PLB33428S0370-2693(17)31019-510.1016/j.physletb.2017.12.042The AuthorsPhenomenology

Fig. 1

Tree-level vector exchange triggering proton decay for V=(3,2)16 and V=(3,2)−56.

Fig. 1

Fig. 2

Proton decay through a dimension five interaction for V=(3,1)23 and V=(3,3)23.

Fig. 2

Fig. 3

Process mediating n−n¯ oscillations via the vector diquark (6,2)−16.

Fig. 3

Table 1

The only leptoquark and diquark models with a triplet or sextet color structure that do not suffer from tree-level proton decay. The primes indicate the existence of dim 5 proton decay channels.

Table 1

FieldSU(3)c×SU(2)L×U(1)Y reps.Scalar leptoquark(3,2)76′Scalar diquark(3,1)23, (6,1)−23, (6,1)13, (6,1)43, (6,3)13Vector leptoquark(3,1)23′, (3,1)53, (3,3)23′Vector diquark(6,2)−16, (6,2)56Table 2

Possible vector color triplet and sextet representations.

Table 2

OperatorSU(3)cSU(2)LU(1)Yp decayQLc‾γμuRVμ32−5/6tree-level6¯2−5/6–QLc‾γμdRVμ321/6tree-level6¯21/6–QL‾γμLLVμ31,32/3dim 5QLc‾γμeRVμ⁎32−5/6tree-levelLLc‾γμuRVμ⁎321/6tree-levelLLc‾γμdRVμ⁎32−5/6tree-leveluR‾γμeRVμ315/3dim 7dR‾γμeRVμ312/3dim 5Baryon number and lepton universality violation in leptoquark and diquark models

Nima

Assad

nima.michael.assad@gmail.com

Bartosz

Fornal

⁎

bfornal@ucsd.edu

Benjamín

Grinstein

bgrinstein@ucsd.edu

Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USADepartment of PhysicsUniversity of California, San Diego

9500 Gilman Drive

La JollaCA

92093

USA⁎Corresponding author.

Editor: J. Hisano

Abstract

We perform a systematic study of models involving leptoquarks and diquarks with masses well below the grand unification scale and demonstrate that a large class of them is excluded due to rapid proton decay. After singling out the few phenomenologically viable color triplet and sextet scenarios, we show that there exist only two leptoquark models which do not suffer from tree-level proton decay and which have the potential for explaining the recently discovered anomalies in B meson decays. Both of those models, however, contain dimension five operators contributing to proton decay and require a new symmetry forbidding them to emerge at a higher scale. This has a particularly nice realization for the model with the vector leptoquark (3,1)2/3, which points to a specific extension of the Standard Model, namely the Pati–Salam unification model, where this leptoquark naturally arises as the new gauge boson. We explore this possibility in light of recent B physics measurements. Finally, we analyze also a vector diquark model, discussing its LHC phenomenology and showing that it has nontrivial predictions for neutron–antineutron oscillation experiments.

Keywords

Proton decayLeptoquarksPati–Salam unificationB anomaliesDiquarksn−n¯ oscillations1

