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JHEP02(2021)211 d , which 2 , Springer ¯ D rare 1 τ Q 3 January 14, 2021 L February 24, 2021 December 11, 2020 : : : Nicolás A. Neill, 0 312 λ a and 1 Accepted Received Published ¯ , D 1 Q 3 L 0 311 [email protected] λ Minakshi Nayak, , Published for SISSA by c https://doi.org/10.1007/JHEP02(2021)211 [email protected] , e,f Juan Carlos Helo, b minakshi@tauex..ac.il [email protected] produced at the Belle II experiment, excellent sensitivity to . 3 , , τ 2012.00438 The Authors. and Zeren Simon Wang Claudio O. Dib, c Phenomenology a

a We consider light neutralinos of mass about 1 GeV, produced from , [email protected] scenario, it can put limits up to two orders of magnitude stronger than the current 0 311 λ Asia Pacific Center forHyoja-dong, Theoretical Physics Nam-gu, (APCTP) Pohang — 790-784, Headquarters Korea SanE-mail: 31, [email protected] [email protected] Avenida Cisternas 1200, La Serena,Instituto Chile de Alta Investigación,Casilla Universidad de 7D, Tarapacá, Arica, Chile Department of Physics, NationalHsinchu Tsing Hua 300, University, Taiwan School of Physics andTel Astronomy, Aviv Tel 69978, Aviv University, Israel Departmento de Física andValparaíso CCTVal, 2340000, Universidad Chile Técnica Federico Santa María, Departamento de Física, Facultad de Ciencias, Universidad de La Serena, b c e d a f Open Access Article funded by SCOAP bounds. Keywords: ArXiv ePrint: induce both the production anda decay displaced of vertex, the we lightestMonte-Carlo require neutralino. simulations at For for the least both reconstruction signal twoexplore of and charged regions background events, in in and the the findthe parameter that final Belle space states. II competitive can We with perform other probes. In particular, for Abstract: decays at Belle II, inand the clean samples context of of R-parity-violatingsuch (RPV) light neutralinos supersymmetry. with With the large exoticon signatures two benchmark of scenarios displaced of vertices single is RPV expected. operators, We focus Abner Soffer Long-lived light neutralinos at Belle II Sourav Dey, JHEP02(2021)211 3 symmetry called R-parity is 2 Z 16 ] for reviews), one can still circumvent 6 7 – 5 ] in an elegant manner. In order to preserve 4 , – 1 – 3 5 ]. Such RPV-SUSY models are equally legitimate 11 9 , 8 ] remains one of the fore-runners for physics beyond the 2 , ]. This has led to increased interest in other BSM signatures. 1 14 13 – symmetry [ 3 10 B 1 15 In either the RPC or RPV scenarios, the LHC has so far not discovered SUSY parti- and in fact offerno a longer rich stable phenomenology and at can colliders. decay into In SM particular, . cles with of RPV, any the kind,squarks LSP but is and only placed [ TeV-scale lower bounds on the masses of the predicted the lightest supersymmetric (LSP)chromodynamics is (QCD) stable. and If quantum theand electrodynamics LSP (QED), sleptons is its neutral decays production under willSUSY in quantum with lead squarks R-parity-violation to (RPV) (seethe large refs. phenomenological transverse issue [ of missing protonthe energy decay by at imposing triality a LHC. different discrete symmetry However, e.g., in than the TeV scale.Geneva, Switzerland, SUSY have searches mostlyto at decay focused the promptly on Large into suchthe SM minimal heavy Collider particles supersymmetric particles, with (LHC) standard whichusually large at model assumed are transverse (MSSM), in CERN expected order momentum. a in to avoid Furthermore, in decay. R-parity conservation (RPC) implies that Supersymmetry (SUSY) [ (BSM). In particular,(SM) by particles, predicting whose contributions for toparticles, the the standard it Higgs model solves the self-energy hierarchytechnical cancel naturalness, problem SUSY those [ predicts of such new the heavy SM particles with masses not much higher 1 Introduction 6 Numerical results 7 Conclusions A decay constants and branching fractions 3 Neutralino production and decay 4 Event selection and background estimate 5 Sensitivity estimation method Contents 1 Introduction 2 Model basics and possible displaced-vertex signatures JHEP02(2021)211 – τ − τ 34 + τ → -factories, − B . e τ + ν -factories, and e B 10 10 × 6 . 4 -boson rare decays via the Z ], and a list of proposed ex- , roughly 50 times that of the 40 1 , − 39 introduces the RPV-SUSY model con- ] considered heavy neutral leptons as the 2 55 – 2 – in which we present the analytic formulas of 3 GeV. The projected total integrated luminosity of ]: 1) pair production in 58 . 44 – decays. ]. Here we focus on the ongoing Belle II experiment in ]. Note that in order to avoid overclosing the Universe [ = 10 41 τ between the Bino and Wino masses is lifted and the dark 45 s ], future lepton colliders [ 33 – 2 √ 38 ], for a variety of models has been investigated extensively (see lepton decays. 27 M τ 49 decays at Belle II, using as a benchmark model a heavy neutral ]). In particular, ref. [ W θ τ 57 2 – 50 tan 5 3 ]. Such light neutralinos are also consistent with both astrophysical and ]. In the literature, two main production mechanisms have been discussed ] at the intensity-frontier, where and beams are colliding 26 = – ] and Belle [ 26 ]. , 1 47 18 , 48 17 25 M – 46 leptons are copiously produced at colliders, including the LHC, 15 This paper is structured as follows. Section The LLP search potential of Belle II and the previous-generation τ The GeV-scale neutralinos are necessarily Bino-like in order to avoid the present In this work, we consider a long-lived, light neutralino LSP within the RPV-SUSY. ], light neutralinos should decay, e.g., in the context of the RPV-SUSY. Moreover, for pairs and the large missing momentum. These properties make Belle II one of the best -charm threshold colliders [ for instance refs. [ LLPs produced from lepton that mixes predominantly with the third-generation active sidered in this work, followed by section previous Belle experiment.events. This The resulting corresponds events toτ are a easily identifiable sample via of facilities the for back-to-back the production study of of the rare BABAR [ τ Japan [ at a center-of-mass energy Belle II, to be collected in the next few years, is 50 ab and studied for thetended LHC programs [ at thesmall LHC [ component, andmesons 2) via single an production RPV in coupling.neutralinos rare produced In decays this in of work, for charm the and first bottom time, we propose