R-Parity Violation and Light Neutralinos at Ship and the LHC

BONN-TH-2015-12 R-Parity Violation and Light Neutralinos at SHiP and the LHC

Jordy de Vries,1, ∗ Herbi K. Dreiner,2, † and Daniel Schmeier2, ‡ 1Institute for Advanced Simulation, Institut fürKernphysik, JülichCenter for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany 2Physikalisches Institut der UniversitätBonn, Bethe Center for Theoretical Physics, Nußallee 12, 53115 Bonn, Germany We study the sensitivity of the proposed SHiP experiment to the LQD operator in R-Parity violating supersymmetric theories. We focus on single neutralino production via rare meson decays and the observation of downstream neutralino decays into charged mesons inside the SHiP decay chamber. We provide a generic list of effective operators and decay width formulae for any λ0 coupling and show the resulting expected SHiP sensitivity for a widespread list of benchmark scenarios via numerical simulations. We compare this sensitivity to expected limits from testing the same decay topology at the LHC with ATLAS.

I. INTRODUCTION method to search for a light neutralino, is via the production of mesons. The rate for the latter is so high, Supersymmetry [1–3] is the unique extension of the ex- that the subsequent rare decay of the meson to the light ternal symmetries of the Standard Model of elementary neutralino via (an) R-parity violating operator(s) can be particle physics (SM) with fermionic generators [4]. Su- searched for [26–28]. This is analogous to the production persymmetry is necessarily broken, in order to comply of neutrinos via π or K-mesons. with the bounds from experimental searches. To solve For a neutralino in the mass range of about 0.5 - 5 GeV, the hierarchy problem [5, 6], the masses of the super- the newly proposed SHiP experimental facility [29] seems symmetric partners of the SM fields should be lighter ideal. It will consist of a high intensity 400 GeV pro- than 1-10 TeV, with a clear preference for lighter fields, ton beam from the CERN SPS incident on a fixed tar- see for example [7–11]. All experimental searches for su- get. 63.8 m down beam line there will a detector for persymmetric particles have hitherto been unsuccessful. long-lived heavy neutral particles. Two of us (HKD, The LHC sets limits on the strongly interacting spar- DS) have performed a preliminary analysis in [30], for ticles, the squarks and gluinos, of order 1 TeV, see for a small set of R-parity violating operators: the decay + 0 + 0 ¯ example [12, 13]. However the limits on the only elec- D → χ˜1 ì via the operator λi21LiQ2D1, and the de- 0 0 ∓ ± 0 troweak interacting particles, such as the sleptons, the caysχ ˜1 → (K ν; K ì ) via λi21,i21. It is the purpose neutralinos and the charginos, are significantly weaker of this paper to extend this to all possible production [14, 15]. In fact, as has been discussed in the literature modes and decay channels, and to determine the search there is currently no lower mass bound on the lightest su- sensitivity of the SHiP experiment. As an example, in persymmetric particle (LSP) neutralino, which is model earlier work [26], the production of B-mesons at NuTeV 0 0 independent [16, 17]. The limits from LEP can easily be was considered. The mesons could decay as Bd,s → χ˜1 ν ± 0 ± 0 avoided, and in particular, a massless neutralino is still or B → χ˜1 ì via λi13. The neutralinos could decay via allowed, for sufficiently heavy selectrons and squarks [18]. the LiQ1D¯ 3 operators, or purely leptonically via LiLjE¯k We discuss this in more detail below, in Sec. III. operators. These scenarios can also be tested at SHiP, We are here interested in the possibilities of searching where a sufficiently large number of B-type mesons is ex- arXiv:1511.07436v1 [hep-ph] 23 Nov 2015 for a light neutralino, up to masses of about 10 GeV. pected and we will show the sensitivity reach to the LQD¯ Already for a slepton mass of 150 GeV there is no sen- operators here. + − 0 0 sitivity via the process e e → χ˜1χ1γ from LEP data This paper is organized as follows. In Sec. II, we briefly [19], see also [20, 21]. Since mass reach is not a factor, review supersymmetry with broken R-parity. We also one might suspect that the high intensity facilities used discuss the bounds on the operators relevant to our anal- as B-factories, would be more sensitive, but this is also ysis. Constraints on the neutralino mass and motiva- not the case [19]. tions for the possibility of a very light neutralino follow In order to avoid over-closing the universe [22–24], a in Sec. III. In Sec. IV we discuss the relevant neutralino 10 GeV or lighter neutralino must decay via R-parity production and decay channels we expect to be most rele- violating operators [25]. If the R-parity violating cou- vant for SHiP observations of R-parity violation (RPV). plings are not too small, in this case, the most promising In Sec. V we set up the effective field theory need to compute the relevant meson and neutralino decays and compute the latter in general fashion. With these re- ∗ [email protected] sults at hand, we specialize to the SHiP set-up in Sec. VI † dreiner@uni–bonn.de and discuss under what circumstances a neutralino could ‡ [email protected]–bonn.de be detected. In Sec. VII we explain the methodology of 2 our numerical study. The discussion of the sensitivity of B. Bounds on R-parity Violating Couplings the SHiP experiment to the existence of light, but unstable, neutralinos in several benchmark scenarios is given The operators LiQjD¯ k which we investigate here all vi- in Sec. VIII. In Sect. IX, we discuss an estimate for pos- olate lepton number; thus there are strict bounds on the sible effects of our benchmark scenarios at the LHC. We 0 coupling constants λijk, see for example the reviews [46– conclude in Sect. X. 50]. We briefly summarize here the existing bounds on the specific operators which we investigate in our benchmark scenarios in Sec. VIII. For our results on the sensi- 0 II. R-PARITY VIOLATION tivity reach of SHiP, we focus on the cases λ1jk, involving 0 final state electrons, and λ31k, which produces tau lep- A. Introduction tons. However, experimentally final state muons should be testable at least as well as electrons. We thus also 0 present the corresponding bounds on the couplings λ2jk, The minimal supersymmetric extension of the SM re- which are typically weaker. We take most of the numbers quires the introduction of an additional Higgs doublet. from the most recent review [50]. The minimal set of pure matter couplings are then en- 0 ms˜r 0 ms˜R coded in the minimal supersymmetric SM (MSSM) su- λ112 < 0.03 , λ212< 0.06 . (5) perpotential 100 GeV 100 GeV m m ˜ ¯ ¯ λ0 < 0.2 c˜L , λ0 < 0.1 dR , (6) WMSSM = (hE)ijLiHdEj + (hD)ijQiHdDj 121 100 GeV 221 100 GeV +(h ) Q H U¯ + µH H . (1) U ij i u j d u m m λ0 < 0.2 c˜L , λ0 < 0.1 s˜R , (7) 122 100 GeV 222 100 GeV Here hE,D,U are dimensionless 3 × 3 coupling matrices. The Higgs mixing µ has mass dimension one. This su- m˜ 0 mt˜L 0 bL perpotential is equivalent to imposing the discrete Z λ131 < 0.03 , λ231< 0.18 , (8) 2 100 GeV 100 GeV symmetry R-parity [31], or the Z6 proton hexality [32], and results in a stable proton. Both are discrete gauge ms˜R λ0 < 0.06 , (9) anomaly-free, ensuring stability under potential quantum 312 100 GeV gravity corrections [33]. A model restricted to the super- m˜ potential W is called R-parity conserving. The ad- 0 tL MSSM λ313 < 0.06 , (10) vantage is that the LSP, usually the lightest neutralino, is 100 GeV 0 an automatic WIMP dark matter candidate [34]. How- The bound on λ231 is from [48], as there is no bound ever, no such dark matter has been observed to-date, quoted in [50]. Assuming an experimental neutrino mass motivating the search for other forms of supersymmetry. bound mν < 1 eV results in a bound of approximately 0 Instead in R-parity violating models, there are further λ122,222 < 0.01 m/˜ 100 GeV [40, 48, 51]. This is how- possible terms in the superpotential ever model dependent, as we know there must be further, possibly off-diagonal,p contributions to the neutrino mass WRPV = WLV + WBV , (2) matrix. ¯ 0 ¯ WLV = λijkLiLjEk + λijkLiQjDk + κiLiHu , (3) 00 ¯ ¯ ¯ WBV = λijkUiDjDk . (4) III. A LIGHT NEUTRALINO

Imposing the discrete Z3 symmetry baryon triality [32, The best lower mass limit on the lightest supersym- 35], the lepton-number violating terms in WLV remain. metric particle (LSP) neutralino is from LEP [17] The proton remains stable, since baryon-number is con- served, however the neutralino LSP is unstable and is m 0 > 46 GeV. (11) χ˜1 no longer a dark matter candidate. This can be solved by introducing the axion to solve the problem of CP- This uses the LEP-II chargino search to restrict the range violation in QCD. The supersymmetric partner, the ax- of µ and the SU(2) soft breaking gaugino mass parameter ino, is then automatically a good dark matter candidate M2 and then assumes the supersymmetric GUT relation [36–39] and light neutrino masses are also generated auto- 5 2 1 matically [40–44]. We thus consider this a well-motivated M1 = tan θW M2 ≈ M2 , (12) 3 2 model to investigate. At a given energy scale the bilinear terms κiLiHu can be rotated away, even for com- where M1 is the U(1)Y soft breaking gaugino mass. The plex κi, µ [40, 41, 45]. This leaves us with the tri-linear LEP-II searches translate into M1 & 50 GeV. Performing 0 couplings λijk and λijk. In this work we focus on the lat- the Takagi diagonalization of the neutralino mass matrix ter couplings as they lead to neutralino production via [52] and scanning the parameters M1,M2, µ, tan β over the decays of mesons. the allowed ranges, results in the bound of Eq. (11). 3

