Probing $R$-parity violation in $B$-meson decays to a baryon and a light neutralino

We propose a search for $B$ meson decays to a baryon plus missing energy at the Belle II experiment to probe supersymmetry with a GeV-scale lightest neutralino ${\tilde{\chi}_1^0}$ and $R$-parity violation (RPV). We perform analytic computations of the signal branching fractions in the framework of effective field theory, with a single nonzero RPV operator $\lambda''_{ij3}\bar{U}_{i}^c\bar{D}_{j}^c\bar{D}_3^c$, where $i,j=1,2$. The hadronic form factors are calculated using an SU(3) phenomenological Lagrangian approach for the proton, as well as several hyperons and charmed baryons. Since the decay of the neutralino is kinematically and CKM suppressed in this theoretical scenario, it decays outside the detector and appears experimentally only as missing energy. We detail the analysis techniques at the experimental level and estimate the background in the ${B^+ \to p {\tilde{\chi}_1^0}}$ search using published results for $B^+\to K^+ \nu\bar\nu$. Our final sensitivity plots are shown for both $\lambda''_{113}$ versus the squark mass $m_{\tilde{q}}$ and $\lambda''_{113}/m^2_{\tilde{q}}$ versus the neutralino mass $m_{{\tilde{\chi}_1^0}}$. We find that the search at Belle II could probe $\lambda''_{113}/m^2_{\tilde{q}}$ down to the order of $10^{-8}$ GeV$^{-2}$ in the kinematically allowed $m_{{\tilde{\chi}_1^0}}$ range. We also obtain current limits on $\lambda''_{123}$ by recasting an existing search interpreted as ${B^0 \to \Lambda^0 {\tilde{\chi}_1^0}}$, and comment about searches for ${B^+ \to \Sigma^+ {\tilde{\chi}_1^0}}$, ${B^0 \to \Sigma^0 {\tilde{\chi}_1^0}}$, ${B^+ \to \Lambda_c^+ {\tilde{\chi}_1^0}}$, and ${B^+ \to \Xi_c^+ {\tilde{\chi}_1^0}}$. In closing, we briefly discuss potential searches at the LHCb and BESIII experiments.


Introduction
Supersymmetry (SUSY) [1,2] has been one of the most sought candidates for physics beyond the Standard Model, in particular because it offers an elegant solution to the hierarchy problem [3,4] in the Higgs sector of the Standard Model (SM). Although no evidence of SUSY has been found so far, the LHC explored wide regions of the SUSY parameter space and has established tight mass bounds on TeV-scale SUSY. The minimal supersymmetric extension of the Standard Model (MSSM) in its general form contains lepton-number violating (LNV) and baryon-number violating (BNV) terms, which can induce undesirable proton decay. One can impose by hand a simple Z 2 symmetry, called R-parity, which forbids these terms. Under R-parity, all SM particles are even, while all their superpartners are odd. If R-parity is conserved, not only is the proton stable but so is also the lightest supersymmetric particle (LSP). Nevertheless, in SUSY with R-parity violation (RPV), one can still prevent proton decay by imposing certain conditions 1 . RPV-SUSY models are thus acceptable and, moreover, provide a rich phenomenology that is still only loosely constrained by colliders (see e.g. Refs. [6][7][8] for reviews of these models). The LHC has established strong lower bounds on the masses of heavy SUSY particles, especially color-charged superpartners [9][10][11][12][13]. Color-neutral superpartners, however, are not so strictly bounded. The lightest neutralinoχ 0 1 can be as light as a GeV or even lighter [14][15][16][17][18][19][20][21][22][23], in which case its production in decays of τ lepton or heavy mesons is kinematically allowed. Such a light neutralino should be dominantly Bino according to current bounds [21][22][23][24][25]. In addition, it has to decay, e.g., via RPV couplings so as to avoid overclosing the Universe [26][27][28][29]. If the RPV couplings are tiny, such light neutralinos can be long-lived so that they appear as displaced signatures or even missing energy in collider experiments. In this context, in a previous work [30] we proposed searching for such a neutralino in τ lepton decays at Belle II. Refs. [31][32][33][34] have estimated the sensitivities of various LHC far detectors, and Ref. [35] investigated the sensitivity of neutrino detectors to atmospheric light neutralinos. In this work we consider a light neutralinoχ 0 1 produced in B-meson decays through the superpotential RPV terms λ ijkŪ iDjDk , where U and D are the up-type and downtype right-handed supermultiplet fields, λ ijk are dimensionless couplings, and i, j, k are generation indices in the quark-mass basis. Such interaction terms not only violate Rparity but also baryon number. However, proton decay does not occur in this scenario, as long as no lepton number violating RPV terms are allowed and the neutralino is not lighter than the proton.
We focus on the case in which only one of the RPV couplings λ 113 , λ 123 , λ 213 , or λ 223 is non-zero and there is no squark mixing. 2 We propose to probe λ 113 via the decay mode B + → pχ 0 1 and to study λ 123 via the decays into strange baryons B 0 → Λ 0χ0 1 , B + → Σ +χ0 1 , and B 0 → Σ 0χ0 1 . Similarly, the decays into charmed baryons B + → Λ + can be used to study λ 223 . We generically refer to all four modes as B → Bχ 0 1 , where B indicates one of the above baryons. These decays can be searched for at an e + e − → BB "B-factory" facility, namely, the currently running Belle II experiment [36,37] or the BABAR and Belle experiments, which are no longer collecting data. The decay of the light neutralino is suppressed, so that it appears as missing energy at the detector level [25]. While these decays violate baryon number, strictly speaking it is not possible to verify experimentally that the neutralino, which escapes detection, does not carry baryon number.