Introduction

Protons have never been observed to decay. Minimal grand unified theories (GUTs) [1,2] predict proton decay at a rate which should have already been measured. The only four-dimensional GUTs constructed so far based on a single unifying gauge group with a stable proton require either imposing specific gauge conditions [3] or introducing new particle representations [4]. A detailed review of the subject can be found in [5]. Lack of experimental evidence for proton decay [6] imposes severe constraints on the form of new physics, especially on theories involving new bosons with masses well below the GUT scale. For phenomenologically viable models of physics beyond the Standard Model (SM) the new particle content cannot trigger fast proton decay, which seems like an obvious requirement, but is often ignored in the model building literature.Simplified models with additional scalar leptoquarks and diquarks not triggering tree-level proton decay were discussed in detail in [7], where a complete list of color singlet, triplet and sextet scalars coupled to fermion bilinears was presented. An interesting point of that analysis is that there exists only one scalar leptoquark, namely (3,2)76 (a color triplet electroweak doublet with hypercharge 7/6) that does not cause tree-level proton decay. In this model dimension five operators that mediate proton decay can be forbidden by imposing an additional symmetry [8].In this letter we collect the results of [7] and extend the analysis to vector particles. This scenario might be regarded as more appealing than the scalar case, since the new fields do not contribute to the hierarchy problem. We do not assume any global symmetries, but we do comment on how imposing a larger symmetry can remove proton decay that is introduced through nonrenormalizable operators, as in the scalar case.Since many models for the recently discovered B meson decay anomalies [9,10] rely on the existence of new scalar or vector leptoquarks, it is interesting to investigate which of the new particle explanations proposed in the literature do not trigger rapid proton decay. Surprisingly, the requirement of no proton decay at tree level singles out only a few models, two of which involve the vector leptoquarks (3,1)23 and (3,3)23, respectively. Remarkably, these very same representations have been singled out as giving better fits to B meson decay anomalies data [11]. An interesting question we consider is whether there exists a UV complete extension of the SM containing such leptoquarks in its particle spectrum.Finally, although the phenomenology of leptoquarks has been analyzed in great detail, there still remains a gap in the discussion of diquarks. In particular, neutron–antineutron (n−n¯) oscillations have not been considered in the context of vector diquark models. We fill this gap by deriving an estimate for the n−n¯ oscillation rate in a simple vector diquark model and discuss its implications for present and future experiments.The letter is organized as follows. In Sec. 2 we study the order at which proton decay first appears in models including new color triplet and sextet representations and briefly comment on their experimental status. In Sec. 3 we focus on the unique vector leptoquark model which does not suffer from tree-level proton decay and has an appealing UV completion. In particular, we study its implications for B meson decays. In Sec. 4 we analyze a model with a single vector color sextet, discussing its LHC phenomenology and implications for n−n¯ oscillations. Section 5 contains conclusions.2

Viable leptoquark and diquark models

For clarity, we first summarize the combined results of [7] and this work in Table 1

, which shows the only color triplet and color sextet models that do not exhibit tree-level proton decay. The scalar case was investigated in [7], whereas in this letter we concentrate on vector particles. As explained below, the representations denoted by primes exhibit proton decay through dimension five operators (see also [8]). We note that although the renormalizable proton decay channels involving leptoquarks are well-known in the literature, to our knowledge the nonrenormalizable channels have not been considered anywhere apart from the scalar case in [8].

2.1

Proton decay in vector models

We first enumerate all possible dimension four interactions of the new vector color triplets and sextets with fermion bilinears respecting gauge and Lorentz invariance. A complete set of those operators is listed in Table 2

[12]. For the vector case there are two sources of proton decay. The first one comes from tree-level diagrams involving a vector color triplet exchange, as shown in Fig. 1

. This excludes the representations (3,2)16 and (3,2)−56, since they would require unnaturally small couplings to SM fermions or very large masses to remain consistent with proton decay limits. The second source comes from dimension five operators involving the vector leptoquark representations (3,1)23 and (3,3)23:(1)1Λ(QLc‾H†)γμdRVμ,1Λ(QLc‾τAH†)γμdRVμA, respectively. Those operators can be constructed if no additional global symmetry forbidding them is imposed and allow for the proton decay channel shown in Fig. 2

, resulting in a lepton (rather than an antilepton) in the final state. The corresponding proton lifetime estimate is:(2)τp≈(2.5×1032years)(M104TeV)4(ΛMPl)2, where the leptoquark tree-level coupling and the coefficient of the dimension five operator were both set to unity. The numerical factor in front of Eq. (2) is the current limit on the proton lifetime from the search for p→K+π+e− [13]. Even in the most optimistic scenario of the largest suppression of proton decay, i.e., when the new physics behind the dimension five operator does not appear below the Planck scale, those operators are still problematic for M≲104TeV, which well includes the region of interest for the B meson decay anomalies.

The dimension five operators can be removed by embedding the vector leptoquarks into UV complete models. As argued in [8] for the scalar case, it is sufficient to impose a discrete subgroup Z3 of a global U(1)B−L to forbid the problematic dimension five operators. They are also naturally absent in models with gauged U(1)B−L.11

We note that the dimension five operators (1) provide a baryon number violating channel which may be used to generate a cosmological baryon number asymmetry.