If the GUT assumption is dropped and M1,M2 are IV. LIGHT NEUTRALINOS AT SHIP independent parameters, setting the determinant of the neutralino mass matrix to zero results in the relation A. The SHiP Setup

2 2 M2MZ sin(2β) sin θW We present some details of the SHiP setup that are M1 = 2 2 . (13) µM2 − MZ sin(2β) cos θW relevant to our analysis. The SHiP proposal [29] is not definitive yet. The plan is to employ the 400 GeV pro- For real parameters this equation can always be solved ton beam at CERN in the fixed-target mode. This yields [16]. Thus for given values of µ, M2, and tan β there is al- a center-of-mass energy of roughly 27 GeV, sufficient to ways a mass-zero singular value [52], and thus a massless produce D and B mesons. Over the lifetime of the ex- neutralino state. A neutralino lighter than O(10 GeV) periment a total of 2·1020 protons on target are foreseen. is dominantly bino and does not couple directly to the Such a large event yield is expected to be achievable by Z0-boson. Thus the bounds on a light neutralino from e.g. a hybrid target consisting of tungsten and titanium- the invisible Z0-width are avoided [19]. In fact to our zirconium doped molybdenum alloy. knowledge all laboratory bounds are avoided [16]. A major motivation for the SHiP experiment is to The strictest mass bounds on a stable, light neutralino observe new, weakly-interacting particles with long life- are astrophysical. Supernova [18, 53] or white dwarf cool- times. Such particles could be produced via proton- ing [54], give a lower mass bound: m 0 250 MeV, for se- target-collisions and propagate for finite distances of the χ˜1 & lectron masses around 320 GeV. In this case, too few neu- order tens of meters before decaying back into Standard tralinos are produced, due to Boltzmann suppression, as Model pairs. For that purpose, a decay volume is located the supernova temperature is about 30 MeV. A massless 68.8 m behind the target. It has a cylindrical shape with neutralino is allowed for selectron masses above about a total length of 60 m, however with the first 5 m ded- icated for background suppression vetoes. Furthermore, 1275 GeV or below 320 GeV. For Me˜ > 1275 GeV also the decay region has an elliptic face front with semi-axes too few neutralinos are produced. For Me˜ < 320 GeV the neutralinos are trapped in the supernova, similar to 5 m and 2.5 m. We sketch this setup in Fig. 2. A spec- neutrinos and must be included in the full supernova sim- trometer and a calorimeter system positioned behind the ulation. Since this has not been done to-date, the super- decay volume can identify the visible final state particles nova does not give a reliable bound in this region. that are potentially produced when a hidden particle decays. Cosmologically the Cowsik-McClelland bound [55] on a very light neutrino translates into the upper neutralino mass bound [16] B. Production and Decay of Neutralinos via R-Parity Violation M 0 < 0.7 eV . (14) χ˜1 Since the goal is to investigate the light supersymmet- The neutralino in this case provides hot dark matter, ric neutralino, one might consider their direct produc- but not enough to negatively affect structure formation. tion. However, in Ref. [26] using the Monte Carlo pro- The observed dark matter density must then originate gram HERWIG 6.2 [60] the pair production of neutralinos elsewhere, for example from the axino. in a proton fixed target experiment Requiring the lightest neutralino to provide the ob- 0 0 p + p → χ˜1χ˜1 + X, (17) served dark matter results in a lower mass bound, the Lee–Weinberg bound [56]. The proper bound is obtained was shown to be three orders of magnitude too low at the by scanning over the allowed supersymmetric parameter NuTeV experiment, for squark masses of 100 GeV. Since space, while dropping the relation in Eq. (12). This is the light neutralino is almost pure bino, the dominant thus an on–going process [22]. The most recent bound production mode is via t-channel squark exchange, and 4 including the Higgs–discovery data and also constraints the cross section goes as 1/mq˜. At SHiP it is planned to from stau searches gives [24, 57–59] have 100 times more protons on target than at NuTeV. However, the lower squark mass bound is now also about a factor of 10 stricter. One would thus expect a fur- Mχ˜0 > 24 GeV . (15) 1 ther two orders of magnitude suppression, making this a hopeless endeavour also at SHiP. Therefore the mass range As in Ref. [26], we instead consider the production of mesons M, which can have a very large production cross 0.7 eV < M 0 < 24 GeV , (16) χ˜1 section. On rare occasions these mesons can decay to the neutralino LSP and a neutral or charged lepton l via is excluded for a stable neutralino LSP as it gives too p + p → M + X much dark matter. A neutralino LSP in this mass range 0 is only allowed if it decays, i.e. R-parity is violated. M → χ˜1 + l. (18) 4

0 c ℓ+ c χ1 c ℓ+

e ℓL dR cL e e + d 0 χ0 d e ℓ χ1 d 1

e e + 0 + FIG. 1. Relevant Feynman Diagrams for D → χe1 + `

At SHiP energies, with a 400 GeV proton beam we ex- A. The Formalism pect (next to the production of light mesons containing up, down, or strange quarks) high production rates 0 ¯ The interaction Lagrangian due to λiabLiQaDb is given for charmed mesons, and somewhat lower rates for B- in terms of four-component fermions by mesons. As we discuss below, for example over the life- 16 ± 0 C ∗ i time of SHiP about 4.8 × 10 D -mesons are expected. L ⊃ λiab (νi PLda)dbR + (dbPLν )daL + (dbPLda)νiL Thus even very rare decays can be probed. Individual 0h C i ∗ i i LiQaD¯ b R-parity violating operators allow for leptonic − λ (u PL` )d + (dbPLua)ìL + (dbPL` )uaL iab a e bR e e decays of mesons. As an example the tree-level Feynman + h.c.h (21)i diagrams for the decay e e e Here, da,b, ua, νi, ì denote the down-like quark, up- like quark, neutrino, and charged lepton fields, respec- D+ → χ0 + `+, i = 1, 2 . (19) 1 i tively. The tilde denote the corresponding supersymmetric scalar partners. The dominant contribution to the are given in Fig. 1, fore a = 2, b = 1. In this specific R-parity violating decay of a meson typically proceeds example the light neutralino can decay via the same R- at tree-level via operators associated with the Feynman parity violating operator: diagrams as shown in Fig. 1. Thus we also need the standard supersymmetric fermion-sfermion-neutralino ver- tices. We assume that the sfermion mixing is identi- 0 0 ¯ 0 χ˜1 → (ν KS/L;ν ¯ KS/L) . (20) cal to the fermion mixing such that their contributions to the gauge couplings cancel. As we consider a domi- 0 nantly bino LSP neutralino,χ ˜1, we only take chirality- Both sets of decays are possible, as the neutralino is a conserving terms into account 0 Majorana fermion. For small values of the coupling λi21 L ⊃ g (χ0P u )u∗ + g (χ0P d )d∗ and given that the neutralino must be lighter than the uãL 1 L a aL dãL 1 L a aL + D meson, the neutralino lifetime can be long enough to 0 ∗ 0 ∗ + g˜l (χ1PLì)ìL + gν˜iL (χ1PLνi)νiL decay downstream in the SHiP detector. iL e e e e ∗ 0 + g (dPLχ1)dR + h.c. (22) deR e e e e We assume that the sfermione masses are significantly larger than the momentume exchange of the process. Thus V. EFFECTIVE INTERACTIONS: the sfermions can be integrated out, resulting at tree- LEPTON-NEUTRALINO-MESON level in the low-energy effective four-fermion Lagrangian for both the production and decay of the neutralino: g∗ In this section, we discuss the R-parity violating ef- 0 debR ¯ 0 C L ⊃ λiab 2 (dbPLχ˜1)(νi PLda) fective interactions between a meson, a lepton, and a m debR neutralino. These interactions are relevant for both the ∗ g g production and the decay of the neutralino and are nec- debR ¯ 0 C deaL 0 − (dbPLχ˜1)(uaPLì) + (χ˜1PLda)(dbPLνi) essary to determine the possible signatures at SHiP, as m2 m2 deb dea 0 ¯ R L in Eqs. (19),(20). We focus on the operators λiabLiQaDb gueaL 0 gνeiL 0 where i denotes the leptonic generation index and a and − 2 (χ˜1PLua)(dbPLì) + 2 (χ˜1PLνi)(dbPLda) mu mν b the quark generation indices. The index b is always as- eaL eiL g sociated with a down-like SU(2) singlet quark, whereas èiL 0 − (χ˜ PLì)(dbPLua) + h.c. (23) the index a can refer to either an up-like or down-like m2 1 ` SU(2) doublet quark. If a is up-like then i corresponds eiL to a charged lepton, i.e. electron, muon, or tau, whereas We have omitted the terms involving pairs of neutralinos, if a is down, i corresponds to a neutrino. which are most likely not relevant at SHiP, see [28]. Sim- 5