The superpotential terms λ ijkŪ iDjDk can also contribute to baryon-antibaron oscillations (n −n) [38,39] and di-nucleon decay (e.g., N N → ππ, N N → KK) [40]. However, in the scenario studied here, with no squark mixing, the contributions of λ 113 and λ 123 to these processes are suppressed by two weak insertions, yielding limits on λ 113 and λ 123 that are much weaker than current LHC bounds and weaker still than the sensitivity of the studies we propose here. Following Eq. (23) in Ref. [40], we can estimate the order of magnitude of the baryon-antibaryon transition amplitude δ BB in terms of the RPV couplings: where κ ∼ 10 −2 GeV 3 [41]. From the LHCb bound on the Ξ 0 b −Ξ 0 b oscillation rate [42], namely ω < 0.08 ps −1 , we extract the bound λ 123 /m 2 q 4 × 10 −4 GeV −2 , where we have taken mχ0 1 = 2.5 GeV in Eq. (1.1). Similarly, the bounds on δ BB from dinucleon decay listed in Table I of Ref. [40] can be translated into bounds on the RPV couplings. The bounds δ (udb) 2 10 −13 , δ (usb) 2 10 −10 , δ (cdb) 2 10 −17 , and δ (csb) 2 10 −14 translate into λ 113 /m 2 q 6 × 10 −4 GeV −2 , λ 123 /m 2 q 2 × 10 −2 GeV −2 , λ 213 /m 2 q 5 × 10 −6 GeV −2 , and λ 223 /m 2 q 2 × 10 −4 GeV −2 , respectively, again taking mχ0 1 = 2.5 GeV in Eq. (1.1). Decays of a B meson to a baryon and missing energy are motivated also in the context of B-mesogenesis models [43][44][45]. Signatures of directly detectable or escaping long-lived particles at B factories have also been discussed in Refs. [30,[46][47][48][49][50][51][52][53][54][55][56][57][58][59] This paper is organized as follows: section 2 contains the theoretical framework of the RPV scenario. This includes the effective Lagrangian with the RPV terms and the calculation of the matrix elements, which we estimate using an effective BNV Lagrangian. In section 3 we explain the experimental analysis technique, estimate the background in the search for B + → pχ 0 1 , and determine the sensitivity of a Belle II measurement of the branching fraction Br(B + → pχ 0 1 ). We also interpret preliminary BABAR results [60] in terms of B 0 → Λ 0χ0 1 and compare the sensitivity of this mode to those of B 0 → Σ 0χ0 1 and B + → Σ +χ0 1 . Furthermore, we discuss the expected sensitivity of decays involving charmed baryons. Our main physics results are provided in section 4, where we extract current limits on λ 123 and estimate the Belle II sensitivity to λ 113 . Finally, section 5 contains our summary, as well as a brief discussion of related measurements that we propose to perform at the LHCb and BESIII experiments. We have included an Appendix with details of the form factor calculation for the hadronic transitions.
2 Theoretical framework: effective Lagrangian and neutralino production and decay In this section we describe our theoretical framework. We start with an effective Lagrangian needed for the description of B → Bχ 0 1 decays at the partonic level. Then we define the effective four-fermion operators that generate the matrix elements for the transition hadronic form factors. Finally, we discuss the computation of the decay rates Γ(B → Bχ 0 1 ) and present the contributions arising from the different intermediate squarks as a function of the neutralino mass.
The effective Lagrangian L eff for the description of B → Bχ 0 1 at the partonic level contains two parts: an R-parity violating part, L RPV , with trilinear couplings of two quarks and one squark, and an R-parity conserving (RPC) part describing the couplings of the neutralinoχ 0 1 with a quark and a squark [61][62][63][64]: where a, b, c are color indices, abc is the color three-dimensional Levi-Civita tensor, P L = (1 − γ 5 )/2 is the chiral projector, the subscript R indicates a right-handed quark or squark, and the upper index C denotes charge conjugation. Since we takeχ 0 1 to be a pure Bino, the coupling gq 1R is given by where θ W is the Weinberg angle with tan θ W 0.54840, g W = e/ sin θ W 0.62977 is the SU(2) weak coupling, and e q is the quark electric charge, i.e. e u = 2/3 and e d = e s = e b = −1/3. In the last line of Eq. (2.1) we explicitly write only the terms with right-chiral fields, since we consider vanishing squark mixing, and only such fields are involved in the RPV interactions considered here. Figure 1. Parton-level diagrams for the decays B → Bχ 0 1 , where B stands for one of the baryons and λ ij3 the corresponding RPV coupling: p, n (for λ 113 ); Λ, Σ + , Σ 0 (for λ 123 ); Λ + c , Σ + c , Σ 0 c (for λ 213 ); and Ξ + c , Ξ 0 c (for λ 223 ).