to consider light a range of valuessignificant of probability of the decaying RPV inside couplings, a(DV) detector. signature GeV-scale at neutralinos Such decays colliders, are lead which long-lived to is a and the displaced-vertex have signature a webounds explore [ here. relation constraint is dropped,massless the [ neutralino cancosmological be constraints as [ light as37 in the GeV scale or even tor”, where theneutral scalar//pseudoscalar/vector lepton/-like portal particle/dark predicts .refs. a [ dark For recent scalar/heavy reviews of LLP searches,Although see there are increasinglythis is stronger not lower the limits case on for the the lightest squark neutralino. and In fact, masses, if the GUT (grand-unified theory) ally defined as particlesin that travel the a SM, macroscopicit such distance is before LLPs decaying. perhaps are In nomodels. common, fact, surprise e.g., even Well that the studied LLPsneutral-naturalness , examples are models, include , predicted and gauge-mediated, , in portal RPV, and a physics and . large that split number connects SUSY of Hence, the models, classes SM of and BSM a “dark sec- One class of signatures of interest involves long-lived particles (LLPs), which are gener- JHEP02(2021)211 - ) 1 1 ¯ Z D M ∗− 1 (2.1) K Q 3 , L − . Finally, K 0 311 6 may ensue. ( , λ k 0 1 ¯ ]. It is worth u ¯ ˜ χ D s 0 1 j 58 ]. Providing the ˜ ¯ χ D i 38 , or ¯ U → , which are relevant 2 00 ijk Z 6= 0 ¯ D λ 1 1 2 Q 0 311 3 + λ L k ¯ D 312 0 j λ . Q , the decay operators are non-vanishing. More · 2 7 i k / L and ¯ Z D j 1 M 0 ijk Q ¯ λ D i 1 L ) in the case of + Q − k 3 0 ijk ρ ¯ L E λ – 3 – , j operator also gives rise to the neutralino decay. − L 311 0 ¯ · π λ D ] estimated the transition form factors for both the i L we explain the signature definition and background 38 LQ is lighter than 4 ijk for first-generation as well, within the expected 1 0 λ ˜ χ 1 2 V where we elaborate on the sensitivity estimate procedure decay produces a charged pseudoscalar (vector) meson f (namely, 5 + operators can lead to the decay of charm and bottom mesons ¯ − u = u d τ ¯ D H T is even smaller. · f refer to any of the three fermion generations. The first three sets of i . The same T LQ L -boson. If f , the only, mediating both the production and decay of a light neutralino. i  Z 6= 0 2 1 ¯ i, j, k = D 1 0 312 Q λ 3 RPV L W 0 312 λ decay. ), with content In this paper we consider the case of one single RPV operator, either In the RPV-MSSM, while the decay of the lightest neutralino can only proceed via one τ ∗ 1 M only or As shown in figure ( in the case of cannot rigorously apply suchit a is method reasonable to to purelydecay first-generation constants follow satisfy flavors, the they samelevel state approach of that and precision to determinedare by assume dominated other that by uncertainties. pseudoscalar theof mesons, tensor In the we and uncertainty our use in benchmark vector the scenarios, same which assumption because the impact into the lightest neutralino. Thisanalytic was expressions proposed of for the thestudied decay first the time widths in sensitivity of ref. of the [ mentioning mesons ATLAS that and and the the light authors neutralinos, fixed-target ofsecond the experiment ref. and authors SHiP [ third [ generations of using the heavy quark formulation. While they or more RPV couplings,despite its the production Bino-like nature can of befor such mediated light coupling neutralinos, via with the different a small mechanisms.This Higgsino component scenario First, allows has beenfactories. studied Second, the in the context of both LHC experiments and future triality, can remove, e.g., the baryon-number-violatingFor terms, simplicity, thus we forbidding assume proton that decay. onlyconcretely, certain we focus onfor the operators where the indices terms are lepton-number-violating, andthe the terms last would set lead violates tosmall. a baryon too number. large As Allowing proton all mentioned decay rate, above, unless the imposing couplings certain were extremely discrete symmetries, such as the baryon We consider a model whereWith the MSSM the is R-parity appended violation, withing the R-parity terms: usual violation MSSM (RPV-MSSM). superpotential is extended with the follow- estimate, followed by section via a Monte-Carlo (MC) simulation.we summarize We present the the work and numeric results offer in an section outlook in section 2 Model basics and possible displaced-vertex signatures and neutralino decays. In section JHEP02(2021)211 ¯ s ]. S d K 60 (2.2) . The 0 1 τ ¯ ˜ , these ¯ d u χ ν d d 0 0 311 λ (s)quark. K s of data [ . As a result, 1 and τ , the DV does − coupling, or τL L 0 − ˜ ˜ τ ν π π 0 311 + < m λ π ′ 311 ′ 311 1 0 ˜ λ λ χ → m (s)quark to an S d 0 1 K − ˜ χ τ ) for the and the momentum vector of decays via the coupling ω 0 1 S search for high-mass resonances . ˜ χ K τ ] for the case in which the virtual ). Except for the 0 1 τ cτ , or 046 ˜ 0 d χ d ν ) τν . 59 1 and η S ¯ τ ¯ u d K , → + 0 η 0 is a pseudoscalar (vector) neutral meson , /m 0 Rk that decays as ) W S ˜ ρ d ∗ TeV ( K ]. The expected bound at the HL-LHC with R R 2 0 , ˜ ˜ p m d d 1 → 0 ( 61 K – 4 – M ′ 311 π λ decays, its mass satisfies TeV: 20 pp ] and the uncertainty in their theoretical predic- . ′ 311 1 τ 0 λ is a 62 decay position. Owing to the relatively short lifetime & < 2 )[ | τ R , where k ˜ M coupling (see figure q ) is calculated in ref. [ 0 1 − ∗ 0 31 m ( ˜ 2 τ χ λ k ]. Since it is only slightly tighter than the present bound of π, K 0 312 | M 0 31 λ can be easily obtained by changing a 59 = λ + 0 312 0 1 P τ τ ¯ λ ˜ ( ¯ u χ ν d ν (forming either a τ ¯ d d d d ) for the P ν 0 ∗ K → L L , ˜ ˜ u d 0 τ 311 ′ cm). Nonetheless, this signature can be used to detect a signal if the neutralino λ K 7 . ′ 311 2 . The parton-level Feynman diagrams for both λ , its decay position can be safely taken to be the interaction point (IP) of the does not point back to the ), we use only the present bound for comparison to our results below. is also given in Ref. [ τ ≈ S 1 2.2 S − An additional bound comes from the current uncertainties in the measured branching The current bound on K K 0 1 − ˜ χ τ cτ A similar search was3 performed ab by CMS [ eq. ( fractions for is a squark with mass This limit was obtaineddecaying by to recasting a a tau lepton and a neutrino, performed at ATLAS with 36.1 fb flight distance is significantlythe larger than of the collider