0 µν ρσ ilarly, we have dropped interactions involving four SM (χ σ ì)(dbσ ua) × fermions, see [46]. ∗ g ˜ 0 gu˜L dR iµνρσ For pure bino interactions, the coupling constants g λ + gµρgνσ − , (33) X e iab 4m2 4m2 2 uã d˜ are family independent [61] L bR ≡GT,` g2 iab g`˜ = g`˜ = +√ tan θW , (24) iL L 2 and their| hermitean{z conjugates.} g2 For pseudoscalar mesons composed of anti-quarksq ¯1 gν˜iL = gν˜L = +√ tan θW , (25) 2 and quarks q2, we can connect the quark bilinear vec- g2 tor currents with external meson fields by defining pseu- guãL = gu˜L = − √ tan θW , (26) 3 2 doscalar meson decay constants fM 5g2 µ 5 µ g ˜ = g ˜ = + √ tan θW , (27) h0|q¯1γ γ q2|M(p)i ≡ ip fM , (34) daL dL 3 2 2g2 where |M(p)i denotes a pseudoscalar meson M with mo- g ˜ = g ˜ = − √ tan θW , (28) dbR dR 3 2 mentum p. The standard current-algebra approximation then predicts

Here, θW denotes the electroweak mixing angle and g2 2 5 mM S the Standard Model SU(2) gauge coupling. h0|q¯1γ q2|M(pM )i = i fM ≡ f , (35) m + m M Using chiral Fierz identities (e.g. [62]), one can rear- q1 q2 range the four-fermion interactions in Eq. (23) such that for the pseudoscalar currents in Eqs. (30) and (31). Here, each term factorizes in a neutralino-lepton current and a mM , mq1 and mq2 are the masses of the meson M and quark-bilinear: the quarks q1, q2, respectively. The tensor structure in Eqs. (32), (33) does not lead 1 0 ¯ ¯ to purely leptonic processes such as M → χ + li or (ψ1PL η2)(¯η1PL ψ2) = − (ψ1PL ψ2)(¯η1PL η2) 0 R R 2 R R χ → M + li, because a pseudoscalar meson only has 1 µν one relevant Lorentz-vector, its momentum. Thus the − (ψ¯1σ ψ2)(¯η1σµν η2) e 4 tensor interactions only contribute to higher multiplicity e 0 0 0 i µνρσ processes such as M → χ + li + M , where M denotes a ± (ψ¯1σµν ψ2)(¯η1σρση2), (29) 8 lighter meson. These are phase space suppressed by two to three orders of magnitude,e and we do not consider µν µ ν 0123 with σ ≡ i/2 [γ , γ ] and = 1. ψ1,2, η1,2 denote them here. four component fermions. Making use of these identi- Vector mesons have two intrinsic Lorentz vectors, their C C µ µ ties and applying ψ PL/Rη = η PL/Rψ in combination momentum p and polarization . The decay constant ∗ with the Majorana identity χC = χ for the neutralino, of a vector meson M with mass mM ∗ , can be defined as Eq. (23) can be written as the sum of the following four µ ∗ V µ h0|q¯1γ q2|M (p, )i ≡ f ∗ mM ∗ . (36) interactions: M Heavy-quark symmetry relates the vector and pseu- 0 (χ PLνi)(dbPLda) × doscalar constants for mesons containing a heavy quark V ∗ fM ∗ ' fM [63]. We use this relation for the B and D g 1 g ˜ 1 g ˜ ∗ λ0 ν˜L − dL − dR , (30) mesons. For lighter mesons, such as K and φ, the re- e iab m2 2 m2 2 m2 ν˜i d˜ d˜ lation is not accurate and instead we follow Ref. [64], L aL bR where the vector decay constants are obtained from ≡GS,ν ∗ + − iab M → e e decays. | {z } Similarly we can define the tensor meson constant 0 × (χ PLì)(dbPLua) µν ∗ T µ ν ν µ ∗ h0|q¯1σ q2|M (p, )i ≡ ifM ∗ (pM − pM ). (37) 1 g 1 g ˜ g˜ λ0 u˜L + dR − `L , (31) e iab 2 m2 2 m2 m2 For mesons containing a heavy quark (c or b), heavy- uã d˜ ν˜i L bR L quark symmetry [65] also relates the vector and tensor de- T V S,` cay constants f ∗ ' f ∗ ' fM . Because the tensor de- ≡Giab M M cay constants are not known in all cases, we also employ | {z } 0 µν ρσ this relation for lighter mesons. The additional uncer- (χ σ νi)(dbσ da) × ∗ tainties entering via these simplifying assumptions hardly 0 2 gd˜ gd˜ iµνρσ affect the SHiP sensitivity curves on λ /m . These range 0 L R f e λiab 2 + 2 gµρgνσ − , (32) e 4m ˜ 4m ˜ 2 O da db over many orders of magnitude and thus an (40%) cor- L R S,T,V rection in a decay constant fM (∗) does not noticeably ≡GT,ν iab change the results presented in the figures in Sec. VIII. | {z } 6

We list the values of the pseudoscalar and vector decay B. Possible Decay Modes constants we use in Table I in Sect. VIII. In general, we find that neutralinos can interact both with pseudoscalar 0 ¯ From Eqs. (30)–(33), a single λiabLiQaDb operator and vector mesons via different but related effective cou- leads to interactions with charged (pseudoscalar or vec- plings. In the following analysis we therefore consider + tor) mesons Mab of flavour content (uadb), as well as both meson types and also show how the inclusion of the 0 neutral mesons M with quark composition dadb, and latter affects the overall sensitivity. ab their respective charge conjugated equivalents. If m 0 < χ˜1 mM − mli , the operator opens a decay channel of the meson into the neutralino plus lepton li. For example, ± 0 ± 0 0 ¯ 0 0 Potentially fine-tuned models with non-degenerate M → χ˜1` , or M → χ˜1ν, M → χ˜1ν¯. Such processes sfermion masses could lead to a complete cancellation serve as the initial neutralino production mechanism here. In addition, for m 0 > m +m the neutralino can of the individual contributions in Eqs. (30), (31). No sen- χ˜1 M li 0 + − − + 0 0 ¯ 0 sitivity would be expected in such a scenario if only pseu- decay viaχ ˜1 → M ` ,M ` orχ ˜1 → M ν, M ν¯. Such doscalar mesons were considered in the analysis. How- decays can potentially be observed in the SHiP detector. ever, for a nonzero RPV coupling the effective operators From the structure of the operators in Eqs.(30)–(33), in Eqs. (30)–(33) can not all vanish simultaneously. We the definition of the effective couplings and the meson hence safely use the simplifying assumption of completely structure constants in Eqs. (35), (37), we obtain the fol- mass degenerate sfermions. lowing unpolarized decay widths1:

1 2 2 2 λ 2 (mM , m 0 , ml ) 0 ab χe1 i S,f 2 S 2 2 2 2 Γ(Mab → χ1 + li) = |G | (fM ) (mM − m 0 − ml ), (38) 3 iab ab ab χe1 i 64πmMab 1 2 2 2 λ 2 (mM ∗ , m 0 , ml ) ∗ e0 ab χe1 i T,f 2 T 2 2 2 2 2 2 2 2 Γ(Mab → χ1 + li) = |G | (fM ∗ ) mM ∗ (mM ∗ + m 0 + ml ) − 2(m 0 − ml ) , (39) 3 iab ab ab ab χe1 i χe1 i 3πmM ∗ ab h i 1 2 2 2 e λ 2 (m 0 , mM , ml ) 0 χe1 ab i S,f 2 S 2 2 2 2 Γ(χ → Mab + li) = |G | (f ) (m 0 + m − m ), (40) 1 3 iab Mab χ1 li Mab 128πm 0 e χe1 1 2 2 2 e λ 2 (m 0 , mM ∗ , ml ) 0 ∗ χe1 ab i T,f 2 T 2 2 2 2 2 2 2 2 Γ(χ1 → Mab + li) = |G | (fM ∗ ) 2(m 0 − ml ) − mM ∗ (mM ∗ + m 0 + ml ) . (41) 3 iab ab χe1 i ab ab χe1 i 2πmχ0 e1 h i e

± ¯ + ∗+ Here, li either denotes ` or νi, depending on whether (ub) = (B ,B ) i λ0 → (44) Mab is charged or neutral. The phase space function i13 ¯ 0 ∗0 1 ( (db) = (B ,B ) , λ 2 (x, y, z) ≡ x2 + y2 + z2 − 2xy − 2xz − 2yz. The co- efficients G are defined in Eqs. (30)–(33). For each of (cd¯) = (D+,D∗+) the above decaysp there exists a charge-conjugated pro- λ0 → (45) i21 ¯ 0 0 ∗0 cess with identical decay width. Here we list the most ( (sd) = (KL,KS,K ) , important mesons Mab that participate in each interac- (cs¯) = D+,D∗+ tion for given a, b 0 s s λi22 → (46) ( (ss¯) = η, η0, φ (ud¯) = (π+, ρ+) 0 ¯ + ∗+ λ → (42) (cb) = Bc ,Bc , i11 ¯ 0 0 λ0 → (47) ( (dd) = (π , η, η , ρ, ω) , i23 ¯ 0 ∗0 ( (sb) = Bs ,Bs (us¯) = (K+,K∗+) 0 0 → ¯ 0 ∗0 λi12 → (43) λi31 (bd) = B ,B (48) ( (ds¯) = (K0 ,K0 ,K∗0) , L S 0 ¯ 0 ∗0 λi32 → (sb) = Bs ,Bs (49) 0 ¯ λi33 → (bb) = ηb, Υ (50) For light neutral pseudoscalar mesons, mass and 1 The neutralino decay width in Ref. [30] contains an erroneous flavour eigenstates do not coincide. For our studies this sign and misses a factor of 2, which is fixed here in Eq. (40). ¯ 0 0 is only relevant for the KL,S, η, and η mesons (we take 7

