$\psi_{i}\to\psi_{j}\ell^{+}\ell^{-}$ decays would constrain the mass HNF
splittings through the inequality $m_{\ell\ell}<\Delta_{ij}m_{j}$. In this
case, displaced vertices would help isolate the different HNF decay cascades.
Finally, with displaced vertices in both ISR and s-channel, the experiment
would be able to extract more information on the boosts of the HNFs. The
boosts will be larger for s-channel HNFs than in ISR.

We now comment on a short-term possibility that can be pursued. Current
published datasets from searches for visible $A^{\prime}$ at KLOE/KLOE-2 [144,
145] and BES-III [146] can shed light on semi-visible $A^{\prime}$ by looking
for a broad invariant mass spectrum of dileptons on top of their smooth
backgrounds. While visible resonances are better constrained due to the
smaller backgrounds in a bump hunt, the smooth but often narrow distributions
of dilepton invariant masses $m_{\ell\ell}$ can still be searched for. We
identify the following channels as promising datasets for a semi-visible
analysis:

  * •

$e^{+}e^{-}\to\gamma(X\to\ell^{+}\ell^{-})$ [144, 10, 194, 195],

  * •

$\varphi\to\eta(X\to e^{+}e^{-})$ [71, 196], and

  * •

$e^{+}e^{-}\to(X\to\ell^{+}\ell^{-})X_{\rm invisible}$ [197].

  * •

$e^{+}e^{-}\to(X\to\ell^{+}\ell^{-})(Y\to(X\to\ell^{+}\ell^{-})(X\to\ell^{+}\ell^{-}))$
[198],

where $X$ and $Y$ are some fully visible resonances. We leave the evaluation
of these constraints to future literature.

Couplings to the $Z$ boson — In addition to the $A^{\prime}$ coupling to the
EM current, one can also explore the SM $Z$ boson coupling to the _dark_
current, shown in Eq. 3. For the large values of kinetic mixing explored here,
$Z$ decays can produce HNFs with branching ratios of

$\displaystyle\mathcal{B}(Z\to\psi_{i}\psi_{j})$
$\displaystyle\simeq\tau_{Z}\frac{|V_{ij}|^{2}G_{F}m_{Z}^{3}}{12\sqrt{2}\pi}\left(\frac{2g_{X}s_{W}\varepsilon}{g}\right)^{2}$
(52) $\displaystyle=|V_{ij}|^{2}\times
10^{-7}\times\left(\frac{\alpha_{D}}{0.1}\right)\left(\frac{\varepsilon}{10^{-2}}\right)^{2},$

where we neglected the small mass of the HNFs. While this BR is too small to
be constrained by invisible $Z$ decays, it can be used to look for lepton
jets, as done at LEP by the DELPHI [189] and L3 [190, 191] collaborations. The
signature considered at LEP was a single HNL decaying to leptons or quarks,
produced alongside a neutrino, $Z\to\nu N$. This search can, in principle,
also be used to constrain semi-visible dark photons, using the channels
$Z\to\psi_{1}\psi_{2}$, where $\psi_{2}$ decays either promptly or displaced
inside the detector and $\psi_{1}$ would constitute missing energy. The limits
will be modified due to the small splitting between parent and daughter HNFs,
which decreases the dilepton energy. Channels like
$Z\to\psi_{2,3,\dots}\psi_{2,3,\dots}$ could be much more common than in the
HNL scenario, where they are doubly suppressed by neutrino mixing. We leave a
detailed study of this interesting probe for future literature.

Neutrino experiments — Neutrino experiments can test semi-visible dark photon
models in two ways. Firstly, in a DM interpretation, the HNFs can be produced
in neutral meson decays and bremsstrahlung at the target, and the DM could
travel to the detector, where it can coherently interact with nuclei
$\mathcal{N}$ to produce its coannihilation partners [140, 52, 199],

$\psi_{1}\,\mathcal{N}\to(\psi_{2,3,\dots}\to\psi_{1}\ell^{+}\ell^{-})\,\mathcal{N}.$
(53)

The decays of the co-annihilators can then produce displaced dileptons. This
displacement is especially interesting for multi-component detectors like
MINERvA and the near detector of T2K, ND280 [200, 201], and can be explored at
future experiments like DUNE and Hyper-Kamiokande [202, 203]. At high
energies, experiments like IceCube and KM3NET can search for the upscattering
signature by using the atmospheric production of DM particles. The DM particle
can then upscatter via deep-inelastic scattering and the subsequent decay of
$\psi_{2,3,\dots}$ would be sufficiently displaced from the vertex to form a
double-bang signature. This has been explored in the context of HNLs in Refs.
[204, 205], but can be adapted to DM particles. Co-annihilators can also be
produced at the target alongside $\psi_{1}$. For small mass splittings, they
would be long-lived and can be constrained using high-energy beam dumps, like
CHARM and NuCAL [52], or searched for at forward or surface detectors at the
LHC [206] Their prompt decays, however, would contribute to the flux of
$\psi_{1}$.

In Ref. [199], the authors study the sensitivity of the SBN program at
Fermilab to the production of iDM. The three Liquid Argon detectors placed
along the Booster Neutrino Beam (BNB) are sensitive to the production of DM in
the BNB, as well as those produced in the NuMI beam, which is located at an
off-axis location. In addition, the BNB has the ability to run in off-target
mode, directing the proton straight into the beam dump. In this way, the flux
of neutrinos going through the detector is minimized due to the absence of
focus from the magnetic horns. The sensitivity of SBN to iDM models can
improve on CHARM and NuCAL constraints [52], especially in the mass range of
interest for our work.

If instead an HNL interpretation is assumed, the decays in flight of the
lightest HNF, $N_{4}=\psi_{1}$ can be searched for. The HNL $N_{4}$ can only
decay via the small mixing with SM neutrinos, through CC, NC, or dark photon
interactions. Typically, for the masses and large values of
$\alpha_{D}\varepsilon^{2}$ considered here, the decays of $N_{4}$ will
proceed predominantly through the dark photon, and will make the branching
ratios into $\psi_{i}\to\nu_{1,2,3}\ell^{+}\ell^{-}$ dominate. The leading
limits on this type of decay come from the T2K experiment [179]. The dark
photon contribution increases the decay rate of $N_{4}$, and enhances the
decay-in-flight event rate, opening new parameter space between cosmological
and laboratory-based limits on the mixing of $N_{4}$ with active neutrinos
[182].

