Scalar-mediated double beta decay and LHC

The decay rate of neutrinoless double beta (0νββ) decay could be dominated by Lepton Number Violating (LNV) short-range diagrams involving only heavy scalar intermediate particles, known as “topology-II” diagrams. Examples are diagrams with diquarks, leptoquarks or charged scalars. Here, we compare the LNV discovery potentials of the LHC and 0νββ-decay experiments, resorting to three example models, which cover the range of the optimistic-pessimistic cases for 0νββ decay. We use the LHC constraints from dijet as well as leptoquark searches and find that already with 20/fb the LHC will test interesting parts of the parameter space of these models, not excluded by the current limits on 0νββ-decay.


Introduction
From the theoretical point of view, neutrinoless double beta decay (0νββ) can be written as a dimension-9 operator: O 0νββ = c 9 Λ 5 LN Vūū ddēē. (1.1) Here, Λ LN V is the scale of lepton number violation (LNV). Many beyond the standard model contributions to this operator have been discussed in the literature, for a review see [1]. Contributions to the decay rate of 0νββ decay can be classified as (i) neutrino mass mechanism; (ii) long-range [2] and (iii) short-range contributions [3]. 1 Particularly interesting is the possibility that all beyond-standard-model particles, appearing in the ultra-violet completions of this operator, are heavy. This corresponds to the short-range part of the 0νββ decay amplitude. In this case, with the current sensitivities of 0νββ decay experiments [6,7] of the order of roughly O(10 25 − 10 26 ) yr, one probes mass scales in the range Λ LN V ∼ (1 − 3) TeV -exactly the range of energy explored at the LHC.
A list of all possible decompositions of eq. (1.1) has been found in [8]. Models fall into two classes, called topology-I (T-I) and topology-II (T-II), see figure 1. In this figures outside lines correspond to the six fermions appearing in eq. (1.1) , while the internal particles can be scalars, vectors or fermions. Just to mention one example for T-I and T-II each: in left-right (LR) symmetric models, right-handed gauge bosons (W R ) and neutrinos (N R ) appear in T-I as W R − N R − W R exchange [9,10], while a T-II type diagram can appear as W R − ∆ ±± R − W R exchange [11] in LR models with right-handed triplets (∆ R ). 1 Neither in the long-range nor the short-range part of the amplitude the neutrino mass does appear directly. However, the ∆L = 2 interactions, present necessarily in all contributions to 0νββ decay, implies Majorana neutrino masses must be non-zero in all possible models contributing to eq. (1.1) [4,5]. Only SSS and VVS can contribute significantly to 0νββ decay [8]. We will concentrate on scalar-only contributions.
The classical LNV signal searched for at the LHC is two same-sign leptons plus jets (lljj), first discussed as a possible signal for left-right symmetric models in [12], see also [13]. This signal is generated from the T-I diagram with right-handed neutrinos. The doubly charged scalar can be searched via vector-boson-fusion, see for example [14,15]. This corresponds to the T-II diagram mentioned above. VBF gives the same final state (lljj), but has different kinematics. We mention in passing that also di-lepton searches can be used to put bounds on LR models [16].
Both ATLAS and CMS have published results for run-I of the LHC. CMS [17] observed an excess in the electron sample around m eejj 2 TeV, 2 but no excess in the muon sample. CMS interprets the excess as a statistical fluctuation. ATLAS used 20.3/fb of pp collision data in their search [18], finding no anomalous events. The experimental collaborations then give limits on heavy Majorana neutrinos in left-right (LR) symmetric models, derived from this data.
However, LNV searches at the LHC do not give bounds only for LR models. In principle, all models that contribute to eq. (1.1) via short-range contributions should lead to a LNV signal at the LHC. For the case of topology-I, the implications of LNV searches at the LHC and their connection to 0νββ decay has been studied in [19,20]. In this paper we will study future LHC constraints on topology-II models. We will concentrate on the case where the non-SM particles are all scalars.
Both, ATLAS and CMS have published searches using dijets, based on √ s = 8 TeV [21,22] and √ s = 13 TeV [23,24] data. No new resonances have been observed in these searches, both collaborations give instead upper limits on σ ×BR as a function of resonance mass. While dijet data of course can not be used to establish the existence of LNV, nonobservation of new resonances in dijet searches at the LHC can be used to obtain limits on 0νββ decay [25]. In our analysis, presented below, we will also estimate the reach of future LHC data and compare it to expectations for the LNV searches.
As discussed below, in many of the models for T-II double beta decay leptoquarks (LQs) appear. Searches for leptoquarks have been carried out at the LHC by both ATLAS