Decay Value Ref. our analysis. Given the number NM of mesons M pro- constant duced at SHiP and the lifetime τM , the expected number ss¯ of initially produced neutralinos is given by fη -142 MeV [28, 66] ss¯ fη0 38 MeV [28, 66] f V 230 MeV [64] prod. 0 φ Nχ = NM · Γ(M → χ1 + l) · τM . (55) fK 156 MeV [17] M V X fK∗ 230 MeV [64] e V fD, fD? 205 MeV [17] As apparent from the previous section, for each operator V fD , f ? 259 MeV [17] s Ds there are both pseudoscalar and vector mesons which can fB 191 MeV [17] produce neutralinos. However, the lifetimes of a pseu-

fBs 228 MeV [68] doscalar and the corresponding vector meson of the same quark composition differ by many orders of magnitude. TABLE I. Values of the pseudoscalar and vector decay con- As an example, for the lightest charged charm meson, stants that are used in the various benchmark scenarios. Def- D±, and its vector resonance partner, D∗±, one finds −9 initions of the constants are given in Eqs. (34) and (36). Ten- τD∗± /τD± ≈ 8 × 10 . Similar ratios appear for kaons sor decay constants are chosen to be equal to the pseudoscalar and even though the lifetime of vector B-mesons is yet decay constants. unknown there is no reason to expect largely different be- haviour. For the RPV decay widths, however, one finds ∗± 0 ± ± 0 ± that Γ(D → χ1` )/Γ(D → χ1` ) depends mainly φ to be a pure (ss¯) state). For the former, we ne- on the ratio of masses and of the effective operator cou- glect any CP-violation and√ define the mass eigenstates ¯ ¯ plings GT /GS.e Thus, it is hardlye larger than 2 orders |KL/Si ≡ (|K0i ± |K0i)/ 2, where |K0i and |K0i are of magnitude unless one chooses a very peculiar setup of ¯ flavor eigenstates (ds¯) and (sd), respectively. We can fine-tuned parameters. As the expected number of initial then read off the decay constants from mesons, NM and NM ∗ , will also be of roughly the same 0 ∗ 0 µ order, we conclude that if both M → χ1f and M → χ1f µ 5 0 ¯ µ 5 0 ip fK h0|sγ¯ γ d|KL(p)i = + h0|dγ γ s|KL(p)i = √ , (51) are kinematically allowed, the contribution from vector 2 meson decays is completely negligible. µ e e µ 5 0 µ 5 0 ip fK h0|sγ¯ γ d|K (p)i = − h0|dγ¯ γ s|K (p)i = √ , (52) In the small mass range mM < m 0 < mM ∗ it might S S χe1 2 only be the vector mesons that can produce neutralinos in the first place, but by the above arguments and from where fK is the decay constant of the charged kaon as defined in Eq. (34). the results below we expect the neutralino event rates to For η and η0 we consider mixing between the η0 and η8 be far too small to be observable. We therefore ignore flavor states. We are only interested in the (¯ss) content any neutralino production via vector meson decays in the of these mesons. We follow Refs. [28, 66] and define following study. For the neutralinos to be observable, a sufficiently high µ 5 0 µ ss¯ h0|sγ¯ γ s|{η, η }(p)i = ip f{η, η0} , (53) fraction must decay within the decay chamber of the SHiP experiment. By summing the widths Γ(χ0 → Mf) and give numerical values in Table I. 1 0 of all allowed channels for a given operator, we can derive For the special cases λijj the radiative neutralino decay the proper lifetime of a neutralino given the parameterse is possible [40, 67] of the RPV supersymmetric model. Given the kinemati- 0 cal distributions of neutralinos produced via meson decay χ˜ → γ + (ν , ν¯ ) . (54) 1 i i and knowing the geometry of the decay chamber, we can 0 This is necessarily relevant for very light neutralinos, be- find the average probability hP [˜χ1 in d.r.]i of a neutralino low the pion mass. We do not know how well this would decaying inside the detectable region. This is explained be visible at SHiP, and do not consider it further here. in more detail below. A neutralino decaying inside the decay chamber is a necessary, but not a sufficient condition, as the final state VI. OBSERVABLE NEUTRALINOS particles have to be observed and traced back to an invisibly decaying new particle. For a charged final state, With the predicted decay widths at hand we investi- e.g. K+e−, one can measure the trajectory of both par- gate how and under which circumstances R-parity viola- ticles, measure their momenta and presumably identify tion can be observed at the SHiP experiment. Each λ0 the neutralino decay vertex. For a neutral final state, coupling causes at least one type of meson to decay into e.g. KLν, one loses both the tracking information and a neutralino and another charged or neutral lepton, pro- the momentum of the second particle. We expect that vided it is kinematically allowed. This process serves as these are hard to be linked to the decay of a neutralino. the initial neutralino production mechanism at SHiP in Thus we only count neutralinos that decay into charged 8

2 16 final state particles. The final number of observed neu- Ncc¯ 9 × 10 tralinos is then −4 σb¯b/σcc¯ 2.1 × 10 obs. prod. 0 0 cc¯ N 0 = N 0 · hP [˜χ1 in d.r.]i · BR(˜χ1 → charged) . (56) n ± 0.53 χ˜1 χ˜1 D cc¯ n ± 0.074 With the above considerations, we thus demand the fol- Ds b¯b lowing for observable neutralino decays via LQD opera- nB± 0.83 b¯b tors at SHiP: nB0 0.80 b¯b 1. A pseudoscalar meson M with mM > m 0 must nB0 0.14 χe1 s have a non-vanishing decay rate into neutralinos. TABLE II. Numerical values used to estimate the number 2. The neutralino must have a non-vanishing decay NM in Eq. (60). Except for Ncc¯ , which is taken from [71], 0± rate into another charged meson M with mM 0± < all numbers are evaluated by simulating 1M events of each m 0 < m . HardQCD type in Pythia. χ˜1 M With these this conditions, it is practically impossible 0 for SHiP to observe R-Parity violation if only one λiab to [71], the number of cc¯ events after 5 years of opera- coupling is nonzero. Eqs. (43)–(50) show which operator 16 tion is expected to be Ncc¯ = 9 × 10 . By simulating 1M leads to which sets of mesons that can decay into the events of type cc¯, we can use Pythia to find the average neutralino or the neutralino can decay into. The only number of produced charm mesons per cc¯ event, i.e. operator related to both a pseudoscalar meson M and 0± 0 a charged meson M with mM > mM 0± is λ112, which cc¯ ± nD ≡ ND/Ncc¯, with D ∈ {D ,Ds} . (58) might be observable via the chain The analogous simulation of b¯b events gives the respective K0 → χ˜0ν, χ˜0 → K±`∓ . (57) L/S 1 1 number for bottom mesons:

However, as |mK± −mK0 | ≈ 4 MeV, the testable range b¯b 0 ± 0 L/S nB ≡ NB/Nb¯b, with B ∈ {B ,B ,Bs } . (59) of neutralino masses is extremely limited and the expected energies of the final state particles are so small The total number of expected b¯b events is taken by scaling that the decays would be very challenging to observe. the known number for cc¯ events by the ratio of total cross 0 0 Thus we require two different operators λiab, λjcd =6 0, sections determined by Pythia, i.e. Nb¯b = Ncc¯ ×σb¯b/σcc¯. with iab and jcd such that the decays fulfill the above re- We therefore combine quirements. This necessary extension leads to a plethora cc¯ of possible combinations. In Sect. VIII we restrict our- nM for charm mesons NM = Ncc¯ · (60) selves to an interesting subset of benchmark scenarios. b¯b ( nM · σb¯b/σcc¯ for bottom mesons

VII. SIMULATION OF RPV SCENARIOS and list the numerical values in Table II. For each benchmark scenario, we simulate 20,000 events of the correct HardQCD type and — to increase Eqs. (55), (56) tell us how to estimate the number of 0 obs statistics — set the branching ratio BR(M → χ˜1f) to observable neutralino decays N 0 for any given opera- χ˜1 100 %. We then scale our results accordingly. As the de- tor combination and parameter values. The total widths caying mesons are scalar particles, the momentum of the and branching ratios into charged final states can be cal- neutralino is chosen to be uniformly distributed in the culated from the general width formulae, Eqs. (38)-(41). rest frame of the decaying meson. We then sum over all We next describe the numerical tools we use to estimate produced neutralinos and determine the average prob- 0 NM and hP [˜χ1 in d.r.]i. ability, i.e. for all possible neutralino momenta, that To get a reliable estimate on the kinematics of the ini- an arbitrary neutralino decays within the SHiP detector. tially produced mesons at SHiP, as well as the result- 0 Given the four-vector of the neutralino (˜χ1)i in spherical ing neutralinos after their decay, we use Pythia 8.175 z coordinates as (Ei, pi , θi, φi) and the distances and an- [69, 70]. In scenarios with initial charm (bottom) mesons MC gles as defined in Fig. 2, the average probability for Nχ˜0 we use the HardQCD:hardccbar (HardQCD:hardbbbar) 1 neutralinos in a given sample generated by a Monte Carlo matrix element calculator within Pythia, which includes program is evaluated as the partonic processes qq,¯ gg → cc¯(b¯b) and select the spe- MC cific meson type for each benchmark scenario. According N 0 χ˜1 0 1 0 hP [˜χ1 in d.r.]i = MC P [(˜χ1)i in d.r.], (61) N 0 χ˜1 i=1 2 X We note that this assumption has not been made in [30] and as z z 0 −Lt→d/λi −Li/λi such, our results for the same scenario slightly differ. P [(˜χ1)i in d.r.] = e · 1 − e . (62) 9