Also in a HNL interpretation, there is a second possibility to produce the
HNLs. As discussed in Section VI.2, the scattering of neutrinos in the beam
with the dirt or in the detector can produce HNLs, which subsequently decay.
Through the exchange of a dark photon, active neutrinos coherently upscatter
on nuclei $\mathcal{N}$,

$\nu_{\alpha}\,\mathcal{N}\to(N_{i}\to N_{j}\ell^{+}\ell^{-})\,\mathcal{N}.$
(54)

The event rate is proportional to the mixing between the HNLs and active
neutrinos. The mixing with the muon flavor is the most relevant in this case
as most of the flux at accelerator and atmospheric experiments is composed of
$\nu_{\mu}$ and $\overline{\nu}_{\mu}$.

While both DIF and upscattering signatures are proportional to parameters that
have no impact on the semi-visible signatures at collider and fixed-target
experiments, they cannot be uniquely determined in the parameter space shown
in Section V. Nevertheless, evidence for either of these would indicate that
HNFs mix with active neutrinos, $\psi_{i}=N_{i}$, and confirm an HNL
hypothesis.

Kaon factories — Kaon decays can further constrain the parameter of semi-
visible dark photons in two ways: i) through the direct production of dark
photons, or ii) through the direct production of the HNFs. The latter
possibility is, in particular, a powerful probe mixing between neutrinos and
the HNFs. As discussed in Section III, dark photons can be produced directly
via kinetic mixing in $K\to\pi A^{\prime}$ as well as
$K^{+}\to\ell^{+}\nu_{\ell}A^{\prime}$. The subsequent semi-visible decays of
$A^{\prime}$ would then lead to multi-lepton final states [207], albeit with
at least two invisible particles. Direct production of $A^{\prime}$ via
kinetic mixing is, however, suppressed by $m_{A^{\prime}}^{2}/m_{K}^{2}$, and
has limited reach (cf. Fig. 2). A much more promising channel, however, is the
direct production of HNLs through their mixing with the electron or muon
flavor. Just as the upscattering signatures discussed in the paragraph above,
direct production of HNFs in kaon decays would provide direct evidence for
their heavy neutral lepton interpretation, $\psi_{i}=N_{i}$. As pointed out in
Ref. [176], NA62 can use a three-track search to look for the production and
the decay products of $N_{i}$,

$K^{+}\to\ell^{+}_{\alpha}(N_{i}\to
N_{j}\ell^{+}_{\beta}\ell^{-}_{\beta}),\text{ where
}\alpha,\beta\in\\{e,\mu\\}.$ (55)

The striking feature of this signature is the reconstruction of the dark
particle masses via the reconstructed quantities,
$m_{N_{i}}^{2}\sim(p_{K}-p_{\ell_{\alpha}})^{2}$ and
$m_{N_{j}}^{2}=(p_{K}-p_{\ell_{\alpha}}-p_{\ell_{\beta}^{+}}-p_{\ell_{\beta}^{-}})^{2}$.
The event rate is proportional to $|U_{\alpha N_{i}}|^{2}$, and the subsequent
primary as well as any secondary decays would provide the additional lepton
tracks at no additional cost to the rate. Displaced vertices in NA62 can be
identified for proper lifetimes of the HNFs as small as $c\tau^{0}\sim 10$ ps
thanks to the $\mathcal{O}(10)$ cm vertex resolution of NA62.

Future fixed target experiments (LDMX) — The next-generation fixed-target
experiment LDMX [208, 209] provides a unique setup to search for semi-visible
signatures. The proposed design is focused on searches for bremsstrahlung-
production of dark sector particles, $e^{-}\mathcal{N}\to
e^{-}A^{\prime}\mathcal{N}$, through the missing-_momentum_ technique.
Differently from NA64, LDMX aims to measure both the energy and transverse
momentum of the recoil electron, having more access to the kinematics of
$A^{\prime}$ production. The proposal considers a primary beam of electron of
$\sim 4-16$ GeV at SLAC impinging on a thin target inside a magnetic field
[210]. The beam would be tracked with low-mass trackers up and downstream of
the target and then stopped by a large detector with ECAL and HCAL components,
where the total energy of the recoil electron can be measured.

Because the primary electrons will not shower until reaching the ECAL, the
production of additional $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, and $\pi^{+}\pi^{-}$
pairs in prompt semi-visible $A^{\prime}$ decays would provide a striking
signature in the experiment. In contrast to NA64, LDMX offers a lower-energy
beam, enlarging its reach in HNF lifetime, and tracking capabilities, allowing
the additional tracks to be seen in association with the recoiling electron.
While a detailed background study is needed, we note that QED processes like
trident production, $e^{-}\mathcal{N}\to e^{-}e^{+}e^{-}\mathcal{N}$, and
hard-photon conversions, $e^{-}\mathcal{N}\to e^{-}(\gamma_{\rm brem}\to
e^{+}e^{-})\mathcal{N}$, would not carry the missing momentum of the semi-
visible signal. In this regard, semi-visible events have an advantage over
fully invisible ones because of the high multiplicity of tracks. Finally, we
note that the $\Delta a_{\mu}$ region of interest overlaps with many vector
meson $V$ resonances. The mass mixing of $V$ and $A^{\prime}$ can provide an
additional and powerful production mechanism of semi-visible HNFs [211].

##  VII Conclusions

Semi-visible dark photons typically arise in rich $U(1)_{D}$ dark sectors with
a non-minimal particle content. If a symmetry distinguishes the DS fields from
the SM fields, the lightest DS particle provides an ideal candidate for dark
matter below the GeV-scale. If no such symmetry is present, the HNFs can mix
with the SM light neutrinos and are identified with heavy neutral leptons. In
this case, the lightest HNL decays to SM particles with long lifetimes.

