Simultaneous B and L Violation: New Signatures from RPV-SUSY

Studies of R-parity violating (RPV) supersymmetry typically assume that nucleon stability is protected by approximate baryon number (B) or lepton number (L) conservation. We present a new class of RPV models that violate B and L simultaneously (BLRPV), without inducing rapid nucleon decay. These models feature an approximate $Z_2^e \times Z_2^\mu \times Z_2^\tau$ flavor symmetry, which forbids 2-body nucleon decay and ensures that flavor antisymmetric $L L E^c$ couplings are the only non-negligible L-violating operators. Nucleons are predicted to decay through $N \rightarrow K e \mu \nu$ and $n \rightarrow e \mu \nu$; the resulting bounds on RPV couplings are rather mild. Novel collider phenomenology arises because the superpartners can decay through both L-violating and B-violating couplings. This can lead to, for example, final states with high jet multiplicity and multiple leptons of different flavor, or a spectrum in which depending on the superpartner, either B or L violating decays dominate. BLRPV can also provide a natural setting for displaced $\tilde{\nu} \rightarrow \mu e$ decays, which evade many existing collider searches for RPV supersymmetry.


Introduction
R-parity violating (RPV) [1,2,3,4,5,6] supersymmetry (SUSY) has become increasingly well motivated, due to comparatively weaker collider bounds on colored sparticle production [7,8], along with null results for MSSM dark matter. If both baryon number (B) and lepton number (L) violating RPV couplings are present, 4-fermion effective operators of the form qqqℓ will induce 2-body nucleon decay [9]. Consequently, Super-Kamionkade bounds on 2-body nucleon decay (τ N →M ℓ 10 34 years) [10] strongly constrain the product of B and L violating couplings for ∼ TeV scale superpartners. In order to avoid these bounds, the canonical approach is to assume that either B or L is approximately conserved. This leads most authors to consider two broad classes of RPV SUSY models: those which violate B (BRPV), and those which violate L (LRPV).
The main purpose of this paper is to establish the existence of a third class of RPV models: those which violate B and L simultaneously. We refer to this class of models using the acronym BLRPV. This possibility arises because λ ijk L i L j E c k superpotential couplings which are antisymmetric in flavor indices 1 i.e. i = k, j = k do not generate dangerous 4-fermion qqqℓ effective operators in the presence of the λ ′′ U c D c D c BRPV couplings.
Instead, the combination of λ ′′ U c D c D c and λ ijk LLE c , i = k, j = k couplings generate 6-fermion effective operators, resulting in the nucleon decay modes N → Kνe ± µ ∓ and n → e ± µ ∓ ν. There have been no recent experimental attempts to search for these decay modes; a discovery in these channels would provide strong evidence for BLRPV. We will show that the experimental constraint τ (N → µ + inclusive) 10 32 years [11] results in the bound |λ ′′ 112 λ ijk | 10 −10 for i = k, j = k assuming ∼ 1 TeV superpartners; this bound weakens to ∼ 10 −4 − 10 −3 for λ ′′ couplings with heavy flavors. Such comparatively weak bounds allow both λ ′′ U c D c D c and λ ijk L i L j E c k , i = k, j = k couplings to be relevant for collider phenomenolgy, without violating nucleon decay bounds. For a consistent model of BLRPV, one expects a symmetry to enforce the flavor antisymmetry of L i L j E c k while suppressing/forbidding all other LRPV couplings. If λ ijk L i L j E c k , i = k, j = k are the only non-vanishing LRPV couplings, there is a Z e 2 ×Z µ 2 ×Z τ 2 flavor symmetry which is exact in the absence of neutrino masses. Under Z τ 2 , both L τ and E c τ are even while all other lepton superfields are odd; Z µ 2 and Z e 2 are similarly defined. This symmetry forbids all effective operators of the form qqqℓ, providing an intuitive explanation for the absence of 2-body proton decay. Z e 2 × Z µ 2 × Z τ 2 must be broken by neutrino masses and mixing angles for realistic neutrino phenomenology [12]; we will argue that the resulting bounds from qqqℓ operators induced by neutrino masses are mild.
The collider phenomenology of BLRPV can give distinct LHC signatures which differentiates it from standard RPV SUSY. Such novel phenomenology occurs because in BLRPV, sparticles can decay via both LRPV and BRPV couplings. For example, if a sparticle's decay rates via BRPV and LRPV are comparable, new signatures from sparticle pair pro-duction arise e.g.qq → qqχ + χ − → 5q eµτ . Such final states will be characterized by high jet multiplicity and multiple leptons of different flavor. Alternatively, different sparticles can decay predominantly via either BRPV or LRPV. For instance, a mostly RH squark can decay predominantly via the U c D c D c coupling, while the LH squark decays to the LSP which subsequently decays through LLE c . We discuss illustrative examples which highlight these qualitative features, saving a more general study for future work. We also review existing LHC searches for RPV SUSY, which are straightforward to apply to BLRPV scenarios.
Finally, we discuss the phenomenology of displacedν → µe decays. Though this decay mode is certainly not unique to BLRPV, it can occur naturally in the BLRPV framework for a τ -sneutrino LSP, due to the flavor structure in LLE c couplings enforced by Z e 2 × Z µ 2 × Z τ 2 . It has been noted in [13] that displacedν → µe decays can mitigate both LEP and LHC constraints on sleptons and charginos decaying via LRPV. Here we discuss how displaced ν τ → µe decays can also decrease the multiplicity and/or p T of jets resulting from squark or gluino cascade decays. This weakens constraints from high jet multiplicity searches, which otherwise provide robust bounds on squarks and gluino masses ( 1.5 TeV) in RPV SUSY [14]. Thus, events such asgg → 4j + 2ν τ → 4j + 2(µe) where theν τ decay is displaced represent a potential blindspot in existing experimental analyses. We urge experimentalists to search for final states with multiple hard jets and displaced leptons to fill this gap. This paper is organized as follows. In Section 2, we compute nucleon decay bounds on |λ ′′ λ ijk | for i = k, j = k. In Section 3, we discuss the Z e 2 × Z µ 2 × Z τ 2 symmetry of BLRPV, and discuss implications of Z e 2 × Z µ 2 × Z τ 2 breaking due to neutrino masses. In Section 4, we discuss the collider phenomenology of BLRPV. In Section 4, we discuss existing collider bounds on BLRPV, and how they can be mitigated by displacedν τ → µe decays. We conclude in Section 6.

