Scalar-mediated double beta decay and LHC

The decay rate of neutrinoless double beta decay could be dominated by short-range diagrams involving heavy scalar particles ("topology-II"diagrams). Examples are diagrams with diquarks, leptoquarks or charged scalars. Here, we compare the discovery potential for lepton number violating signals at the LHC with constraints from dijet and leptoquark searches and the sensitivity of double beta decay experiments, using three example models. We note that already with 20/fb the LHC will test interesting parts of the parameter space of these models, not excluded by current limits on double beta decay.

decay [8]. We will concentrate on scalar-only contributions.
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 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 short-range contributions to double beta decay and we take into account these constraints in our numerical analysis. The rest of this paper is organized as follows. In section II 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 III, we present our numerical results. We then close with a short summary and discussion.

II. 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 III. 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.

A. Topology-II decompositions
Considering only the unbroken SU(3) C and U(1) Q there are only five possible decomposition of eq. (1) for topology-II. These are listed in table I. Note that in some cases there is more than one possibility for colour. There are six scalar states in these decompositions: (i) charged scalars, S + and S −− ; (ii) diquarks, S 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 . 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 I. 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].

B. 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 III.

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) 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 ǫ 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 [32]. 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. (3) (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 ǫ S 1 is given by and the matrix element is given by: Again, for further definitions and numerical values see [1, 3].

III. 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 [33], for the leptoquark and the singly charged scalar CalcHEP [34]. We have compared our results with the literature [35] 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 [20] [17] the main background can be traced to tt events. We then do a simple estimate which considers that the tt production cross section is very roughly about a factor 3 higher at √ s = 13 TeV than at √ s = 8 TeV. Thus, we scale the original fit to √ s = 8 TeV data with a simple constant and scale the background function from L = 3.6 fb −1 to future expected 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 [37]. For both, dijet and lljj 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. For future LQ searches at the LHC, we calculate LQ pair production cross sections as a function of LQ mass. We simply define the reach of the LQ search then as the mass for which there are less then 10 signal events in 20/fb (300/fb) at the LHC (before cuts). This results in the simple estimate of m LQ > ∼ 1.3 TeV (m LQ > ∼ 1.8 TeV) as the near (far) future limit. Thus, our results should be considered only rough estimates. For more exact results 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 [40,41] 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 [42]. 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). In this model the three components of the triplet diquark, the scalars S  fig. 2. 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 fig. 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) and are a function of the leptoquark mass m LQ and the (unknown) parameters µ and g 2 .
In Fig. 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 figs (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 fig. (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. (4). Thus, in general LHC and 0νββ decay probe complementary parts of parameter space. This can also be seen in fig. (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 Fig. 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 = m DQ 6 (left) and µ = m DQ 50 (right). Grey and blue regions show again the sensitivity of 0νββ decay current and future. The solid lines correspond to future LHC limits from dijet (black curves) and dilepton plus jets (red curves). The red curves start at m DQ = 2 × m LQ and stop at masses of the DQ, for which there are less than 5 signal events expected in L = 300 fb −1 .
For these choices of parameters, dijet searches can probe larger masses, but the lljj search probes smaller values of the coupling g 1 . Again, for larger choices of µ the branching ratio for the lljj final state is larger and the lljj search becomes more sensitive. Negative results from the dijet searches would exclude large part of the parameter space explorable by future 0νββ decay experiments. However, for large values of µ there is always a corner of parameter space for large couplings and DQ masses, where 0νββ decay is more sensitive.
Finally in Fig. 5 we plot a comparison of sensitivities of 0νββ decay and the dilepton plus jets (Fig. 5 right) and dijet (Fig. 5 left) searches at LHC for the three different models discussed in section II: 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). Fig. 5 shows also current and future limits from 0νββ decay for the respective models in consideration. The gray area is the currently excluded 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 (5) and (9). LHC is least sensitive for the singly charged scalar case, the S Finally, we briefly comment on other T-II models. As shown in table I, 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.

IV. 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 nonobservation 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 [43,44].