Ld 0 χe1 values for the lepton ﬂavor indices. It is clear that we can

ρi φ not investigate all of these cases in detail. In order to an- Li i

A R alyze the sensitivity at SHiP, we have thus focussed on

θi Lt→d R a subset which we propose as, hopefully representative, B Decay Chamber benchmark scenarios. In choosing the benchmark scenarios, we took the fol- Target lowing points into consideration. For the sensitivity, to first order, it does not matter if we consider electrons or muons. We thus restrict ourselves to electrons.3 We FIG. 2. Schematic overview of the SHiP detector geometry and definition of distances and angles used in text. have one benchmark with final state taus, as their con- siderably larger mass affects the accessible decay phase space and the respective total widths. z The meson production rates can differ substantially. The mean decay length λi of the neutralino in the lab frame is given by Thus we consider various scenarios where the neutralinos are produced via neutral or charged D- or B-mesons. z z 0 λi = βi γi/Γtot(˜χ1), (63) To estimate the production rates we use Pythia, as discussed above. We do not consider the production of z z 0 0 βi = pi /Ei, (64) neutralinos via λi11, λi12 as the production of the corresponding light mesons are not well simulated in for- γi = Ei/m 0 . (65) χ˜1 ward direction with Pythia. We postpone this to future z 0 work. For kinematic reasons, we restrict the final state βi is the z-component of the relativistic velocity of (˜χ1)i, 0 mesons in the neutralino decays to K and D mesons. γi the corresponding Lorentz boost factor. Γtot(˜χ1) is We also do not consider decays into pions and the asso- the total decay width of the neutralino LSP, which only ciated vector resonances, as we expect sizable deviations depends on the model parameters and not on the kine- 0 from our approximations. However, from the results of matics of an individual candidate (˜χ1)i. Li denotes the the benchmark scenarios discussed below, an estimate for distance in z-direction a neutralino can travel inside the pion final states can be derived easily, by letting the neu- decay chamber before leaving it in radial direction. It tralino mass range down to the pion mass of about 135 can be determined as, see also Fig. 2, MeV, instead of the kaon mass. The pion and kaon decay constants are related by SU(3) flavor symmetry. Of 0 if ρ cot θ < L , i i t→d course, a neutralino decay into pions requires turning on 0 Li =  Ld if ρi cot θi > Lt→d + Ld,(66) the coupling λi11 where i = 1, 2. Note that in this case  it does matter if i = 2 as now mµ ' mπ.  − ρi cot θi Lt→d else, To be precise we consider the following cases, which   2 2 we propose as benchmarks. ρi = RARB/ (RB cos φi) + (RA sin φi) . (67) ρ is the radius ofp the ellipse in the direction φ . θ is the i i i A. Benchmark Scenario 1 angle between the flight direction of the neutralino and the central axis of the detector; the polar angle, φi, is the azimuthal angle of the neutralino momentum 3-vector. 0 0 λP for production λ121 RA denotes the semi-minor axis, and RB the semi-major λ0 for decay λ0 axis of the elliptical face of the detector. Lt→d denotes D 112 the distance from the target to the front of the detector. produced meson(s) D± (∗)± ∓ Ld denotes the length of the detector along the central visible final state(s) K e 0 0 0 ∗ axis. As explained in Sec. IV A, we use the numerical invisible final state(s) via λP (KL,KS ,K ) + (ν, ν¯) 0 0 0 ∗ values Lt→d = 68.8 m, Ld = 55 m, RA = 2.5 m and invisible final state(s) via λD (KL,KS ,K ) + (ν, ν¯) RB = 5 m. TABLE III. Features of Benchmark Scenario 1.

VIII. RESULTS FOR VARIOUS BENCHMARK We begin with scenarios where the neutralino is pro- SCENARIOS duced via the RPV decay of a D meson and subsequently decays into a kaon plus lepton. This scenario has al- In Eqs. (43)-(50) we have listed the twenty-seven oper- ready been studied in some detail (by two of us, HD, 0 0 ators λiab together with the corresponding mesons they DS) in Ref. [30]. We turn on two RPV couplings λ121 couple to. For a ﬁxed lepton ﬂavor there are 36 possible combinations for production and decay of the neutralinos, if we assume distinct operators. The number of pos- 3 Although the bounds on λ0 are typically weaker than on λ0 . sibilities exceeds 100 if one in addition tests all possible 2jk 1jk 10

a) b)

FIG. 3. SHiP sensitivity curves for Benchmark Scenario 1. In a), the two couplings are set equal. The maximum sensitivity reach, corresponding to ≥ 3 events, is shown in bright blue, with a solid curve edge. The blue area corresponds to ≥ 3 · 103 events and dark blue to ≥ 3 · 106 events. The dashed curve extending just below the light blue region denotes the extended sensitivity if the neutral mesons from neutralino decays are also visible. The horizontal hashed lines correspond to the existing limits on the RPV couplings, for three different sfermion masses: 250 GeV, 1 TeV, 5 TeV. In b), the sensitivity reach is shown as a function of the two independent RPV couplings for three fixed neutralino masses: 600 MeV (bright blue), 1200 MeV (blue), 1800 MeV (dark blue). For the x-axis we always choose the coupling responsible for the production of the neutralinos, here 0 2 0 2 λ121/mf˜. For the y-axis we choose the coupling responsible for the decay of the neutralinos, here λ112/mf˜. The existing bounds on the RPV couplings are again shown as solid hashed lines for 3 different sfermion masses: 250 GeV, 1 TeV, 5 TeV.

0 and λ112. Neutralino production then occurs via the de- mf˜, cf. Sect. II B. ± 0 ± 0 2 cay D → χ˜1 + e which is proportional to (λ121) . The We see that the SHiP experiment can improve the cur- 0 2 same coupling also leads to neutralino decay via the pro- rent bounds on λ /m ˜ by one to three orders of mag- cessχ ˜0 → (K0 ,K0 ,K∗0) + ν which contains no charged f 1 L S nitude depending on m ˜. The kinematically accessible particles in the final state and will therefore be difficult f range for m 0 is dictated by the requirements that the χ˜1 to observe. However, these decays do impact the neu- neutralino must be lighter than the D-meson, yielding an tralino lifetime. The relevant information is summarized upper bound, while at the same time being heavier than in Table III. the K-meson, thus giving a lower bound. We find that 0 Because we have turned on the coupling λ112 the neu- the discovery region is mostly independent of m 0 , i.e. 0 ± ∗± ∓ χ˜1 tralino can furthermore decay viaχ ˜1 → (K ,K )+e , the lower solid curve edge of the discovery range is fairly which is possible to detect at SHiP. This coupling also flat, as long as the mass lies within this kinematically leads to the same invisible decay to neutral kaons and allowed range between 500 and 1900 MeV. 0 neutrinos as λ121. The invisible decays are important to The additional small lighter shaded region marked by include in the computation. the dashed line indicates the extended sensitivity if the We now present our results. The expected num- SHiP detector could detect neutral kaons in the final ber of events depends on three independent parameters: state, as well. In this particular scenario, this barely 0 2 0 2 λ /m , λ /m , and the neutralino mass m 0 . We 121 f˜ 112 f˜ χ˜1 affects the exclusion contours because the branching ra- find it convenient to present our results in two different tios to visible and invisible final states are roughly the ways. At first, we assume the RPV coupling constants same. However, as we see below, in other scenarios the 0 0 0 to be equal, λ121 = λ112 ≡ λ . In Fig. 3a) we show the difference can be more substantial. number of expected visible neutralino decays in the SHiP 0 2 −5 −2 If λ /m ˜ becomes larger than 10 GeV , the neu- detector as event rate iso-curves which are functions of f 0 2 tralinos decay too fast, in fact mostly before reaching λ /m and mχ˜0 . The bright blue area bounded by a thick f˜ 1 the detector. We note that this parameter region is solid line shows the expected maximum sensitivity curve subject to large numerical uncertainties in our Monte for the SHiP experiment. This area in parameter space Carlo simulation approach, and as such the exclusion gives rise to ≥ 3 neutralino decays within the detector lines show fluctuations with no underlying physical cause. 20 into charged final state particles for 2 · 10 protons-on- In Fig. 3b) we remove the restriction that the couplings target. The corresponding expected meson production 0 0 λ121 and λ112 are equal. Instead, we present the SHiP rates are listed in Sec. VII. To show how the event rate sensitivity depending on the separate couplings for three 0 2 increases with λ /m ˜ and mχ˜0 , we also show areas for representative values of m 0 . We choose a light mass f 1 χ˜1 ≥ 3 · 103 (blue) and ≥ 3 · 106 (dark blue) observable de- close to the lower kinematic threshold (600 MeV, bright cays. The horizontal hashed lines depict existing bounds blue), a heavy one close to the higher kinematic thresh- on the couplings for various values of the sfermion mass old (1800 MeV, dark blue) and one halfway between the 11

a) b)