Nucleon Decay Phenomenology of BLRPV
The R-parity violating superpotential is given by 2 : To simplify terminology, we will refer to the U c D c D c operator as the BRPV operator, and the other lepton number violating operators as LRPV operators.
If λ ′′ is non-vanishing, the presence of non-vanishing λ ′ , λ nkk or κ i will induce 2-body nucleon decay modes such as p → π + ν and p → K + ν via the 4-fermion effective operators depicted in Figure 1. Assuming a common superpartner mass scale M SU SY , the Super- Kamiokande constraint τ N →M ℓ 10 34 years [10] results in the bounds: where m ℓ j is a SM lepton mass e.g. m ℓ 3 = m τ .
In RPV scenarios, the canonical approach is to satisfy (2) by assuming that either the BRPV or LRPV violating couplings are negligible. However, if the only non-vanishing LRPV couplings are λ ijk L i L j E c k for i = k, j = k in the lepton mass eigenstate basis, the diagrams of Figure 1 vanish (absent flavor-changing slepton mass insertions). In this limit, the leading diagrams which induce nucleon decay are those similar to Figure 2, resulting in 6-fermion effective operators and the 4-body decay modes In Section 2.1, we will compute Γ(N → Kν i ℓ + j ℓ − k ) using chiral Lagrangian techniques, and obtain the experimental bound |λ ′′ 112 λ ijk | 10 −10 for i = k, j = k, assuming a common superpartner mass scale of 1 TeV. In Section 2.2, we compute analogous bounds for λ ′′ ijk couplings with at least 2 heavy (c, b, t) quark flavors, which induce nucleon decay via penguinlike loop diagrams with flavor changing W ± , H ± , χ ± exchange [17]. The resulting bounds are a factor of 10 6 − 10 7 weaker than the corresponding bounds on λ ′′ 112 . Figure 2: An example of the diagrams which generate 6-fermion effective operators in the presence of BRPV and λ ijk , i = k, j = k couplings. Such operators induce the 4-body nucleon decay modes N → Kν i ℓ − j ℓ + k .

Constraints on |λ ′′
A representative example of the diagrams relevant for nucleon decay in the presence of λ ′′ and λ ijk , i = k, j = k couplings is depicted in Figure 2. There are also analogous diagrams involving virtuald c ,ũ c ,l andl c exchange, corresponding to different permutations of the external quark and lepton legs. Taking into account all the relevant diagrams, integrating out sfermions in the limit of vanishing LR mixing leads to the following 6-fermion effective operators: where A 1 = 1/(2m 2 ν i ) + 1/m 2 ℓ Rk and A 2 = 1/(2m 2 ν i ) − 1/(2m 2 ℓ Lj ). The couplings which enter into (3) are evaluated at a renormalization scale near the superpartner mass scale.
These ∆S = 1 effective operators will induce the decay modes , which induce the 3-body decay modes p → ν i ν j ℓ + k and n → ν i ℓ − j ℓ + k . The coefficients of these operators are flavor suppressed, such that the 3-body decays provide subdominant bounds if the squark flavor-changing mass insertions are not large. This is discussed in more detail in Appendix A.
In order to compute rates for nucleon decay, we match the effective operators in (3) to operators in the chiral Lagrangian [18]. To do this, note that the operators in (3) transform as elements in the (1,8) representation of SU(3) L × SU(3) R . To simplify the calculation, we assume sfermion mass degeneracy and set A 2 = 0 in (3). Adopting 2-component spinor The shaded box represents the B and L violating vertex in the chiral Lagrangian.
notation [19], the corresponding operators in the chiral Lagrangian are [18,20,21]: where Ψ B = (B, B c † ) is the Dirac fermion corresponding to the baryon octet, ξ = exp (iM/f π ) where M are the meson fields, andF ′ ,F ′′ are flavor projection matrices defined in [21]. Parenthesis denote contraction of spinor indices. β is related to the 3-quark annihilation hadronic matrix element i.e. 0|(u R d R )u R |p ≈ βP R u where u is the spinor associated with the proton in Dirac notation. Lattice calculations give β ≈ 0.0118 (GeV) 3 [22].
Matching the operators in (3) and (4) fixes C 1 = C 2 = 3A λ ′′ 112 λ ijk g 2 Y /MBm 4 , wherem is the degenerate sfermion mass and A ≈ 0.22 accounts for QCD effects which renormalize the effective operators in (3) from Q = M SU SY to Q = Λ QCD [23]. Expanding the chiral Lagrangian along with the terms in (4) to first order in 1/f π then gives the necessary terms to compute nucleon decay amplitudes at tree level. This procedure was explicitly demonstrated in [18,20] and carried to completion in [21] so we will not reproduce it here, though we have independently verified the relevant results. Although [21] focuses exclusively on 2-body nucleon decay, the only difference between the present case and the amplitudes computed in [21] is the addition of 2 leptons ν † i ℓ † j see (4) at each B and L violating vertex. Neglecting lepton masses, the p → K + ν i ℓ + j ℓ − k and n → K 0 ν i ℓ + j ℓ − k decay rates are given by 3 : where λ(x, y, z) ≡ (x 2 + y 2 + z 2 − 2xy − 2xz − 2zy) 1/2 and 3 The amplitudes for p → K + ν i ℓ + j ℓ − k and n → K 0 ν i ℓ + j ℓ − k are related by approximate isospin invariance.
The baryon masses M Σ , M Λ enter in (6) via diagrams with virtual Σ, Λ exchange as shown schematically in Figure 3; we neglect chiral symmetry breaking terms in the chiral Lagrangian.
In computing (5) and (6), we have neglected terms in the amplitude proportional to q 2 = (p − k) 2 , where p and k are respectively the nucleon and kaon 4-momenta. This approximation is justified as ; a more precise calculation can be performed by including the q 2 terms which are given in [21]. We use measured values for the chiral Lagrangian parameters D = 0.8 and F = 0.47.
Before proceeding, we discuss the relationship between the results in (5)-(7) and previous work. Proton decay bounds on |λ ijk λ ′′ | were briefly discussed in [26], which did not emphasize the antisymmetry condition i = k, j = k and focused on the subdominant p → ννℓ + decay mode (see Appendix A for further discussion). References [27,16] first noted that λ ijk L i L j E c k , i = k, j = k induces p → K + νe ± µ ∓ in the presence of B violating couplings. However, the dimensional analysis estimate of Γ(p → K + νe ± µ ∓ ) in [27,16] did not account for hadronic matrix elements or phase space factors, resulting in a significantly stronger bound on |λ ′′ 112 λ ijk | than obtained here.