In this paper, we have systematically studied a range of models with
increasing fermionic content in the dark sector. In particular, we have
discussed the role of a charge-conjugation symmetry $C$ in the dark sector,
which ensures that the $A^{\prime}i\overline{\psi}_{i}\gamma^{\mu}\psi_{j}$
interactions are predominantly off-diagonal in the $i$ and $j$ indices,
generalizing the idea behind the popular iDM model. This is necessary to
suppress dark photon decays into two $\psi_{1}$ particles, as such decays
contribute to the invisible branching ratio of the dark photon, which is
severely constrained. As an addition to iDM, we propose the three fermion
mixed-iDM model, where a mostly-sterile Majorana DM $\psi_{1}$ particle co-
annihilates with a mostly-dark and heavier Dirac fermion, $\Psi_{2}$. Due to
the $C$ symmetry, the DM can only couple to $\Psi_{2}$ through a small mixing
$\alpha$, while $\Psi_{2}$ can have $\mathcal{O}(1)$ self-couplings. This
possibility has not been explored before in the context of DM. Within the
three fermion scenario, we also consider more general models of three Majorana
particles, with and without enforcing $C$ symmetry. These models typically
favor more pronounced mass hierarchies and a HNL interpretation over that of
DM, since coannihilations are strongly suppressed.

We follow this with a discussion of the exact Dirac limit of a four Weyl-
fermion model, recovering the inelastic Dirac dark matter (i2DM) scenario
[45]. In this case, a $U(1)_{D}$-charged Dirac fermion mixes with a sterile
Dirac particle. In contrast to the mixed-iDM case, the Dirac DM particle now
has self-interactions, albeit suppressed by a small mixing angle $\beta$.

If $A^{\prime}$ is heavier than the HNFs, its decay to pairs of HNFs can be
followed by their subsequent decays, $\psi_{i}\to\psi_{j}\ell^{+}\ell^{-}$ or
$\psi_{i}\to\psi_{j}\pi^{+}\pi^{-}$. As the decays of the $A^{\prime}$ do not
lead to any visible resonances, resonance searches in the invariant mass
spectra of dilepton pairs are not constraining. For large couplings and
kinetic mixing, these HNF decay lengths can be much smaller than the size of a
typical particle detector. The presence of visible particles and missing
energy within the detector vetoes these semi-visible decays, modifying
existing constraints on invisible dark photons.

In order to quantitatively assess these effects, we develop our own fast MC
simulation of dark photon production and decay at BaBar and NA64, recasting
the bounds from these experiments on the parameter space of semi-visible
$A^{\prime}$ models.

We find a significant modification of the bounds on kinetic mixing in the
region of $m_{A^{\prime}}\sim 0.3-1.3$ GeV. This opens up new parameter space
at large values of $\varepsilon\sim\mathcal{O}(10^{-3}-10^{-2})$, which has
been fully excluded for visible and invisible $A^{\prime}$. This region is of
particular interest as dark photons with masses in this range can explain the
discrepancy between dispersive calculations and the measurement of the
anomalous magnetic moment of the muon, $\Delta a_{\mu}^{\rm disp}$. In both
iDM (cf., Fig. 6) and mixed-iDM scenarios (cf.. Fig. 10), we find that the
lightest HNF can also constitute a thermal DM candidate. In i2DM, however, the
mixing-suppressed self-annihilations of DM are still significant at late
times, so CMB constraints exclude the entire $\Delta a_{\mu}$ region (cf.,
Fig. 13).

We point out that in the newly-open parameter space, the bremsstrahlung
production of $A^{\prime}$ in the fixed target of NA64 can still probe the
semi-visible dark photon without the need for displaced decays, the method
employed in Ref. [24]. By aggregating the additional energy of $e^{+}e^{-}$
pairs produced by short-lived HNFs to the energy of the primary electron beam
in the electromagnetic calorimeter, NA64 would be sensitive to the missing
energy carried away by stable or long-lived HNFs. Our sensitivity curves show
that this new signal definition could cover most of the open $\Delta
a_{\mu}^{\rm disp}$ parameter space in an iDM model. A companion paper derives
new NA64 limits and future sensitivity curves for iDM and i2DM models based on
this method [38]. Our projections are in good agreement with the experimental
results given in Ref. [38], where a sophisticated detector simulation is used.

In addition, the newly-open regions also provide a realistic target for
$e^{+}e^{-}$ colliders. Re-analyses of existing BaBar, KLOE/KLOE-II and BESS-
III data can target HNF production with multiple leptons associated, with or
without initial state radiation. This also includes LEP, where the $Z$ boson
coupling to the dark current can be constrained using
$Z\to\psi_{1}\psi_{2,3,\dots}$ events. In the near future, monophoton searches
at Belle-II can improve on the BaBar limits on invisible $A^{\prime}$. The
veto on additional leptons will be even more important in this case due to the
improved hermeticity of the detector. Dedicated searches, such as those
discussed here and in Refs. [30, 31, 193], will be essential in constraining a
semi-visible $A^{\prime}$.

Following a possible detection of HNFs in the semi-visible decays of
$A^{\prime}$ in fixed-target or collider experiments, a key question would
arise on whether they are a DM or HNL particles, revealing the presence or
absence of additional symmetries in the theory. Direct detection of non-
relativistic DM particles would be challenging due to the large mass
splittings and inelastic interactions. Boosted DM, produced via the DM
upscattering by cosmic rays, could provide an interesting detection avenue to
be further investigated, in particular, the potential to exploit large
neutrino detectors with low energy thresholds. On the other hand, signatures
associated with the HNF mixing with neutrinos would provide decisive evidence
for the HNL interpretation and a possible connection to the origin of neutrino
masses. The three most promising experimental strategies for this scenario are
decay-in-flight searches for $\psi_{1}\to\nu\ell^{+}\ell^{-}$, neutrino
upscattering to $\psi_{2,3,\dots}$, and direct production of $\psi_{i}$
particles in leptonic kaon decays.

