A Novel Proton Decay Signature at DUNE, JUNO, and Hyper-K

: Proton decay, although unobserved so far, is a natural expectation when attempting to explain the baryon asymmetry of the universe. p


Introduction
Despite its many successes, the Standard Model (SM) of particle physics cannot be viewed as an exhaustive description of Nature.One of the essential puzzles from the cosmological perspective is embodied by the prevalence of matter over antimatter in the observable Universe [1,2].Baryogenesis [3] appears as a viable explanation of this obvious discrepancy between SM and observations, provided that baryon number (B), in particular, is violated in particle interactions.In the SM, this quantity is accidentally conserved at the perturbative level, along with lepton number (L); here, 'accidental' means that the most general renormalizable gauge-invariant (perturbative) interactions automatically preserve these quantum numbers.Violation of the baryon number through non-perturbative effects in the SM, such as instantons and sphalerons [4][5][6][7], proves insufficient to account for the oberved asymmetry [8][9][10][11].
In this paper, we focus on a proton desintegration dominated by kaons in the final states, with modes such as p → K + ν and p → K + χ0 1 , where χ0 1 is a light exotic neutral particle with m χ0 1 ≤ m p − m K + ≈ 445 MeV . (1.1) This scenario can be easily embedded within the phenomenology of the Minimal Supersymmetric Standard Model with R-parity-violating terms (RPV-MSSM), and we shall employ this model as a predictive framework for our study.However, one could also consider instead the decay to a heavy neutral lepton (N R ): p → K + N R , see our Appendix A, and for example Ref. [38].
Originally motivated by the Hierarchy Problem [39,40], SUSY extensions of the SM [13] have far-reaching phenomenological consequences at low-energy, with features such as dark matter candidates, gauge-coupling unification, or neutrino-mass generation.We refer the reader to e.g.Refs.[16,18,41] for an overview.In contrast to the SM, baryon-and leptonnumber violating interactions are naturally present in these models at the renormalizable level, unless an additional discrete symmetry, often (but not imperatively) R-parity [42], is explicitly requested [18,26,43,44].There exists no deep theoretical motivation to exclude such terms via R-parity [45][46][47], but experimental evidence for a generally B-and L-conserving phenomenology at low-energy implies that they remain comparatively suppressed.In particular, they may trigger proton decay, which is the feature in which we are interested here.Nevertheless, we stress that R-parity conservation does not forbid Band L-violation in higher-order operators, so that proton instability can also be expected in the R-parity conserving MSSM when the latter is considered as an effective field theory [26,45,48].
Numerous aspects of proton disintegration in the RPV-MSSM have been studied in the literature in the past: the reader may consult e.g.Refs.[41,[49][50][51].Two-body final states with only SM particles imply a violation of both B and L: spin-conservation indeed dictates the presence of an odd number of fermions among the decay products and, in the SM, these can only be leptons, then produced in association with a meson.Sfermions mediate such processes at tree-level in the RPV-MSSM [52,53], provided both B-and Lviolating couplings are simultaneously present.More involved topologies have also been considered [54][55][56][57][58], leading to bounds on a wide set of products of RPV couplings.Nevertheless, proton disintegration involving only B-violation is a viable scenario as well, as long as a light exotic fermion χ0 1 is available in the spectrum.In particular, the cases involving a light photino [54,59], a light gravitino or axino [60,61] have been considered in the literature.In this paper, we identify χ0 1 with the lightest, bino-like neutralino, which is the only phenomenologically viable candidate within the strict limits of the MSSM spectrum.A low-mass particle of this nature indeed evades existing experimental constraints [62,63].Astrophysical constraints are also satisfied [64,65].Cosmological and astrophysical limits however demand that it is unstable, which, in view of the mass and suppressed couplings of this particle, typically makes it long-lived.We assume that the bino decays are dominated by L-violating channels, triggering final states with leptons and mesons.More exotic decay channels involving e.g.axinos or gravitinos are an alternative.Searches of the light neutralino at colliders have been recently proposed in [66][67][68][69][70][71][72], in the more general context of searches for long-lived particles [73][74][75][76][77][78][79].
p → K + χ0 1 appears as the simplest proton decay mode violating only B in the context of the RPV-MSSM.On a superficial level, it shares many similarities with the canonical mode p → K + ν, for which Super-Kamiokande provides the (currently strictest) limit τ p→K + +ν > 5.9 × 10 33 yrs [80].The light long-lived bino might distinguish itself from the neutrino through its mass, reducing the kaon momentum in experiments.In addition, for decay lengths comparable to the detector size, the neutralino could decay within the detector, leading to a distinct signature.We analyze both possibilities in detail and reinterpret the Super-K bound in this context.We also calculate the sensitivities of the upcoming experiments DUNE, JUNO, and Hyper-K to such a proton-decay mode.
The paper is organized as follows: in Section 2, we introduce the RPV-MSSM, as well as the light-neutralino scenario.We discuss proton decay modes in such a setup, including the resulting signatures.In Section 3, we briefly discuss the experimental setups at present and upcoming proton-decay search facilities: Super-K [81], Hyper-K, DUNE and JUNO.We further discuss the prospects for detecting a light exotic fermion, such as a neutralino in these detectors, as well as what happens when a proton decays inside a nucleus with large mass number A. We conclude this section with a description of our simulation procedure for estimating the sensitivities at the above experiments.In Section 4, we outline our numerical analysis and present our results.We conclude in Section 5.In Appendix A, we discuss the close connection between the light neutralino we have discussed extensively here and the related heavy neutral lepton scenario.
2 The Signature: Proton Decay followed by Neutralino Decay We first discuss the decay of the proton in our model and subsequently the various decays of the light neutralino.