The Lagrangian of Eq. (2.1) generates the parton-level diagrams shown in Fig. 1 for the decays B → Bχ 0 1 , as long as the decays are kinematically allowed 3 . To simplify the discussion, we first focus on the B + → pχ 0 1 process, i.e., the diagrams that have a d quark in the final state.
For simplicity, we assume the squark masses to be degenerate and no squark mixing. In the limit that the squark mass mq is large, the squark propagators are approximately 1/m 2 q . Therefore, the d-quark diagrams in Fig. 1 correspond to the four-fermion effective Lagrangian the effective operators are defined as and we have used the definitions where C = iγ 0 γ 2 is the charge conjugation matrix. Following Refs. [65][66][67][68][69], we define the invariant matrix element for the B + → pχ 0 1 decay as whereū p (p , s ) and vχ0 1 (q, s) are the spinors of the final-state proton and neutralino, respectively; p, p , and q = p − p are the momenta of B + meson, proton, and neutralino, respectively; m B , m p , and mχ0 1 are the masses of these particles; and W LL 0 (q 2 ) and W LL 1 (q 2 ) are form factors that parametrize the hadronic matrix elements of the BNV three-quark operator between the proton and B + meson states: The form factors W LL 0 (q 2 ) and W LL 1 (q 2 ) can be calculated using an SU(3) phenomenological Lagrangian (see Refs. [70] and [65]- [69]) applied to the BNV matrix elements involving a proton and light pseudoscalar mesons (π, K, η) and extending it to the bottom sector. The contribution to the form factors of each of theb,ũ, andd squarks is the sum of a direct contribution from the B + → pχ 0 1 process and a pole contribution induced by the bottom baryon resonances Λ 0 b and Σ 0 b , leading to the virtual two-step process Here, we estimate our matrix elements by extrapolating the strange flavor in p|O LL |K + to the bottom flavor. In Refs. [71][72][73][74][75] detailed discussion of possibilities to extend the phenomenological Lagrangians from the SU(3) sector to the heavy quark sector have been provided. Derived effective Lagrangians involving light and heavy mesons and baryons have been successfully applied to different tasks. More concretely, one replaces the decay constant of the Kaon by that of the B meson, f K → f B , and the masses of the K + meson and the Λ 0 and Σ 0 hyperons by the corresponding masses of B + , Λ 0 b and Σ 0 b . Our analytical results for the corresponding form factors are shown in Appendix A. We also demonstrate in the appendix that the same results for the transition form factors can be obtained using an effective phenomenological BNV Lagrangian constructed in terms of the fields of the B + , proton, Λ 0 b , and Σ 0 b , which contains some adjustable couplings fixed from matching of matrix elements p|O LX |B + in the two approaches.
The B + → pχ 0 1 decay width is calculated according to the formula where is the kinematical triangle Källen function. The last line in Eq. (2.10) shows that the matrix element squared pol |M| 2 is manifestly positive because of the kinematical constraint m B > m p + mχ0 1 . We estimate the rate of the decay B + → Σ +χ0 1 in the same way (see details in Appendix A). In particular, we replace the d quark by the s quark, λ 113 → λ 123 , p(duu) → Σ + (suu), and the intermediate baryons Λ 0 b and Σ 0 b are replaced by the Ξ 0 b and Ξ 0 b , which have a (bus) quark-flavor content with an antisymmetric light spin-0 diquark [us] 0 and a symmetric light spin-1 diquark {us} 1 , respectively. We also replace the couplings α and β that define the matrix elements of the three-quark operators between the proton and vacuum state (see Eq. (A.1)) by the respective couplings that define the matrix elements for the case of the Σ + baryon. See the discussion in Appendix A for more details, as well as for the form-factor calculation for B 0 → Λ 0χ0 1 and B 0 → Σ 0χ0 1 .
p mode  4 , as functions of the neutralino mass.
In Fig. 2 we compare the four branching fractions of interest, normalized by G 2 = λ 1j3 2 × (1 TeV/mq) 4 , as functions of mχ0 1 . One can see that Br(B 0 → Λ 0χ0 1 ) is significantly suppressed relative to Br(B + → pχ 0 1 ). This is explained by the compensation of the relativẽ b * ,ũ, ands contributions in Br(B 0 → Λ 0χ0 1 ), whose magnitudes are shown in Fig. 3. In particular, at small values of mχ0 1 , the ratio (A.17)), and is given by In the case of the Σ production modes, the corresponding ratio R 2 = Br(B 0 → Σ 0χ0 1 )/Br(B + → Σ +χ0 1 ) 1/2 is explained by the isotopic flavor factor 1/ Next we extend our analysis to a charmed baryon produced in the B-meson decay. In particular, we consider the modes with two possible configuration of two light quarks in the charmed baryons B(cq 1 q 2 ), where q 1 , q 2 = u, d, s, -antisymmetric [q 1 q 2 ] with spin zero and symmetric {q 1 q 2 } with spin 1. For detailed classification of charmed baryons see, e.g., Refs. [76][77][78][79]. In particular, in our calculations we consider two states with antisymmetric light-quark configurations -  [80,81]. For the couplings β defining the matrix elements of threequark operators between respective charmed baryon and vacuum we use the predictions of the QCD sum rules approaches [82][83][84][85]. In particular, from results of Refs. [82][83][84][85] we deduce the following values of the β couplings: Before moving on to the experimental aspects of the proposed search, we briefly discuss the decay of the lightest neutralino. For the considered RPV couplings, the neutralino mass range of interest, and the absence of squark mixing, the neutralino decay is kinematically suppressed by 3 off-shell propagators: a squark, a bottom quark, and a W boson, and is also CKM suppressed. This renders the lifetime of the lightest neutralino so long that it appears only as missing energy in the detector.