beams. (forming a mesons decay promptly into chargeddisplaced , decay enabling the position. identificationnot of When the identify neutralino the decay( position of the neutralino, because of the long lifetime of the diagrams for the coupling Since the neutralino is producedit from can only decaywith into quark content Figure 1 JHEP02(2021)211 and (3.4) (3.1) (3.2) (3.3) (2.3) (2.4) (2.5) (2.6) 2 g , . , , , W     couplings, θ ], we use the kR kR 01)% 004)% R R ˜ ˜ 0 ijk 2 d 2 d . . ˜ ˜ 63 ∗ ∗ d d tan λ 0 0 g g m m , , 2 4 4 2 ± ± 2 g event selection may √ − 2 + + 3 h.c. ) 0 1 ) 0 1 gauge coupling 10 ˜ 696 722 j χ − ˜ . . + jL jL χ ˜ u L L ˜ 2 d 2 u P × ˜ ˜ d  u P  = L g g m m P R 4 4 → = (0 = (0 k ˜ → d 011 2 SU(2) 2 d . ], µνρσ     µνρσ τ τ )( ( SM i i lifetime in ref. [ i 38 . ) EXP 0 ijk B ` 0 ijk ) , g = 0 τ ) τ − τ λ L λ − τ ) W τ P θ P ν = Kν = 0 νσ νσ Kν g e m. We ignore the fact that this bound → Kν g χ τ ( → ( 1 T,` ijk tan → µρ T,ν ijk µρ → B τ g g τ S,` ijk ( 2 fs. From the combined experimental and > σ (  2 τ B  2 G ) ( √ g , so that the ) 0 1 B j j ˜ 3 τ χ 5) ,G ,G B d . . u 0 − m cτ ) +   )   , σ ρσ – 5 – j ρσ 0 1 2 σ ± d = iL ˜ σ L kR R χ − ˜ k 2 ` ˜ ˜ ˜ ` , L ∗ k 2 d d 3 L , ) d g P . ˜ g d P d 10 m τ , m k )( g )( i d i 1 2 × → − ν are proportional to the corresponding ` P ν = only for )( τ i − µν ( : µν L 05)% 027)% 6 kR R ν . . → ˜ ˜ 057 ˜ u σ ∗ = (290 2 d d approaches σ B S,T ijk . are given by the electroweak L W 0 0 0 jL L g 0 ˜ τ ˜ θ 2 d d τ 0 1 m P e ˜ G e χ ( f χ ˜ τ ± ± 0 g χ ( ( g 1 2 m = 0 B e , g χ m ( ) 1 2 82 90 T,ν ijk T,` ijk + W τ . . θ G G S,ν ijk − πν jL L G ˜ ˜ 2 u + + → u iL L τ tan g ˜ 2 ν ˜ ν ( m = (10 = (10 = g 2 B 2 m 1 2 σ g L √ SM     ) EXP τ = ) 0 ijk τ 0 ijk L λ πν λ ˜ ν πν candidate sample, depending on the analysis criteria. Therefore, we show this g = = → ], 0 1 = → ˜ χ τ 63 S,` ijk S,ν ijk ( L τ ˜ P ` ( B G G g B → and the coupling constants the electroweak mixing angle where the effective couplings We extract the effective interaction Lagrangian from ref. [ bound with our results inbecomes section inaccurate when reject the resulting soft pseudoscalar. 3 Neutralino production and decay This limit is relevant onlydaughter for particles large are neutralino lifetimes. inτ For principle short visible lifetimes, in the the neutralino detector and might be rejected from the theoretical uncertainties in we extract the following 95% confidence level bounds on where for the SMcurrent values, PDG quoted average value as a function of the tions [ JHEP02(2021)211 0 311 ], λ , are (3.5) (3.6) (3.7) (3.8) 0 38 η meson, , . For the events as i − 1 ) a 0 1 , and − A ˜ ) 2 χ τ 0 1 η ˜ 2 χ m + . ˜ τ 2 f m + , ∗ i 2 + → ) /m 0 2 M 0 1 ˜ − 2 M λ 2 χ e m ( m m + ∗ 2 , e ) − + 2 M 2 2 τ 2 τ m 2 M m m ( m − 2 mesons, namely, 2 | + | 0 1 ∗ 1 − 1 ˜ 2 4 χ ∗ 1 T 0 1 S are given in appendix M M ˜ m 2 M 2 χ M f f | ∗ 2 | 1 2 m h 2 m | T | ( M ( 2 ∗ 2 f 1 | jk jk | ∗ 2 T,` 3 S,` 3 2 , 2 M 1 T M S G M G | m | S f M f | ) | ) . Note that the charge-conjugate channel for f 2 0 1 − 2 0 1 | | ˜ 2 χ ˜ 2 χ τ 2 yz τ ) 2 – 6 – jk jk 0 1 , m T,ν 3 , m S,ν 3 ˜ 2 χ − ∗ 1 1 3 τ , this mode does not contribute to visible DVs in G G 3 τ η | m | 2 M 2 M xz 2 πm 0) − 0) πm , , m , m , − 2 τ ∗ 2 2 2 < m 2 τ 2 τ is ]. Therefore, the background suppression and estimation 128 0 1 m 2 M 2 M 1 m m xy ˜ ∗ 0 1 3 χ 1 a ( ( 2 55 ˜ 2( 3 χ 2 2 h , m , m m / / M − 0 1 0 1 1 1 πm 2 ˜ ˜ × 2 χ 2 χ πm λ λ − z we show the rates for the different neutralino production and 2 τ m m 128 + 3 ( ( 2 2 2 m ) = ) = / / y 1 0 1 0 1 1 ˜ ˜ χ χ + λ λ and ∗ 1 1 2 ]. This selection leaves the backgrounds from 2 x M M ) = ) = ]. In particular, the selection of signal events will begin with the typical 64 . Since τ τ we do not consider neutralino production associated with the , → selection criteria. These include a single track that recoils against the rest of → ) = ν ν − 1 55 ∗ 2 τ 2 a 55 τ − 0 1 , enabling a study of the sensitivity in terms of the ratio lists the different decay modes that we consider for the cases of non-zero τ ˜ M or a M ˜ f Γ( χ Γ( 6= 0 + 1 1 τ x, y, z m → → ( → M . In figures decays is implied. For simplicity, we assume degenerate sfermion masses, henceforth 0 1 λ 0 311 0 1 → 0 1 λ − χ χ Subsequently, the main LLP requirement involves selection of two identified charged Table 0 312 ˜ χ τ − λ Γ(˜ e Γ(˜ + the event in the oppositemomentum hemisphere, [ accompanied by large missingthe energy dominant and source, transverse efficiently rejecting all other sources ofpions background. that originate from a high-quality vertex that is significantly displaced from the The search proposed here is(HNL) experimentally very search similar proposed to in that ref.methods, of [ the as heavy neutral well lepton those as of conclusions ref. about [ e the major sources of background, are similar to this scenario and henceneutralino is branching fractions irrelevant into in finalneeded this states for estimate. with reconstruction at of Also the least shown DV two and charged in pions, its the position. which4 figures are are the Event selection and background estimate decay modes, asIt well is as evident the thatby decay neutralino far branching decays dominant, ratios into and ofcase pseudoscalar hence the constitute neutralino the best toi.e. search visible strategies. modes. For this reason, in the where the denoted or where the scalar anddecays tensor of decay the neutralino constants in our scenario, we reproduce the expressions given in ref. [ meson From the effective Lagrangian, the neutralino production rate in association with a scalar JHEP02(2021)211 . ) τ τ ν ν ) 2 ∗ ( 2 M ∓ , M − K 0 → ∗ π ∗± ± → 0 1 + π K K π 0 1 χ 0 312 , ˜ λ 0 χ ± → mesons. Γ(˜ → , 0 K K ) ∗ Scenario 2 2 S ∗ ( 1 K K M M 0 1 ˜ χ , ) ) ) 0 → γ γ π ) τ − − − − π π π π + + + + π π π π , ) → → 0 → → 0 π η decays: η π ρ η ( ( − ( ( − 0 τ 0 − π γ τ π π π ν π 0 + scenario. (Top left) Branching fractions 0 0 + η + ) π → π π π ∗ , π 0 ( 2 η η ± → → 6= 0 → , → ρ , M → 0 η ) 0 ω , 0 311 0 η η 0 η , ( , η λ ± → , π 0 311 , γ ρ − ) , π ) 0 1 − λ ) − , π 0 Scenario 1 ˜ χ 0 π π − γγ π + , π + π 3 + π ) π ∗ – 7 – π + → ( 1 π → → η → ( → M 0 η ( η → − η ω 0 1 ( − π ˜ χ ω γ π + ( + γ π → → π τ 0 → → η 0 0 → η η 0 η ) 1 ) M 2 M decays into a meson and a neutralino. (Top right) Decay rates τ decay ( 0 1 production ( ˜ χ for 0 1 particles ˜ ) χ 0 1 ˜ χ decays with charged 1 . Neutralino production and decay in the . Benchmark scenarios for neutralinos produced from 2 (production and decay) M 0 M Mesons in λ → Mesons in τ ( Figure 2 B for neutralino decays intodecays a into final meson states and with a charged neutrino. particles, accounting for (Bottom) the Branching decays fractions of for the neutralino Table 1 JHEP02(2021)211 . , ) ]. 