0 0 FIG. 4. Search sensitivity for Benchmark Scenario 2. The labelling is as in Fig. 3, except for the couplings λ112, λ122. other two (1200 MeV, blue). In the same manner as be- If SHiP is sensitive also to neutral final states, the fore, existing limits on the two couplings are plotted for above results also apply to the cases three representative values of m . Again we see that the f˜ 0 0 6 contours are fairly insensitive to the neutralino mass. In Production: λi21, Decay: λj21, i, j = 3 . (69) all cases, SHiP probes a new region of parameter space The sensitivity curve in this case would be very similar even for 5 TeV sfermion masses. The shape of the sensi- to the shaded region in Fig. 3a). tivity regions are due to both couplings defining the neu- 0 tralino lifetime, but only λ121 leads to the production and 0 only λ112 to the observable decay of the neutralinos. To B. Benchmark Scenario 2 0 0 avoid confusion, let us call the couplings λP and λD re- 0 2 spectively in the following discussion. If λP /mf˜ becomes 0 0 too small, too few neutralinos are produced in the first λP for production λ122 0 0 place. This leads to an overall minimum requirement on λD for decay λ112 0 2 λP /mf˜. produced meson(s) Ds 0 2 visible final state(s) K±e∓,K∗±e∓ For increasing λP /mf˜, more neutralinos are produced 0 0 and thus the allowed neutralino lifetime to observe 3 invisible final state(s) via λP (η, η , φ) + (ν, ν¯) 0 0 0 ∗ events at SHiP can be reached for increasingly larger invisible final state(s) via λD (KL,KS ,K ) + (ν, ν¯) 0 2 ranges of λD/m ˜. As before, too small/large couplings f TABLE IV. Features of Benchmark Scenario 2. lead to too many neutralinos decaying after/before the 0 0 detector. Furthermore, smaller λD/λP ratios lead to more invisibly decaying neutralinos. In Fig. 3b) this cor- This is similar to the previous benchmark, except the responds to the slanted edge running from the upper left- production of the neutralinos is via Ds mesons. The ob- hand corner to the lower right-hand corner. servable charged final states are the same. There are Once λ0 /m2 becomes too large, the lifetime induced however further invisible neutral final states, which are P f˜ kinematically accessible: (η, η0, φ) + (ν, ν¯). The details by this operator is already too small and neutralinos de- of this benchmark scenario are summarized in Table IV. cay mostly before reaching the detector, regardless of The results for the SHiP sensitivity are presented in λ0 /m2 . This explains the sensitivity limitations on the D f˜ Fig. 4a). The reach is extended to higher neutralino right edge of Fig. 3b. masses, as MDs > MD. The lower edge is still given 0 2 The analysis described here applies to the cases: by m 0 ≈ MK . The sensitivity in λ /m is slightly χ˜1 f˜ 0 0 Production: λi21, Decay: λj12, i, j =6 3 . (68) weaker for two reasons. First, the production rate Ds mesons is lower than D± mesons, see Table II. Second, Note that for i = 2, neutralinos are produced via D± → 0 ± the neutralino branching ratio to an observable charged χ1 + µ and the extra muon would shift the upper kine- final state is smaller. Correspondingly the sensitivity is ≈ matical limit of all regions in Fig. 3 by mµ 100 MeV enhanced more in this scenario if neutral mesons are ob- toe the left. Analogously, the case j = 2 would move the servable at SHiP, shown by the dashed line in the bottom lower kinematical limit of all but the shaded regions by of Fig. 4b). the same amount to the right. Within this scenario, the Analogously to Benchmark Scenario 1, the analysis de- cases i = 3 and/or j = 3 would not be observable as scribed here applies to the cases there would not be enough phase space to produce a τ 0 0 lepton. Production: λi22, Decay: λj12, i, j =6 3 . (70) 12

a) b)

0 0 FIG. 5. Search sensitivity for Benchmark Scenario 3. The labelling is as in Fig. 3, except for the couplings λ112, λ131. Also in b) the neutralino masses are 1000 MeV (light blue), 3000 MeV (blue), and 5000 MeV (dark blue). Note that the x-axis range for ﬁgure a) has been changed compared to previous cases, as the initial state B meson allows for a larger kinematical reach.

If the neutral final-state mesons are visible, it also applies to right, much more so than in Figs. 3a) and 4a). This ef- to the cases fect is due to the presence of the final-state vector mesons ∗ 0 0 0 (K ), which are more important for the heavier neutrali- Production: λi22, Decay: λj21, λj22, i, j =6 3 . (71) nos, accessible in B-meson decays. This is discussed in more detail in Section VIII G, below. The added sensitivity due to possible neutral final states is marginal, as C. Benchmark Scenario 3 the branching ratios are comparable. As can be seen from the numbers in Table II, the B-

0 0 meson production rate is roughly four orders of magni- λP for production λ131 tude smaller than the D-meson production rate. As the λ0 for decay λ0 0 2 2 D 112 neutralino production is proportional to (λ /m ˜) , the 0 ¯0 f produced meson(s) B , B curves in Fig. 5b) are shifted by almost two orders of ± ∓ ∗± ∓ visible final state(s) K e ,K e magnitude to the right, compared to the corresponding 0 invisible final state(s) via λP none results, e.g. Fig. 3b), of the previous benchmark scenar- 0 0 0 ∗ invisible final state(s) via λD (KL,KS ,K ) + (ν, ν¯) ios. The new sensitivity reach of SHiP is thus smaller 0 here. However, since λ131 does not induce any invisi- TABLE V. Features of Benchmark Scenario 3 ble decays of the neutralino, the sensitivity regions in Fig. 5b) are not bounded on the right, in contrast to anal- In this scenario the neutralino production is via B 0 2 ogous regions of previous scenarios. Increasing λ131/mf˜ mesons. The decay of the neutralino via the coupling then always leads to an increased number of expected 0 λ112 leads to charged K-mesons and electrons, which are neutralinos and hence always improves the sensitivity to readily visible. The coupling λ0 also leads to neutral 0 2 112 λ112/m ˜. We note that Fig. 5b) has the same character- neutralino decays to K0 mesons and neutrinos. There f istic shape as Fig. 6 in Ref. [26]. are no additional kinematically accessible invisible neu- 0 The analysis described here applies to the cases tralino decay modes through the coupling λ131. This information is summarized in Table V. Production: λ0 , λ0 , Decay: λ0 , i, j =6 3 . (72) The results of the simulation in this scenario are pre- i31 i13 j12 sented in Fig. 5. The kinematically accessible neutralino Again, if neutral final state mesons can be observed it mass range is M ± < m 0 < M 0 . This is reflected in K χ˜1 B also applies to the cases the shape of the sensitivity region in Fig. 5a), which is cut 0 0 0 off on the left at a neutralino mass of about 500 MeV and Production: λi31, λi13, Decay: λj21, i, j =6 3 . (73) on the right just under 5.3 GeV. In the top-right corner, 0 2 where λ /m and m 0 are large, the neutralino lifetime Although as we discuss below in Section VIII D, in this f˜ χ˜1 becomes very short. The neutralinos then overwhelm- case there are additional charged decay modes via D- ingly decay before the detector. Since so few neutralinos mesons. reach the detector, we are here probing the extreme tail of the exponential decay distribution. Consequently, the top-right part of the curve is jagged, due to lack of statistics in this regime. The lower curve slopes downward left 13

a) b)

0 0 FIG. 6. Search sensitivity for Benchmark Scenario 4. The labelling is as in Fig. 3 except for the couplings λ131, λ121. Also in b) the neutralino masses are 2000 MeV (light blue), 3500 MeV (blue), and 5000 MeV (dark blue).

D. Benchmark Scenario 4 E. Benchmark Scenario 5

0 0 0 0 λP for production λ313 λP for production λ131 λ0 for decay λ0 λ0 for decay λ0 D 312 D 121 0 ¯0 ± ∓ 0 ¯0 produced meson(s) B , B ,B (+ τ ) produced meson(s) B , B ± ∓ ∗± ∓ ± ∓ ∗± ∓ visible final state(s) K τ ,K τ visible final state(s) D e ,D e 0 0 invisible final state(s) via λP none invisible final state(s) via λP none 0 0 0 ∗ invisible final state(s) via λ0 (K0 ,K0 ,K∗) + (ν, ν¯) invisible final state(s) via λD (KL,KS ,K ) + (ν, ν¯) D L S

TABLE VI. Features of Benchmark Scenario 4 TABLE VII. Features of Benchmark Scenario 5. At the end of the third row we emphasize that the charged B-meson decay to the neutralino is accompanied by a tau. In this scenario the neutralinos are also produced via 0 B-mesons and the coupling λ131. However, now the decay Here the production goes via B-mesons and the cou- 0 0 is into D mesons via the coupling λ121. There are kine- pling λ313. We thus consider the third generation lepton matically accessible invisible decays to neutral K mesons index. The features of this benchmark scenario are sum- 0 and neutrinos via λ121 This is summarized in Table VI. marized in Table 7. At the end of the third row, the The results of the simulation for this scenario are dis- (+ τ ±) indicates that the charged B-meson decay to a played in Fig. 6. In the left panel, we show the search neutralino is accompanied by a tau, sensitivity as a function of the neutralino mass and of a B± → χ˜0 + τ ± . (75) 0 2 0 2 ≡ 0 2 1 common coupling λ113/mf˜ = λ112/mf˜ λ /mf˜. The mass sensitivity range is again mainly fixed kinemati- Therefore the charged B± meson can only contribute for cally: M < m 0 < M , which is narrower than in the restricted neutralino mass range mχ˜0 < MB − mτ . D χ˜1 B 1 Fig. 5 due to the larger D-meson mass. The corresponding neutral B-meson decay has neutri- When allowing for neutral final states the sensitivity is nos in the final state and is thus allowed for the larger range m 0 < MB. In Fig. 7a) this leads to a kink in dramatically increased to lower neutralino masses, cor- χ˜1 responding to the kinematic range MK < m 0 < MB. the solid curves surrounding the sensitivity regions at χ˜1 6 mχ˜0 = MB − mτ . The region corresponding to ≥ 10 This is shown in Fig. 6a) by the very light blue region 1 bounded by the dashed line. events is also cut off to higher neutralino masses compared to Fig. 6a). In Fig. 6b) the sensitivity range is similar to Fig. 5b), The neutralino decay proceeds via the coupling λ0 . as it is dominated by B-meson production. The dif- 312 0 ± ∓ ferences in the curves are mainly due to the different χ˜1 → K + τ . (76) neutralino masses that are considered: 2000 MeV (light blue), 3500 MeV (blue), and 5000 MeV (dark blue). Thus the visible final states involve charged K-mesons and tau leptons, which might be difficult to detect, es- The analysis described here also applies to the cases pecially in the hadronic decay mode. This also requires m 0 > MK + mτ ' 2300 MeV. This is the cutoff on the 0 0 0 χ˜1 Production: λi31, λi13, Decay: λj21 i, j =6 3 . (74) left of the blue regions in Fig. 7a). 14

a) b)

0 0 FIG. 7. Search sensitivity for Benchmark Scenario 5. The labelling is as in Fig. 3 except for the couplings λ313, λ312. Also in b) the neutralino masses are 2750 MeV (light blue), 3750 MeV (blue), and 5000 MeV (dark blue).