Experimental sensitivity estimation
The proposed search for B + → pχ 0 1 is experimentally very similar to those of of B + → K + νν and B + → π + νν, previously performed at the B factories. In all these cases, the signal involves a B decay to a single charged particle and unobserved, "missing" particles. We describe the basic analysis techniques in Sec. 3.1 and use previous searches for B + → K + νν to estimate the sensitivity for B + → pχ 0 1 in Sec. 3.2. In Sec. 3.3 we discuss the relative sensitivities of the modes B 0 → Λ 0χ0 1 , B + → Σ +χ0 1 , and B 0 → Σ 0χ0 1 . The sensitivity of the more promising modes involving charmed baryons, In our sensitivity estimates we assume that the search is performed with the full Belle II data set, which is projected to have an integrated luminosity of 50 ab −1 , corresponding to the production of about N BB = 55×10 9 BB pairs.

Analysis technique
Searches for B + → K + νν and B + → π + νν have been published by the CLEO [86], BABAR [87][88][89], Belle [90][91][92], and Belle II [93] collaborations. Using e + e − → B + B − data collected at the Υ(4S) resonance, these searches were performed by reconstructing, or "tagging", the decays of one of the B mesons, referred to as the tag B. A single kaon is the only visible particle associated with the signal B candidate, which is assumed to undergo the signal decay of interest. Tagging is performed via one of three methods: hadronic, semileptonic, or inclusive. Hadronic tagging has the highest signal purity and lowest efficiency, and the opposite is true for inclusive tagging. In what follows we briefly describe the three tagging methods for use in the decay B + → pχ 0 1 . In hadronic tagging, one attempts to reconstruct the tag B in up to thousands of decay modes involving the Cabibbo-favored quark-level processes b → cūd and b → ccs. Background is suppressed with selections based on the variables ∆E = E B − √ s/2 and where E B and p B are the measured energy and momentum of the tag B in the center-of-mass (CM) frame of the e + e − collision, and √ s is the CM energy. Signal events peak at ∆E = 0 and M bc = m B , while background events are more broadly distributed. Hadronic tagging provides also the squared invariant mass of the unobserved neutralino candidate, the so-called missing mass Here p ee is the known 4-momentum of the beam particles [94], and p tag and p p are the measured 4-momenta of the tag B and of the proton, respectively. The presence of a signal would appear as a peak in the m miss distribution centered at m miss = mχ0 1 . In semileptonic tagging, the tag B is reconstructed in B − → D ( * ) −ν decays, where is an electron or muon. Because of the unobserved neutrino, the values of ∆E and M bc are not physically meaningful. Rather, one applies the constraint where E D ( * ) , p D ( * ) , and m D ( * ) are the CM-frame energy, momentum, and invariant mass of the D ( * ) − system. For signal events reconstructed without measurement errors, cos θ B−D ( * ) is the cosine of the angle between the true momenta of the tag B and the D ( * ) system. Therefore, signal events are distributed mostly in the range −1 < cos θ B−D ( * ) < 1, with tails arising from the finite measurement resolution and the spread in √ s, while background events reach greater negative values. The unobserved neutrino makes it impossible to determine p tag . Nonetheless, the neutralino mass can be calculated with the assumption that the two B mesons are at rest in the CM frame, i.e., p tag = 0 in Eq. (3.1). The resulting m miss signal peak is significantly broader than for hadronic tagging. Inclusive tagging is similar to semileptonic tagging in that the tag B decay products are not all observed. This may arise from particles that are outside the angular or momentum acceptance of the detector, poor resolution and identification capability for K L mesons and neutrons, particles with overlapping detector signatures, accelerator background, electronic noise, and limitations of the detector or the reconstruction algorithms. In Ref. [93], the variable used to discriminate between signal and background was the output of a multivariate algorithm, which was trained with simulated events. Previous uses of inclusive tagging, beginning with Ref. [95], used kinematic variables such as M bc for background discrimination.
In all tagging methods, one would require that the event contain only one chargedparticle track that is not associated with the tag B. For B + → pχ 0 1 , this track should be identified as a proton using the detector's hadron-identification capabilities. In addition, the energy deposition E extra in calorimeter clusters that are not associated with the proton or the tag-B decay products must be small in order to suppress background from neutrons and K 0 L mesons. As in B + → K + νν searches, this requirement can be applied directly on E extra , or by including this variable as input to a multivariate algorithm.
In both B + → K + νν and other studies of rare B decays with final-state neutrinos [96] the three methods have similar sensitivities for a given data set. Therefore, all can be used for the search of B + → pχ 0 1 , yielding similar sensitivities. In what follows, we use hadronic tagging for our sensitivity estimation.