Constraining Heavy Flavor λ ′′ Couplings
In Section 2.1, we focused on finding bounds on |λ ′′ 112 λ ijk | for i = k, j = k. Similar bounds on λ ′′ 113 , λ ′′ 123 , λ ′′ 212 and λ ′′ 312 arise via tree-level diagrams involving flavor changing squark mass insertions; these bounds will be weaker than (7) by a model dependent factor. Depending on these flavor violating parameters, different decay modes such as p → ν i ν j ℓ + k or n → ν i ℓ + j ℓ − k may provide the dominant bound for these λ ′′ couplings, but this is a model dependent question which we will not address here. Figure 4: Loop diagrams which generate proton-decay inducing effective operators for λ ′′ couplings with 2 heavy flavor indices [17]. The blob with / L denotes the L violating part of the diagram, akin to the right-hand side of Figure 2.
However, λ ′′ couplings with at least 2 heavy flavor (c, b, t) indices i.e. λ ′′ 213 , λ ′′ 223 , λ ′′ 313 and λ ′′ 323 will not contribute to nucleon decay at tree level, as tree-level diagrams analogous to Figure 2 will contain at least 2 heavy quark external legs. Instead, the relevant 6-fermion effective operators involving light quarks will be induced by flavor changing loop diagrams involving W ± , charged Higgs and chargino exchange, as depicted in Figure 4. Diagrams of this sort were first discussed in [17], though explicit formulae have yet to appear in the literature.
Upon computing the coefficients of the 6-fermion operators from Figure 4, we follow the procedure outlined in Section 2.1 to compute nucleon decay rates. The λ ′′ 231 and λ ′′ 331 couplings generate effective operators which induce n → ν i ℓ − j ℓ + k , while the λ ′′ 232 and λ ′′ 332 couplings generate effective operators which induce p(n) → K + (K 0 )ν i ℓ − j ℓ + k . The resulting computation is slightly different from Section 2 due to the operator structure of the diagrams in Figure 4. The effective operators corresponding to the diagrams in Figure 4 are: where The L ijk are loop functions determined by summing the loop amplitudes and are derived in Appendix B; the result is given in (21).
The operators in (8) transform as elements in the (3, 3) representation of SU(3) L × SU(3) R , so in the degenerate sfermion (A 2 = 0) limit, the corresponding operators in the chiral Lagrangian are: Here α is defined as 0|(u L d L )u R |p ≈ αP R u in Dirac notation; lattice calculations give α ≈ β ≈ 0.0118 (GeV) 3 [22]. Matching the operators in (8) to the chiral Lagrangian gives : wherem is again the degenerate sfermion mass. The operator with coefficient C ′ 2 induces the decay modes N → Kν i ℓ − j ℓ + k whose rate (neglecting lepton masses) is given by equation (5), with The operator with coefficient C ′ 1 induces the 3-body neutron decay mode n → ν i ℓ − j ℓ + k , whose rate is given by: The bound on the partial width for n → νe + µ − from IMB-3 is similar to the bound on N → µ + inclusive, τ (n → νe + µ − ) 10 32 years [28]. Comparing the computed decay rates with these constraints, we obtain bounds on |λ ′′ lmn λ ijk | , i = k, j = k, for (l, m, n) = (2, 3, 1), (3, 3, 1), (2,3,2) and (3,3,2), which are plotted in Figure 5 as a function of a common superpartner mass scale M SU SY . These bounds are significantly weaker than (7), as the diagrams in Figure 4 are suppressed by quark Yukawa couplings and off-diagonal V CKM elements, along with the usual 1/(16π 2 ) loop factor. We remind the reader that Figure 5 only applies to λ 132 and λ 231 ; bounds on λ 123 are weaker for reasons mentioned above.
We close this section by remarking that discovery of the nucleon decay modes N → Ke ± µ ∓ ν and n → e ± µ ∓ ν without a similar discovery in same flavor lepton modes e.g. N → Ke + e − ν, n → e + e − ν would provide strong evidence for BLRPV. Note that bounds quoted above on τ (N → µ + + anything) [11] and τ (n → νe + µ − ) [28] are more than a decade old. We urge experimentalists to continue searching for these decay modes, as the nucleon decay signatures of BLRPV are significantly more robust and model independent than the collider signatures discussed below (see Section 4).