JHEP12(2016)130
and CMS. Lower limits on the masses of first generation LQs from pair production in the √ s = 8 TeV data are now roughly of the order of 1 TeV [26,27]. ATLAS has published first limits from √ s = 13 TeV data with only 3.2/fb, which already give very similar limits [28] despite the smaller statistics. Searches for singly produced LQs, published by CMS [29], give more stringent limits, albeit only for large values of the LQ coupling to quarks and leptons. Also these limits and results of future searches can be used to constrain shortrange contributions to double beta decay and we take into account these constraints in our numerical analysis.
While the observed non-zero neutrino masses are the main motivation to study TeVscale LNV extensions of the SM [4,5,[32][33][34], in this paper we do not do an explicit fit to all oscillation data for the different example models we study. The reason for this is simply that both 0νββ and LHC are mostly sensitive to first generation quark couplings, while for all models considered below, one expects that third generation quark couplings give the dominant contributions to the neutrino mass matrix. Thus, no definite cross-check can be made between neutrino data and the signals that we are interested in, unless one were to make the (unjustified) assumption that the couplings are (quark) generation independent. However, we have checked that the regions of couplings that LHC and 0νββ decay can probe are allowed by experimental data on neutrino masses.
The rest of this paper is organized as follows. In section 2 we discuss different T-II contributions to 0νββ decay. We give the Lagrangian and necessary definitions for three example models. These models cover the optimistic/pessimistic cases for 0νββ decay. In section 3, we present our numerical results. We then close with a short summary and discussion.

General setup
In this section we will first recall the general setup of the topology-II contributions to 0νββ decay. We will then give a few more details for those three concrete example models, that we will study numerically in section 3. These examples, chosen from the full list of possible scalar models given in [8], allow us to cover both the most optimistic and the most pessimistic cases for the sensitivity of future double beta decay experiments.

Topology-II decompositions
Considering only the unbroken SU(3) C and U(1) Q there are only five possible decomposition of eq. (1.1) for topology-II. These are listed in table 1. Note that in some cases there is more than one possibility for colour. There are six scalar states in these decompositions: LQ . We define scalar diquarks as particles coupling to a pair of same-type quarks and leptoquarks as particles coupling to a quark and a lepton.
Depending on the chirality of the outer fermions, the diquarks could come either from electro-weak (EW) singlets or triplets, while the leptoquarks could either be members of singlets or doublets. We have examples for each in the three selected models below. The singly charged scalar S + necessarily has to be a member of an SU(2) L doublet: S 1,2,1/2 . Table 1. List of decompositions for topology II from [8]. Only the electric and colour charges of the internal bosons are given here. All listed possibilities give short-range contributions. For the colour charges in some cases there exist two possible assignments.

JHEP12(2016)130
Here and everywhere else in this paper the subscripts give the transformation properties under the SM group in the order SU(3) C × SU(2) L × U(1) Y . Finally, S −− could either come from an EW singlet or a triplet. Considering the full SM group, overall [8] gives 27 different combinations ("models") for the five decompositions shown in table 1. All of these generate Majorana neutrino masses, from tree-level masses for decompositions with S 1,3,−1 to 4-loop neutrino masses for the diagram containing S 3,1,−1/3 −S 3,1,−1/3 −S6 ,1,2/3 [4]. Our three examples correspond to two 2-loop and one 1-loop model, see below. This is motivated by the fact that for 2-loop neutrino mass models one can expect that the short-range part of the amplitude for 0νββ and the mass mechanism can give similar contributions to the overall decay rate [4].

Selected example models
Here, we will give the basic Lagrangian terms of three decompositions of the d = 9 0νββ decay operator taken from [8]. These examples correspond to T-II-2 BL # 11, T-II-4 BL # 11 and T-II-5 BL # 11 in the notation of [8]. Constraints on other short-range T-II decompositions will be very similar to these examples, as we will also discuss in section 3.