A semi-visible option for GeV-scale dark photons keeps the door open for large
kinetic mixings that can be directly probed at low-energy experiments. This
scenario provides a last chance for the kinetically-mixed dark photon
interpretation of the muon $\Delta a_{\mu}^{\rm disp}$, which has already been
ruled out in the visible and invisible options. The class of semi-visible dark
photon models is certainly within present experimental reach and may give us a
clue as to whether nature prefers a rich and complex dark sector.

###### Acknowledgements.

We want to thank Martina Mongillo, Paolo Crivelli, Laura Molina Bueno, and
Benjamin Banto Oberhauser for in-depth discussions about the NA64 experiment.
We also thank Maxim Pospelov and Yue Zhang for illuminating discussions. This
project has received partial funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No. H2020-MSCA-ITN-2019/860881-HIDDeN (ITN HIDDeN) and from the
European Union’s Horizon Europe research and innovation programme under the
Marie Sklodowska-Curie Staff Exchange grant agreement No. 101086085 -
ASYMMETRY. The research at the Perimeter Institute is supported in part by the
Government of Canada through NSERC and by the Province of Ontario through the
Ministry of Economic Development, Job Creation and Trade, MEDT. This work is
also supported by the Fermi National Accelerator Laboratory, managed and
operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy.

##  Appendix A Simulations and event distributions

In this appendix, we show some more details of the BaBar and NA64 simulations.
The kinematical distributions of the final state leptons in semi-visible
events at BaBar and NA64 are shown for representative benchmarks in Fig. 17 to
Fig. 22. Each panel shows the truth-level distribution of events in total
energy ($E_{e^{+}}+E_{e^{-}}$), energy asymmetry
($|E_{e^{+}}-E_{e^{-}}|/(E_{e^{+}}+E_{e^{-}})$), the lutput is actepton angle
with respect to the beam pipeline ($\theta_{e^{\pm}}$), the lepton pair
invariant mass ($m_{ee}$), as well as two-dimensional distributions. We also
show the total dark photon branching ratios in terms of the final state
particles in a two-dimensional grid.

##  Appendix B New physics in the anomalous magnetic moment of the muon

Several BSM theories have been put forward to explain $\Delta a_{\mu}$. Among
them are supersymmetric models [212], leptoquarks [213, 214], including those
capable of explaining other flavor anomalies [215, 216, 217], and several
other theories with new, heavy degrees of freedom. One interesting
possibility, however, is that the new physics lies below the EW scale.

At the one-loop level, a new vector or scalar mediator can contribute with the
correct, positive sign to $\Delta a_{\mu}$. In this article, we focus on a
dark photon kinetically mixed with the SM hypercharge [9, 28]. Like several
other low-scale new physics solutions to $\Delta a_{\mu}$, the minimal dark
photon model is excluded. Scalars mixed with the Higgs face severe constraints
from meson decay, beam dump, and collider experiments, and are ruled out as an
explanation of $\Delta a_{\mu}$ in minimal realizations. Particles with muon-
specific couplings are still allowed, including $U(1)_{L_{\mu}-L_{\tau}}$
models [218] and leptonic Higgs portal scalars [219, 220, 221]. Parity-
violating interactions may also contribute to $\Delta a_{\mu}$ via one-loop
vertex corrections. These come with a negative sign and can co-exist with
scalar or vector contributions, which could still remain sufficiently large.
Parity-violating gauge bosons as part of a solution to $\Delta a_{\mu}$ have
been discussed in the context of the proton radius puzzle [222, 223] and arise
naturally in the so-called dark-Z ($Z_{d}$) models [224, 225, 226]. In the
case of axion-like-particles, Barr-Zee diagrams [227] may dominate the
contribution to $\Delta a_{\mu}$, even though such diagrams are effectively a
two-loop effect with heavy charged fermions $f$ loops [228]. This contribution
is positive and enhanced with respect to the one-loop result by
$(m_{f}/m_{\mu})^{2}$. Models with axion-like-particles coupled to a dark
force [229, 230] have also been proposed as an explanation of $\Delta a_{\mu}$
with the added assumption of a direct coupling of $a$ to the muon [231].

Figure 17:  The kinematical distributions dark photon decay products for the
initial-state-radiation signatures at BaBar. In a clockwise order, we show the
distribution of the total $e^{\pm}$ energies, their energy asymmetry,
invariant mass $m_{ee}$, the 2D distribution of total energy and separation
angle, the 2D distribution of each electron and positron energies, the
distance traveled by the HNF between its production point and its decay
position the angle of $e^{\pm}$ with respect to the beam pipe, and the opening
angle of $e^{\pm}$ pairs.  Figure 18:  The kinematical distributions dark
photon decay products for bremsstrahlung $A^{\prime}$ in NA64. In a clockwise
order, we show the distribution of the total $e^{\pm}$ energies, their energy
asymmetry, the angle of $e^{\pm}$ with respect to the beam pipe, the 2D
distribution of separation angle and total energy, the 2D distribution of each
electron and positron energies, the invariant mass $m_{ee}$, and the branching
ratios of the dark photon decays to the HNFs.  Figure 19: Same as Fig. 17 but
for BP2a.  Figure 20: Same as Fig. 18 but for BP2a.  Figure 21: Same as Fig.
17 but for BP5.  Figure 22: Same as Fig. 18 but for BP5.

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# Supplementary Information for Antiferromagnetic Spin Fluctuations in the
Spin-Triplet Superconductor Candidate CeRh2As2

Tong Chen<EMAIL_ADDRESS>Hasan Siddiquee  Zack Rehfuss  Shiyuan Gao  Chris
Lygouras  Jack Drouin  Vincent Morano  Keenan E. Avers  Christopher J. Schmitt
Andrey Podlesnyak  Sheng Ran<EMAIL_ADDRESS>Yu Song<EMAIL_ADDRESS>Collin Broholm<EMAIL_ADDRESS>
††preprint: APS/123-QED

##  I Sample alignment, and X-ray Laue Pattern

High-quality single crystals of CeRh2As2 were synthesized using Bi flux
methods, as described in [1]. Representative resistivity plots are shown in
Figures S1(a,b). These single crystals typically manifest as octagonal plates
with dimensions of 1.0 mm $\times$ 1.0 mm $\times$ 0.2 mm. Natural edges align
along either (110) or (100) directions, while the out-of-plane direction
corresponds to (001), the c-axis (Fig. S1(c)). A total mass of 0.9 g,
comprising approximately 120 pieces of CeRh2As2 single crystals, was co-
aligned on two oxygen-free high conductivity (OFHC) copper plates using CYTOP
for attachment. Thermally shielded by OFHC copper foil secured to the OFHC
base, the sample mount was attached to the cold finger of a dilution
refrigerator with 80 $\mu$W cooling power at 100 mK (Fig. S2).