There are possible additional neutral final states involving neutral K-mesons and neutrinos and no tau lepton. When these are included the sensitivity reach is dramatically extended to lower neutralino masses as can be seen in Fig. 7a), just as in Fig. 6a). Fig. 7b) shows the sensitivity region as a function of the 0 2 0 2 two now independent couplings λ313/mf˜ and λ312/mf˜ for the three neutralino masses 2750 MeV (light blue), 3750 MeV (blue), and 5000 MeV (dark blue). These are slightly modified compared to Scenario 4, because of the tau mass, leading to slightly different curves. The analysis described here applies to the cases

0 0 0 Production: λi13, λi31, Decay: λj12 (77)

0 with either i or j or both equal to 3. In case of λi31 only neutral B-mesons can decay into a neutralino and FIG. 8. Sensitivity curve for enchmark Scenario 3 if decays corresponding neutrino such that the sensitivity curves into vector mesons are ignored. The pink region shows the are independent of i. The curves look similar to those sensitivity curve including vector mesons which corresponds in Fig. 7, but the kink around MB − mτ ' 3500 MeV to the “SHiP sensitivity reach” of Fig. 5a) does not appear. The case j = 1 leads to similar limits . as the dashed region drawn in Fig. 7b), whereas similarly to previous scenarios it is shifted by mµ to the right for j = 2. Similarly the decay in Scenarios 4 and 5 are easily ex- 0 0 tended to the cases λD = λ122, where Ds mesons appear in the final state. This gives rise to very similar sensi- F. Related Scenarios tivity curves, the only difference being a slightly larger lower-mass reach for the neutralino. As we have seen the sensitivity curves are largely shaped by the kinematics. There are thus some related scenarios which we briefly describe here. Benchmark sce- G. Relevance of Vector Mesons narios 3, 4 and 5 are easily extended to the cases As discussed at the end of Sec. V A, the inclusion of Production: λ0 , λ0 , (78) i32 i23 vector mesons in the final state leads to a reduced sensitivity to potential fine-tuning of the SUSY parame- by considering the production via Bs mesons. As can be ters. As discussed at the end of Sec. V A, the inclu- seen from Table II, Bs production is suppressed compared b¯b b¯b sion of vector mesons in the final state lead to com- to B-meson production by roughly a factor nB0 /nB0 ≈ s plementary dependence on SUSY paramaters. In ad- 0.18. The sensitivity to the coupling λ0/m2 is reduced √ f˜ dition, the final state vector mesons lead to an inter- by roughly a factor 4 0.18 ≈ 0.65. esting kinematical enhancement of the neutralino de- 15 0 χ˜1 Ld 15 Ncc¯ 1.5 × 10 0 −3 ATLAS Detector Li Li σb¯b/σcc¯ 8.6 × 10

O cc¯

R nD± 0.59

I ¯ θi R bb Primary Vertex nB± 0.87 b¯b nB0 0.87

TABLE VIII. Numerical values used to√ estimate the number FIG. 9. Schematic overview of the ATLAS detector geometry NM in Eq. (60) for the LHC with s = 14 TeV and an −1 and definition of distances and angles used in text. integrated luminosity of 250 fb . All numbers are evaluated by simulating 1M events of each HardQCD type in Pythia. cay width. This enhancement can be understood from the decay width formulæ in Eqs. (40) and (41). When Benchmark Scenarios 1 and 5. These involve the observ- m 0 m , where m is the mass of the final state able decay chains χ˜1 M M meson, the decay to scalar mesons is proportional to ± 0 ± 0 ± ∓ 0 0 S 2 2 2 2 2 D → χ˜1e , χ˜1 → e K via λ121, λ112, (79) (fM ) mχ˜0 ' fM mM mχ˜0 , whereas the decay to vector 1 1 ± 0 0 ± 0 ± ∓ 0 0 T 2 4 2 4 B /B → χ˜ τ /ν, χ˜ → τ K via λ , λ . (80) mesons is proportional to (fM ) m 0 ' fM m 0 . The de- 1 1 313 312 χ˜1 χ˜1 cay into vector mesons is thus enhanced by roughly a 2 2 To compare like with like, we estimate the neutralino factor m 0 /mM . χ˜1 event rates analogously to Sec. VII: we simulate these This is mainly relevant for cases such as our Bench- scenarios using Pythia 8.175 [69, 70], and find a pro- mark Scenario 3, where neutralino production occurs via duction cross section for cc¯ at 14 TeV of 6 · 1012 fb and B-meson decays. Then the neutralino can be significantly −3 σb¯b/σcc¯ = 8.6 × 10 . We consider an integrated lu- heavier than the final-state meson, here the kaon. To il- minosity of 250 fb−1, which roughly corresponds to the lustrate the enhancement, in Fig. 8, we repeat the anal- expected value for a high-energy LHC running for 5 ysis for Benchmark Scenario 3, but exclude final state years. We determine the other parameters of interest vector mesons. The sensitivity contour of the same anal- as in Sec. VII and list them in Table VIII. ysis including vector mesons is shown by the pink shaded As an example we consider the ATLAS detector setup area. This region is identical to the sensitivity area shown as sketched in Fig. 9. Here we assume the detectable in Fig. 6a). We show it again here to make the difference region to approximately range from RI = 0.0505 m, the between the two cases easier to see. The inclusion of vec- beginning of the inner detector, to RO = 11 m, the end of tor mesons is barely visible for neutralino masses smaller the muon chambers. The detector has cylindrical shape than 1 GeV, but the enhancement is clearly visible in the with a total length of 2LD = 43 m. The probability for 2-5 GeV mass range. the neutralino to decay within this range is then, similarly to Eq. (62),

IX. LHC ESTIMATE z 0 z 0 −Li/λi −Li/λi P [(˜χ1)i in d.r.] = e · 1 − e , (81)

As we have seen, the sensitivity at SHiP in the pro- Li ≡ min(Ld, |RI / tan θi|) (82) 0 duction coupling results from an interplay between the Li ≡ min(Ld, |RO/ tan θi|) − Li (83) length scales of the target- detector-distance Lt→d, the z length of the detector Ld, the meson production rate NM , with angles and distances defined in Fig. 9 and λi as the boost γi of the neutralinos and their azimuthal angle defined in Eq. (63). In similar manner as for SHiP, we θi. The LHC operates at much higher energies in the use Pythia to simulate 20,000 events, force all mesons center-of-mass frame and the detectors are built right of the right type to decay into neutralinos and average 0 at the collision points. Thus all the above parameters the results for P [(˜χ1)i in d.r.] over all these Monte-Carlo change and we would expect a different sensitivity when neutralinos to find the overall probability that an LHC- comparing to SHiP4. To estimate the net result of these produced neutralino decays inside the detectable region effects, we thus here briefly discuss the sensitivity for our of the ATLAS detector. The number of observable neu- scenarios at the LHC. tralino decays is then determined by considering the RPV To allow for an easy comparison, we consider two ex- branching ratio of the initially produced mesons and the ample cases which correspond to our earlier discussed branching ratio of neutralinos into charged final states, according to Eqs. (55), (56). The expected sensitivity regions are shown in Fig. 10. For easy comparison we show the same information as 4 We thank Jesse Thaler for drawing our attention to this point. in their corresponding SHiPs analogues Figs. 6a) and See also [72] on a related discussion on dark photons. 7a) and focus on the 3 event threshold which would be 16

a) b)