0 τ τ ν ν π 67 and 2 L − – 0 p M π K π 64 0 mesons p + π → π 0 − 0 1 π π χ to mesons. Γ(˜ L → decays [ 2 , for which the K 0 d decay candidate − τ M π τ 0 − π π − + , a conceivable source π 0 π + π π − ]. Therefore, even if one → π 64 L → + and 12.5%, respectively. The π K ] is that the neutralino decay η 3 in detail. We then qualitatively − 55 0 10 π scenario. (Top left) Branching fractions − × π 2 -factories [ 6= 0 . + 4 B . A fake π 2 0 312 , λ → , followed by 3 , which is rarely produced in η 0 − = 1 – 8 – , η τ i 10 or ην × η 7 . → 1 decay that occurs far from the interaction point will be 0 1 , with 0 i d ˜ , followed by the decay of the long-lived χ η γ τ , ν 0 1 or L ˜ χ , respectively. The that are produced in the η − K and d π − i is reconstructed in the final state p π is greatly suppressed, given that the masses of these mesons differ γ η → and η → − p τ − decays into a meson and a neutralino. (Top right) Decay rates τ τ for ) 0 1 ˜ χ mass resolution is less than 4 MeV at 1 . Neutralino production and decay in the 0 M decay faking an π We estimate that the largest source of background will arise from Naïvely, since the − → L decays are denoted π τ K ( 0 + d which has a branchingbranching fraction fraction of is 12.5%.with For the clarity, subscripts we denoteπ the prompt and displaced The branching fractions ofπ these two decays are accounts for the resolutiona degradation in the caseby of about a 50 MeV. displaced decay, the likelihood of Requiring the presence of a very effective at suppressing thefor background. the In signal what case followsextend we our estimate conclusions the to background the other signal modes. of background is for neutralino decays intodecays a into final meson states and with a charged neutrino. particles, accounting for (Bottom) the Branching decays fractionsIP. of for the Furthermore, neutralino a keyproduces difference a narrow with hadron, namely, respect a to ref. [ Figure 3 B JHEP02(2021)211 τ cm, 80 and the candidate i 0 background ]. Similarly, m π < r < τ 62 decay position ν 10 L L ], we require the K K 0 p 64 π − or a fake π ] Monte-Carlo generator 0 p π . Each photon is required 68 → 2 d and away from the correct γ mass spectra are broad and candidate and the displaced i − 1 d decay position, which will be τ γ 2 0 m γ π L 1 K γ ], we require the − 55 π + π and either the leptons decays via this background decay ± ] has only limited ability to determine the π GeV and for τ 46 1 . – 9 – and the decay vertex. Following ref. [ = 1 2 d 0 1 γ − ˜ 1 χ p π γ m + π candidate formed from the where one of the η ], we further apply the constraints of the signal to − ], and thus leads to a more robust and conservative background candidate to be within 15 MeV of the known value [ 55 τ 0 64 center-of-mass frame. The distributions of these variables are shown + π ] or have a lab-frame energy smaller than 100 MeV. This requirement τ ], since our signal simulation lacks final-state radiation. The signal − , this allows us to compute the calculated neutralino mass e 46 candidate from two photons other than 2 55 . The two remaining photons would, if detected, indicate that this is → , + 0 j d cm. The background events have an acceptance of 15% relative to this e γ π − i p e decays [ = 1 γ + i 120 τ e for signal events with in the i 4 E pair is required to be within 15 MeV of the known value, corresponding to about < z < As detailed in ref. [ We further require that each of the additional photons in the event be unobservable We form a To estimate the level of this background, we use the EvtGen [ − distributions for signal, which peak at the generated mass, are missing the radiative distributions, which peak at half the center-of-mass energy, show tails that result from π i 40 i + m tails seen in ref.E [ the simulated initial-state radiation.those The background of distributions the are very signal: different from events accumulate at high values of determine the neutrino 4-momentum up to aindicated 2-fold by ambiguity. For each of theenergy two solutions, in figure events. The background plots contain 300 events, illustrating the full Belle II dataset. The with conservative requirements, namely, escapein the the detector calorimeter through [ theretains endcap 3.9% openings of thetotal background of events about selected up 300 background to events this in point. the As Belle a II whole, dataset. we expect a π 3 times the resolution.smooth, the lack Since of a the We detector find simulation that does these not requirements significantlyfiducial retain impact region. our 2.5% rough of estimate. the background events that are within the we take the directionthe of calorimeter, the assuming photon thatexperimentally it momentum known originated from from from its the the hit of the position onthe the invariant mass surface of of the to have a lab-frame energyselection of in at least 100 MeV.estimate. This requirement The is Belle tighterflight than II the direction crystal usual of calorimeter a [ photon, and we conservatively ignore this ability altogether. Thus, to produce chain. No detector simulation isto used. be Following in ref. [ the fiducial− volume defined by theselection, radial which and longitudinal is ranges motivatedsuppress by background the from long prompt tracks lifetime and of from the material-interaction background. neutralino and by the desire to formed from not a signal event.or However, otherwise they go may undetected, escapephoton typically via since background the that they endcap originates are openings from too in the soft the collider to calorimeter itself. be clearly identified above can then be reconstructed with the displaced JHEP02(2021)211 ) τ ρ ν L . 6 5 4 3 2 1 0 8 7 6 5 4 3 2 1 0 0 K π . This signal, 0 p 0 − ) π channel. η 2 π π − − τ + π π π background, ην + (GeV) → τ 1 energy (bottom π → ν → E (GeV/c τ S − L 1 0 1 → τ ˜ K m χ K GeV (left plots) and L − 1 . K π , and to some extent, = 1 → η 1 0 − ˜ χ we present results with the τ m 6 is available to fake the bgd bgd 0 0 L 0 L π K K 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 signal channel is 1 8 6 4 2 0 0