Figure S1:  (a) Temperature-dependent resistivity of CeRh2As2. The inset
highlights the approximate 0.3 K superconductivity offset. (b) Temperature-
dependent resistivity of CeRh2As2 up to 300 K. (c) X-ray Laue pattern of a
CeRh2As2 single crystal, displaying the crystallographic orientation within
the ab-plane.  Figure S2:  (a) Photo depicting the assembly aligned in the
$(hk0)$ scattering plane for CNCS. (b) Photo showing the assembly with an
oxygen-free high conductivity (OFHC) copper shielding attached to the cold
finger of a dilution refrigerator.

##  II  Absolute Normalization

The monitor normalized scattering intensity obtained from initial data
reduction is related to the scattering cross-section as follows:

${\cal I}({\bf Q},\omega)=N{\cal
C}\frac{k_{i}}{k_{f}}\overline{\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E_{f}}}({\bf
Q},\omega)$ (1)

Here $N{\cal C}$ is a configuration-dependent normalization constant that is
proportional to the number of unit cells in the sample ($N$). The bar
indicates convolution with the instrumental resolution function:

$\overline{\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E_{f}}}({\bf
Q},\omega)=\int\mathrm{d}^{3}{\bf
Q}^{\prime}\hbar\mathrm{d}\omega^{\prime}{\cal R}_{{\bf Q},\omega}({\bf
Q-Q^{\prime}},\omega-\omega^{\prime})\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E_{f}}({\bf
Q^{\prime}},\omega^{\prime})$ (2)

${\cal R}_{{\bf Q},\omega}$ is sharply peaked when its arguments vanish and
unity normalized:

$\int\mathrm{d}^{3}{\bf Q^{\prime}}\hbar\mathrm{d}\omega^{\prime}{\cal
R}_{{\bf Q},\omega}({\bf Q-Q^{\prime}},\omega^{\prime})\equiv 1$ (3)

Absolute normalization of the scattering cross-section was achieved using the
integrated intensity of the (110) nuclear Bragg reflection. This has the
advantage of being unaffected by sample absorption for the elastic channel.
The Bragg scattering cross section per unit cell of volume $v_{N}$ is given by
(Fig. S3(a)):

$\displaystyle\left(\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E_{f}}\right)_{\rm
el}^{\rm
nuc}=\frac{(2\pi)^{3}}{v_{N}}\sum_{\bm{\tau}}|\mathbb{F}_{N}(\bm{\tau})|^{2}\delta({\bf
Q}-\bm{\tau})\delta(\hbar\omega),$ (4)

where $\mathbb{F}_{N}(\bm{\tau})$ is the structure factor given by:

$\displaystyle\mathbb{F}_{N}(\bm{\tau})=\sum_{\bf d}b_{\bf
d}\textup{exp}(-2W_{\bf d}({\bf Q}))\textup{exp}(i\bm{\tau}\cdot{\bf d}).$ (5)

Here $b_{\bf d}$ is the bound coherent scattering length and
$\textup{exp}(-2W_{\bf d}({\bf Q}))$ is the Debye-Waller factor for the atom
on the site $\bf d$ within the unit cell with volume $v_{N}$. The scattering
intensity near the $\bm{\tau}=$(110) Bragg peak is thus given by

${\cal I}_{N}({\bf Q},\omega)=N{\cal
C}\frac{(2\pi)^{3}}{v_{N}}|\mathbb{F}_{N}(\bm{\tau})|^{2}{\cal
R}_{\bm{\tau},0}({\bf Q-\bm{\tau}},\omega).$ (6)

Using the normalization property of the resolution function we find

$N{\cal C}=\int_{{\bf Q\approx\bm{\tau}},\omega\approx 0}\mathrm{d}^{3}{\bf
Q}\hbar\mathrm{d}\omega{\cal I}_{N}({\bf
Q},\omega)\left(\frac{(2\pi)^{3}}{v_{N}}|\mathbb{F}_{N}(\bm{\tau})|^{2}\right)^{-1}$
(7)

We thus obtain the following expression relating the measured neutron
scattering intensity to the resolution smeared absolute normalized scattering
cross-section per unit cell:

$\frac{k_{i}}{k_{f}}\overline{\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E_{f}}}({\bf
Q},\omega)=\frac{{\cal I}({\bf Q},\omega)}{\int_{\bm{\tau}}\mathrm{d}^{3}{\bf
Q}\hbar\mathrm{d}\omega{\cal I}_{N}({\bf
Q},\omega)}\frac{(2\pi)^{3}}{v_{N}}|\mathbb{F}_{N}(\bm{\tau})|^{2}$ (8)

If we denote by $\overline{{\cal I}_{N}}(110)$ the average (110) nuclear Bragg
intensity throughout a volume of ${\bf Q}-\omega$ space that exceeds the
resolution volume and has dimensions $\Delta h,\Delta k,\Delta l$ in
reciprocal lattice units and $\Delta\hbar\omega$ in energy, this expression
simplifies to

$\frac{k_{i}}{k_{f}}\overline{\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E_{f}}}({\bf
Q},\omega)=\frac{{\cal I}({\bf Q},\omega)}{\overline{{\cal I}_{N}}(110)\Delta
h\Delta k\Delta l\Delta\hbar\omega}|\mathbb{F}_{N}(\bm{\tau})|^{2}$ (9)

for orthogonal reciprocal lattice vectors.

