Baryon Number and Lepton Universality Violation in Leptoquark and Diquark Models

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.


I. 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 fourdimensional 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) 7 6 (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 paper 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) 2 3 and (3, 3) 2 3 , 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 paper is organized as follows. In Sec. II 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. III 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. IV we analyze a model with a single vector color sextet, discussing its LHC phenomenology and implications for n −n oscillations. Section V contains conclusions. For clarity, we first summarize the combined results of [7] and this work in Table I, 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 paper 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].

A. 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 II [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) 1 6 and (3, 2) − 5 6 , 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) 2 3 and (3, 3) 2 3 : 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: 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 10 4 TeV, 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 Z 3 of a global U(1) B−L to forbid the problematic dimension five operators. They are also naturally ab-  Ultimately, as shown in Table I, there are only five color triplet or sextet vector representations that are free from treelevel 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.

B. Leptoquark phenomenology
The phenomenology of scalar and vector leptoquarks has been extensively discussed in the literature [14][15][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 with respect to the SM expectations has been reported [9,10]. A detailed analysis of the anomalies can be found in [17][18][19][20][21][22]. Several leptoquark models have been proposed to alleviate this tension and are favored by a global fit to R K ( * ) , R D ( * ) 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) are naturally free from any tree-level proton decay, since for the scalar leptoquark (3, 2) 1 6 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 II, both vector models (3, 1) 2 3 and (3, 3) 2 3 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) 1 6 . However, as we will see in Sec. III, only in the case of the: there exists a very appealing SM extension which intrinsically contains such a state in its spectrum, simultaneously forbidding dimension five proton decay.
In Sec. IV we close the gap in diquark phenomenology by discussing the implications of a vector diquark model for n−n oscillation experiments.

III. VECTOR LEPTOQUARK MODEL
As emphasized in Sec. II B, the SM extended by just the vector leptoquark (3, 1) 2 3 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.

A. 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) 2 3 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) 2 3 state arises in composite models [52][53][54]. The third, perhaps the most desirable option, is that the vector leptoquark (3, 1) 2 3 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: at higher energies [55]. 2 In this case the vector leptoquark (3, 1) 2 3 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 Interestingly, the theory is free from tree-level proton decay via gauge interactions. The explanation for this is straightforward. Since SU (4)  The quark and lepton mass eigenstates are related to the gauge eigenstates through n f × n f unitary matrices, with n f = 3 the number of families of quarks and leptons. Expressing the interactions that couple the (3, 1) 2 3 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: The SU(4) gauge coupling constant, g 4 , is not an indepen-

C. B meson decays
In the SM, flavor-changing neutral currents with ∆B = −∆S = 1 are described by the effective Lagrangian [58,59]: where the ellipsis denote four-quark operators, O 7 and O 8 are electro-and chromo-magnetic-moment-transition operators, and O 9 , O 10 and O ν are semi-leptonic operators involving either charged leptons or neutrinos: 3 Chirally-flipped (b L(R) → b R(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, in the form of scalar operators: Tensor operators cannot arise from short distance NP with the SM linearly realized and, moreover, under this assumption C P = −C S and C P = C S [60]. Exchange of the (3, 1) 2 3 vector leptoquark gives tree-level contributions to the Wilson coefficients at its mass scale, M : The recent B meson decay measurements [9,10] show an excess above the SM background in the ratios R K ( * ) = Γ(B → K ( * ) µµ) / Γ(B → K ( * ) ee). Those anomalies are best fit by ∆C 9 = −∆C 10 ≈ −0.6 [17], which requires (g 2 4 /M 2 )L d * bµ L d sµ ≈ 1.8 × 10 −3 TeV −2 . Assuming L d * bµ L d sµ = 1 2 , which is the largest value allowed by unitarity, we obtain the previously quoted leptoquark mass of M ≈ 16 TeV. Limits on ∆C 9,10 and ∆C ( ) S can be accommodated by adjusting R sµ and R bµ . [61], severely constrain theoretical models for those anomalies. As seen above, the (3, 1) 2 3 vector leptoquark evades this constraint by giving no correction at all to C ν ; the result holds generally for this type of NP mediator at tree level [11,62]. It has been pointed out that generally the condition ∆C ν (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 ∆C ν down to the electroweak scale. The only contribution comes from the renormalization by exchange of SU(2) gauge bosons that mixes the singlet operator (qγ µ P L e)(ēγ µ P L q) into the triplet, (q τ a γ µ P L e)(ē τ a γ µ P L q), resulting in where S = V † L u = L d U . The vector contribution to the rate does not interfere with the SM, which implies R K ( * ) ν − 1 = 1 3 ij |C ij ν /C SM ν | 2 ; using R K ( * ) ν < 4.3 [64], we obtain the condition M > 0.8 TeV. Since ln(M/M W ) is not large for M ≈ 16 TeV, 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 L γ µ ν 2 L )(ν 3 L γ µ b L ), while the SM always involves a sum over the same neutrino flavors ∼ jν j L γ µ ν j L . 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 3 ν 2 ) while there are three configurations in the SM case (ν j ν j , j = 1, 2, 3).

D. 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 1000 TeV scale [82][83][84][85][86]. Surprisingly, all of the kaon decay bounds can be avoided if the unitary matricesL d andR are of the form (see also [86]): 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 40 TeV [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 19 TeV.
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 and does not require introducing any new fermion fields beyond the SM particle content and a RH neutrino: Such a model predicts rates for B + → K + e ∓ µ ± and µ → e γ, 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]. 4

IV. VECTOR DIQUARK MODEL
In this section we discuss the properties of a model with just one additional representation -the vector color sextet: 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: 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. IV B, are mediated by the dimension five terms:

A. LHC bounds
Several studies of constraints and prospects for discovering vector diquarks at the LHC can be found in the literature. Most of the analyses have focused on the case of a sizable diquark coupling to quarks [47][48][49], although an LHC fourjet 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 λ ij ≈ 1 (i, j = 1, 2) those searches result in a bound on the vector diquark mass M λ≈1 8 TeV .
Lowering the value of the coupling to λ ij ≈ 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 37 fb −1 of data by the ATLAS experiment [94] with no evident excess above the SM background constrains the vector diquark mass to be 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.

B. 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 ∆B = 1 are indeed suppressed or do not occur in Nature at all, the next simplest case involves ∆B = 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 ∆B = 2 would be accompanied by ∆L = 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. II C. 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 O 1 in Eq. (27) involving two vector diquarks (6, 2) − 1 6 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. 5 With the simplifying assumption that c 1 ≈ 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 O 1 is: Combining this with the results of [7,101], we obtain an estimate for the n −n transition matrix element, 6 where |n is the neutron state at zero momentum. Current experimental limit [102] implies, assuming λ 11 ≈ 0.01 (which is well below the LHC bound from dijet searches, as discussed earlier), that M 2.5 TeV 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: 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 ∼ 175 TeV. This is especially interesting since models with TeV-scale diquarks tend to improve gauge coupling unification [45,46].

V. 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 ap-propriate UV completion to remain consistent with experiments. This is especially relevant for the Standard Model extension involving the vector leptoquarks (3, 1) 2 3 or (3, 3) 2 3 , 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) 2 3 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 paper we focused on a model with a vector diquark (6, 2) − 1 6 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.