12 10

0.8 1.8 0.6 0.4 0.2 1.6 1.4 1.2

τ

2

2 momentum vector be inconsistent with ) (GeV/c m E

(GeV) ην 2 S ]. K → 55 0 1 signal events with ˜ χ , as well as the fact that the dominant decays 0 0 – 10 – 400 350 300 250 200 150 100 50 0 90 80 70 60 50 40 30 20 10 0 channel suffers from η π − τ cm. π ) ν decay vertices. The neutralino flight length must be 2 + S 10 S π K channel is much smaller than in the K > → τ -decay background that leads to the same final state as the (GeV) 0 1 → 1 ν τ ˜ η χ 0 E 0 1 (GeV/c η , and 1 ˜ cτ χ m ην − → τ . In this case, only the prompt 0 1 → ˜ ν χ 0 1 ∓ . The ˜ χ ` τ provide an additional intermediate hadron ( , ± background events (right plots). 0 1 ην π ˜ χ τ ργ ν − → → L π case, there is no 0 1 K L signal. Therefore, the background for this channel is much smaller than our and ˜ 0 p χ → channel only for 0 312 Signal Signal . Therefore, we conclude that the analysis is essentially background-free, with K π τ i − 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 λ − τ − 1 ν 8 0 6 4 2 0 π E τ ν

0 12 10

π

. The two solutions for the computed neutralino mass (top plots) and 0.8 1.8 0.6 0.4 0.2 1.6 1.4 1.2