Ultimately, as shown in Table 1, there are only five color triplet or sextet vector representations that are free from tree-level proton decay, two of which produce dimension five proton decay operators. In the scalar case, as shown in [7], there are six possible representations with only one suffering from dimension five proton decay.2.2

Leptoquark phenomenology

The phenomenology of scalar and vector leptoquarks has been extensively discussed in the literature [14–16] and we do not attempt to provide a complete list of all relevant papers here. For an excellent review and many references see [12], which is focused primarily on light leptoquarks.Low-scale leptoquarks have recently become a very active area of research due to their potential for explaining the experimental hints of new physics in B meson decays, in particular B+→K+ℓ+ℓ− and B0→K⁎0ℓ+ℓ−, for which a deficit in the ratios RK(⁎)=Br(B→K(⁎)μ+μ−)/Br(B→K(⁎)e+e−) with respect to the SM expectations has been reported [9,10]. A detailed analysis of the anomalies can be found in [17–22]. Several leptoquark models have been proposed to alleviate this tension and are favored by a global fit to RK(⁎), RD(⁎) and other flavor observables. Surprisingly, not all of those models remain free from tree-level proton decay.The leptoquark models providing the best fit to data with just a single new representation are: scalar (3,2)16 [23], vector (3,1)23 [11,24] and vector (3,3)23 [25]. Among those, only the models with the vector leptoquarks (3,1)23 and (3,3)23 are naturally free from any tree-level proton decay, since for the scalar leptoquark (3,2)16 there exists a dangerous quartic coupling involving three leptoquarks and the SM Higgs [7] which triggers tree-level proton decay.Interestingly, as indicated in Table 2, both vector models (3,1)23 and (3,3)23 suffer from dimension five proton decay and require imposing an additional symmetry to eliminate it. An elegant way to do it would be to extend the SM symmetry by a gauged U(1)B−L. Actually, such an extended symmetry would eliminate also the tree-level proton decay in the model with the scalar leptoquark (3,2)16. However, as we will see in Sec. 3, only in the case of the:

►vector leptoquark (3,1)23

there exists a very appealing SM extension which intrinsically contains such a state in its spectrum, simultaneously forbidding dimension five proton decay.2.3

Diquark phenomenology

The literature on the phenomenology of diquarks is much more limited. It focuses on scalar diquarks [26] and predominantly looks at three aspects: LHC discovery reach for scalar diquarks [27–40], n−n¯ oscillations mediated by scalar diquarks [41–44,34,45,7,46] and baryogenesis [42,7,34,44–46]. Studies of vector diquarks investigate only their LHC phenomenology [47,48,35,49–51], concentrating on their interactions with quarks.In Sec. 4 we close the gap in diquark phenomenology by discussing the implications of a vector diquark model for n−n¯ oscillation experiments.3

Vector leptoquark model

As emphasized in Sec. 2.2, the SM extended by just the vector leptoquark (3,1)23 is a unique model, which, apart from being free from tree-level proton decay, has a very simple and attractive UV completion, automatically forbidding dimension five proton decay operators.3.1

Pati–Salam unification

A priori, any of the leptoquarks can originate from an extra GUT irrep, either scalar or vector. In particular, the vector leptoquark (3,1)23 could be a component of the vector 40 irrep of SU(5). Nevertheless, this generic explanation does not seem to be strongly motivated or predictive. Another interpretation of the vector (3,1)23 state arises in composite models [52–54].The third, perhaps the most desirable option, is that the vector leptoquark (3,1)23 is the gauge boson of a unified theory. Indeed, this scenario is realized if one considers partial unification based on the Pati–Salam gauge group:(3)SU(4)×SU(2)L×SU(2)R at higher energies [55].22

Similar unification models have recently been constructed based on the gauge group SU(4)×SU(2)L×U(1) [56,57], in which the new gauge bosons also coincide with the vector leptoquark (3,1)2/3.

In this case the vector leptoquark (3,1)23 emerges naturally as the new gauge boson of the broken symmetry, and is completely independent of the symmetry breaking pattern. It is also interesting that the Pati–Salam partial unification model can be fully unified into an SO(10) GUT.The fermion irreps of the Pati–Salam model, along with their decomposition into SM fields, are(4)(4,2,1)=(3,2)16⊕(1,2)−12(4¯,1,2)=(3¯,1)13⊕(3¯,1)−23⊕(1,1)1⊕(1,1)0. Interestingly, the theory is free from tree-level proton decay via gauge interactions. The explanation for this is straightforward. Since SU(4)⊃SU(3)c×U(1)B−L, this implies that B−L is conserved. However, after the Pati–Salam group breaks down to the SM, the interactions of the leptoquark (3,1)23 with quarks and leptons have an accidental B+L global symmetry. Those two symmetries combined result in both baryon and lepton number being conserved in gauge interactions, thus no proton decay can occur via a tree-level exchange of (3,1)23.In addition, there are no gauge-invariant dimension five proton decay operators in the Pati–Salam model involving the vector leptoquark (3,1)23. This was actually expected from the fact that SU(4)⊃SU(3)c×U(1)B−L and, as discussed in Sec. 2.2, a U(1)B−L symmetry is sufficient to forbid such operators.3.2