##  III Absorption Correction

When a neutron beam travels through the sample, the decrease in scattering
intensity due to the absorption cross-section is quantified by the
transmission coefficient:

$\displaystyle A=\frac{1}{V}\int{\rm exp}(-\mu T){\rm d}V,$ (10)

where $\mu$ is the attenuation length, and the integration is over the volume
of the crystal. Here, $T$ denotes the path length of the neutron beam within
the crystal, encompassing the cumulative path lengths of both the incident and
outgoing neutrons [2, 3].

For the scattered neutron intensity within the equatorial plane of a cylinder
with radius $R$ in the neutron beam, the expression for the transmission
coefficient is:

$\displaystyle
A(\sigma_{r},\rho,\lambda_{i},\lambda_{f},R,\theta)=\frac{1}{\pi
R^{2}}\int_{0}^{R}\int_{0}^{2\pi}{\rm exp}($
$\displaystyle-\mu(\lambda_{i})\\{[R^{2}-r^{2}{\rm
sin}^{2}(\phi+\theta)]^{1/2}+r{\rm cos}(\phi+\theta)\\}$ (11)
$\displaystyle-\mu(\lambda_{f})\\{[R^{2}-r^{2}{\rm
sin}^{2}(\phi-\theta)]^{1/2}-r{\rm cos}(\phi-\theta)\\})r{\rm d}r{\rm d}\phi,$
$\displaystyle\mu(\lambda)=\rho*\sigma(\lambda)=$
$\displaystyle\rho*\sigma_{r}\frac{\lambda}{1.7982\ {\rm\AA}},$

where $\rho$ is the atomic number density of the sample and $\sigma_{r}$ is
the absorption cross-section for the reference wavelength of 1.7982 Å. Thus,
the transmission coefficient $A$ is a function of $\sigma_{r}$, $\rho$, the
wavelengths of the incident and outgoing neutrons $\lambda_{i}$ and
$\lambda_{f}$, the radius of the cylinder sample $R$, and the scattering angle
$\theta$. The calculated transmission coefficients agree with those obtained
through numerical integration [4]. For CeRh2As2,

$\displaystyle\rho=\frac{M}{V}\frac{N_{\rm A}}{m}=9.05~{}{\rm
g/cm}^{3}*\frac{6.023\times 10^{23}~{}{\rm mol}^{-1}}{492.109~{}{\rm
g/mol}}=0.01108~{}{\rm\AA}^{-3},$ (12)

where $\frac{M}{V}$ is the density, $N_{\rm A}$ is Avogadro’s constant, and
$m$ is the molecular weight. From the data on the NIST website [5, 6], we have
$\sigma_{r}=299.23$ barn/f.u. and in our experimental setup, the sample
assembly resembles a cylinder with $R=3.5$ mm. With $\sigma_{r}=299.23$
barn/f.u., $\rho=0.01108~{}{\rm\AA}^{-3}$, $\lambda_{i}=4.964$ Å, and $R=3.5$
mm, the transmission coefficient $A$ (Eq. 11) depends on $\theta$ and
$\lambda_{f}$, which is thus a function of Q and $\omega$.

Since we performed the absolute normalization using the (110) nuclear Bragg
peak, the absorption correction is conducted by comparing the absorption
coefficient $A(\textbf{Q},\omega)$ with that of the (110) peak. After the
absorption correction, Equation 9 becomes:

$\displaystyle\frac{k_{i}}{k_{f}}\overline{\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\Omega\mathrm{d}E_{f}}}({\bf
Q},\omega)=\frac{{\cal I}({\bf Q},\omega)}{\overline{{\cal I}_{N}}(110)\Delta
h\Delta k\Delta
l\Delta\hbar\omega}\frac{A(\bm{\tau},0)}{A(\textbf{Q},\omega)}|\mathbb{F}_{N}(\bm{\tau})|^{2},$
(13)

where $A(\bm{\tau},0)=A(\bm{\tau}=(110),\omega=0)=0.0671$.

##  IV Upper Bound of Ordered Moment

While our experiment did not detect elastic magnetic Bragg diffraction in
CeRh2As2, the small sample neutron absorbing sample means that this does not
set a particularly strict limit on the staggered magnetization of possible
magnetic ordering in the sample. To determine the limits on any ordered moment
set by our experiment we start by the following general expression for the
elastic magnetic scattering cross section.

$\displaystyle\left(\frac{\textup{d}^{2}\sigma}{\textup{d}\Omega\textup{d}E_{f}}\right)_{\rm
el}^{\rm mag}$ $\displaystyle=(\frac{\gamma r_{0}}{2})^{2}|gF({\bf
Q})|^{2}\sum_{\alpha\beta}(\delta_{\alpha\beta}-\hat{Q}_{\alpha}\hat{Q}_{\beta})\frac{1}{N}\sum_{RdR^{\prime}d^{\prime}}\langle
S_{Rd}^{\alpha}\rangle\langle S_{R^{\prime}d^{\prime}}^{\beta}\rangle e^{i{\bf
Q}\cdot(\bm{R}+\bm{d}-\bm{R}^{\prime}-\bm{d}^{\prime})}\delta(\hbar\omega)$
(14) $\displaystyle=(\frac{\gamma r_{0}}{2})^{2}|gF({\bf
Q})|^{2}(1-\hat{Q}_{z}^{2})\frac{1}{N}\sum_{RdR^{\prime}d^{\prime}}\langle
S_{Rd}^{z}\rangle\langle S_{R^{\prime}d^{\prime}}^{z}\rangle e^{i{\bf
Q}\cdot(\bm{R}-\bm{R}^{\prime}+\bm{d}-\bm{d}^{\prime})}\delta(\hbar\omega)$

Here $\gamma=1.913$ is the ratio between the neutron dipole moment and the
nuclear Bohr magneton, $r_{0}=2.8179$ fm is the classical electron radius,
$|F({\bf Q})|$ is the cerium form factor, $\hat{Q}_{z}$ is the $z$-component
of the unit scattering vector $\bm{\hat{Q}}$. The factor $1/N$ where $N$ is
the number of unit cells of volume $v_{N}$ ensures consistent normalization
with Eq. 4. For simplicity, we assume that any magnetic order is uniaxial and
polarized along the $\bf c$ axis. Other assumptions will modify the inferred
limits by factors of order unity.