The Flavor Symmetry of BLRPV
Having seen that a hierarchy between λ ijk L i L j E c K , i = k, j = k and all other LRPV couplings avoids the 2-body proton decay bounds in (2), we now consider the question of whether such a hierarchy can be obtained naturally. This might seem difficult, because once lepton number is violated, there are usually no remaining symmetries which can distinguish the lepton chiral multiplets L i from the Higgs multiplet H d . As a result, the presence of LRPV couplings typically induces wavefunction renormalization mixing of L and H d [2,29], which, for example, radiatively generates a λ ′ QLD c coupling proportional to the down-type Yukawa couplings.
However, even if lepton number is violated, there can still be flavor symmetries acting in the lepton sector which forbid certain LRPV operators. Neglecting neutrino masses, if λ ijk L i L j E c k for i = k, j = k are the only non-vanishing LRPV couplings in the basis where Y E is diagonal, the theory enjoys a global Z e 2 × Z µ 2 × Z τ 2 flavor symmetry. Under Z τ 2 , both L τ and E c τ are even while all other lepton fields are odd; Z µ 2 and Z e 2 are similarly defined. Thus absent neutrino masses, the vanishing of all other L violating couplings is protected by this global symmetry. Z e 2 × Z µ 2 × Z τ 2 also forbids all 4-fermion operators of the form qqqℓ, explaining the absence of 2-body proton decay from an effective operator point of view.
In models with a realistic neutrino sector, Z e 2 ×Z µ 2 ×Z τ 2 will be broken by neutrino masses and mixing angles [12]. Thus we expect the presence of non-vanishing neutrino masses to induce effective 4-fermion qqqℓ operators 4 whose coefficients are proportional to m ν . Such operators are indeed generated, by loop diagrams which induce lepton-gaugino/higgsino mixing. The dominant diagram of this sort is depicted in Figure 6, resulting in the following effective operator: which induces the decay mode n → K ± ℓ ∓ . M SU SY is taken to be the common superpartner mass scale. We have verified that other similar diagrams give qqqℓ operators whose coef-ũ mW ficients are suppressed with respect to (13) by lepton mass insertions, LR squark mixing, and/or powers of spacetime derivatives. Given the bound on |λ ′′ λ ′ | in (2), the resulting bound on |λ ′′ 112 λ ijk | from (13) is weaker than (7) for M SU SY 100 TeV for m ν ∼ 0.1 eV.
Thus if neutrino masses are the only source of Z e 2 × Z µ 2 × Z τ 2 breaking, bounds from 2-body proton decay are mild. However, the same dynamics which breaks Z e 2 × Z µ 2 × Z τ 2 and generates neutrino masses might also regenerate other dangerous LRPV operators. This is a model dependent issue, which depends on the UV dynamics responsible for neutrino mass generation. As an illustrative example, we analyze a particular right-handed neutrino model, (3) N flavor symmetry that is broken by lepton and neutrino Yukawa couplings (see [30] for a similar analysis). Bounds from 2-body proton decay will constrain the right-handed neutrino sector, but can still allow for right-handed neutrinos above the TeV scale, assuming non-holomorphic contributions to the superpotential are sufficiently suppressed. The details of this model and the resulting analysis are presented in Appendix C.