∗

+ S 2

2

) (GeV/c m E

(GeV) → 2 K K ηπ − In the Background in the A related background channel in the τ → → → 1 0 1 0 0 ˜ ˜ estimate for which can be suppressed bythe requiring direction that between the the long enough for this cutχ to be effective. Therefore, in section This arises from theη higher mass offor the which a mass cut can be used for background rejection. χ followed by resulting in a factor ofbackground 3 is suppression relative further to suppressed thelepton. background to Overall, from the its sub-percent contribution is level expected by to rejecting be DVs smaller formed than by that a of values of less background than even in the case of ref. [ Figure 4 plots) for for JHEP02(2021)211 , ) ) . or ∗ ) ) ( 1 cm. 0 (5.3) (5.4) (5.5) (5.1) (5.2) case, π M 0 1 decays. ˜ 120 χ visibles 6= 0 S visibles visibles → K → τ 0 311 → → 0 λ < z < 0 ρ ω ( η ( ( , 40 ) ) ·B ·B − ·B ) ) ) ) 0 0 ρ η ωγ τ τ cm, ν ν visibles visibles , . → ) visibles . 80 → 0 3 → ∓ ) η → → 0 1 π − ( 0 1 0 χ → ], we use that reference to de- ± π S χ B ∗ (˜ (˜ 0 + K B K 55 ρ K B π ( , < r < ( ( , ) ) + ) , → γ ·B 0 ) ) + → ·B ·B 10 ) + 0 ]. Relying on particle-reconstruction − ) π ∗ ) ) S π S 0 − γ 55 ∗ K K + 0 π K ( ( ρ π visibles τ + K B B ν τ π visibles visibles visibles ν → → → 0 → → η η ) = ) = → → ( → η events. First, we consider the visible branching 0 1 – 11 – → ( ( ω B 0 0 1 χ ω ( B η − (˜ ( π χ ( ·B B τ ( (˜ B ) ) + + ], we take the efficiency to find any additional B ·B ) + ·B + 0 η ·B τ visibles visibles η ) 0 ) + ) π ) − η ω 70 π − 0 − , 0 → τ τ π ) = π π π → → ν ν π + τ 69 + + + 0 S ν π e π ∗ π → → → − K 0 K 0 1 0 1 ( → e → → ( → η χ χ 0 B 0 1 ( visibles (˜ (˜ B ω η η χ B ( ( ( B B (˜ , , B B B + → B + + -factories [ 0 1 B χ (˜ ) = 1 ) = ) = ) = ) = 0 ) = B signal using limit rejects most DVs that arise from material interaction and τ scenario, the ‘visible’ decay branching ratio of the neutralino is ν ) visibles visibles visibles visibles visibles visibles 10 ∗ ( 2 6= 0 → → → → → M → 0 r > 0 0 η 0 1 ω For a DV within the acceptance volume, the efficiency to reconstruct a signal event Next, we consider the signal acceptance and efficiency. Given the similarity between 0 312 η ( ρ π ( → ( χ λ ( ( B (˜ B B 0 1 B B B ˜ reconstruction, given the size of the Belle II trackingis systems. 12%, includingof the particle-identification impact criteria of onefficiencies reconstructing both at the the tracks two [ prompt tracks and application the search proposed herefine and the the event-selection one criteriareconstructed discussed if and in its their position ref. related is [ defined within efficiencies. an by acceptance the volume, In radial which principle,The we and take a longitudinal to requirements be DV effectively canThe be other requirements ensure a sufficient number of hits for adequate track and vertex A summary of the meson branching fractions is given in table with In where the visible decay branching ratios of the mesons are In what follows we estimateχ the sensitivity of thefraction Belle of II the experiment neutralino, forat defined the to least be two charged thewe pions, total have which branching fraction are into necessary final for states finding with the DV. In the 5 Sensitivity estimation method JHEP02(2021)211 , O i S z and K (5.7) (5.8) (5.9) (5.6) 0 1 (5.10) (5.11) ˜ χ events, τ , + τ det. momentum −  using a MC τ · S , we calculate k K → acc. 0 31 acc.  λ  + · e ) − are calculated as de- e ) 4 leptons undergo the sig- 10 simulated neutralino, ob- τ decays into a single track, visibles × τ th 5 i . , visibles . 1 I i → ) z z i 0 1 ∼ , are the longitudinal edges of the χ → /λ − with (˜ is the reconstruction efficiency dis- , O i ) 0 ) i acc. 0 1 of the | z MC | i  i ·B χ − i θ acc. N < (˜ θ , e ) θ  det. 0 1 ) MC B  i ˜ χ ∗ =1 − θ ( N i X 1 tan tan 2 c τ and the relevant coupling / mode is susceptible to background such as (1 / M i and I O 0 1 · 0 1 tan γ ˜ S χ ˜ MC χ z i R R ) z i 1 | | ) ] event generator. We use the Pythia module K – 12 – m N β /λ ∗ ( I → i 1 to produce 2 Z, Z, z ≡ ( ( = 72 τ − M , ( z i direction of the neutralino given the lifetime = 0 1 e cm for 2 λ ˜ χ . We estimate 71 z min min ·B M = 3 40 ) obtained from the Pythia simulation. acc. ≡ ≡ →  i is the branching fraction for γ I i i acc. O τ i cm are the inner and outer radii of the acceptance volume,  z z i ( z and , because of the significant proper flight distance of the β B S 85% 0 depends on the neutralino boost, lifetime, and travel direc- , exclusively, according to the relative branching ratios computed 1 prong K ≈ = 80 1 > 1 = ) i acc. → O  2 θ τ R denotes the acceptance, and ( M tan ·B coordinate at which the neutralino enters the acceptance volume, and acc. cm,  1 prong + . This background will be suppressed by requiring that the z τ τ − cm. As a result, the ν is the acceptance for the polar angle τ → S 7 = 10 cm for N . K τ I is the ( i acc. −  B R I i = 2 π are listed in table = 2 z -direction boost factor S The acceptance For each value of the neutralino mass While generally, the position of the DV is the decay position of the neutralino, this is ) z S → ∗ = 120 N ( 1 K − acceptance volume. The factor is the average flight distancethe in the where Z tained from Here is the distance traveled inside the acceptance volume: where simulation with the PythiaWeakSingleBoson:ffbar2ffbar(s:gm) 8.243 [ including simulation of initial-state radiation.nal All decays listed the in simulated table with the formulas given in section where M scribed above, cussed above. tion, as well as the geometry of the acceptance volume. We estimate does not point back to the IP. The efficiencythe of total this number requirement of is signal taken events to observed be in 90%. the experiment, of charged pions to be 85%. not the case when cτ τ photon that is part of the signal decay to be 70%, and the efficiency for an additional pair JHEP02(2021)211 , ˜ 2 f ˜ 2 f are /m ˜ case, /m GeV, 2 f k k 0 31 . Three 55 0 31 /m . λ 0 1 = 2 λ k ˜ 0 χ k 0 31 cτ vs. > λ 0 1 0 1 ˜ χ ˜ χ m and detector effi- m ) , taking 100% for the , requiring only that ) 0 1 0 1 ˜ χ ˜ χ ) cτ ∗ ( 1 visibles M shown for the Belle (light blue, , as a function of → GeV. The ratios 0 1 0 1 → ˜ χ 3 det. χ .  (˜ τ m 1 ( GeV, and in steps of 0.001 GeV in B B < vs. 28 . 0 1 ˜ 2 f 1 ˜ χ , for 49 points in total. 2 < signal events at Belle II with a data sam- − /m in steps of 0.01 GeV for 0 1 < m ˜ χ 0 311 0 1 ˜ λ χ . Curves show the parameter values that yield ob- = 3 GeV 28 4 . m S – 13 – 1 − < m 6= 0 N 5 10 . ) data sets. The orange dashed curves show the proper 0 311 0 )) for squark masses of 1 TeV and 5 TeV. 1 × λ − 8 2.2 ab GeV and and 50 (eq. ( (blue), 1.0 (orange), and 1.5 GeV (green). The red dot-dashed line GeV we take the finer steps of 0.001 GeV. In the 2 case, we scan R 499 − 6 ˜ . 2 q . 55 0 . 0 /m visibles and the detector efficiency , taking the branching fraction = 0 < k = 1 GeV ) < 0 1 ) 0 1 0 31 ). This constraint is relevant only for large lifetimes, and is drawn only for ˜ ∗ χ k 10 → ˜ 0 1 χ ( λ in a rectangular grid of points in the parameter space 1 , corresponding to the data already collected by the Belle experiment, ˜ , and use the efficiency estimates described above. Both types of limits χ − ” m 1 0 1 2.6 0 1 S M ˜ ˜ − χ χ 10 0 1 : model-dependent results on N , of the neutralino in intervals of 100 m. The red dashed horizontal lines denote < m ˜ m χ 0 1 ab < m for a background expectation of close to zero events, corresponding to 95% ˜ χ 1 , eq. ( in steps of 0.01 GeV for 1 49 → ) cτ . Right . In the to be 100%. These limits are shown as a function of τ − 0 1 0 547 τ ˜ 2 ν . χ ( , 0 ab − . Sensitivity reach for the case m. m B π det.  decay within the acceptance region. The model-dependent limits are given in terms 1 ) and Belle II (dark blue, 50 = 1 → 1 0 1 > We evaluate ˜ − χ k : model-independent limits on the branching fraction − 0 1 τ ˜ ab χ ( as a function of are obtained by requiring observationple of of confidence-level limits. Theluminosity model-dependent of limits are presented also for an integrated We proceed todependent discuss limits. the The numericalfraction model-independent results limits for areciency both quoted model-independent in andthe terms of model- theof branching the ratio between the RPV coupling and the universal squark mass squared, the ranges scanned between 6 Numerical results for while for we scan represents the current constraint thatB arises from the experimentalcτ and theoretical uncertainties in 1 decay length, the current limits on Figure 5 servation of 3 signal events,Left corresponding to 95% confidence-level limitsbranching for fraction negligible background. curves correspond to JHEP02(2021)211 , ) ) S ˜ 2 f cm X K )) is 2.2 + /m = 10 0 1 2.6 ˜ 2 χ m only. that we 0 311 > λ ˜ 1 M 2 f 0 1 → ˜ also shows χ (eq. ( is irrelevant & τ ) /m 5 cτ ( m. The curves 0 1 τ k ˜ B ν χ 1 cτ 0 31 − − m, in which this λ > π π and its theoretical 1 for 0 1 root of the ratio of τ ˜ χ → for details. Unlike in ν → τ < cτ th − − 5 cm. − 4 τ 0 1 π ( τ Kν ˜ χ B 20 → cτ – → − 10 τ τ channel only for of the neutralino in the shown ∼ S . 0 1 4 0 1 where K ˜ ) ˜ χ χ ˜ 2 f 0 1 cτ cτ ˜ χ /m m 0 311 . The model-independent limits are given λ scenario, treating separately the . See caption of figure ( and ˜ 2 f 6= 0 6= 0 cm, because of the high background at short flight 6= 0 – 14 – /m . This is because in the large decay length limit, ). We find that Belle II may probe 0 311 10 0 312 6 0 312 . λ λ 0 311 2 λ 2.6 > λ 0 1 , its comparison to our results depends on the assumed ∼ ˜ ˜ χ f 4 cτ / m 1 displays the model-dependent limits on 50 5 channel suffers from large background at short distances. We S K for the benchmark squark masses of 1 and 5 TeV. Since the limit of R ˜ 2 q /m cases. We overlap the limits derived from are shown only for we show the limits for 0 , with the best sensitivity obtained for 0 311 ∗ τ τ 5 gives the results for the . The upper bound derived from the uncertainty on λ ν ˜ K f m S 6 ) for sfermion masses of up to about 5 TeV. The right plot of figure m K = . Sensitivity reach for the case ) has a different power of ∗ 2.2 , the red dot-dashed curve that represents the constraint from 2 → 5 below 0 1 M 2.2 Figure The right panel of figure In figure ˜ χ 0 1 ˜ χ parameter space. and As mentioned above, the incorporate this by showing the limits expected for the value for here, as it correspondsbound does to not values apply. of Weof find eq. that ( the searchorange we propose dashed is isocurves more of sensitive than the the proper limit decay length values many orders of magnitudem smaller than the present experimental upper bound, for expect with the full Belleon and Belle the II ratio data sets.eq. Also ( shown is the upper bound of eq. ( in the left panel1.5 for GeV (green) three using neutralino the masscurrent full uncertainties values: in Belle 0.6 the GeV II measured (blue), branchingprediction data. fraction 1.0 (red for dot-dashed GeV Also lines), (orange), plotted eq. and ( is the limit extracted from the and twice the integrated luminosity ofthe BABAR. Belle We note limits that aretheir for weaker small integrated than values luminosities, of the Bellethe number II of limits signal by events about is proportional the to Figure 6 figure relevant also for thefor model-dependent limits shown indistances. the right panel for JHEP02(2021)211 , τ 0 6 → π τν − TeV, pp π and lepton → + 5 0 channel, τ π = 1 , and also S W ) R ∗ ˜ q → ( K 1 → m L M K pp -factory experiments with B τ ν are orders of magnitude L ) 0 1 K ˜ χ − π are comparable with the energy, we expect the effective X ˜ , is non-vanishing. Each