Flavor structure

The quark and lepton mass eigenstates are related to the gauge eigenstates through nf×nf unitary matrices, with nf=3 the number of families of quarks and leptons. Expressing the interactions that couple the (3,1)23 vector leptoquark to the quark and the lepton in each irrep of Eqs. (4) in terms of mass eigenstates, one must include unitary matrices, similar to the Cabibbo–Kobayashi–Maskawa (CKM) matrix for the quarks in the SM, that measure the misalignment of the lepton and quark mass eigenstates:(5)L⊃g42Vμ[Liju(u¯iγμPLνj)+Lijd(d¯iγμPLej)+Rij(d¯iγμPRej)]+h.c.. The SU(4) gauge coupling constant, g4, is not an independent parameter but fixed by the QCD coupling constant at the scale M of the masses of the vector bosons of SU(4); to leading order g4(M)=4παs(M)≈0.94 at M=16 TeV, where, as will become clear later, this is the maximum value of M consistent with the RK(⁎) anomaly in this model. The unitary matrices Lu and Ld, the CKM matrix V and the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix U satisfy Lu=VLdU.3.3

B meson decays

In the SM, flavor-changing neutral currents with ΔB=−ΔS=1 are described by the effective Lagrangian [58,59]:(6)L=−4GF2Vtb⁎Vts⁎(∑k=710CkOk+∑i,jCνijOνij+...), where the ellipsis denote four-quark operators, O7 and O8 are electro- and chromo-magnetic-moment-transition operators, and O9, O10 and Oν are semi-leptonic operators involving either charged leptons or neutrinos33

We have assumed the B anomalies in RK(⁎) arise from a suppression in the μ channel relative to the SM. The Pati–Salam (3,1)2/3 vector leptoquark model can equally well accommodate an enhancement in the electron channel, but a suppression in the muon channel is preferred by global fits to data that include angular moments in B→K⁎μμ and various branching fractions in addition to RK(⁎).