For three-dimensional magnetic orders, we obtain

$\displaystyle\left(\frac{\textup{d}^{2}\sigma}{\textup{d}\Omega\textup{d}E_{f}}\right)_{\rm
el}^{\rm mag}$ $\displaystyle=(\frac{\gamma r_{0}}{2})^{2}|F({\bf
Q})|^{2}(1-\hat{Q}_{z}^{2})(m_{z}/\mu_{B})^{2}\frac{(2\pi)^{3}}{v_{m}}\frac{N_{m}}{N}\sum_{\tau_{m}}|\mathbb{F}_{m}(\bm{\tau}_{m})|^{2}\delta^{3}({\bf
Q}-\bm{\tau}_{m})\delta(\hbar\omega)$ (15)

Here $N_{m}$ is the number of magnetic unit cells of volume $v_{m}$,
$\bm{\tau}_{m}$ are reciprocal lattice vectors of the magnetic unit cell,
$m_{z}$ is the ordered magnetic dipole moment, and $\mathbb{F}_{m}({\bf Q})$
is the scalar magnetic structure factor:

$\mathbb{F}_{m}({\bf Q})=\sum_{{\bf d}_{m}}\textup{exp}(-i{\bf Q}\cdot{\bf
d}_{m}).$ (16)

Here ${\bf d}_{m}$ indicates position vectors for magnetic ions within the
magnetic unit cell and we have suppressed the Debye-Waller factor for
simplicity. From the analogous forms of the magnetic and nuclear Bragg
diffraction cross-section we readily obtain the following:

$m_{z}^{2}=\frac{\int_{\bm{\tau}_{m}}\mathrm{d}^{3}{\bf
Q}\hbar\mathrm{d}\omega{\cal I}_{m}({\bf
Q},\omega)}{\int_{\bm{\tau}}\mathrm{d}^{3}{\bf Q}\hbar\mathrm{d}\omega{\cal
I}_{N}({\bf
Q},\omega)}\cdot\frac{A(\bm{\tau},0)}{A(\bm{\tau_{m}},0)}\cdot\left(\frac{v_{m}}{v_{N}}\right)^{2}\cdot\frac{|\mathbb{F}_{N}(\bm{\tau})|^{2}}{(\frac{\gamma
r_{0}}{2})^{2}|F(\bm{\tau}_{m})|^{2}(1-\hat{\tau}_{mz}^{2})|\mathbb{F}_{m}({\bm{\tau}_{m}})|^{2}}\mu_{B}^{2}.$
(17)

Here we have used $N_{m}v_{m}=Nv_{N}$ and introduced the sample absorption
correction. In Figure S3, we show the elastic scattering along the $(hh)$ and
$(h1)$ directions, from which we calculate
$\int_{\bm{\tau}_{m}}\mathrm{d}^{3}{\bf Q}\hbar\mathrm{d}\omega{\cal
I}_{m}({\bf Q},\omega)$ for putative magnetic orders at
$(\frac{1}{2}\frac{1}{2})$ and $(01)$, respectively.

Figure S3:  (a) Schematic of the $(hk)$ plane and the cut along the $(hh)$
direction. (b) Elastic scattering along $(hh)$. Since no discernible
difference was observed between 0.08 K and 0.8 K, data are averaged to enhance
statistics. (c) Temperature difference (0.08 K - 0.8 K) of the elastic
scattering along $(hh)$. Data in (b,c) are averaged along $(k\bar{k})$ and $l$
in windows of $\pm 0.10$ and $\pm 0.5$ r.l.u., respectively. The energy window
is from -0.2 meV to 0.2 meV. Gaussian peaks are fitting for the (110) nuclear
peak (b) and putative magnetic scattering at $(\frac{1}{2}\frac{1}{2})$ (c).
(d) Schematic of the $(hk)$ plane and the cut along the $(h1)$ direction. (e)
Elastic scattering along $(h1)$. 0.08 K and 0.8 K data are averaged. (f)
Temperature difference (0.08 K - 0.8 K) of the elastic scattering along
$(h1)$. Data in (e,f) are averaged along $(0k)$ and $l$ in windows of $\pm
0.15$ and $\pm 0.5$ r.l.u., respectively. The energy window is from -0.2 meV
to 0.2 meV. Gaussian peaks are fitted for the (110) nuclear peak (e) and
putative magnetic scattering at $(01)$ (f).

For quasi-two-dimensional magnetic order, there are no magnetic correlations
between cerium planes and the diffraction cross-section takes this form

$\displaystyle\left(\frac{\textup{d}^{2}\sigma}{\textup{d}\Omega\textup{d}E_{f}}\right)_{\rm
el}^{\rm mag}$ $\displaystyle=(\frac{\gamma r_{0}}{2})^{2}|F({\bf
Q})|^{2}(1-\hat{Q}_{z}^{2})(m_{z}/\mu_{B})^{2}\frac{(2\pi)^{2}}{a_{m}}\frac{N_{m}}{N}\sum_{\tau_{m}}|\mathbb{F}_{m}(\bm{\tau}_{\parallel
m})|^{2}\delta^{2}({\bf Q}_{\parallel}-\bm{\tau}_{\parallel
m})\delta(\hbar\omega)$ (18)

Here $N_{m}$ is the number of 2D magnetic unit cells with area $a_{m}$ and
$\mathbb{F}_{m}$ is the corresponding scalar magnetic structure factor. Note
that $\frac{1}{2}N_{m}a_{m}c=Nv_{n}$ where the factor of a half arises because
there are two cerium layers per nuclear unit cell.