Proton Decay to a light Neutralino in the RPV-MSSM
As explained in the introduction, we work, for convenience, within the framework of the RPV-MSSM model with a light neutralino.The renormalizable superpotential may be expressed in the notation of Ref. [44]: where W MSSM is the MSSM superpotential, and violate lepton-and baryon-number, respectively.In W LNV we have dropped the bilinear term, as it can be rotated away at a fixed energy scale [54,82].ϵ ab is the two-dimensional Levi-Civita symbol, and the Latin indices a, b ∈ {1, 2} are the SU(2) L gauge indices in the fundamental representation.ε αβγ is the three-dimensional Levi-Civita symbol and the Greek indices α, β, and γ ∈ {1, 2, 3} denote the SU(3) C gauge color indices.λ ′′ ijk is a dimensionless Yukawa coupling and i, j, k ∈ {1, 2, 3} denote the generation indices.We employ the summation convention.L i and Q i are the SU(2)-doublet lepton and quark chiral superfields; Ēi , Ū i , and Di are SU(2)-singlet electron-, up-and down-type quark chiral superfields, respectively.
The bino, which we will consider further below, is the spin-1/2 SUSY partner of the hypercharge gauge field.As such, its coupling to matter are dictated by the gauge interaction.As reminded in the introduction, such a particle might exist with an almost arbitrarily low mass without violating collider, astrophysical and cosmological constraints, as long as it is unstable.
In the RPV-MSSM, as well as in any viable model of new physics, B-(and L-) violation manifests itself at low-energy (the proton mass-scale) through the mediation of high-energy fields (taking mass at the SUSY-breaking scale, comparable to or larger than the electroweak scale).In such configurations with a sizable hierarchy of scales, it is always desirable to disentangle long-and short-distance effects through the definition of a low-energy effective field theory (EFT), allowing for the resumation of large logarithmic corrections.The impact of new-physics for low-energy particles is then summarized within contributions to operators of higher-dimension built out of the light fields.The operators of lowest dimension relevant for nucleon decay are of dimension 6: we refer the reader to Ref. [51] for a recent overview of this specific EFT and associated calculation techniques.
The operators that are relevant for the production of a neutral SU(2) L -singlet fermion χ0 1 (i.e.neglecting contributions that require an electroweak vacuum expectation value, which receive further mass-suppression from new-physics) read: , where P R is the right-handed projection operator.u, d, s are the four-component quark spinors with c denoting charge conjugation.χ0 1 denotes the 4-component neutralino spinor, or any other exotic neutral fermion.
Contributions to the operators of Eq.(2.4) emerge at tree-level in the RPV-MSSM through the mediation of a squark, provided a B-violating trilinear coupling non-vanishing in the superpotential.The corresponding Wilson coefficients can be obtained by matching of the EFT and read: where g ′ is the U (1) Y gauge coupling and the masses in the denominator refer to scalar SUSY partners of the right-handed quarks mediating the transition: see Fig. 1. η QCD (µ R ) accounts for the renormalization group evolution of the operators driven by the strong interaction: its expression can be found in Eq.(6) of Ref. [51].η QCD (2 GeV) ≈ 1.4.
Naturally, the difficult step consists in extracting hadronic matrix elements from these partonic operators.For any operator Ω ∈ { Q1 , Q′ 1 , Q2 } of Eq.(2.4), one can derive the following general form [83] (we employ the + − −− metric), where W p→K + 0,Ω (q 2 ) and W p→K + 1,Ω (q 2 ) represent the form factors associated with the operator Ω for the transition p → K + , depending on the squared momentum-transfer, q 2 (= m 2 χ0 1 in our case).u p and v χ0 1 denote the four-component spinors for the proton and neutralino respectively.
Lattice evaluations of the p → K + form-factors [83][84][85] focus on the limit q 2 → 0 (hence on W p→K + 0,Ω ), corresponding to the neutrino final-state.This limit does not necessarily apply in the case of a neutralino.Alternatively, it is possible to compute the form factors in chiral perturbation theory and determine chiral parameters from the lattice [83], thus retaining full momentum dependence (at leading order in chiral perturbation theory, in practice).
The decay amplitude for the proton decay may now be expressed as follows: leading to the partial decay width: ) .
The decay width normalized to |λ ′′ 112 | 2 /m 4 f , where m f represents a universal value for the squark masses, is shown in Fig. 2 for various lattice inputs.Here, we use the results of the two lattice approaches presented in Refs [84,85].The solid lines employ the form-factors retaining full momentum-dependence, while the dashed lines have been obtained under the approximation . The dotted lines account for the quoted lattice uncertainties at 1σ.We observe that the momentum dependence in the form-factors can be neglected in view of the large uncertainty originating in the lattice modelization.Obviously, no better than an order of magnitude can be set on the actual size of the hadronic matrix elements, while the variations due to momentum dependence typically remain under 10%.
Assuming that the neutralinos would behave as an invisible particle at Super-K, one may exploit the limit of this experiment for the decay rate p → K + ν.In the massless neutralino limit, this led Ref. [51] to the limit: Aoki, 2017 Yoo, 2021 , where m f represents a universal value for the squark masses.Two different lattice evaluations are used for the numerical values of the form factors: from Aoki, 2017 [84] and Yoo, 2021 [85].The dashed line represents the case where lattice form-factors at q 2 = 0 are used, whatever the neutralino mass, while the solid lines represent results with form factors determined according to chiral perturbation theory [83].The bands around the dashed lines denote the approximate error in the lattice calculation of the form factors. updating older estimates.This bound can naively be extended to massive neutralinos through a rescaling by the square-root of the the quantity depicted in Fig. 2.
Nevertheless, this simplistic picture holds only under the approximation where the modified kaon kinematics (due to the neutralino mass) and subsequent neutralino decays have a negligible impact on the experimental strategy.The purpose of this paper exactly consists in demonstrating how these two effects may leave their imprint on the experimental results, allowing, under favorable conditions, to disentangle the decay mode involving the neutralino from the more classical channels with a(n anti)neutrino in the final state.
Considering the kinematical effect first, we write the kaon momentum in the rest-frame of the proton: This quantity is shown in Fig. 3.In the massless neutralino limit (or in the case of a neutrino in the final state), |⃗ p K + | ≈ 339 MeV.