3.2 Background and sensitivity estimate for Br(B + → pχ 0 1 ) We estimate the m miss -dependent background in the hadronic-tagging analysis from Fig. 8 of Ref. [89], which shows the m miss distribution in the BABAR B + → K + νν search, performed with a data sample of 0.471 × 10 9 BB pairs. The background estimate is 0.2 events per 100-MeV-wide bin for 1 < m miss < 2.5 GeV, thereby rising linearly to 3.5 events per bin at m miss = 4.2 GeV. We scale the background estimate to the Belle II data set by multiplying it by the ratio R L = N BB (Belle II)/N BB (BABAR) = 55 × 10 9 /0.471 × 10 9 of the numbers of e + e − → BB events in the final Belle II data set and in Ref. [89], respectively.
The estimated background is reduced relative to that in B + → K + νν by the fact that the proton production rate in B + decays is about R p/K ≈ 1/16 of K + production. Specifically, Br(B + → K + + anything) = (66 ± 5)% [97]. While this was measured by ARGUS [98] and CLEO [99] for a nearly equal admixture of B + and B 0 , it is not expected to depend significantly on the isospin state of the B meson. Therefore, we take this branching fraction to apply to the B + . By contrast, Br(B →p/p+anything) = (8.0±0.4)% [97], which is about 8 times smaller. Furthermore, correlating the charge of the proton with the charge of the tag B will reduce the background by a factor of two, given that such a correlation was not performed in the Br(B →p/p + anything) measurement. Baryon production is expected to be similarly suppressed in the non-BB background, specifically e + e − → qq, where q = u, d, s, c is a light quark. Furthermore, this background is reportedly smaller than the BB background in the B + → K + νν searches [48,92,93]. Thus, multiplication by R p/K gives the expected background yield N 100 b for the B + → pχ 0 1 search at Belle II in each 100-MeV-wide m miss bin. From this, we calculate the number of background events in a m miss bin with a width of 2-standard-deviation, The m miss resolution σ(m miss ) is dominated by the resolution on the transverse momentum p T of the proton for most of the m miss range. For the Belle experiment, this has been measured to be [94] σ(p T ) where β is the charged-particle velocity, which we calculate on average assuming a uniform angular distribution for the protons. We ignore the fact that the p T resolution is expected to be somewhat better for Belle II, which has a larger drift chamber. A smaller contribution to σ(m miss ) arises from the spread of the collider center-of-mass energy, which we take to be 5 MeV. The value of σ(m miss ) as a function of mχ0 1 is shown in Fig. 5. . The missing-mass resolution σ(m miss ) (left) and the expected Belle II 90% confidencelevel upper limit on the branching fraction Br(B + → pχ 0 1 ) as functions of the neutralino mass mχ0 1 . The upper limit decreases with the decreasing value of σ(m miss ) but increases for mχ0 1 > 2.5 GeV due to the increasing background. The kink at 2.5 GeV results from the simple parameterization of the background governed by the limited data available in Ref. [89].
The 90% confidence-level (CL) upper limit on the B + → pχ 0 1 branching fraction at Belle II, assuming the absence of a signal, is calculated from where ≈ 95 × 10 −5 is the total reconstruction efficiency [89]. 4 We check this branchingfraction sensitivity estimate against an inclusive-tagging study of B + → K + a, where a is an axion-like particle that is long lived enough to always escape the detector [48]. Fig. 10 of Ref. [48] gives the 90% CL limit Br(B + → K + a) < 10 −7 . Scaling this by R p/K yields Br(B + → pχ 0 1 ) < 2.5 × 10 −8 , in agreement with our hadronic-tagging-based estimate.

Sensitivity estimate for Br(B
1 ) The Belle collaboration has published a search for B 0 → Λ 0 ψ D , where ψ D is an invisible dark-matter particle [100], obtaining limits on the branching fraction in 500 MeV-wide steps of m miss . A preliminary result from BABAR [60] shows tighter limits in the range 1 < m miss < 4.2 GeV. In Sec. 4 we reinterpret the BABAR search in terms of B 0 → Λ 0χ0 1 and extract limits on λ 123 .
Considering the B 0 → Σ 0χ0 1 signal, we note that the dominant decay mode of the Σ 0 is Σ 0 → Λ 0 γ. Since the Σ 0 − Λ 0 mass difference is only 77 MeV, the photon is soft, so its reconstruction incurs lower efficiency and higher background than for B 0 → Λ 0χ0 1 reconstruction. The efficiency loss may be avoided by foregoing the photon reconstruction altogether. However, this results in an additional contribution of order the photon energy to the m miss resolution. Furthermore, while the production rate of Σ baryons in B-meson decays has not been measured, one can assume that it is similar to that of Λ 0 . As a result, background from true Σ baryons is not smaller than that arising from true Λ barons. Overall, although B 0 → Σ 0χ0 1 appears theoretically more promising than B 0 → Λ 0χ0 1 in Fig. 2, these experimental difficulties mean that it does not offer an advantage over B 0 → Λ 0χ0 1 . In the search for B + → Σ +χ0 1 , one must reconstruct the Σ + in the final state pπ 0 , which has a branching fraction of 51.6% [97]. About 48.3% of the Σ + decays are to nπ + . However, since the neutron interacts only hadronically, the reconstruction of this mode is much less practical. The efficiency for reconstruction of the soft π 0 is in the low tens of percent [101]. This approximately offsets the advantage of the B + → Σ +χ0 1 mode seen in Fig. 2. Hence we focus on these only. Since experimental information involving similar modes is lacking, our sensitivity estimates in this case are crude.