Since the HYSPEC instrument used in this work only covers a relatively small
range of wave vector transfer perpendicular to the scattering plane, we can
approximate the resolution function with a factorized form:

${\cal R}_{{\bf Q},\omega}(\delta{\bf Q},\delta\omega)\approx{\cal
R}^{\parallel}_{{\bf Q}_{\parallel},\omega}(\delta{\bf
Q}_{\parallel},\delta\omega){\cal R}^{\perp}(\delta{\bf Q}_{z}),$ (19)

where both ${\cal R}^{\parallel}$ and ${\cal R}^{\perp}$ are unity normalized
over their domain. Convoluting Eq. LABEL:quasi2D with this form of the
resolution function, we obtain the following approximate form for the
resolution smeared diffraction cross-section:

$\displaystyle\left(\overline{\frac{\textup{d}^{2}\sigma}{\textup{d}\Omega\textup{d}E_{f}}}\right)_{\rm
el}^{\rm mag}$ $\displaystyle=(\frac{\gamma r_{0}}{2})^{2}|F({\bf
Q})|^{2}(1-\hat{Q}_{z}^{2})(m_{z}/\mu_{B})^{2}\frac{(2\pi)^{2}}{a_{m}}\frac{N_{m}}{N}|\mathbb{F}_{m}(\bm{\tau}_{\parallel
m})|^{2}{\cal R}^{\parallel}_{{\bf Q}_{\parallel},\omega}({\bf
Q}_{\parallel}-\bm{\tau}_{\parallel m},\omega)$ (20)

This finally leads to an expression for the staggered magnetization:

$m_{z}^{2}=\frac{\int_{\bm{\tau}_{m}}\mathrm{d}^{3}{\bf
Q}\hbar\mathrm{d}\omega{\cal I}_{m}({\bf Q},\omega)(|F({\bf
Q})|^{2}(1-\hat{Q}_{z}^{2}))^{-1}}{\int_{\bm{\tau}}\mathrm{d}^{3}{\bf
Q}\hbar\mathrm{d}\omega{\cal I}_{N}({\bf
Q},\omega)}\cdot\frac{A(\bm{\tau},0)}{A(\bm{\tau_{m}},0)}\cdot\left(\frac{a_{m}c}{v_{N}}\right)^{2}\cdot\frac{(c^{*}/\Delta
Q_{z})|\mathbb{F}_{N}(\bm{\tau})|^{2}}{(\frac{\gamma
r_{0}}{2})^{2}2|\mathbb{F}_{m}({\bm{\tau}_{\parallel m}})|^{2}}\mu_{B}^{2}$
(21)

Here $\Delta Q_{z}$ is the range of integration along the $\bf c$ direction
for the $\bf Q$ integration in the numerator and we have used
$N_{m}/N=2(v_{N}/a_{m}c)$. Table S1 shows $a_{m}$, $c_{m}$, polarization
factor $(1-\hat{Q}_{z}^{2})$, magnetic form factor $|F(\textbf{Q})|$,
$|\mathbb{F}_{m}|$, and corresponding $m_{z}^{2}$ and upper bounds for
putative magnetic orders at specific wavevectors. Note that there are magnetic
peaks at both $l=-0.5$ and $0.5$ for $(\frac{1}{2}\frac{1}{2}\frac{1}{2})$ and
$(01\frac{1}{2})$ order, but this factor is coincidentally compensated by the
fact that the $l$ coverage is -0.5 to 0.5 r.l.u.

Table S1: Magnetic ordering and upper limit of moment determined from the corresponding analysis of our diffraction data. wavevector ${\bf Q}$ | $A_{N}/A(\textbf{Q},\omega)$ | $a_{m}$ | $c_{m}$ | $(1-\hat{Q}_{z}^{2})$ | $|F(\textbf{Q})|$ | $|\mathbb{F}_{m}|^{2}$ |  $m_{z}^{2}$($\mu_{\rm B}^{2}$) |  $m_{\parallel,max}$ ($\mu_{\rm B}$)
---|---|---|---|---|---|---|---|---
($\frac{1}{2}\frac{1}{2}0$) | 2.46 | 4a | c | 1 | 0.937 | 64 | 0.00(1) | 0.10
($\frac{1}{2}\frac{1}{2}\frac{1}{2}$) | 2.46 | 4a | 2c | 0.914 | 0.932 | 128 | -0.01(3) | 0.14
(010) | 1.65 | a | c | 1 | 0.880 | 4 | 0.01(1) | 0.14
(01$\frac{1}{2}$) | 1.65 | a | 2c | 0.955 | 0.875 | 8 | 0.01(3) | 0.22
quasi-2D ($\frac{1}{2}\frac{1}{2}$) | 2.46 | 4a | c | $>0.914$ | $>0.932$ | 16 | -0.01(3) | 0.14

##  V Scattering from CYTOP

We used the fluoropolymer-based adhesive CYTOP to attach the crystals. CYTOP
has a powder-like diffraction peak around 1.1 Å, which practically coincides
with ${\bf Q}_{\rm AF}=(\frac{1}{2}\frac{1}{2})$ in CeRh2As2, as shown in
Figure S4(a). Therefore, it is crucial to examine whether the peak at ${\bf
Q}_{\rm AF}$ could be from CYTOP. We can rule this out because scattering from
CYTOP is approximately isotropic and independent of sample rotation while
magnetic scattering is four-fold rotational symmetric.

To compare the scattering from CYTOP with the excitation spectrum, we scaled
the $(hh)$ cut of the CYTOP scattering to that of the peak at 0.3 meV and
constructed the $(k\bar{k})$ and $(0k)$ cuts for CYTOP (Fig. S4(b,c)). The
CYTOP scattering is wider than the observed excitations centered at
$\textbf{Q}_{\rm AF}=(\frac{1}{2}\frac{1}{2})$ in the $(k\bar{k})$ cut. In
addition, CYTOP also shows a peak in the $(0k)$ cut, which is absent in the
measured spectrum.

Figure S4:  (a) Constant energy cuts along the longitudinal $(hh)$ direction
at 0.3 meV. The averaging windows along $(k\bar{k})$ and $l$ directions are
$\pm 0.15$, and $\pm 0.5$ r.l.u., respectively. (b) Constant energy cuts along
the transverse $(k\bar{k})$ direction at 0.3 meV. The averaging windows along
$(hh)$ and $l$ directions are $\pm 0.1$, and $\pm 0.5$ r.l.u., respectively.
(c) Constant energy cuts along the longitudinal $k$ direction at 0.3 meV. The
averaging windows along $h$ and $l$ directions are $\pm 0.2$, and $\pm 0.5$
r.l.u., respectively. The energy window is $\pm 0.1$ meV. The horizontal error
bars represent the resolution width.

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