Neutralino Decay
The second experimental handle on the decay channel involving the neutralino rests with the possible observation of neutralino decay products within the detector, which obviously only  .applies if the neutralino is sufficiently short-lived.The decay modes of a light neutralino have been recently reviewed in Ref. [86] and we briefly present the channels relevant for our study.Under the assumption that neutralino decays involving lighter exotic fermions, such as axinos or gravitinos, in the final state are absent or subdominant, neutralino disintegrations necessarily violate lepton number, as the only accessible lighter fermions (needed for the conservation of angular momentum) are muons, electrons or neutrinos.Interactions of this type are possible in the RPV-MSSM, see Eq. (2.2).The first possibility is that neutralino decays are controlled by LQ D operators.Couplings of this type lead to semi-leptonic disintegrations.Considering the mass-regime relevant here, m χ0 1 ≲ 445 MeV, a pion would be the only kinematically accessible meson.The expressions for the partial decay widths Γ χ0 1 → π ±/0 + ℓ ∓ i /ν i in the limit of a pure bino state can be found in Ref. [66].The RPV couplings involved here (discarding CKM mixing, see [87,88]) are λ ′ i11 , i = 1, 2, 3. 1 In addition, we stress that, with both λ ′′ 112 and λ ′ i11 nonvanishing, protons may directly employ the decay mode p → K + νi (see e.g.Ref. [51]).For small λ ′ i11 (which is the relevant regime with a long-lived bino), the associated decay width is correspondingly suppressed, however, with respect to the p → K + χ0 1 mode (provided the latter is kinematically open).Thus, these two proton decay modes can only compete close to the neutralino production threshold, or for λ ′ i11 ≈ 1. Purely leptonic decays of the neutralino can be mediated by operators of the LL Ētype.In the pure bino approximation, neglecting all mixing effects in the sfermion sector and exploiting m χ0 1 ≪ m f , where m f represents a universal sfermion mass, one obtains [89]: (2.11) Finally, both LQD and LLE operators produce the radiative decays [54,70,86,90,91], where λ if f is the relevant trilinear coupling (L i Q j Dj or L i L j Ēj ), f / f the associated fermions/sfermions running in the one-loop diagram, e f their electric charge in units of e, m f /m f their masses, N f c their color number (1 or 3).Once again, simplifying assumptions have been used as to the scalar sector.
Both the tree-level and radiative decay modes can be relevant for light neutralinos [70].Depending on the neutralino mass the tree-level decay modes might be kinematically inaccessible while the radiative mode has, essentially, no threshold.Even in scenarios where both modes are possible, the radiative decay becomes particularly important for very light neutralinos as the mass dependence makes evident: f for the tree-level three-body decay into leptons, cf.Eq. (2.11).

Proton Decay Experiments
In this section, we present the proposed proton decay experiments that we focus on in this study: DUNE, JUNO, and Hyper-K.We summarize the important technical features for each detector in Table 1.

Detectors
DUNE, or the Deep Underground Neutrino Experiment [31][32][33][34], currently under construction, consists of a far detector (FD), situated 1.5 km underground at the Sanford Underground Research Facility (SURF), about 1300km west of the Fermi National Accelerator Laboratory (FNAL).The FD is divided into four cuboidal detector volumes.Each has the dimensions 58.2 m×14 m× 12 m and consists of modular Liquid Argon Time Projection Chambers (LArTPCs) with a fiducial mass of 10 kt.Charged particles ionize the medium and also produce scintillation light as they drift along the chamber.At the planned start of the beam run at FNAL, two FDs will be deployed, in two separate cryostat chambers.The first detector module is scheduled to be operational by 2026.In this work, we consider four chambers with a combined fiducial volume of 40 kt to derive our results, and assume that proton decays in one chamber are searched for in the same chamber, i.e. we do not consider the possibility that decays from one chamber are detected in another and DUNE [31].We include the characteristics of Super-K [81] for comparison.ϵ inv. is the approximate detection efficiency of the charged kaon.can, in principle, reduce the background below single-event level for key nucleon decay channels [31].
The Jiangmen Underground Neutrino Observatory, or JUNO [35,36], is a proposed spherical liquid scintillator detector of inner diameter 35.4 m with 20 kt fiducial mass, situated 700 m underground at Kaiping in Southern China.It is expected to start operations in 2024.The current choice for the liquid scintillator material is linear alkylbenzene.The detector is submerged in a water pool to protect it from radioactivity from the surrounding rock and air.The expected background for p → K + + ν is 0.5 events per 10 years, and the expected detection efficiency reaches about 36.9% [92].
The Hyper-Kamiokande observatory, or Hyper-K [37], is the successor to the Super-Kamiokande (Super-K) experiment [81], and is a proposed large-scale underground water Cherenkov neutrino detector in Japan.Its proposed start of operations is around the year 2025.It consists of a cylindrical vertical tank with a diameter of 74 m and height of 60 m.The fiducial volume contains highly transparent purified water with a fiducial mass of about 187 kt.While most of the background is scattered by the surrounding water accross the fiducial volume, or rejected through veto-detectors, neutron and kaon backgrounds from cosmic rays persist.Detection of the process p → K + + ν is achieved through the reconstruction of the pionic or muonic decay modes of the kaon.The former, owing to lower background, leads to stricter bounds.Its efficiency is expected to be about 10.8 ± 1.1%.The corresponding expected background is 0.7 ± 0.2 red events per Mt per year.

