Lepton number violation at the LHC with leptoquark and diquark

We investigate a model in which tiny neutrino masses are generated at the two-loop level by using scalar leptoquark and diquark multiplets. The diquark can be singly produced at the LHC, and it can decay into a pair of leptoquarks through the lepton number violating interaction. Subsequent decays of the two leptoquarks can provide a clear signature of the lepton number violation, namely two QCD jets and a pair of same-signed charged leptons without missing energy. We show that the signal process is not suppressed while neutrino masses are appropriately suppressed.


I. INTRODUCTION
The Standard Model (SM) gauge symmetry of elementary particles based on the SU(3) C × SU(2) L × U(1) Y has been tested very accurately. On the other hand, the existence of the neutrino masses has been established [1][2][3][4][5][6]. This is clear evidence of the new physics beyond the SM because neutrinos are massless in the SM. Since neutrinos are electrically neutral, they can be Majorana particles unlike the other SM fermions [7]. The reason why neutrino masses are very different from those of the other SM fermions might be the Majorana property of neutrinos.
The most familiar utilization of the Majorana property to generate tiny neutrino masses is the so-called Type-I Seesaw mechanism in which SU(2) L -singlet right-handed neutrinos mediate in the tree diagram [8]. Because of the suppression by mass scales of new heavy particles, naturally light neutrinos can arise. Another typical prescription to obtain tiny Majorana neutrino masses is the so-called radiative seesaw mechanism [9][10][11][12][13], where neutrino masses are induced at the loop level.
In these models, the suppression of neutrino masses can be achieved by the loop suppression factor and/or a combination of new coupling constants even if new particles are not very heavy. The masses of charged leptons involved in the chirality flipping loop provide further suppression of the neutrino masses in some of such models [9][10][11].
Although the lepton number is conserved in the SM, the addition of the Majorana mass term of neutrinos breaks the lepton number conservation by two units. The measurement of the lepton number violating (L#V) processes such as the neutrinoless double beta decay [14,15] is extremely important because it gives evidence that neutrinos are Majorana particles. Such processes are naively expected to be very rare because neutrino masses are very small. This is true for the Type-I seesaw model with very heavy right-handed neutrinos because light Majorana neutrino masses are unique lepton number breaking parameters at the energy scale which is experimentally accessible.
However, in radiative seesaw models, a trilinear coupling constant for light (e.g. TeV-scale) scalars can be more fundamental than light neutrino masses as the L#V parameter at the accessible energy scale. Then, L#V processes via the trilinear coupling constant can be significant at the TeV-scale even if the neutrino masses are suppressed enough.
New particles related to the neutrino mass generation are usually produced via the electroweak interaction, and therefore the production cross sections are not so significant at the LHC. However, new particles in the loop of the radiative seesaw models can be charged under the SU(3) C [16][17][18].
Such a colored particle can easily be produced at hadron colliders. In these models, decay patterns of new colored particles could be related to the form of the neutrino mass matrix constrained by the neutrino oscillation data [16][17][18][19] (See also [20]).
In this paper, we investigate a radiative seesaw model with a scalar leptoquark multiplet and a scalar diquark multiplet. Majorana masses of neutrinos are induced via the two-loop diagram where colored particles are involved in the loop. The lepton number violation is caused by the trilinear coupling constant of the leptoquarks and diquark, which can produce a characteristic signature at the LHC. The signature consists of two QCD jets and a pair of same-signed charged leptons without missing energy, which would be easily observed at the LHC.
This paper is organized as follows. In Sec. II, we present the model. Section III is devoted to discussion on the collider phenomenology and the low energy constraints for the leptoquark and the diquark in the model. Conclusions are given in Section IV.