:(7)O9(10)=e2(4π)2[s¯γμPLb][μ¯γμ(γ5)μ],(8)Oνij=2e2(4π)2[s¯γμPLb][ν¯iγμPLνj]. Chirally-flipped (bL(R)→bR(L)) versions of all these operators are denoted by primes and are negligible in the SM. New physics (NP) can generate modifications to the Wilson coefficients of the above operators, and, moreover, it can generate additional terms in the effective Lagrangian,(9)ΔL=−4GF2Vtb⁎Vts⁎(CSOS+CPOP+CS′OS′+CP′OP′) in the form of scalar operators:(10)OS(′)=e2(4π)2[s¯PR(L)b][μ¯μ],(11)OP(′)=e2(4π)2[s¯PR(L)b][μ¯γ5μ]. Tensor operators cannot arise from short distance NP with the SM linearly realized and, moreover, under this assumption CP=−CS and CP′=CS′ [60].Exchange of the (3,1)23 vector leptoquark gives tree-level contributions to the Wilson coefficients at its mass scale, M:(12)ΔC9(M)=−ΔC10(M)=−2π22GFM2g42e2Lbμd⁎LsμdVtb⁎Vts⁎,(13)ΔC9′(M)=ΔC10′(M)=−2π22GFM2g42e2Rbμ⁎RsμVtb⁎Vts⁎,(14)ΔCS(M)=−4π22GFM2g42e2Lbμd⁎RsμVtb⁎Vts⁎,(15)ΔCS′(M)=−4π22GFM2g42e2Rbμ⁎LsμdVtb⁎Vts⁎,(16)ΔCνij(M)=0.The recently measured B meson decays [9,10] show an excess above the SM background in the ratios RK(⁎)=Γ(B→K(⁎)μμ)/Γ(B→K(⁎)ee). Those anomalies are best fit by ΔC9=−ΔC10≈−0.6 [17], which requires (g42/M2)Lbμd⁎Lsμd≈1.8×10−3TeV−2. Assuming Lbμd⁎Lsμd=12, which is the largest value allowed by unitarity, we obtain the previously quoted leptoquark mass of M≈16TeV. Limits on ΔC9,10′ and ΔCS(′) can be accommodated by adjusting Rsμ and Rbμ.Experimental bounds on RK(⁎)ν=Br(B→K(⁎)νν¯)/ Br(B→K(⁎)νν¯)SM=13∑ij|δij+Cνij/CνSM|2, where CνSM≃−6.35 [61], severely constrain theoretical models for those anomalies. As seen above, the (3,1)23 vector leptoquark evades this constraint by giving no correction at all to Cν; the result holds generally for this type of NP mediator at tree level [11,62]. It has been pointed out that generally the condition ΔCν(M)=0 is not preserved by renormalization group running of the Wilson coefficients [63]. Because of the flavor structure of the interaction in Eq. (5) there are no “penguin” or wave function renormalization contributions to the running of ΔCν down to the electroweak scale. The only contribution comes from the renormalization by exchange of SU(2) gauge bosons that mixes the singlet operator (q¯γμPLe)(e¯γμPLq) into the triplet, (q¯τaγμPLe)(e¯τaγμPLq), resulting in(17)ΔCνij(MW)=−342GFM2g42sin2⁡θwln⁡(MMW)SsjSbi⁎Vtb⁎Vts⁎, where S=V†Lu=LdU. The vector contribution to the rate does not interfere with the SM, which implies RK(⁎)ν−1= 13∑ij|Cνij/CνSM|2; using RK(⁎)ν<4.3 [64], we obtain the condition M>0.8 TeV. Since ln⁡(M/MW) is not large for M≈16TeV, the leading log term is subject to sizable (≈100%) corrections. However, a complete one-loop calculation is beyond the scope of this work.We pause to comment on the remarkable cancellation of the interference term between the SM and NP contributions to the rate for B→K(⁎)νν¯ and the absence of a sum over generations in the pure NP contribution to the rate. These observations hold generally for any vector leptoquark model that couples universally to quark and lepton generations. This can be easily seen by not rotating to the neutrino mass eigenstate basis, a good approximation for the nearly massless neutrinos. Vector leptoquark exchange leads to an effective interaction with flavor structure (s¯LγμνL2)(ν¯L3γμbL), while the SM always involves a sum over the same neutrino flavors ∼∑jν¯LjγμνLj. There are never common final states to the SM and the NP mediated interactions and therefore no interference. Moreover, there is a single flavor configuration in the final state of the NP mediated interaction (ν¯3ν2) while there are three configurations in the SM case (ν¯jνj, j=1,2,3).It has been suggested that the vector leptoquark (3,1)23 may alternatively be used to account for the anomaly in semileptonic decays to τ-leptons [65,11]. Defining, as is customary, RD(⁎)=Br(B→D(⁎)τν)/Br(B→D(⁎)ℓν), the SM predicts [66] (see also [67–70]) RD=0.299(3) and RD⁎=0.257(3). These branching fractions have been measured by Belle [71–73], BaBar [74,75] and LHCb [76], and the average gives [77] RD=0.403(47) and RD⁎=0.310(17). The effect of leptoquarks on B semileptonic decays to τ is described by the following terms of the effective Lagrangian for charged current interactions [78,79]:(18)L⊃−4GF2Vcb[(Uτj+ϵLj)(c¯γμPLb)(τ¯γμPLνj)+ϵsRj(c¯PRb)(τ¯PLνj)]+h.c. with(19)ϵLj=g4242GFM2LcjuLbτd⁎Vcb,ϵsRj=−g4222GFM2LcjuRbτ⁎Vcb. The Bc lifetime [80] and Bc→τν branching fraction [81] impose severe constraints on ϵsR; these are accommodated by abating Rbτu. Hence,(20)RD(⁎)RD(⁎)SM≈1+2Re[∑jϵLcjuLbτd⁎]≈1+2Re[ϵ(VLd)cτLbτd⁎]≈1+2Re[ϵLsτdLd⁎bτ]≤1+ϵ, where,(21)ϵ=g4242GFVcbM2≈0.1(2TeVM)2.3.4