Prospects for the Detection of the Neutralino Final State
Hyper-K offers a significantly higher fiducial volume, compared to the other detectors, but it is limited by its detection principle: Hyper-K (and Super-K) cannot measure the momentum for particles below the Cherenkov threshold of water, which for kaons, is about 750 MeV [80]. 3 This is a serious disadvantage for distinguishing between proton decay modes through a measurement of the momentum of the outgoing kaon, since the kaons from p → K + + ν are produced with a momentum of about 340 , and the value is even lower in the case of a massive neutralino in the decay p → K + + χ0 1 , cf.Fig. 3. Thus, Hyper-K cannot distinguish the case of a kaon produced in association with a detectorstable (or invisibly decaying) massive neutralino from that of a decay involving a neutrino.Despite their smaller dimensions, DUNE and JUNO are very competitive in this respect: being scintillation detectors, they can measure the kinetic energy of the kaon down to a threshold of 50 MeV.
Turning to the detection strategy based on measuring the neutralino decay, we assume zero background and a neutralino detection efficiency, ϵ vis., of 100% in our simulations.The characteristic signature includes visible objects from the kaon decay, as well as from the neutralino decay, with a likely displaced vertex structure.Timing-coincidence effects should provide us with various methods of implementing high-efficiency cuts leading to a clean signal-to-background ratio.However, the subsequent decay products of the neutralino, e.g. the muons and pions, might be invisible to the Super-K and Hyper-K detectors, again due to the threshold for Cherenkov radiation. 4The momenta of the muons and pions are fully determined by the mass and momentum of the parent neutralino.Depending on this momentum distribution, only a small fraction should induce Cherenkov radiation.Furthermore, due to the energy loss of the muons or pions, these should come to a stop inside the detector volume (in about 2 m) and produce characteristic subsequent decay products.
The muon, as well as the pion, can both be identified by a Michel electron.Thus, the event can be distinguished from competing processes by the identification of its final states.The preceding proton decay can be used as a trigger to reject background events.On the other hand, DUNE and JUNO can directly identify secondary particles in their tracking chambers.A precise estimate of their performance, taking into account the exact shape of the fiducial volume, would however require detailed simulations for each detector.

Proton Decay in Nuclei
In our discussion so far, we have considered free protons.Yet most of the protons in the detector materials are in bound states.Super-K and Hyper-K use water (H 2 O) with most of the protons in the oxygen nucleus, DUNE uses liquid argon, and JUNO employs linear alkylbenzene (C 6 H 5 C n H 2n+1 ) as proton sources.We now briefly discuss the impact of this distinction for our analysis.
Effects such as Fermi momentum, correlations between nucleons, and nuclear binding energy can alter the decaying proton's momentum, and thus the kaon momentum given in Eq. (2.10), in the lab frame.These effects can be analyzed through various approachesfor instance, by modeling the nucleus as a local Fermi gas [93], or through the so-called approximated spectral function approach [94,95].In addition, the produced kaon can rescatter on the surrounding nucleons, further affecting its momentum, as it exits the nucleus.This can be modeled as a Bertini cascade [96][97][98], implemented in GEANT4 [99,100].
Due to the above effects, the kaon momentum is no longer fixed at the value of Eq. (2.10), but smeared out instead.A significant fraction of kaons lose energy, although some may also gain energy due to collisions with high-momentum nucleons.A detailed discussion for Super-K considering the 16 O nucleus can be found in Ref. [80].A simulation for the case of argon can be found in Ref. [93].A study for liquid scintillators can be found in Ref. [101], according to which we expect a comparable behavior with linear alkylbenzene.
For our study, the smearing effects are not expected to be of critical importance, since we do not attempt to simulate momentum measurements of the kaon, but focus instead on the search where the kaon is reconstructed through the observation of its decay products.This choice has a minor impact on the detection efficiency.For the momentum measurements, we will directly employ the sensitivity estimates claimed by the DUNE and JUNO colaborations.In the case of Super-K and Hyper-K, the kaon reconstruction relies on the kaon decaying at rest.Due to the effects discussed above, some kaons become energetic enough that they do not stop in the water before decaying.However, Super-K estimates this number to be only 11% [80].
In the case of DUNE, kaon tracks with a momentum higher than 180 MeV can be measured with a detection efficiency of 90% [34].If the kaon momentum is below this threshold, the track cannot be reconstructed due to the short decay length of the kaon (βγτ K ).However, the reconstruction algorithms used in the experiment can still identify the kaon through its dominant decay chain, K → µ → e, with an efficiency higher than 90%. 5or JUNO the search strategy relies on the observed time coincidence and well-defined energies of the kaon decay products [35].Additionally, the liquid scintillator detects a prompt signal coming from the kaon.We assume that the efficiency for kaon reconstruction with this approach is constant for all momenta.A later study [92] indicates that the best reconstruction can be achieved by a time-correlated triple coincidence, composed of the energy deposit of the kaon, a short-delayed deposit of its decay daughters and the energy deposit of the Michel electron from the muon decay.A detailed simulation of detector effects and proton momentum distribution for DUNE and JUNO would allow for an enhanced efficiency in the considered proton decay mode.However, this goes beyond the scope of this work.
Thus, we shall neglect any nuclear effects.However, we wish to emphasize that, in order to determine the mass of the neutralino from the kaon decay, a precise momentum measurement of the kaon is required.In that case, an accurate description of nuclear effects is crucial.
Before concluding this subsection, we briefly mention one more effect.A proton decaying inside a nucleus can leave the nucleus behind in an excited state.The latter de-excites promptly into the ground state by the emission of a gamma ray.One can estimate the energy of this photon and search for it as a coincidence signal.With this method, it is possible to reduce background events from cosmic ray muons and radioactivity of materials around the detector wall.Indeed, Super-K has searched for such gamma rays [80], and Hyper-K will be able to as well.JUNO can exploit the time-coincidence between the prompt signal from kaons hitting the liquid scintillator and a delayed signal from its decay products, to reduce background events.However, DUNE indicates that measuring time differences on the scale of the kaon lifetime is difficult [34].