The most favorable final state for Λ + c decay is pK − π + , with a branching fraction of about 6.3% [97]. We take the reconstruction efficiency for this mode to be 70% the efficiency of B + → pχ 0 1 , due to the additional tracks and particle-identification requirements. To estimate the background, we note that a Λ + c orΛ − c is produced in 3.6% of Bmeson decays [97]. This arises mostly from the Cabibbo-favored quark-level transition b →cud, which leads toΛ − c production from a B 0 or B + decay. By contrast, our signal involves Λ + c production, which can proceed via the Cabibbo-favored decayb →ccs. In this case, the decays with the lightest final-state hadron combination are B + → Λ + cp −D+ s and B 0 → Λ + cn D − s for which the sum of hadron masses is 5.193 GeV and 5.194 GeV, respectively. Since the B + and B 0 masses are only 5.279 and 5.280 GeV, respectively, these decays are highly phase-space suppressed. The other alternative for Λ + c production in B decays isb →ūcs, which is suppressed by V ud /V cb ≈ 0.08 [102]. Thus, we expect that Br(B → Λ + c + anything) 0.4% which is 20 times less than Br(B + → p + anything). The background yield is further reduced by Br(Λ + c → pK − π + ) and the 70% efficiency hit, thus being 450 times smaller than in the search for B + → pχ 0 1 . However, the dominant source of background is combinatorial, i.e., random combinations of pK − π + that happen to have invariant mass within the Λ + c signal peak. A rough estimate of the level of this background can be obtained from published studies of 3-track B-meson decays where tag-B reconstruction was performed. We use the study of B + → ρ 0 + ν conducted by Belle [103], where Fig. 2 shows the background m 2 miss distribution for a data sample of 711 fb −1 . The background is highest at m 2 miss ≈ 2 GeV 2 , approximately 300 events per 0.1 GeV 2 -wide bin. This corresponds to a ∼ 0.35 GeV-wide m miss bin around mχ0 1 ≈ 1.4 GeV. For simplicity, we also take σ(m miss ) ≈ 0.35 GeV at mχ0 1 ≈ 1.4 GeV. This estimate is based on Fig. 5 above, but is conservative, since the resolution for three soft tracks is better than for the single hard proton used in Fig. 5. Thus, the initial background estimate per typical σ(m miss )-wide bin is 300 events for 711 fb −1 , which is 21000 per 50 ab −1 . This is then modified by the following factors.
First, the invariant mass of the π + π − + system in Fig. 2 of Ref. [103] can take any value up to m B , while that of the pK − π + in our search must equal the known Λ + c mass up to the Λ + c resolution, which is about 5 MeV [104]. This leads to a background reduction of order R mass ≈ 5 MeV/5 GeV = 10 −3 . Second, the area of the π + π − + Dalitz plot in B + → ρ 0 + ν can take values up to order 0.5 m 2 This increases the background estimate by R DP ≈ 6. Lastly, we multiply by the ratio R p/ ≈ 4%/21% of the proton and lepton abundances in B decays and by the ratio R K/π ≈ 13%/180% of the K − and π − abundances [97]. Therefore, the final background estimate per σ(m miss ) region of missing mass is 1.7 events at Belle II. We stress that this is to be taken only as an order of magnitude. Nonetheless, it is similar to or somewhat larger than the background estimated for B + → pχ 0 1 , which ranges between 0.25 and 1.4 events per σ(m miss ) bin.

Results for the RPV parameters
We proceed to calculate the sensitivity to the trilinear RPV couplings from the experimental sensitivity to the decay branching fractions of B mesons into a charged baryon and a light neutralino. As stated above, we take only one RPV coupling to be non-vanishing at a time, either λ 113 or λ 123 , and assume degenerate squark masses and no squark mixing. In Fig. 6 we show the limits on λ 123 /m 2 q vs. mχ0 1 that we extract from the BABAR search for B 0 → Λ 0 ψ D [60]. These limits are around λ 123 /m 2 q < 1 × 10 −6 GeV −2 , with fluctuations that reflect the BABAR results. The BABAR search is effectively backgroundfree in mχ0 1 regions in which the m miss difference between background events is significantly larger than the experimental m miss resolution. With more than 100 times more data, the Belle II search will probably not have a background-free region, unless further analysis improvements are implemented. Therefore, the Belle II sensitivity to λ 123 is naively expected to be better than that of BABAR by about 1/ √ R L for most regions of mχ0 1 . Related to this, as a result of the nonvanishing background, the Belle II sensitivity is expected to become poorer as the m miss resolution decreases for low values of mχ0 1 .   Figure 7.