FIG. 10. Expected LHC search sensitivity for a) Benchmark Scenario 1 and b) Benchmark Scenario 5. The labelling is as in Figs. 3,7. required for a significant observation. We restrict the The large center-of-mass energy of the LHC leads to 0 2 discussion to the m 0 –λ /m –plane as the limits in the an even larger average boost, hγi ≈ 55, which however χ˜1 f˜ λ0 /m2 –λ0 /m2 –plane can be easily deduced. has a larger spread, resulting in many neutralinos with prod. f˜ dec. f˜ boost of O(1) and a few with boost O(1000). As shown For both scenarios, the structure of the plots do not in Fig. 11, the peak of the boost distribution for the LHC largely differ between ATLAS and SHiP. The testable is located at γ = 2.5 and the resulting large fraction of kinematic regions are obviously identical for the same 0 2 unboosted neutralinos improve the overall probability to scenario and it is only the required value for λ /mf˜ to ob- observe a decay within ATLAS. serve enough neutralino decays which changes. At SHiP Combining the above effects, we find that we found that the expected sensitivity quickly drops if 0 hP [˜χ1 in d.r.]i at ATLAS is greater by a factor 4 the neutralinos decay too promptly, that is before they than at SHiP. Taking into account the much larger reach the decay chamber at roughly 70 m behind the meson production yield of SHiP, it is expected to ob- target. This resulted in an upper limit on the couplings serve approximately 25 times more events than ATLAS, SHiP would be sensitive to. However, as the detectable leading√ to an improved sensitivity on the coupling of region at ATLAS already starts at O(cm) distances from 4 25 ≈ 2.2. the primary vertex, this upper limit is pushed to higher For Scenario 5, we compare Figs. 7a) and 10b) and values. interestingly find very comparable expected sensitivities. Comparing the results for Scenario 1, Figs. 6a) and All the effects discussed for the previous scenario equally 10a), we find that LHC has a comparable but still by a apply here and lead to approximately similar results. 0 2 factor 2 weaker expected sensitivity on λ /mf˜ than the Therefore one would still expect SHiP to observe roughly expected value from SHiP. From comparing the respec- 25 times more decays. However, this scenario requires tive values for Ncc¯ in Tables VIII and VIII, one expects B-mesons to be initially produced. The larger center-of- SHiP to produce almost 100 times more neutralinos than mass energy at LHC leads to an increased relative pro- the LHC in a comparable time frame. This is partially duction yield σb¯b/σcc¯ of approximately 40 (see Tables II, compensated by the effect that neutralinos which are pro- VIII). This results in a roughly 60 % larger overall ex- duced at large angles θ can be observed at the almost pected event rate at the LHC, which however is a negli- 0 2 spherical ATLAS detector but miss the decay chamber gible improvement when translated into a limit on λ /mf˜. at SHiP. It is clear that the results we show can just serve as Furthermore, the boost distribution of the two exper- a very approximate comparison. As we do not know the iments largely differ. The fixed target setup of SHiP efficiency with which SHiP would be able to detect a neu- causes most produced mesons to have a large boost which tralino decay and distinguish it from Standard Model, we is inherited by the daughter neutralinos they decay into. did not take it into account for our LHC discussion ei- Contrarily, the center-of-mass collision for the LHC will ther. However, it can be expected that the final state effi- lead to most mesons to be produced at rest. We show the ciencies for the two experiments differ significantly, most boost distribution of the neutralinos we get with Pythia likely with a significant penalty on the ATLAS side. SHiP in Fig. 11. For SHiP, the distribution shows an expec- will be specifically designed to observe rare decays of new, tation value of hγi ≈ 30 and a maximum probability for long-lived particles. It can therefore be expected that the γmax ≈ 7.5. This leads to an increased lifetime in the neutralino decays will have a large probability to actu- lab frame which reduces the detection probability if cτχ ally be measured by the detector. The ATLAS detector, is larger than the size of the detector, see e.g. Eq. (62). however, is not designed for this purpose. The combined 17

√ FIG. 11. Boost distribution for neutralinos produced at a s = 14 TeV LHC and at SHiP, determined with Pythia. We show the same results for two different choices of axis scaling. The fluctuations for γ & 300 are caused by limited Monte Carlo statistics. efficiencies to trigger on the event, to reconstruct the fi- have classified benchmark scenarios for different combi- nal state particles, to identify the significantly displaced nations of the generation indices i, j, a, b, c, d. Although vertex and to distinguish it from Standard Model mesons many different combinations of couplings exist, we have decays will most likely lead to a significant reduction of argued that most of them can be captured in this rel- the final event yield, potentially by orders of magnitude. atively small set of benchmark scenarios. We highlight Still, we find it an interesting observation that when just here a number of conclusions and caveats of our findings. considering the geometry of the setup, the meson produc- • We find no feasible scenario where SHiP is sensitive tion yield and the expected kinematics of the neutralinos, 0 SHiP and ATLAS seem to have comparable sensitivity to to only a single λiab coupling. The main obstacle the discussed decay scenario. Thus we conclude that the for such a scenario is that the final state decay prod- final state reconstruction efficiency will play a crucial role ucts will consist of a neutrino and a neutral meson in determining the importance of the SHiP experiment from which it is difficult to reconstruct the neutralino decay. An example of such a scenario would with regards to the search for light neutralinos. 0 be a nonzero λi21 coupling, which would lead to the ± 0 ± production and decay channels D → χ˜1 + li and 0 X. SUMMARY χ˜1 → KS,L + ν. In order to get an observable final state we thus always require two distinct nonzero λ0 couplings. We have found, however, that includ- In this work we have studied the sensitivity of the pro- ing the invisible final states is mandatory as they posed SHiP experiment to the R-parity violating pro- influence the neutralino lifetime. duction and decay of neutralinos whose masses lie in the range of 0.5 − 5 GeV. As discussed in Sect. III, a neu- • That being said, we have shown that the SHiP tralino in this mass range is only allowed if R-parity is experiment has the potential to significantly im- violated. We have focused on the semi-leptonic R-parity prove the constraints on various combinations of violating operators λ0LQD¯, but our work is easily ex- R-parity violating couplings. This is clearly illus- tended to the purely leptonic case λLLE¯. The basic idea trated in the figures in Sect. VIII. For instance, pursued in this work is that a small fraction of the D and 0 2 constraints on λ112/mf˜ can be strengthened by B mesons produced in the SHiP experiment, can poten- one to three orders of magnitude (see Fig. 3), de- tially decay into a neutralino plus lepton. Because the pending on the sfermion mass. Similar improve- R-parity violating couplings are expected to be small, the ments are found for third generation couplings neutralino can have a sufficiently long lifetime to travel a such as λ0 λ0 /m4 . We have presented sensi- distance of 63.8 m to the ShiP detector where it can sub- 131 121 f˜ tivity curves for each of the benchmark scenarios: sequently decay into a, presumably detectable, meson- [λ0 , λ0 ], [λ0 , λ0 ], [λ0 , λ0 ], [λ0 , λ0 ], lepton pair. 121 112 122 112 131 112 131 121 [λ0 , λ0 ]. These curves can be used to estimate For neutralinos in the mass-range of 0.5 − 5 GeV, the 313 312 the sensitivity of the SHiP experiment to various SHiP experiment is sensitive to several combinations of combinations of R-parity violating interactions, as R-parity violating couplings. In general, the number of outlined in the text. neutralino decays in the SHiP detector is proportional to 0 0 2 8 (λiabλjcd) /mf˜, where i, j denote the lepton generation • We have focused on neutralino production via the indices and a, b, c, d the quark generation indices. We decay of B and D mesons. Very light neutralinos 18

could be produced in kaon decays and be detected roughly 4 % of the number of events expected for 0 2 by their subsequent decay into pionic final states. SHiP, leading to an expected limit on λ /mf˜ which We have not included this in this work as the pro- is weaker by roughly a factor of 2. This is caused by duction of light mesons is not well simulated in the a combination of a larger meson flux expected for forward direction with Pythia. We aim to study SHiP and a higher detection probability for long- this in future work as it would extend the sensi- lived neutralinos at ATLAS. For initially produced tivity to the neutralino mass range 0.1 − 0.5 GeV. B mesons, the expected sensitivities are very sim- Similarly, we plan to include neutralino production ilar, as the large LHC energies will produce rela- in the decays of ¯bb mesons, which would give a sen- tively more b-quarks. It is therefore the final state 0 sitivity to λi33 and to higher-mass neutralinos. reconstruction efficiency which will be the decisive factor. • We have found that including vector mesons in the final state leads to an enhanced sensitivity to neutralinos at the higher end of the allowed mass range. This enhancement can be understood from Note added the decay-width formula presented in Sect. V B as discussed in Sect. VIII G. In addition, the neu- While completing this work, a related study appeared tralino decay into vector mesons is proportional to [73]. They consider the production via B-mesons as in a different combination of SUSY parameters than [26] and the decay to kaons or purely leptonically. They the corresponding decay into scalar mesons. The make extensive use of the formulæ in the original study processes are therefore complementary. as presented by two of us (HKD and DS) in [30], and thus employs the error we made there, as discussed here • Our analysis did not include possible uncertainties in footnote 1. arising from hadronic matrix elements. The decay constants used are not in all cases known to high precision. Nevertheless, the SHiP sensitivity curves ACKNOWLEDGMENTS range over many orders of magnitude and we do not expect that changes in the decay constants drasti- cally change our conclusions. This work (JdV) is supported in part by the DFG and the NSFC through funds provided to the Sino-German • We determined the expected sensitivity of the AT- CRC 110 “Symmetries and the Emergence of Structure LAS detector at a 14 TeV LHC with an integrated in QCD” (Grant No. 11261130311). We thank Christoph luminosity of 250 fb−1. To do a fair comparison, Hanhart for several useful discussions. We thank Jesse we did not take into account the final state re- Thaler for drawing our attention to the possibility of construction efficiency and only determined the ex- searching for these scenarios at the LHC. One of us (HD) pected number of neutralino decays inside the de- thanks the Galileo Galilei Institut in Florence for hospi- tector region of ATLAS. We found that in scenarios tality, where a significant part of this work was com- with initially produced D mesons, ATLAS expects pleted.

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