of these 2 f τ 2 → → ¯ D /m − 1 τ search for squark mass τ ( Q τ 0 312 B 3 λ L τν GeV. Overall, comparison of figures → lepton, e.g., long-lived particles produced in 0 312 9 0 . We study two benchmark scenarios in which . signal channel. After using the constraints of λ τ 0 π 0 W – 15 – m or < plus a tau neutrino. We propose to conduct the π decay into a charged hadron plus a neutrino. The − − 0 1 1 ) → τ ˜ ∗ χ . τ π ¯ ( D 2 are up to two orders of magnitude tighter than the + 1 pp m m ˜ M π 2 f Q 6= 0 events, and the ongoing Belle II experiment is scheduled 3 is probed with greater sensitivity, by almost an order of → L /m − 0 312 τ η λ TeV. The limits on 0 311 + 6= 0 -factories, which are particularly suitable for the study of rare , 0 311 τ λ τ λ B ], this work exemplifies the usefulness of 0 311 ην = 5 → λ 55 R − TeV. We note that the bounds obtained from the ˜ q → e 0 1 events in the coming years. + m decay into the neutralino plus an accompanying meson ˜ e χ = 1 τ 10 9 ˜ f 10 10 m × × 4 6 . for . 1 4 τ τν is roughly between 0.5 GeV and Together with ref. [ Using our estimated acceptance, efficiency, and estimated negligible background yield, For both RPV operators, we consider in detail the possible decays of the → 0 1 0 ˜ decays. Specifically, the BABAR and Belle experiments have collected a combined data decays. We propose that further scenarios be pursued along this direction. χ search are relevant only whenalso the probe RPV the process case involves a of virtual a squark, virtual while stau our andfor results sneutrino. searching for new physicsτ related to the tighter than the current bounds stemminguncertainties in from the the branching combined ratio experimental and for model-dependent theoretical limits on limits obtained by recasting [59]and the comparable for W background yield in thisother channel major to channels, be the negligible expected even background in is even the smaller. we full present Belle numerical II results sample. forboth both In scenarios, model-independent the the and model-dependent model-independent limits. limits In on propose to study. Correspondingly, weleading perform background Monte-Carlo and simulations toused the determine to signal the estimate events the acceptance.which largest impacts source the In of particular,the background, signal simulation decay is to also obtain the neutralino mass and the τ sample of to collect and of the neutralino, defining the final states and displaced-vertex signatures that we m one of the RPVoperators operators, induces the neutralino decay intosearch a at meson electron-positron In recent years, searches forfor long-lived probing particles (LLPs) new havephysics physics, become is R-parity-violating an particularly (RPV) important supersymmetry at with means neutralinos light, the long-lived may neutralinos. be LHC. Such produced atIn A colliders this and work, well-motivated lead we scenario to develop an a for exotic method new displaced-vertex for signature. studying the scenario in which the neutralino mass its advantage is that it isshows sensitive to that the scenario magnitude, than the scenario 7 Conclusions (magenta and purple curves in the two panels). Despite this shortcoming of the JHEP02(2021)211 and (A.1) π SU(3) used in 8 η M . f ρ f . and are not known = 3 0 ∗ η 1 T a M f f , M f ] 2 q ] ] ] ] ] ] ] mesons, the decay constants are 75 , m 76 77 62 62 76 73 73 0 . Note that for the neutral 2 M η 74 ] + M meson decay constant find different m f 1 38 q ≈ 0.893 0.426 0.289 0.228 0.692 0.665 and 0.0153 0.2292 0.0422 0.3941 0.3268 0.0262 m i ∗ η T M = (1260) f 1 E its valence quarks. The values of 209 [ 209 [ 230 [ Branching fraction a 130.2 [ 155.7 [ 195.3 [ 49.42 [ 111.43 [ ]. For definiteness, we chose 2 M – 16 – q | 2 0 ∓ 0 η 0 75 − γ q η π , π π − π 0 5 − π . For the − − ± 0 − π π π γ + and π 74 ργ π ωγ π π γγ 0 + 1 + K π + 0 ¯ q 1 + π + π . Because the tensor decay constants π | π π q A.1 → π → π → 0 → → 2 0 0 Decay → ], we assume → D ± ∗ η → η ± 0 → η → 0 1 S 0 → ¯ ¯ → dd dd ∗ 0 π ρ a K K ω η η ω η 38 η η ≡ η Constant Value [MeV] Ref. f f f f f f f f K η ω K in eq. S M 2 . Values of the decay constants used in this work. f √ / ]. ], which takes into account the mixing between the π,ρ decay constant [ 62 73 f ρ Table 2 = 0 ,ρ 0 π f is the meson mass and are summarized in table . Branching fractions for the meson decays considered in our analysis. The values are 3 M A summary of the relevant meson branching fractions is given in table mesons, ρ obtained from ref. [ flavor states. Distinctvalues around calculations the of the where section for all mesons, following ref. [ A Meson decay constants and branchingThe fractions decay constant for the pseudoscalar current is [ Table 3 reproduced from ref. [ JHEP02(2021)211 ]. 368 SPIRE ]. 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Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. entist Training Asia Pacificfor Economic Theoretical Physics. Cooperation programNo. C.O.D. 1170171 acknowledges of and support the ANID fromfrom (Chile) Asia FONDECYT grant PIA/APOYO Pacific FONDECYT (Chile) AFB (Chile) Center 180002. Grant No.der 1201673. J.C.H. the N.A.N. acknowledges grant was support ANID supported REC by Convocatoria ANID Nacional (Chile) Subvención un- a Instalación en la Academia We would like to thank JordySjöstrand de for Vries, Marcela useful González, Sergey discussions.and Kovalenko, and Technology Torbjörn Z.S.W. (MoST) is ofpartly supported Taiwan by partly with the by grantGovernment, Ministry the number and of Ministry MoST-109-2811-M-007-509, the Science, of Gyeongsangbuk-do and Provincial Science ICT Government & through Future the Planning Young Sci- of Korea, the Pohang City Acknowledgments JHEP02(2021)211 ]. 220 Rept. Phys. ]. 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