T-II-4, BL#11
Our first example model contains two new particles: a scalar diquarks and a leptoquark. In the context of 0νββ decay, diquark contributions were first discussed in [30]. We define scalar diquarks as particles coupling to a pair of same-type quarks. We choose the example T-II-4, BL # 11 in the notation of [8]. This model generates neutrino masses at 2-loop order [4], which means the TeV scale is the natural scale to fit to neutrino data. One expects therefore that this model is testable at the LHC. Note that a possible SU(5) embedding of this model has been recently discussed in [31].
The new beyond the SM states in this model are: The interaction Lagrangian of the model is given by: Here we introduced the notationŜ DQ = S DQ,a (T6) a IJ , with I, J = 1 − 3 and the color triplet indexes and a = 1 − 6 the color sextet indexes. g 1 and g 2 are dimensionless Yukawas and µ has dimension of mass. The symmetric 3 × 3 matrices T 6 and T6 can be found in ref. [8]. Note that eq. (2.1) violates lepton number by two units.
The inverse half-life for 0νββ for the diagram of figure 2, is given by [8]: where G 01 is a phase space integral and DQ is defined by and the nuclear matrix element is: Here M 1,2 are defined in [3], numerical values for 136 Xe can be found in [1].

T-II-5, BL#11
As a second example we discuss another model with a scalar diquark. However, this diquark couples only to down-type quarks. This model was first discussed in [34]. It corresponds to the example T-II-5, BL# 11 from the list of decompositions of the d = 9 0νββ decay operator [8]. Also this model generates neutrino masses at 2-loop order as discussed in [4]. This particular case introduces a singlet diquark S 2/3 DQ = S6 ,1,2/3 and a singlet leptoquark S 1/3 LQ = S3 ,1,1/3 . With these new fields, the Lagrangian contains the interactions: Here, as before, by definitionŜ DQ,a (T 6 ) a IJ . The inverse half-life for the short-range 0νββ decay in this model has the same form as eq. (2.2) (with some obvious replacements). In particular, it depends in the same combination of nuclear matrix elements.

T-II-2, BL#11
Finally, we will discuss a model with a singly charged scalar. We choose the example T-II-2, BL#11 from the list of [8]. This model generates neutrino masses at 1-loop order [4].
In this model, we add the following states to the SM particle content: With these new fields, the relevant Lagrangian is: The inverse half-life for 0νββ (short-range part of the amplitude) can be written as: where S 1 is given by and the matrix element is given by: Again, for further definitions and numerical values see [1,3].

JHEP12(2016)130 3 Numerical results
In this section we present our numerical results. We estimate the sensitivity of current and future 0νββ experiments and compare them with the sensitivity of dijet, leptoquark and dilepton plus jets searches at LHC at √ s = 13 TeV. For definiteness we assume two values for the accumulated luminosity L: L = 20/fb and L = 300/fb.
For the calculation of the cross sections of the diquark scalar resonances we use Mad-Graph5 [35], for the leptoquark and the singly charged scalar CalcHEP [36]. We have compared our results with the literature [37] and found good agreement with published values, whenever available. Plots for the cross sections can be found in our previous work on T-I contributions for 0νββ decay [20].
From the cross sections we then estimate the future LHC sensitivity as follows. For the LNV signal (lljj) we first take a simple fit function [20] to the background of existing data of the CMS analysis [38] based on 3.6 fb −1 at √ s = 8 TeV. We checked this fit against the CMS analysis [17] based on 19.7 fb −1 of data at √ s = 8 TeV, published later, and found good overall agreement. In the CMS analysis [17] the main background can be traced to tt events. Since the tt production cross section is very roughly about a factor 3 higher at √ s = 13 TeV than at √ s = 8 TeV, one can expect that also the background for the lljj signal for √ s = 13 TeV should be larger by a similar factor. Thus, we scale up the original fit function for √ s = 8 TeV with a simple overall constant factor to account for the larger expected backgrounds and then scale from L = 3.6 fb −1 to future expected luminosities of L = 20/fb and 300/fb. Similarly we estimate the LQ background fitting the background of the current search at L = 2.6 fb −1 and √ s = 13 TeV by CMS [39]. We scale this background simply from L = 2.6 fb −1 to future luminosities of L = 20/fb and 300/fb. For the estimation of the future dijet background we use the fit of the SM dijet distribution fitted to Monte Carlo simulation given in [40]. For the dijet, lljj and LQ analysis we then estimate backgrounds as dicussed above and define the sensitivity reach as either the simple square root of the background (times two for 95 % c.l.) or 5 signal events, whichever is larger.
We find that for the case of the LQ and LNV signals the final reach in the masses are dominantly determined by the signal cross sections, since the expected backgrounds at the largest invariant masses are low. We have checked that requiring only 3 signal events (instead of 5) would lead to final estimated mass reaches larger by roughly ∆M ∼ 0.1 (0.2) TeV for leptoquarks (diquarks). Our results should therefore be considered only rough estimates for the true mass reach of the LHC. For more exact numbers, cross sections at NLO and a full MonteCarlo simulation including detector effects would be necessary.
For double beta decay we use the current limit of T 0νββ 1/2 ( 136 Xe) ≥ 1.1 × 10 26 yr from the KamLAND-Zen collaboration [7]. 3 Several experimental proposals aim at half-life sensitivities of the order of 10 27 yr. We will use the estimated sensitivity of the nEXO proposal [43,44] of T 0νββ 1/2 ( 136 Xe) 6 × 10 27 yr for our calculation of the future limits. We convert half-life limits into limits on masses and couplings, using the equations discussed in the previous section. We take into account the QCD corrections to the Wilson coefficients, calculated recently in [45]. In particular for the model with the singly charged scalar QCD corrections have been found to be very important numerically.
We will first discuss the case of our example model 1, see the Lagrangian in eq. (2.1). In this model the three components of the triplet diquark, the scalars S We have assumed for simplicity that the Yukawa couplings g 1 and g 2 are different from zero for the first quark and lepton generations only. As is shown in figure 2, the scalar diquark S (4/3) DQ can only decay through two possible channels: dijets (jj) and dilepton plus two jets (lljj). The respective branching ratios can be calculated directly from the Lagrangian (2.1) and are a function of the leptoquark mass m LQ and the (unknown) parameters µ and g 2 .