II. THE MODEL
The particle contents of the colored radiative seesaw model are shown in Table I. The model is briefly mentioned in Ref. [16]. The model includes a scalar leptoquark multiplet (S LQ ) whose lepton number and baryon number are 1 and 1/3, respectively. Under the SM gauge group, the S LQ is assigned to the same representation of right-handed down-type quarks; a 3 representation of SU(3) C , a singlet under SU(2) L , and hypercharge Y = −1/3. We also introduce a scalar diquark multiplet (S DQ ) which has a baryon number 2/3. We take S DQ as a 6 representation of SU(3) C , a singlet under SU(2) L , and a Y = −2/3 field. The diquark of a 6 representation can be expressed in a symmetric matrix form as The baryon number conservation is imposed to the model such that the proton decay is forbidden. We introduce the soft-breaking term (see the next paragraph) of the lepton number conservation to the scalar potential in order to generate Majorana neutrino masses. The Yukawa interactions with the leptoquark and diquark, which preserve both of the lepton number and the baryon number, are given by where σ a (a = 1-3) are the Pauli matrices, α and β (= r, g, b) denote the color indices; for example, S rr DQ corresponds to S DQ1 in Eq. (1). We choose the diagonal bases of mass matrices for the charged leptons and down-type quarks. Then, the SU(2) L partner of d iL is described as While Y L and Y R are general complex matrices, Y s is a symmetric matrix (Y T s = Y s ). Note that neutrinos interact with the leptoquark only through Y L , and we will see later that Y R is irrelevant to the neutrino mass at the leading order.
In the scalar potential of the model, we introduce the following three-point interaction: The coupling constant µ softly breaks the lepton number conservation by two units while the baryon number is conserved. There is no other possible soft-breaking term of the lepton number and/or the baryon number. We can take the µ parameter as a real positive value by using the rephasing of S DQ . Considering radiative corrections to m LQ and m DQ via the µ parameter, perturbativity requires µ min(m LQ , m DQ ) as discussed in Ref. [21] for the Zee-Babu model (ZBM) [10].
The neutrino mass term 1 2 (M ν ) ℓℓ ′ ν ℓL (ν ℓ ′ L ) c in the flavor basis is generated by a two-loop diagram in FIG. 1 including the leptoquark and the diquark. The mass matrix is calculated as where the loop function I ij is defined as The diagram is similar to the one in the ZBM although SU(3) C -singlet particles in the loop are replaced with colored particles. Thus, we refer to this model as the colored Zee-Babu model (cZBM).
See, e.g., Refs. [21][22][23] for studies about the ZBM for comparison with the cZBM. Note that Y R does not contribute to the two-loop diagram. 1 In the ZBM, at least one massless neutrino is predicted because of the antisymmetric Yukawa coupling matrix. In contrast, all of three neutrino masses can be non-zero in the cZBM because Y L is not an antisymmetric matrix. Since new colored scalars should be much heavier than the SM fermions, the loop function can be reduced to [22] whereĨ Hereafter, we restrict ourselves to the simplest scenario where Y R is small enough to be ignored. 2 A benchmark point in the parameter space of the cZBM is shown in Appendix A. 1 The YR contributes to Majorana neutrino masses at the higher loop level. The four-loop contribution is utilized in the model of Ref. [24] where YL is ignored. 2 If we extend the model as a two-Higgs-doublet model, we can eliminate the YR term by using a softly-broken Z2 symmetry (e.g., u iR (or ℓR) and the second Higgs doublet are Z2-odd fields) which is also required to avoid the flavor changing neutral current at the tree level. Another example to eliminate the YR term is the case where the leptoquark is not an SU(2)L-singlet but a triplet.