The expected upper bounds on λ 113 vs. mq at Belle II for different values of the neutralino mass mχ0 1 , extracted from the proposed search for B + → pχ 0 1 assuming the sensitivities given in Sec. 3.2. We also plot current collider limits on the scalar squark masses under the hypotheses of eight mass-degenerate light-flavor squarks (mq 1850 GeV) and only one light-flavor squark state (mq 1300 GeV) [106].
single squark belonging to the first or second generation. These limits were obtained by the ATLAS collaboration from a search for pp →qq,q → qχ 0 1 in a signature of jets plus missing transverse momentum in a data sample with an integrated luminosity of 139 fb −1 [106]. The squark mass limits assume Br(q → qχ 0 1 ) = 1. In our theoretical scenario this is a good approximation only for small values of λ 113 , for which the competing decay channelq → qq is suppressed. As λ 113 increases, the ATLAS mq limits weaken relative to the values shown in Fig. 7. As mentioned in Sec. 1, current bounds on λ 113 and λ 123 arising from di-nucleon and n −n oscillation are much weaker than bounds shown in Fig. 7 and are hence not included in the figure. Fig. 8 shows our estimated Belle II sensitivity in terms of bounds on λ 113 /m 2 q vs. the neutralino mass. The bound is between 1.5 and 2 × 10 −8 GeV −2 for most values of mχ0 1 .

Discussion and Conclusions
A GeV-scale, Bino-like lightest neutralinoχ 0 1 is allowed by current bounds, provided it decays, e.g., via R-parity-violating (RPV) interactions, and is heavier than the proton, so as not to mediate rapid proton decay. In this work we consider such a lightχ 0 1 in the minimal realization of RPV supersymmetry. We propose to probe this scenario with a search for B meson decays intoχ 0 1 plus a baryon, specifically p, Λ, Σ + , or Σ 0 . The decays are mediated by the RPV operators λ ijkŪ iDjDk . We study the case where only λ 113 or λ 123 is nonzero in the absence of squark mixing and with the simplifying assumption of mass-degenerate squarks. We compute the corresponding B-meson decay rates in the framework of an effective phenomenological BNV Lagrangian. We note that the lightest neutralino in our benchmark scenarios is so long-lived that it appears as missing energy at the detector level. Experimentally, the presence of theχ 0 1 and its mass can be determined from the missing mass at an e + e − B factory, namely, BABAR, Belle, or Belle II. We interpret a BABAR search for B 0 decay into a Λ baryon plus missing energy [60] to obtain current limits on λ 123 /m 2 q around 1 × 10 −6 /GeV 2 , where mq is the mediating squark mass. Relative to this search, we discuss the sensitivities of the modes B + → Σ +χ0 1 and B 0 → Σ 0χ0 1 , as well as subtleties of the extrapolation to the final integrated luminosity of Belle II. Furthermore, we estimate the background level in a future analysis of B + → pχ 0 1 and the resulting sensitivity to the branching fraction of this decay. From this we obtain the sensitivity of Belle II in terms of the upper limit on λ 113 /m 2 q , which is between 1.5 and 2×10 −8 GeV −2 for most kinematically allowed values of mχ0 1 . For the currently allowed squark mass of about 1850 GeV (1300 GeV) in the case of eight (one) degenerate squark states, this corresponds to bounds on λ 113 as low as 0.05 (0.025). The bounds are relevant also for multi-TeV squarks beyond the reach of direct detection at LHC.
We estimate the sensitivity to λ 213 to be weaker than that of λ 113 by a factor of 15 − 35. The sensitivity to λ 223 is anticipated to be weaker still, by about 2.5. Due to lack of sufficient experimental information, these estimates are to be taken as order of magnitude only. Nonetheless, they demonstrate that the proposed searches for B + → Λ + cχ 0 1 and B + → Ξ + cχ 0 1 can provide higher sensitivities than the bounds obtained from dinuclen decays for the scenario under study here.
Lastly, we note that similar studies can be performed with other processes, exploiting the capabilities of other experiments. At LHCb, one can use the decay B + → pπ + π −χ0 1 , with the B + produced in the decay B * 0 s2 → B + K − , which has already been used to study semileptonic B decays [107]. Owing to the unobserved neutralino, the 4-momenta of thẽ χ 0 1 , B + , and B * 0 s2 are unknown. However, they can be determined up to a 2-fold ambiguity from the 12 constraints of the signal decay chain: 4-momentum conservation in the B * 0 s2 and B + decays, the known B * 0 s2 and B + masses, and the flight direction of the B + determined from the position of the pπ + π − vertex. Solution of the constraint equations results in two values of the neutralino mass, m ± , similarly to Refs. [30,47]. Signal events peak more significantly than background in the m ± distribution, except for expected peaks around masses of (mostly charmed) baryons that decay to only neutral particles. The feasibility of this approach depends, among other factors, on the m ± measurement resolution. The decay chains 1 may be used in a similar way. We note that at e + e − B factories, B * + s2 production is not feasible, and final staes with additional pions are probably somewhat disadvantageous, having lower efficiency and potentially higher combinatorial background than single-baryon final states. One may also probe λ 212 in the decays D + → Σ −χ0 1 and D 0 →Λ 0χ0 1 for a smaller range of the neutralino mass. Specifically, the D can be produced in ψ(3770) → DD at BESIII, with full reconstruction of theD in a manner identical to the Belle II analysis we propose here.