Simulation Procedure
We now describe the simulation procedure that we use in order to estimate the sensitivities of the various experiments for detecting a proton decaying into a neutralino, possibly followed by the decay of the neutralino.
The total number of produced neutralinos (or kaons) in a given experimental setup may be written as, where Γ(p → K + + χ0 1 ) is calculated in Eq. (2.8), N p is the total number of protons in the detector volume and t is the runtime of the experiment.If the neutralino is not explicitly looked for, the total number of observed events can then be estimated by simply multiplying the above expression by the efficiency of kaon detection, cf.discussion in the previous section.
In the case where the neutralino may decay visibly into a final state X, however, we also need to determine the number of such decays that can be reconstructed within the detector volume.We estimate them as, Here the function ⟨P [ χ0 1 in d.r.]⟩ represents the average probability of the neutralino to decay within the fiducial volume or the detectable region (d.r.) of the detector.This probability is dependent on the neutralino's lifetime, kinematics, point-of-origin within the detector, and the geometry of the detector itself.ϵ vis., in the above, is the detection efficiency for the visible state X, which we shall ultimately set to 100% in this work, cf.Section 3.2.
In order to estimate ⟨P [ χ0 1 in d.r.]⟩ for each detector, we run a Monte Carlo simulation with N MC χ0 1 neutralinos of a given mass and with fixed RPV couplings. 6The neutralinos originate at random points within the detector, and travel in random directions.We discuss the geometry details for each detector below.Then, we estimate, where, is the individual probability for the i th simulated neutralino to decay inside the detector.L i is the distance between the point where the neutralino originates and the detector boundary along its direction-of-travel.The mean decay length λ (independent of i) is given by, with Γ tot the total decay width of the neutralino and, In the above, m χ0 1 and E are the neutralino mass and energy, respectively.Thus, L i is the only geometry-dependent factor.We now describe how we calculate it for the considered detectors.
Hyper-K: Hyper-K is a vertical cylindrical-shaped detector with a radius of R = 37 m and height H = 60 m.Let the i th neutralino be generated inside the Hyper-K volume at a point (r, φ, z) in a cylindrical coordinate system with origin at the center of the bottom surface (z = 0) of the detector.Let its three-velocity, ⃗ v, be at azimuthal angle φ v with components v z and v ⊥ = ⃗ v 2 − v 2 z along the z-axis and in the polar plane, respectively.Then, we have, where, and t 1 > t 2 if v z = 0. Here, t 2 is given by, Super-K can be modeled analogously.
DUNE: DUNE has four cuboid-shaped FDs with dimensions L = 58.2m, W = 14.0 m, and H = 12.0 m.In a rectangular coordinate system with the origin at the bottom corner of the detector, for the i th neutralino generated at (x, y, z) traveling towards the direction denoted by the three-vector ⃗ v = (v x , v y , v z ), we define, and t x > t y , t z if v x = 0, and analogous expressions for t y and t z , depending on W, H, respectively.L i is then given by,

.11)
JUNO: JUNO has a spherical geometry with a radius of R max = 17.7 m.For a neutralino produced at a random point ⃗ r = (r i , θ i , φ i ) inside the detectable region and with velocity ⃗ v, flying in the direction given by the angles (θ j , φ j ), one can calculate the angle θ between the two vectors: cos θ = ⃗ r•⃗ v |⃗ r||⃗ v| .A coordinate transformation is performed in order to eliminate the dependence of θ on φ i , φ j , such that θ = θ i −θ j ∈ [0, 2π].The final distance is calculated to be: As an illustration, we depict ⟨P [ χ0 1 in d.r.]⟩ in Fig. 4 as a function of the neutralino mass but for fixed decay length, cτ , for the four detectors.The neutralino momentum p χ0 1 at production depends on the neutralino mass m χ0 1 as in Eq. (2.10).Thus for increasing m χ0 1 , |⃗ v χ0 1 | decreases, and the neutralino is more likely to decay in the detector for a fixed lifetime.

Numerical Analysis
We now present benchmark scenarios, which, we believe, capture the bulk of the phenomenology accessible at DUNE, JUNO and Hyper-K for a proton decaying to a lighter neutralino.In all the considered cases, the proton disintegration is controlled by the parameter λ ′′ 121 , as in Eq. (2.8).Thus, we only consider cases for which m χ0 1 ≲ 445 MeV, cf.Eq. (1.1).The produced neutralino either escapes the detector as missing energy or decays into visible modes via an RPV operator λ D ijk .The benchmarks we study are presented in Section 4.1 and summarized in Table 2.We present the corresponding numerical studies in Section 4.2.For each scenario, we assume that the listed couplings are the only non-negligible RPV couplings; the relevant current bounds are also shown in the table.Table 2: Details of the benchmark scenarios.The bounds on λ D ijk are taken from Refs.[102,103] while the one on λ ′′ 121 is from Ref. [104].Product bounds are obtained from Ref. [41] for SUSY masses of 1 TeV, except in the case of λ ′ 311 λ ′′ 121 , where it is the bound on p → K + + ν from [80] reinterpreted for B5.The mass ranges conform (up to some rounding) to the discussion in the text.An additional limit originates from Super-K: |λ ′′ 121 | < 3.9 × 10 −31 m q GeV 2 ."Min.cτ χ0 1 " refers to the minimal decay length, i.e. the decay length at the maximal allowed RPV coupling and neutralino mass within the scenario.