Collider Phenomenology of BLRPV
In this section, we discuss how the collider phenomenology of BLRPV can be qualitatively different from that of canonical BRPV or LRPV. In BLRPV, sparticles can have both BRPV and LRPV decay modes, in addition to the usual R-parity conserving decay modes. The resulting collider phenomenology then depends on branching ratios sparticles to BRPV and LRPV final states.
For clarity, we separate the novel phenomenology of BLRPV into two distinct scenarios: • Scenario A:X → BRPV,X → LRPV. A given sparticleX decays to both BRPV and LRPV final states with comparable branching fractions.
The dichotomoy between Scenarios A and B is somewhat artificial, as the branching ratios to LRPV and BRPV final states for each sparticle can take on a continuum of values in an arbitrary model. Nevertheless we discuss each case separately, and highlight how each scenario can give rise to novel collider phenomenology from sparticle production. We then discuss implications of existing collider constraints on these BLRPV scenarios.
Scenario A:X → BRPV,X → LRPV In this scenario, pair production ofX can lead to novel final states if oneX decays via LRPV and the otherX decays via BRPV. This can be realized if for instanceX is a neutralino/chargino 5 (co)LSP, and the antisymmetric LRPV and BRPV couplings are similar in magnitude.
SupposeX is a Wino-like LSP, in which case there is a nearly mass degenerate chargino NLSP χ + . For approximately degenerate sfermions and sufficiently large 10 −5 BRPV and LRPV couplings, χ + will have a sizeable branching fraction to both χ + → qqq and χ + → eµτ if λ ′′ ∼ λ. This leads to new final states from colored sparticle production: (14) Such final states with large jet multiplicity, 3 hard leptons of different flavors and no MET can differentiate BLRPV from both the R-parity conserving and BRPV/LRPV MSSM. This example illustrates the general point that sparticle pair production in Scenario A leads to final states with large jet and lepton multiplicity, where the leptons will be of different flavors due to the antisymmetric flavor structure in the LRPV couplings.
Scenario B:X 1 → LRPV,X 2 → BRPV There are numerous qualitatively distinct possibilities in Scenario B, depending on the mass spectrum and the magnitudes of the LRPV and BRPV couplings. In this work we will focus on four particular examples involving the simplified spectra depicted in Figure 7. These examples are meant to illustrate generic features of models which fall into this scenario; we save a discussion of the more general case for future work. The four examples in Figure 7 all feature a squark decaying via BRPV, aτ L NLSP, and aν τ LSP which decays via the antisymmetric LRPV couplings toν τ → µe. Our choice of aν τ LSP is motivated by the fact that displaced decays to a µe pair can mitigate constraints from existing RPV searches [13]. We return to this point in Section 5.
In discussing the examples of Figure 7, we assume that λ ′′ 323 = 0.1, and all other BRPV couplings are negligible. This ensures that 3-body decay modes such as asq → q ′ τν τ ,q → Spectrum 3 : q ν τντ can be subdominant to squark BRPV decay modes, regardless of the virtual neutralino/chargino masses. The dominant constraint 6 on λ ′′ 323 is |λ ′′ 323 | 1, which arises from requiring perturbativity up to the GUT scale [32,33]; our fiducial value is well below this bound. We now discuss each example case by case: • Spectrum 1: Light Stop. This example is fairly self-explanatory. Depending on the magnitude of stop mixing, either one or both stop mass eigenstates decay via BRPV, • Spectrum 2: Light Bino. In this example, we assume large stop mixing i.e. sin θt ∼ 0.5 and Mt − MB m t . The R-parity conserving modet → tB is kinematically forbidden, so the stops decay predominantly via BRPV tot → b s.
• Spectrum 3: Light Wino. In this example, we assume vanishing LR stop mixing. 6 Preserving a primordial matter asymmetry imposes a much stronger constraint on λ ′′ 323 [31]. However these constraints can be avoided, for example if baryogenesis takes place below the electroweak phase transition.
The dominant contributions to the R-parity conserving decay modest R → tW 0 , bW + arise from Higgsino-Wino mixing. Taking the µ ≫ M Z , M 2 limit, the relevant mixing angles are √ 2M W (µc β + M 2 s β ) /µ 2 and M Z cos θ W / √ 2µ in the chargino and neutralino sectors. Taking µ 1 TeV, M 2 ∼ 200 GeV and tan β ∼ 10, the BRPV decay modẽ t R → bs dominates over the R-parity conserving decay modes fort R .
• Spectrum 4: Light Higgsino. For this example we assume moderate tan β such that Y s 10 −2 . The dominant R-parity conserving decay mode fors R iss R → sH 0 with a contribution from Higgsino-Bino mixing with mixing angle M Z sin θ W / √ 2M 1 in the M 1 ≫ M Z , µ limit. Thus for M 1 1 TeV, the BRPV modes R → t b dominates. The BRPV decay mode fors L is suppressed by LR mixing, sos L will decay predominantly via R-parity conserving channels.
In these examples, sparticle pair production and associatedgq production 7 yield final states identical to that of pure BRPV or pure LRPV, depending on the sparticle produced. Thus to distinguish Scenario B from standard RPV, an experimental discovery in at least two different channels would be required. For instance, if any of the Spectrums 1-3 with vanishing stop mixing is realized in nature, the discovery of a RH stop decaying via BRPV along with a LH stop decaying via LRPV would give conclusive evidence for BLRPV.

Collider Constraints on BLRPV
We now review existing LHC searches for standard RPV scenarios, which are also sensitive to BLRPV scenarios. For BRPV, ATLAS and CMS have recently released searches for purely hadronic final states [34,35] which place gluino mass bounds of ∼ 800 (950) GeV, assuming decoupled (light) neutral/charged-inos and decoupled squarks. Squarks decaying to 2 jets via BRPV are still weakly constrained, though lighter squarks can push the gluino mass limit up to 1.5 TeV [14]. For LRPV, CMS and ATLAS searches in multilepton final states [36,37,38,39] have placed bounds of 1 TeV for stop masses, 1.5 TeV for gluino and other squarks masses, and 750 GeV for charginos, assuming LSP decays give prompt, isolated leptons. The quoted bounds are for simplified models; more complicated spectra which produce additional hard objects through cascade decays are significantly more constrained by these searches [40,41,42].
It is straightforward to interpret these searches for the BLRPV scenarios discussed above. For Scenario A,X pair production will yield events where bothX particles decay via the same BRPV or LRPV decay mode. Thus bounds onX production will be similar to bounds on standard BRPV/LRPV scenarios, albeit slightly weaker due to branching ratio suppression of the effective cross section. For Scenario B withX 1 → LRPV andX 2 → BRPV, the above searches are also straightforward to apply, as pair production of eitherX 1 orX 2 gives signatures identical to that of standard RPV.
Though we have focused on how BLRPV can give unique phenomenology, it is also possible for BLRPV to give collider signals identical to that of standard RPV scenarios. In this case, one would have to rely on the nucleon decay phenomenology discussed in Section 2 to differentiate BLRPV from standard RPV scenarios.