Towards a viable UV completion

We note that the simplest version of the model is heavily constrained by meson decay experiments. For generic, order one entries of the flavor matrices, the leptoquark mass is forced to be above the 1000TeV scale [82–86]. Surprisingly, all of the kaon decay bounds can be avoided if the unitary matrices Lˆd and Rˆ are of the form (see also [86]):(22)Lˆd≈(001L21dL22d0L31dL32d0),Rˆ≈(001R21R220R31R320), where it is actually sufficient that the entries labeled as zero are just ≲10−4. Although current τ decay constraints are irrelevant for our choice of the leptoquark mass, with unsuppressed right-handed (RH) currents the B meson decay bounds require M≳40TeV [86]. Interestingly, we find that if a mechanism suppressing the RH currents is realized in nature, the present bounds from B meson decays are much less stringent and require merely M≳19TeV.A possible way to suppress the RH currents is to associate them with a different gauge group than the left-handed (LH) ones, and to have the RH group spontaneously broken at a much higher scale than the LH group. A simple setting is offered by the gauge group(23)SU(4)L×SU(4)R×SU(2)L×U(1) and does not require introducing any new fermion fields beyond the SM particle content and a RH neutrino:(24)(4,1,2,0)=(3,2)16⊕(1,2)−12,(1,4,1,12)=(3,1)23⊕(1,1)0,(1,4,1,−12)=(3,1)−13⊕(1,1)−1. Such a model predicts rates for B+→K+e∓μ± and μ→eγ, among others, just above the experimental bounds reported in [87,88]. The details of the model along with an analysis of the relevant experimental constraints will be the subject of a future publication [89].44

Shortly after the results of our work became public, two other papers appeared proposing the explanation of B decay anomalies by introducing new vector-like fields either within the Pati–Salam framework [90] or in a model with an extended gauge group [91].

Vector diquark model

In this section we discuss the properties of a model with just one additional representation – the vector color sextet:(25)Vμ=(VuVd)μαβ=(6,2)−16, which is obviously free from proton decay. Although in the SM all fundamental vector particles are gauge bosons, we can still imagine that such a vector diquark arises from a vector GUT representation, for instance from a vector 40 irrep of SU(5) [92]. The Lagrangian for the model is given by:(26)LV=−14(D[μVν])†D[μVν]+M2Vμ†Vμ−[λij(QLc‾)αiγμ(dR)βj(V†)μαβ+h.c.], where α,β=1,2,3 are SU(3)c indices, i,j=1,2,3 are family indices and there is an implicit contraction of the SU(2)L indices. We assume that the mass term arises from a consistent UV completion.Among the allowed higher dimensional operators, n−n¯ oscillations, as we discuss in Sec. 4.2, are mediated by the dimension five terms:(27)O1=c1ΛVμαα′ϵVνββ′(uRc‾)δσμνdRδ′ϵαβδϵα′β′δ′,O2=c2Λ[∂μ(Vμ)αα′ϵVνββ′][(Vν)δδ′ϵH]ϵαβδϵα′β′δ′.4.1

LHC bounds

Several studies of constraints and prospects for discovering vector diquarks at the LHC can be found in the literature. Most of the analysis have focused on the case of a sizable diquark coupling to quarks [47–49], although an LHC four-jet search that is essentially independent of the strength of the diquark coupling to quarks has also been considered [35].There are severe limits on vector diquark masses arising from LHC searches for non-SM dijet signals [93,94]. For a coupling λij≈1 (i,j=1,2) those searches result in a bound on the vector diquark mass(28)Mλ≈1≳8TeV.Lowering the value of the coupling to λij≈0.01 completely removes the LHC constraints from dijet searches and, at the same time, does not affect the strength of the four-jet signal arising from gluon fusion. Using the results of the analysis of four-jet events at the LHC presented in [35], the currently collected 37fb−1 of data by the ATLAS experiment [94] with no evident excess above the SM background constrains the vector diquark mass to be(29)Mλ≪1≳2.5TeV.Additional processes constraining the vector diquark model include neutral meson mixing and radiative B meson decays [95,96]. The resulting bounds in the case of scalar diquark models were calculated in [7,97,98] and are similar here.4.2