JHEP12(2016)130
In figure 3 we show a comparison between 0νββ decay and dijet, LQ and dilepton plus jets searches at LHC in the plane m DQ vs m LQ , for two fixed choice of g 1 = g 2 (bottom: g 1 = g L , top: g 1 = 0.2) and two values for the accumulated luminosity: L = 20/fb (left) and L = 300/fb (right). Here, g L is the SU(2) L coupling. µ is chosen as µ = m DQ /6 (bottom) and µ = m DQ (top). The vertical black line corresponds to future limits from dijet searches at the LHC, the horizontal purple line is for leptoquark searches and the triangular red curve covers the region probed by the lljj search. The dashed line shows the kinematic limit for the lljj signal, where m DQ = 2×m LQ . For masses m DQ < 2×m LQ , one of the LQs goes off-shell and the branchig ratio for the final state lljj drops to unmeasurably small values.
As the figures 3 on the left show, LHC searches will significantly constrain parameter regions of LNV models contributing to 0νββ decay already with moderate luminosities. The lljj signal depends very sensitively on the choice of µ, while the dijet signal depends mostly on the value of g 1 . Smaller values of µ reduce the branching ratio for the lljj final state, reducing its reach. However, in this case the branching ratio for the dijet final states increases, making the dijet search more powerful, as the figure shows. We stress again, that while dijet searches can be used to exclude parameter regions of LNV models contributing to 0νββ decay, to establish a direct relation between 0νββ and LHC, a positive result from the LNV search (lljj) at the LHC would be necessary.
For L = 300/fb, see figure 3 on the right, the LHC can probe up to DQ masses of the order of 8 − 9 TeV (for g 1 ≥ 0.2). Whether dijet or LNV signal are more constraining depends on the exact value of µ We have chosen the value of µ = m DQ /6, because, as the figure on the bottom right shows, negative results from LHC LQ and dijet searches would rule out partial 0νββ decay half-lives in this model below the current experimental limit for µ = m DQ /6, assuming g 1 = g 2 = g L . For µ ≤ m DQ /50 negative searches from the LHC would rule out partial 0νββ decay half-lives below the future bound of T 1/2 = 6 × 10 27 ys. 0νββ decay depends on the mean of the couplings and masses, see eq. (2.3). Thus, in general LHC and 0νββ decay probe complementary parts of parameter space. This can also be seen in figure 3: for large values of µ and/or large values of g 1 and g 2 there is always a region in parameter space for large values of the DQ mass, where double beta decay is more sensitive than the LHC.
In figure 4 we show the comparison between the 0νββ decay and dijet and dilepton plus jets searches at LHC in the plane g 1 − m DQ . The LQ mass was chosen as m LQ = 1.8 TeV, roughly the expected future bound from LHC. g 2 = g L , µ = (right). The gray region is the current lower limit in 0νββ decay half-life, the blue one the estimated future sensitivity of T 1/2 = 6×10 27 ys. For more details see text.  Figure 5. Future limits for the LHC at √ s = 13 TeV and L = 300 fb −1 compared with current and future double beta decay experiments. The gray region is the current lower limit in 0νββ decay half-life whereas the blue region represents the parameter region accessible in near future 0νββ experiments. The colored lines shows sensitivity limits for the LHC for dijet (left) and dilepton plus jets (right) searches for production of three different scalar bosons S +1 (red), S DQ 2/3 (purple) and S DQ 4/3 (black). These limits were calculated using g 2 = g L and m LQ = 1.8 TeV and µ = m DQ 6 . For more details see text.