define the matrix elements of three-quark operators between nucleon and vacuum (they are essential ingredient of our calculation of the transition form factors): and X, Y = L, R. In our calculations we deal only with O LL q 1 q 2 q 3 operators. Therefore, we need only β couplings and form factors W LL 0,1 (q 2 ). In addition, for completeness, we also discuss and show results for the case of the O LR q 1 q 2 q 3 operators, i.e. for α couplings and W LR 0,1 (q 2 ) form factors. The nucleon couplings α and β have been calculated using various QCD-motivated approaches and lattice QCD (see, e.g., detailed compilation of the predictions in Table 5 of Ref. [67] and recent lattice results in Ref. [69]). One can see that results of QCD models and lattice QCD for the α coupling range from −0.003 GeV 3 to −0.03 GeV 3 and from −0.006 GeV 3 to −0.03 GeV 3 , respectively. In our analysis, we take recent lattice result α = −0.01257 −0.0126 GeV 3 of the USQCD Collaboration [69] (continuum extrapolation) as central value and take into account its variation as: α = −0.0126 +0.0174 −0.0096 GeV 3 . In case of the β coupling mainly lattice QCD results are available, which are distributed from 0.01 GeV 3 to 0.0127 GeV 3 . Therefore, for the β coupling we will use the following predictions with lattice QCD approaches, of the central value from a recent paper of the USQCD Collaboration [69] and take into account error occurring in other lattices calculations [67,69]: β = 0.0108 ± 0.007 GeV 3 .
For the case of the transition B + → Σ +χ0 1 we need the Σ + coupling β Σ + defining the matrix element Σ + |O LL uus |0 (for completeness we also calculate the coupling α Σ + defining the matrix element Σ + |O LR uus |0 ): The couplings α Σ + and β Σ + can be fixed using results for the so-called vector λ V and tensor λ T couplings estimated in Ref. [108] using QCD sum rules: are the vector V and tensor T three-quark currents (operators). Also we will need for further calculations the pseudoscalar P and scalar S three-quark operators: (q 1 q 2 q 3 ) P = ε a 1 a 2 a 3 q a 1 1 (q a 2 2 Cγ 5 q a 3 3 ) , (q 1 q 2 q 3 ) S = ε a 1 a 2 a 3 γ 5 q a 1 1 (q a 2 2 Cq a 3 3 we can express the matrix elements of V and T currents through the matrix elements of the O RR uus operators and, therefore, relate then the pair of the couplings λ V Σ + , λ T Σ + with couplings α Σ + and β Σ + : Taking λ V,T Σ + couplings from Ref. [108] with the sign conventions as in Ref. [67] we get: For the transitions B 0 → Λ 0χ0 1 and B 0 → Σ 0χ0 1 we need the Λ 0 and Σ 0 couplings β Λ 0 and β Σ 0 defining the following matrix elements The β Λ 0 coupling was calculated in Ref. [108] using QCD sum rules: As for β Σ 0 we accept isospin-symmetry relation: β Σ 0 = β Σ + . As we stressed above, the form factors W LX 0 (q 2 ) and W LX 1 (q 2 ) defining matrix elements p|O LX |B + can be calculated using SU(5) extension of the SU(3) version of the phenomenological Lagrangian proposed and used in Refs. [70] and [65][66][67][68][69]. At the leading order form factors W LX 0 (q 2 ) and W LX 1 (q 2 ) are contributed by direct BNV couplings of proton and pseudoscalar mesons and pole contributions because of exchange of baryon resonances. In particular, using results for the BNV matrix elements p|O LX |K + and making replacements f K → f B , m Λ 0 → m Λ b , and m Σ → m Σ b , one obtains: Note that direct BNV couplings of proton/Σ + and B + meson occur on the partonic level only in the case of the subprosses induced by theb andũ squarks and can not occur in the case of thed squark for the proton mode ands for the Σ + mode. In particular, direct BNV coupling of proton and Σ + with B + means that the latter transits into two three-quark currents. One current corresponds to the proton current J P or Σ + hyperon current J Σ + and another one to the BNV operator J BNV . The BNV operator has a form of scalar diquark [q 1 q 2 ] 0 coupled to the third quark q 3 . We have three possibilities for the J BNV (for simplicity we drop color indices): for the Σ + mode. It means that the B + meson having spin J = 0 can decay into light baryons having scalar quark configuration. In our case, proton and Σ + hyperon could be formed as quark-scalar diquark bound state: We have no direct transitions induced by thed squark for the proton mode and by thes squark for the Σ + mode, because it requires production of the pair of scalar diquark contributing to the J BNV current and vector diquark (d{uu} 1 or s{uu} 1 ) contributing to the current of light baryon, which is suppressed by spin conservation. Now we turn to discussion of the allowed pole transitions induced by bottom baryon resonances. We have two types of bottom baryon resonances with quark content of the BNV operators: baryons Λ b and Ξ b with antisymmetric light quark configuration, whose three-quark currents could have a form of light spin-0 diquark [ud] 0 and [us] 0 , respectively, coupled to the b quark 26) or vector diquarks composed of light quark u, d, or s and bottom quark b: 27) and baryons Σ b = b{ud} 1 and Ξ b = b{us} 1 with symmetric light quark configuration, whose three-quark currents could have a form of light spin-1 diquark {ud} 1 and {us} 1 , respectively coupled to the b quark 28) or scalar diquarks composed of light quark u, d, or s and bottom quark b: By analogy, we consider, in addition, decay modes of the neutral B 0 mesons B 0 → Λ 0χ0