Benchmark Scenarios
In the first benchmark scenario, B1, we assume that the neutralino cannot be observed at DUNE, JUNO or Hyper-K, either due to a long lifetime or to invisible decay products.(We set L-violating couplings to 0 in practice.) In the second benchmark, B2, neutralino decays are controlled by the trilinear coupling λ ′ 333 .In this case tree-level decay modes [ χ0 1 → (τ − t b, ν τ b b)] are kinematically prohibited, under the assumption of negligible generation mixing [87,88].The radiative channel, mediated by (s)bottom loops, then appears as the dominant one.Existing limits on λ ′ 333 imply an already sizable decay length (see the last column of Table 2), as compared to the dimensions of the experiments (listed in Table 1).We briefly comment on alternative choices of dominant λ ′ ijj couplings controlling the radiative decay mode of the neutralino.Due to the scaling with the fermion mass squared [see Eq. (2.12)], we can dismiss the cases j = 1, 2 as resulting in very long lifetimes.The choice of a dominant λ ′ 133 offers little competition as well, due to severe experimental bounds [102,103]: should perform comparably to λ ′ 333 retained here.Benchmark B3 also involves radiative neutralino decays, now controlled by the trilinear coupling λ 233 leading to a tau/stau loop.In this case, significantly shorter decay lengths are accessible (see Table 2), as compared to B2.Other choices of λ ijj would yet lead to very long decay lengths again, if j = 1, 2, while λ 133 is severely constrained experimentally.
The two last scenarios involve tree-level decay modes of the neutralino.Here, we dismiss leptonic decays, since the widths associated with Eq. (2.11) lead to very long-lived neutralinos, escaping the detectors in the considered mass regime.Similarly, a very strict experimental limit on λ ′ 111 [103] leaves only λ ′ 211 and λ ′ 311 in a position to generate semileptonic disintegrations of the neutralino that are observable at DUNE, JUNO, or Hyper-K.For B4, a non-vanishing λ ′ 211 potentially opens the semi-leptonic modes: χ0 1 → π 0 +ν µ , π 0 + νµ , with kinematical limits on the neutralino mass reading The SU(2) Lrelated charged channel is kinematically closed.In both cases, the radiative decay mode mediated by a down/sdown loop, although open, results in negligible widths due to the Yukawa suppression.