On Displacedν τ → µe Decays
We now discuss the phenomenology of displacedν τ → µe decays, where cτ (ν τ → µe) ∼ O(mm) − O(m). This might seem unrelated to our discussion of BLRPV thus far, as thẽ ν τ → µe decay mode only involves LRPV couplings. However, the Z e 2 × Z µ 2 × Z τ 2 flavor symmetry of BLRPV gives a natural setting for displacedν τ → µe decays if the LSP is a τ -sneutrino, due to the antisymmetric flavor structure in the LLE c couplings. Thus although much of the following discussion is not unique to BLRPV, models with displacedν τ → µe decays can arise naturally in BLRPV, compared to other RPV scenarios that do not motivate such a flavor structure in the LRPV couplings.
In general LRPV scenarios where the LSP has a has a macroscopic decay length i.e. cτ ∼ O(mm) − O(m), leptons resulting from LSP decay will fail impact parameter cuts imposed by the LRPV searches [36,37,38,39] referenced in Section 4. Consequently, the strict kinematic bounds from these searches will no longer apply if the LSP decay is displaced [43]. Displacedν τ → µe decays are particularly noteworthy, as they also evade dedicated LHC searches for displaced final states [44,45,46] 8 , and for cτ (ν τ → µe) 10 cm can even evade bounds from LEP searches [13].
Displacedν τ → µe decays can also decrease the multiplicity and/or p T of jets resulting from squark and gluino production, compared to other RPV scenarios. This is illustrated in Table 1, which compares gluino NLSP decay modes in different simplified RPV models. For a given gluino mass, the jet p T spectrum is softened due to the momenta carried by thẽ ν τ → µe decay products. If an intermediary chargino/neturalino is also present (e.g. Spectra 2-4 in Figure 7), the jet p T spectrum can be further softened by a small chargino/neturalinosneutrino mass splitting. We assume small slepton mass splittings, such that inter-slepton decays (e.g.τ → ν τ τν τ ) do not introduce additional collider objects in the final state. The displaced µ and e still deposit energy respectively in the muon spectrometer and electromagnetic calorimeter, and will not contribute to missing energy 9 apart from errors in momentum reconstruction.
The above observation implies that compared to other RPV scenarios, events with displacedν τ → µe decays are less constrained by multijet searches [34,48,49,50], which rely on high (≥ 6) jet multiplicity and hard jet p T cuts for background discrimination. These searches provide otherwise robust bounds on squark and gluino masses ( 1.5 TeV) in RPV

RPV Couplingsg Decay Modes
Jets fromgg productioñ models [14], which are largely insensitive to whether or not the LSP decay is displaced. Thus although most RPV scenarios are well covered by some combination of dedicated RPV and/or multijet searches, events such asgg → 4j + 2(τ or ν τ ) + 2ν τ → 4j + 2(τ or ν τ ) + 2(µe) represent a potential blindspot 10 in the existing experimental literature (if theν τ decay is displaced). We urge experimentalists to perform dedicated searches for events with high jet multiplicity and displaced leptons in order to fill this gap.
Our discussion of displacedν τ → µe decays has ignored the potential presence of BRPV couplings, which induce decay modes such asν τ → τ d d d through virtual neutralino/chargino and squark exchange. Focusing on the spectra in Figure 7, we compute theν τ → τ b b s decay width in Appendix D, which is the dominant BRPV decay mode if mν τ m t . For λ ′′ 323 = 0.1 and mν = 100 GeV, we see that there are large regions of parameter space with mt, µ 1 TeV and c/Γ(ν τ → τ b b s) 1 meter, particularly for small tan β. Thus if cτ (ν τ → µe) ∼ O(cm), the branching ratio toν τ → τ b b s can be subdominant, even for the large BRPV couplings (λ ′′ 323 = 0.1) assumed in Figure 7. To the best of our knowledge the calculation of Appendix D, which is also relevant for studying pure BRPV with a slepton LSP, has not appeared elsewhere in the literature.

Conclusion
We have established here a class of RPV models which violate B and L simultaneously (BLRPV), without inducing unacceptable nucleon decay. BLRPV requires an approximate Z e 2 ×Z µ 2 ×Z τ 2 flavor symmetry in the lepton sector, which forbids 4-fermion effective operators leading to 2-body nucleon decay. This symmetry also forbids all LRPV operators aside from λ ijk L i L j E c k for i = k, j = k, significantly reducing the number of free parameters usually associated with the LRPV superpotential. Nucleons are predicted to decay through the decay modes N → Kνe ± µ ∓ and n → µ ± e ∓ ν. A discovery of nucleon decay in these modes, without discoveries in similar modes with same flavor leptons, would give a smoking gun signature for BLRPV.
Current nucleon lifetimes bounds on BRPV and λ ijk L i L j E c k , i = k, j = k couplings are rather weak, allowing both to be relevant for collider phenomenology. Novel phenomenology arises in BLRPV because sparticles can decay via both LRPV and BRPV couplings. Exotic final states can arise from sparticle pair production, if one sparticle decays through BRPV while the other through LRPV. These final states are characterized by large jet multiplicity and multiple leptons of different flavor, e.g.qq → 2qχ + χ − → 5qeµτ . Alternatively, different sparticles could decay predominantly via either BRPV or LRPV, allowing both pure BRPV and pure LRPV signals to manifest within the same spectrum.
Due to the flavor structure in LLE c couplings enforced by Z e 2 × Z µ 2 × Z τ 2 , BLRPV provides a natural framework for displacedν τ → µe decays to occur, providedν τ is the LSP. This decay mode allows sleptons and charginos to evade constraints from both LEP and LHC searches for LRPV [13]. We have argued above that events which involve squarks or gluinos decaying into a long livedν τ that subsequently decays to µe represent a blindspot in existing experimental searches. This can be remedied by performing searches for events with high jet multiplicity and multiple displaced leptons; such events should have very little SM background. Alternatively, proposals for detecting displaced µe resonance decays without relying on additional collider objects have been given in [13].
Although we have focused on RPV supersymmetry in this work, our analysis here (particularly in Section 2) illustrates the general constraint that 6-fermion operators of the form qqqℓℓℓ/Λ 5 must satisfyΛ 100 (10) TeV for couplings to u, d, s (c, b, t) quarks. Therefore, interactions of TeV scale particles can violate B and L 11 without violating nucleon decay bounds, provided some structure is in place to suppress 4-fermion operators of the form qqqℓ/Λ 2 ; a similar observation was made by Weinberg in [52]. Figure 8: Diagrams involving λ ′′ 11m and λ ijk , i = k, j = k which result in effective operators that mediate 3-body nucleon decay. Decay modes of the form p → ℓ + i ℓ + j ℓ − k are kinematically forbidden, as the antisymmetry condition on λ ijk would imply a τ in the final state.

