e}^{2}}\left(m_{V}^{3}+2m_{V}m_{e}^{2}\right)},$
(16) $\displaystyle\frac{{\rm
BR}_{V\rightarrow\tilde{G}\tilde{\chi}_{0}}}{{\rm BR}_{V\rightarrow
e^{+}e^{-}}}\\!=\\!\cos^{2}\theta_{W}$
$\displaystyle\frac{(m_{V}^{2}-(m_{\tilde{G}}+m_{\tilde{\chi}_{0}})^{2})\sqrt{\left(-m_{V}^{2}+m_{\tilde{G}}^{2}+m_{\tilde{\chi}_{0}}^{2}\right)^{2}-4m_{\tilde{G}}^{2}m_{\tilde{\chi}_{0}}^{2}}}{8\pi
F_{\mathrm{SUSY}}^{2}\alpha_{\text{EM}}\sqrt{m_{V}^{2}-4m_{e}^{2}}\left(m_{V}^{3}+2Mm_{e}^{2}\right)}\times$
$\displaystyle\times\left(2m_{V}^{2}\left(m_{\tilde{G}}^{2}+m_{\tilde{G}}m_{\tilde{\chi}_{0}}-m_{\tilde{\chi}_{0}}^{2}\right)+m_{V}^{4}+(m_{\tilde{G}}-m_{\tilde{\chi}_{0}})^{2}\left(3m_{\tilde{G}}^{2}+m_{\tilde{\chi}_{0}}^{2}\right)\right),$

where ${\rm BR}_{V\rightarrow e^{+}e^{-}}$ is the branching ratio
corresponding to decays into $e^{+}e^{-}$ [112], which we took from the PDG
[112].

###  B.2 Pseudoscalar meson decays

The following formulas describe the differential branching ratios of the
pseudoscalar meson decays into $\gamma(p_{1})$ and
$\mathrm{LSP}(p_{2})$-$\mathrm{NLSP}(p_{3})$ pair. We use the form
particularly useful for Monte Carlo simulations, where
$q^{2}=(p_{2}+p_{3})^{2}$ is the momentum squared of the off-shell photon
mediating the decay, and $\theta$ is the angle between the LSP momentum in the
rest frame of the off-shell photon and the momentum of the off-shell photon in
the meson rest frame.

$\displaystyle\frac{d{\rm
BR}_{P\\!\to\\!\gamma\tilde{a}\tilde{\chi}_{0}}}{dq^{2}d\cos\theta}={\rm
BR}_{P\rightarrow\gamma\gamma}\cos^{2}\theta_{W}$
$\displaystyle\\!\times\\!\\!\Bigg{[}\frac{\alpha_{\mathrm{EM}}^{2}}{512\pi^{4}f_{a}^{2}m_{P}^{6}q^{6}}\left(q^{2}-m_{P}^{2}\right)^{3}\sqrt{\left(m_{\tilde{\chi}_{0}}^{2}+m_{\tilde{a}}^{2}-q^{2}\right)^{2}-4m_{\tilde{\chi}_{0}}^{2}m_{\tilde{a}}^{2}}$
(17)
$\displaystyle\times\left((m_{\tilde{\chi}_{0}}+m_{\tilde{a}})^{2}-q^{2}\right)\left(\cos(2\theta)\left((m_{\tilde{\chi}_{0}}-m_{\tilde{a}})^{2}-q^{2}\right)+3(m_{\tilde{\chi}_{0}}-m_{\tilde{a}})^{2}+q^{2}\right)\\!\Bigg{]},$
$\displaystyle\frac{d{\rm
BR}_{P\\!\to\\!\gamma\tilde{G}\tilde{\chi}_{0}}}{dq^{2}d\cos\theta}={\rm
BR}_{P\rightarrow\gamma\gamma}\cos^{2}\theta_{W}$
$\displaystyle\\!\times\\!\\!\left[\frac{1}{64\pi^{2}F_{\mathrm{SUSY}}^{2}m_{P}^{6}q^{6}}(m_{P}^{2}-q^{2})^{3}(m_{\tilde{\chi}_{0}}^{2}-q^{2})^{4}(\cos(2\theta)+3)\\!\right],$

where $m_{P}$ is the mass of pseudoscalar meson and ${\rm
BR}_{P\rightarrow\gamma\gamma}$ is the branching ratio of the decay into two
photons, which we took from the PDG [112].

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