Results
We may now discuss the experimental prospects for each of the proposed benchmark scenarios.For commodity, we dismiss the comparatively broad theoretical uncertainties examplified by Fig. 2, i.e. work with an absolute prediction of the decay rates, as obtained with the hadronic input of Ref. [84], and strictly focus on a comparative analysis of the various experiments in view of identifying a signal of the studied kind.This means that the (prospective) bounds on the parameter space of the RPV-MSSM shown below should not be understood as absolute, but still require a proper account of the theoretical uncertainties before any attempt at correlating them with input from other observables.
In the case of B1, with a stable, i.e. invisible, neutralino at the detector scale, we present in Fig. 5 the projected parameter space coverage achieved by DUNE [34], JUNO [92] and Hyper-K [37] at 90% confidence level sensitivity, after a ten-year run time.Here, we recast the projections quoted for the p → K + + ν mode in the technical reports, under the assumption that these remain valid with the modified kaon kinematics.For Hyper-K, where the kaon momentum is not measurable, the cases of a massive or massless neutralino are indeed indistinguishable7 and our working hypothesis amounts to modeling the signal efficiencies and experimental backgrounds as roughly constant in the considered kinematical window, as discussed in Section 3. The current Super-K bound [80] can be exploited in the same fashion, and is included in the plot.For DUNE and JUNO, however, the searches implement specific cuts on the kaon momentum in order to reduce the background due to atmospheric neutrinos.This implies that the actual bounds in the massive case would be weaker than the naive one, due to eroded signal efficiency.Nevertheless, it might be possible to optimize the momentum cuts in searches dedicated to the massive neutralino case and counteract this diminished sensitivity.A more precise detector simulation goes beyond the scope of the current paper.Fig. 5 shows the parameter space explored by the various experiments in the plane spanned by the neutralino mass m χ0 1 and λ ′′ 121 /m 2 f .The proton decay rate is kinematically suppressed as the neutralino mass approaches the threshold, m χ0 1 < 445 MeV; this weakens the reach in λ ′′ 121 /m 2 f as compared to the massless case.JUNO, DUNE and Hyper-K are all expected to improve on the coverage achieved at Super-K, probing values of λ ′′ 121 /m 2 f as much as 1.3, 1.5 and 2.5 smaller, respectively.Thus, despite Hyper-K's advantage in terms of fiducial mass, JUNO and DUNE remain competitive thanks to their superior efficiencies and high background-rejection rates.In addition, we stress that momentum measurements of the produced kaon, needed to correlate the rate with the neutralino mass, will only be possible at these two experiments.While the current level of information presented in technical reports and the difficulty of modeling the macroscopic effects leading to momentum smearing make it impossible for us to realistically simulate this search, one should keep in mind that this observable would be simultaneously available with the kaon detection search, and might allow DUNE or JUNO to distinguish the scenario with a neutralino from its counterpart with a neutrino, even if both these final states result in missing energy at colliders.We now turn to the scenarios with a visible neutralino decay.In this case, the relevant parameter space is (at least) three-dimensional, consisting of m χ0 1 , λ ′′ 121 /m 2 f (controlling the proton decay, and thus the production rate of the neutralinos), and λ (′) ijk /m 2 f (controlling the neutralino decay together with the neutralino mass).For commodity, we focus on two projections in this parameter space: • First, we examine the plane λ ′′ 121 /m 2 f vs. λ • Then, we consider the plane λ D ijk /m 2 f vs. m χ0 1 , with λ ′′ 121 /m 2 f set to its value saturating the reinterpreted bound from Super-K.The corresponding values of λ ′′ 121 /m 2 f as a function of the neutralino mass would correspond to the edge of the gray area in Fig. 5 in the case of a purely invisible decay of the neutralino, but are actually shifted when a shorter lifetime makes the invisible search less relevant.
In both cases, we also indicate the neutralino decay lengths, as this observable is directly correlated with the considered neutralino decay coupling.Again, we assume an accumulated ten years of observed data at DUNE, JUNO and Hyper-K.As we neglect any sort of background, the reach corresponding to 95% confidence level exclusion is determined by the 3-event isocurves, which we display in our plots. 8Existing constraints of Table 2 are also depicted in gray (single-bounds) or cyan (product-bounds), where relevant.
We then consider the benchmark B2 with a still long-lived neutralino decaying radiatively via bottom / sbottom loops.The corresponding results are shown in Fig. 6.Experiments are sensitive to this scenario from two directions.First, most of the neutralinos decay outside the detectors, due to their long lifetime, hence appear as missing energy, similarly to the scenario of Fig. 5.In fact, consulting the right plot of Fig. 4, we observe that this happens for over 90% of the neutralinos, on average, even for the shortest available lifetimes and a mass as large as m χ0 1 = 400 MeV.The corresponding bound hardly depends on λ ′ 333 /m 2 f (as long as one is deep in this long-lived regime) and produces vertical boundary lines in the plane spanned by λ ′′ 121 /m 2 f and λ ′ 333 /m 2 f , as displayed on the left-hand side of Fig. 6.Larger values of λ ′′ 121 /m 2 f can be probed (or, in case of Super-K, are already excluded) by the invisible neutralino search.A subsidiary region is probed by the visible search, where only the 3-event line associated with Hyper-K appears on the plot (the corresponding boundaries for DUNE and JUNO would be far to the right).We should stress, here, that our estimates for the efficiencies achieved in the visible search are very optimistic and that the reach of this observable may be significantly more reduced than what is presented in Fig. 5.The number of visible decays scales like |λ ′ 333 λ ′′ 121 /m 4 f | 2 in the limit of long neutralino lifetimes, resulting in a sloping coverage limit.The large volume of Hyper-K is a clear advantage for detection sensitivity. 8Where relevant, we also depict 30-and 90-event isocurves.In the right plot of the figure, we show the sensitivity reach of the visible mode in the λ ′ 333 /m 2 f vs. m χ0 1 plane, where λ ′′ 121 /m 2 f is always set at the exclusion limit of Super-K. 9he sensitivity improves as the neutralino mass increases due to (a) the lifetime becoming shorter, cf.Eq. (2.12), and (b) the neutralinos having lower momentum: the decay length of the neutralino (βγcτ ) χ0 1 is indeed shorter, resulting in an increased average decay sensitivity ⟨P [ χ0 1 in d.r.]⟩, as shown in Fig. 4. The physics situation of benchmark B3 is largely comparable to that of the former benchmark, with a radiatively decaying neutralino.Consequently, the parameter space coverage in proton decay experiments follows the same trends, as can be observed in Fig. 7. Shorter lifetimes are accessible nonetheless, resulting in an increased relevance of the visible search channel, although only Hyper-K has viable detection prospects in this mode.The left-hand plot of Fig. 7 distinctly exhibits a region in the higher range of λ ′ 233 /m 2 f where both types of detection are competitive, as well as a tiny region only accessible to the visible search.Once again, we point at the generous efficiencies assumed for the visible search here: this region accessible only to the visible search would likely shrink with more realistic estimates.
In the case of benchmarks B4 and B5, the tree-level decay modes of the neutralino may result in comparatively short lifetimes when the RPV couplings are set to their maximal values compatible with existing bounds.As a consequence, the visible search strategy is expected to perform more competitively than for the previous scenarios.Our results are shown in Fig. 8 and Fig. 9. There, a portion of the parameter space left open by Super-K is accessible to visible searches at DUNE or JUNO, although Hyper-K remains the experiment most sensitive to this mode (as well as to the invisible search).Conversely, the invisible search channel becomes less competitive in the higher range of λ ′ 211 /m 2 f due to fewer neu-Figure 9: Sensitivity reach/Super-K limit for benchmark B5.The existing single-bound on λ ′ 311 from Table 2 is shown in gray, the product-bound is in light blue (both with m f = 1 TeV), while the bound on λ ′′ 121 lies outside the scale of the plot.Left: As in left plot of Fig. 6 but for benchmark B5.Right: As in right plot of Fig. 5 but for benchmark B5.The dashed, solid, and dot-dashed lines correspond to 3-, 30-and 90-event isocurves, respectively.