A Constraints on |λ ′′
11m λ ijk | from 3-body Nucleon Decay The simultaneous presence of λ ′′ 11m and λ ijk , i = k, j = k can at tree-level lead to 3-body nucleon decay modes p → ννℓ + , n → ℓ + ℓ − ν, as noted in [26]. Because λ ′′ 111 = 0 due to SU(3) C invariance, the leading tree level contributions to these 3-body decay modes require some source of quark/squark flavor changing. In this section, we argue that bounds on |λ ′′ 11m λ ijk | , i = k, j = k from 3-body nucleon decay modes are subdominant to the bounds from N → Kℓ + i ℓ − j ν k discussed in Section 2.1, due to flavor suppression of the relevant treelevel diagrams.
The relevant diagrams are shown in Figure 8, giving rise to the following 6-fermion effective operators: The effective operators with coefficients C a and C b induce the decay mode p → ν i ν j ℓ + k , while the operator with coefficient C c induces n → ν i ℓ + j ℓ − k . Following the procedure outlined in Section 2, it is straightforward to compute the nucleon decay rates induced by C a and C c : For the decay rate induced by the derivative interaction corresponding to C b , we instead use an approximate expression obtained by dimensional analysis: As discussed in Section 2, β and α correspond to hadronic matrix elements, which from lattice calculations are α ≈ β ≈ 0.0118 (GeV) 3 [22].
, the constraints from 3-body decay modes are subdominant.
Consider Figure 7a; for degenerate SUSY masses, where M SU SY is the common superpartner mass scale. We have assumed that the left-right squark mixing term m d ℓ X d ℓ is such that X d ℓ = M SU SY , and that Wino-Higgsino mixing angles are O(1). As discussed in Section 2 A ≈ 0.22 accounts for the renormalization of these operators from Q ≈ M SU SY to Q ≈ Λ QCD [23]. Comparing (16) and (5) then gives: for M SU SY 100 GeV. Note that we have omitted a diagram analogous to Figure 7a with Higgsino exchange, as it is suppressed with respect to Figure 7a by a lepton Yukawa coupling.
Applying a similar analysis to Figure 7b gives C b ∼ A λ ′′ 11m λ ijk V * 1ℓ g 2 m d ℓ /M SU SY 7 and thus: for M SU SY 100 GeV. Finally, we consider Figure 7c, which gives where δ RR d,1m represents the flavor changing squark mass insertion. Thus: Note that FCNC constraints require δ RR d,1m 0.1 for ∼ 1 TeV squarks and gluinos [53].

B Loop Function L ijk Calculation
In this appendix, we calculate the loop functions L ijk which determine the coefficients of the effective operators in (8). This requires computing the loop diagrams in Figure 4. In the following calculations, we ignore LR mixing in the squark sector.
The diagram in 4a with W exchange gives the amplitude: where V ij is the CKM matrix, v = 174 GeV, and we use 2-component notation [19] for the final state spinors. In the second line we have written the loop integral in terms of the wellknown Passarino-Veltman 3-point functions [54], omitting terms proportional to the external momenta. Note that C 0 (m 2 1 , m 2 2 , m 2 3 ) in our notation corresponds to C 0 (0, 0, m 2 1 , m 2 2 , m 2 3 ) in the notation of [54]; we use a similar notation for C 24 .
The diagram in 4b with charged Higgs exchange gives the amplitude: The chargino exchange diagram 4c (which involves only scalar 3-point integrals) contributes the amplitude: where we again omit terms proportional to the external momenta. U, V are the chargino diagonalization matrices [55] and A ′′ ijk is a soft breaking trilinear defined as This result vanishes in the SUSY limit, as expected. Note the divergent pieces of M W ijk and M H ijk which are contained within the C 24 term cancel in L ijk . The scalar 3-point function C 0 is given by (see e.g. [56]):