Neutron–antineutron oscillations

In the light of null results from proton decay searches [6], the possibility of discovering n−n¯ oscillations has recently gained increased interest [99]. The most important reason for this is that the matter-antimatter asymmetry of the Universe requires baryon number to be violated at some point during its evolution. If processes with ΔB=1 are indeed suppressed or do not occur in Nature at all, the next simplest case involves ΔB=2, a baryon number breaking pattern that may result in n−n¯ oscillations without proton decay. Moreover, the SM augmented only by RH neutrinos can have an additional U(1)B−L gauge symmetry without introducing any gauge anomalies. Through this symmetry, processes with ΔB=2 would be accompanied by ΔL=2 lepton number violating ones, which in turn are intrinsically connected to the seesaw mechanism generating naturally small neutrino masses [100].Models constructed so far propose n−n¯ oscillations mediated by scalar diquarks, as mentioned in Sec. 2.3. Those oscillations proceed through a triple scalar vertex, so that the process is described at low energies by a local operator of dimension nine. Below we show that n−n¯ oscillations can also be mediated by vector diquarks, in particular within the model with just one new representation discussed in this section.The least suppressed channel is via a dimension five gauge invariant quartic interaction O1 in Eq. (26) involving two vector diquarks (6,2)−16 and the SM up and down quarks, as shown in Fig. 3

, ultimately leading to n−n¯ oscillations through a low energy effective interaction local operator of dimension nine, as in the case of scalar diquarks.55

We note that n−n¯ oscillations occur at the renormalizable level in a model involving the vector diquarks (6,2)−1/6, (6,2)5/6, and the scalar diquark (6,1)−2/3, and proceed through a triple sextet interaction.

With the simplifying assumption that c1≈1 and neglecting other operators contributing to the signal we now estimate the rate of n−n¯ oscillations in this model. The effective Hamiltonian corresponding to the operator O1 is,(30)Heff≈−λ112M4Λ(dc‾)αγμdα′(uc‾)βγνdβ′(uc‾)δσμνdδ′×(ϵαβδϵα′β′δ′+ϵα′βδϵαβ′δ′+ϵαβ′δϵα′βδ′+ϵαβδ′ϵα′β′δ)+h.c.. Combining this with the results of [7,101], we obtain an estimate for the n−n¯ transition matrix element,66

The single particle state normalization adopted here is:(31)〈n(p→)|n(p→′)〉=Em(2π)3δ(3)(p→−p→′).

(32)|〈n¯|Heff|n〉|≈10−4|λ112|M4ΛGeV6, where |n〉 is the neutron state at zero momentum. Current experimental limit [102] implies, assuming λ11≈0.01 (which is well below the LHC bound from dijet searches, as discussed earlier), that(33)M≳2.5TeV(108TeVΛ)1/4.An interesting limit on the vector diquark mass is derived if we assume that the physics behind the triple diquark interaction with the SM Higgs is related to the physics responsible for providing the diquark its mass, i.e., for Λ≈M. In such case the n−n¯ oscillation search provides the bound:(34)M≳90TeV, much stronger than the LHC limit. If there is new physics around that energy scale, it should be discovered by future n−n¯ oscillation experiments with increased sensitivity [99], which are going to probe the vector diquark mass scale up to ∼175TeV. This is especially interesting since models with TeV-scale diquarks tend to improve gauge coupling unification [45,46].5

Conclusions

We have shown that lack of experimental evidence for proton decay singles out only a handful of phenomenologically viable leptoquark models. In addition, even leptoquark models with tree-level proton stability contain dangerous dimension five proton decay mediating operators and require an appropriate UV completion to remain consistent with experiments. This is especially relevant for the Standard Model extension involving the vector leptoquarks (3,1)23 or (3,3)23, since those are the only two models with a single new representation that do not suffer from tree-level proton decay and can explain the recently discovered anomalies in B meson decays.The property which makes the vector leptoquark (3,1)23 even more appealing is that it fits perfectly into the simplest Pati–Salam unification model, where it can be identified as the new gauge boson. If such an exciting scenario is indeed realized in nature, the B physics experiments can be used to actually probe the scale and various properties of grand unification!In the second part of the letter we focused on a model with a vector diquark (6,2)−16 and showed that neutron–antineutron oscillations can be mediated by such a vector particle. The model is somewhat constrained by LHC dijet searches; however, it can still yield a sizable neutron–antineutron oscillation signal, that can be probed in current and upcoming experiments. It can also give rise to significant four-jet event rates testable at the LHC.It would be interesting to explore whether a vector diquark with a mass at the TeV scale can improve gauge coupling unification in non-supersymmetric grand unified theories, similarly to the scalar case [103], providing even more motivation for upgrading the neutron–antineutron oscillation experimental sensitivity.

Acknowledgements

We thank Aneesh Manohar for useful conversations and Pavel Fileviez Pérez for comments regarding the manuscript. This research was supported in part by the DOE Grant No. DE-SC0009919.

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