Conclusions
The search for proton decay is tightly related to the fundamental and experimentally wellestablished question of matter-antimatter asymmetry.Experimental and phenomenological activity in this respect thus appears strongly motivated.In this work, we tried to emphasize a less standard but very realistic decay mode of the proton, p → K + + χ0 1 .Here χ0 1 denotes a light and long-lived exotic neutral particle, e.g. a bino-like neutralino in the RPV-MSSM, which can possibly decay within the detector.We demonstrate the strong potential of the upcoming DUNE, JUNO and Hyper-K experiments to investigate such a scenario.This analysis was performed under the assumption that the experimental searches considering a massless neutrino can be reinterpreted in terms of a massive exotic particle, or that experimental cuts can be adjusted in order to address this situation.Several signatures can be looked for, depending on the lifetime of the neutralino and its decay channels.Measurement of the kaon momentum (at DUNE or JUNO) or observation of a displaced vertex from the neutralino decay would provide means to distinguish such a scenario from the more traditionally considered p → K + + ν.We have illustrated these features with the help of a collection of benchmarks.A significant coverage of the parameter space can be achieved by DUNE, JUNO and, most especially, Hyper-K, clearly improving on the current limits obtained at Super-K.More detailed collider simulations would be necessary to precisely delimit the reach of these experiments.In any case, we stress the necessity to keep the scope of the experiments as broad as possible and allow the reinterpretation of results in non-standard scenarios.through the funds provided to the Sino-German Collaborative Research Center TRR110 "Symmetries and the Emergence of Structure in QCD" (DFG Project ID 196253076 -TRR 110).

A Heavy Neutral Lepton
Throughout our work we have focussed on decays to and of a supersymmetric neutralino in a model with broken R-parity.As explained in Section 2, the low-energy (proton-scale) EFT's of Refs.[51,86] provide a natural framework to study such processes.The imprint of the high-energy (in our case SUSY) model is then encoded within the Wilson coefficients, determined by the matching procedure and the Renormalization Group Equations.In this fashion, we were always able to present our results in terms of the parameters of the more fundamental model, namely the neutralino mass m χ0 1 , the RPV couplings λ ′′ ijk , λ ′ ijk , λ ijk , and the universal scalar fermion mass m f .In the production of the neutralino via proton decay baryon-number is violated.In the decay of the neutralino lepton-number is violated.Here, we have associated no lepton-or baryon-number with the (Majorana) neutralino.
As outlined in the introduction a right-handed neutrino, N R , potentially light, has the same SM gauge quantum numbers as the light (dominantly bino) neutralino.Associating the lepton number 1 with such a Majorana particle, N R , thus appears as a largely formal distinction in the L-violating context and the expected phenomenology in proton disintegrations is formally unchanged.In particular, the proton-scale EFT's of Refs.[51,86] (with the neutralino replaced by N R ) remain fully operational intermediaries between high-energy scales and hadronic physics.Differences of the UV completion in the neutralino case (RPV-SUSY) and any potential UV completion for the HNL model will emerge at the level of the matching.
The phenomenology of the right-handed neutrino can be studied in the context of νSMEFT [105][106][107], itself a low-energy EFT valid at the electroweak scale.Obvious contributions to the proton-decay operators are generated by the operators listed in Eq. (7) in Ref. [105].We insist upon the fact that these contributions from νSMEFT potentially involve low-energy operators beyond Q′ 1 , Q1 , Q2 of Eq. (2.4).Indeed, SUSY (through the holomorphicity condition of the superpotential) favors baryon-number violation involving right-handed [i.e.SU(2)-singlet] quarks, while such a prejudice need not apply in νSMEFT. 10imilarly, the νSMEFT operators listed in Eq. ( 6) in Ref. [105] and denoted O LN Le , O LN Qd , and O QN Ld straightforwardly contribute to the decays of the right-handed neutrino.

Figure 1 :
Figure 1: Feynman diagrams depicting the contributions to the four-fermion operators, which in turn contribute to proton decay.The solid black circle shows the RPV vertex.

Figure 2 :
Figure 2: Proton decay width normalised to |λ ′′ 112 | 2 /m 4 f, where m f represents a universal value for the squark masses.Two different lattice evaluations are used for the numerical values of the form factors: from Aoki, 2017[84] and Yoo, 2021[85].The dashed line represents the case where lattice form-factors at q 2 = 0 are used, whatever the neutralino mass, while the solid lines represent results with form factors determined according to chiral perturbation theory[83].The bands around the dashed lines denote the approximate error in the lattice calculation of the form factors.

Figure 3 :
Figure 3: Momentum of the final state kaon as a function of the neutralino mass in the process p → K + + χ0 1 ..

Figure 4 : 1 =
Figure 4: Average neutralino decay probabilities as a function of the neutralino mass for fixed neutralino decay length: cτ = 10 m (left) and cτ = 1000 m (right).These plots have been generated with a sample size N MC χ0 1

Figure 5 :
Figure 5: Sensitivity reach for the single coupling scenario of benchmark B1.The reinterpreted bound from Super-K is shown in gray.The bound from Table 2 lies above the scale of the plot.The results for Hyper-K, DUNE, and JUNO are for a run-time of 10 years.

3 Figure 6 :
Figure 6: Sensitivity reach/Super-K limit for benchmark B2.The existing single-bound on λ ′ 333 from Table 2 is shown in gray (with m f = 1 TeV), while the product-bound lies outside the scale of the plot.Left: The limits in the coupling-vs.-couplingplane m χ0 1 = 400 MeV.The contours correspond to the visible mode (blue dashed, downward sloping line) and invisible mode (vertical lines); see discussion in the text.Right: The limits in the coupling-vs.-massplane for the visible mode with λ ′′ 121 /m 2 f fixed at the threshold of the Super-K bound of Fig. 5.

Figure 7 :
Figure 7: Sensitivity reach/Super-K limit for benchmark B3.The existing single-bounds from Table 2 are shown in gray while the product-bound is shown in blue (all with m f = 1 TeV).Top Left: As in left plot of Fig. 6 but for benchmark B3.Top Right: Zoomed-out version of the top-left plot.Bottom: As in right plot of Fig. 5 but for benchmark B3.The dashed and solid lines correspond to 3-and 30-event isocurves, respectively.