Finally in figure 5 we plot a comparison of sensitivities of 0νββ decay and the dilepton plus jets (figure 5 right) and dijet (figure 5 left) searches at LHC for the three different models discussed in section 2: T-II-2 BL # 11 (singly charged scalar), T-II-4 BL # 11 (triplet diquark) and T-II-5 BL # 11 (singlet diquark). The double beta decay and LHC limits were calculated using the parameters µ = m DQ /6, m LQ = 1.8 TeV and g 2 = g L . The LHC is most sensitive for the case of the triplet diquark model (T-II-2 BL # 11), black curve. This is simply because the cross section of the resonance production of the scalar diquark S (4/3) DQ is larger than the one for the diquark S (−2/3) DQ (purple curve) and the singled charged scalar S 1 (red curve). Figure 5 shows also current and future limits from 0νββ decay for the respective models in consideration. The gray area is the currently excluded JHEP12(2016)130 part of parameter space from non observation of 136 Xe decay with T 1/2 > 1.1 × 10 26 yr and the blue one the estimated future sensitivity, as before. The full lines are for the two diquark models (which have the same nuclear matrix elements, see above). The dashed lines are for the singly charged scalar model (T-II-2 BL # 11), which has a different nuclear matrix element, compare eqs. (2.4) and (2.8). LHC is least sensitive for the singly charged scalar case, the S (−2/3) DQ is intermediate between the other two. Finally, we briefly comment on other T-II models. As shown in table 1, all T-II decompositions contain either a diquark or a charged scalar (in one case two different diquarks). The three example models, which we used in the numerical analysis, covers the cases with the largest and smallest cross sections at the LHC. It also covers the models with the largest and smallest matrix elements for the 0νββ decay. Thus, our sensitivity estimate for the future covers the extreme cases, both optimistic and pessimistic, and all other models should lie somewhere in between.
In case of a discovery in the future at the LHC, one important question to ask is, which of the different model possibilities is the one realized in nature. As in the case of T-I [20], this might be achieved by investigating mass peaks in different variables and by the measurememt of the "charge asymmetry", i.e. the measurement of the number of events in l − l − jj relative to l + l + jj.

Discussion and summary
We have discussed how future LNV and dijet searches at the LHC can be used to constrain scalar short-range contributions to neutrinoless double beta decay (topology-II diagrams). We have concentrated on three LNV models, chosen from the full list of possible scalar short-range contributions to 0νββ decay given in [8]. Two of these models contribute to 0νββ decay through short-range diagrams mediated by diquark scalars and one of them by a singly charged scalar. For these models we have shown that the future LNV and dijet searches at the LHC will provide stringent constraints on the parameter space of the models, complementary to 0νββ decay experiments. Except for small parts of the parameter region of these LNV models, a 0νββ decay signal corresponding to a half life in the range T 1/2 < 10 27 ys should imply a positive LNV or dijet signal at the LHC. On the other hand, the non-observation of a positive signal at the LHC would rule out most of the parameter region measurable in 0νββ decay. We note that, while we have concentrated on three particular examples, similar constraints will apply to any scalar short-range contributions to 0νββ.
Finally, we mention that the observation of lepton number violation at the LHC and/or in double beta decay will have important consequences for high-scale models of leptogenesis [46,47]. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.