C A Flavor Model for Right-Handed Neutrinos
In this Appendix, we illustrate the constraints which can arise if the dynamics that breaks Z e 2 ×Z µ 2 ×Z τ 2 to generate neutrino masses also generates dangerous LRPV operators. For concreteness, we focus on a right-handed neutrino model involving a spurious SU (3) Table 2. Note that the same SU(3) N representation content was considered in [30]; we will use some results obtained by these authors in the following analysis.
The SU(3) ℓ × SU(3) N invariant superpotential to lowest order in spurions is: Here raised/lowered i, j, k represent fundamental/antifundamental indices transforming under SU(3) ℓ , and a, b, c represent fundamental/antifundamental SU(3) N indices. We treat Λ R as an undetermined mass scale. Y E can be thought of as a symmetric matrix which transforms under SU(3) ℓ as Y E → U T Y E U, so without loss of generality, we take (23) to be in the basis where Y E is diagonal.
The non-zero vev of Y ν breaks Z e 2 × Z µ 2 × Z τ 2 and regenerates dangerous qqqℓ operators. To estimate the coefficients of these dangerous operators and the resulting bounds on the neutrino sector, we perform a spurion analysis focusing on SU(3) N singlets which transform as non-trivial irreps of SU(3) ℓ . Let us first consider holomorphic products of Y N , Y ν . As shown in [30], there are only two irreducible holomorphic SU(3) N singlets which can be formed from the Y ν and Y N spurions: where ǫ acdỸ ab = ǫ bef Y ce Y df . The non-holomorphic SU(3) N singlets which transform as non-trivial SU(3) ℓ irreps are given to leading order in Y N , Y ν by [30]:  (24), (25).
To obtain bounds on Y ν and Y N from 2-body nucleon decay, we consider SU(3) ℓ singlets formed out of the spurions (24)-(25) which generate LRPV operators constrained by 2-body nucleon decay 12 . The leading holomorphic SU(3) ℓ × SU(3) N singlet involving a single lepton field is given by: where we have assumed an anarchic flavor structure for Y ν , Y N . There is also a singlet of the form (Y 2 ) 2 (Y E ) 3 L, but it is higher order in Y N compared to (26). The leading nonholomorphic singlet involving a single lepton field is given by: There are similar non-holomorphic terms generated by V 1 and V 2 , but they are higher order in Y N , Y ν . There are also Kahler potential terms involving non-holomorphic spurions which can induce flavor-changing slepton mass insertions: Thus to leading order in the spurion analysis, the dangerous Z e 2 × Z µ 2 × Z τ 2 violating operators are given by: where x, y represent arbitrary quark flavor indices. m sof t is generated by Planck suppressed Kahler potential operators upon SUSY breaking, i.e. m sof t ≈ F X /M pl where X is a SUSY breaking spurion. The non-holomorphic spurions will also generate trilinear RPV terms upon SUSY breaking; we assume that SUSY breaking renders these contributions negligible i.e. F X /M 2 pl ≪ 1. We now discuss bounds on the neutrino sector in light of (29) and the 2-body nucleon decay bounds (2). Assuming an anarchic flavor structure for Y N and Y ν , the SM neutrino masses are given by m ν ∼ Y ν 2 v 2 sin 2 β/M R where M R ≡ Y N Λ R ; we use this relation to fix Y ν . We also assume that some mechanism enforces κ 0 µ.
Let us first consider contributions from the holomorphic term. Taking α xy ∼ O(1), the holomorphic spurion W i generates an effective QLD c term 13 with λ ′ ≈ W i . Combining (26) and (29), the constraint on λ ′ in (2) can be mapped into a constraint on M R : Thus taking typical parameter values and imposing the di-nucleon decay constraint |λ ′′ 112 | 10 −6 (M SU SY /TeV) 5/2 [25]), bounds from the holomorphic spurion are rather mild and allow for M R to lie well above the TeV scale.
Let us now consider the non-holomorphic superpotential term, which generates effective κ i L i H u couplings with κ i ≈ m sof t K i . Combining (27) and (29) with (2), the bound on M R due to K i is: If m sof t ∼ µ, this constraint is much stronger than the constraint from the holomorphic spurion, and will not allow M R to lie above the weak scale. However, the ratio µ/m sof t depends on SUSY breaking dynamics. Taking µ ∼ M SU SY and m sof t ≈ F X /M pl ∼ Λ M M SU SY /M pl where Λ M is the mass scale of SUSY breaking messengers, we obtain µ/m sof t ∼ M pl /Λ M . Thus if the messenger scale is comparable to the Planck scale (as in gravity mediation), M R is constrained to lie below a GeV or so. However, if the messenger scale is hierarchically smaller than M pl (as in gauge mediation), the resulting suppression of m sof t can allow M R to lie above the TeV scale while still satisfying (31).
Finally, we consider bounds due to flavor-changing slepton mass insertions generated by V 3 which in conjunction with λ ′′ 112 and λ ijk , i = k, j = k results in diagrams similar to b) c) Figure 9: Diagrams which contribute toν τ → τ b b s; we neglect the diagram with pure χ − exchange.

D Sneutrino 4-Body BRPV Decay Rate
If BRPV couplings are non-vanishing, a tau sneutrino LSP has the 4-body BRPV decay modesν τ → τ d i d j d k andν τ → ν u i d j d k , which occur via virtual neutral/charged-ino and squark exchange. In this appendix, we compute this BRPV decay rate, focusing on the spectra in Figure 7 in which λ ′′ 323 = 0.1 is the only non-vanishing BRPV coupling. We also assume mν τ m t , such thatν τ → ν τ tbs is phase space suppressed. This allows us to consider ν τ → τ b b s as the only relevant BRPV decay mode.
The relevant diagrams are depicted in Figure 9; we neglect the diagram with pure χ − exchange which is proportional to y b y τ . The resulting amplitude is given by: where we have used 2-component spinor notation [19]. We have defined z 1 = 2p·k 1 /m 2 ντ , z 2 = 2p · k 2 /m 2 ντ , z 12 = 2k 1 · k 2 /m 2 ντ and rt a = mt a /mν τ ; A, B, C denote color indices. Here p is theν τ 4-momentum, and the label i on the outgoing spinors and momenta k i increases from left to right with respect to Figure 9 (k 1 is the τ momenta). The coefficients c a j , d a j and f a j correspond respectively to Figure 9.a, Figure 9.b and Figure 9.c: Note that y t , y b , y τ are supersymmetric Yukawa couplings i.e. y t = m t /v sin β, y b = m b /v cos β.
We have defined r χ j = m χ ± j /mν τ . U, V are the chargino diagonalization matrices [55], and R t , L t are stop mixing angles defined by: In computing the decay rate, we neglect final state fermion masses, allowing us to neglect interference terms between the diagrams in Figure 9. The decay rate is then given by: where the sums on a, b, j, k run from 1 to 2. We perform the 4-body phase space integration using the RAMBO method [58], which is particularly straightforward to implement for massless final state particles.
The result for mν τ = 100 GeV is shown in Figure 10, which plots contours of constant cτ (ν τ → τ b b s) in the (µ, mt) plane where we take mt to be the degenerate stop mass. The left (right) plot is for tan β = 2 (10); M 2 = 300 GeV and vanishing stop mixing is assumed for both plots. Note from (34) that in the limit of vanishing stop mixing and vanishing Wino-Higgsino mixing, c a j = 0, d a j = 0, and only the pure Higgsino portion of f a j is non-vanishing. Thus for M 2 , µ ≫ M Z , the M 2 dependence is weak while the tan β dependence is strong, as evident in Figure 10.