A. Leptoquark
The main production channel of leptoquarks at hadron colliders would be the pair-creation from gg and qq annihilation [25]. The associated production of S LQ with a lepton from qg coannihilation could also be possible [26]. The pair-production cross section is determined only by QCD interaction at the leading order [25], while the associated production mechanism highly depends on the Yukawa coupling constant of the leptoquark [26]. The associated production mechanism is negligible at a  [27]; the pair-production of scalar leptoquarks is assumed as well as a hundred percent decay branching ratio into the first (second) generation quarks and leptons. See also Refs. [28,29] for the ATLAS results with 1.03 fb −1 integrated luminosity. The analysis of the decay into third generation fermions would be performed in near future. The search strategies for the third generation leptoquarks have been studied in Ref. [30].
The leptoquark induces various LFV processes. At the tree level, four-fermion operators (two left-handed leptons and two left-handed quarks) are generated by integrating leptoquarks out. The constraints on such operators have been extensively studied in Ref. [31]. Tables 3,   4, 12, and 13 in Ref. [31] are relevant to the cZBM. Especially, operators (e L γ µ µ L )(u L γ µ u L ) and (ν ℓL γ µ ν ℓ ′ L )(d L γ µ s L ) are strongly constrained by the µ-e conversion search and the K meson decay measurement, respectively. For the benchmark point given in Appendix A, we have At the loop level, effects of leptoquarks on charged lepton transitions, i.e., ℓ i → ℓ j γ, have also been studied [32]. Since we assume that S LQ has the Yukawa interaction only with the lefthanded quarks (namely Y R = 0), the contribution from the top quark loop does not give a large enhancement of ℓ i → ℓ j γ. 3 Then, the branching ratio of µ → eγ is calculated as 3 It is known that the similar process b → sγ (induced by the uncolored charged Higgs boson) in the Type-II two-Higgs-doublet model is enhanced by the top quark loop [33].
Since we take Y R = 0, the sign of the leptoquark contribution to the leptonic g − 2 cannot be changed. It is worth to mention that the contribution of the leptoquark has an appropriate sign (the plus sign) 4 to compensate the difference between the measured value and the SM prediction for the muon g − 2. The preferred size of Y L is (Y L Y † L ) µµ ∼ 1 for m LQ ∼ 1 TeV. In order to satisfy LFV constraints with this size of Y L , a simple ansatz is that Y L is a diagonal matrix. Note that we must take care about the constraint on (ν eL γ µ ν µL )(d L γ µ s L ) (see Table 12 in Ref. [31] 12 1 because Y L is related to Y s through the neutrino mass matrix. We could not find any viable example of such a parameter set although it might exist with more complicated structures of Y L and Y s .

B. Diquark
At the LHC, the diquark S DQ in the cZBM would be singly produced by the annihilation of two down-type quarks. 5 The single production mechanism has an advantage to search for the relatively heavy diquark due to the s-channel resonance. The single production cross section is determined by (Y s ) 11 , which is evaluated in Ref. [36] as a function of the diquark mass with a fixed Yukawa coupling constant. The (Y s ) 11 in the cZBM is less constrained by the neutrino oscillation data because its contribution to neutrino masses is suppressed by m 2 d /m 2 DQ . If we assume (Y s ) 11 = 0.1 and m DQ = 4 TeV, the single production cross section σ(dd → S DQ ) is about 5 fb at the LHC with √ s = 14 TeV [36]. Note that the CMS experiment at √ s = 7 TeV with 1 fb −1 integrated luminosity excludes diquark masses between 1 TeV and 3.52 TeV at 95 % confidence level by assuming the diquark decay into two QCD jets for the E 6 diquark which couples with an up-type quark and a down-type quark [37]. See also Refs. [38][39][40].
The diquark induces flavor changing neutral current processes in the down-type quark sector.
Especially, it gives tree-level contributions to K 0 -K 0 , B 0 d -B 0 d and B 0 s -B 0 s mixings, resulting in strong constraints on Y s . By using the notations in Ref. [41], the benchmark point in Eqs. (A1) gives These values satisfy the constraints obtained in Ref. [41] (see also Refs. [42,43]).
The diquark in the cZBM decays into not only a pair of the down-type quarks but also a pair of leptoquarks. The fraction of fermionic and bosonic decay modes is calculated as This formula is the same for the other diquarks because of the SU(3) C symmetry. We focus on the case where the ratio in Eq. (9) is less than about unity such that the branching ratio for where c ij (s ij ) denotes cos θ ij (sin θ ij ). We use the following values: sin 2 2θ 23 = 1, sin 2 2θ 13 = 0.1, sin 2 2θ 12 = 0.87, δ = 0, ∆m 2 31 = 2.4 × 10 −3 eV 2 > 0, and ∆m 2 21 = 7.6 × 10 −5 eV 2 . Matrices Y L and Y s in Eqs. (A1) are constructed by assuming the following structures: It is easy to see that M ν with these matrices results in m 1 = x 2 m 2 d C(Y s ) 11 , α 21 = 0, and α 31 = π. We use x = 10 −4 , y = 0.07, z = 0.1, m d = 5 × 10 −3 GeV, m s = 0.1 GeV, m b = 4.2 GeV and (Y s ) 11 = 0.1. Note that the benchmark point gives (M ν ) ee ≃ 1.5 × 10 −3 eV, which is the effective mass relevant for the neutrinoless double beta decay.