Scalar-mediated double beta decay and LHC
aa r X i v : . [ h e p - ph ] J un IFIC/16-41
Scalar-mediated double beta decay and LHC
L. Gonzales, ∗ J.C. Helo, † and S.G. Kovalenko ‡ Universidad T´ecnica Federico Santa Mar´ıa,Centro-Cient´ıfico-Tecnol´ogico de Valpara´ıso,Casilla 110-V, Valpara´ıso, Chile
M. Hirsch § AHEP Group, Instituto de F´ısica Corpuscular – C.S.I.C./Universitat de Val`enciaEdificio de Institutos de Paterna, Apartado 22085, E–46071 Val`encia, Spain
Abstract
The decay rate of neutrinoless double beta (0 νββ ) decay could be dominated by short-rangediagrams involving heavy scalar particles (“topology-II” diagrams). Examples are diagrams withdiquarks, leptoquarks or charged scalars. Here, we compare the discovery potential for leptonnumber violating signals at the LHC with constraints from dijet and leptoquark searches and thesensitivity of 0 νββ decay experiments, using three example models. We note that already with20/fb the LHC will test interesting parts of the parameter space of these models, not excluded bycurrent limits on double beta decay.
Keywords: double beta decay; neutrino masses and mixing; LHC ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: mahirsch@ific.uv.es . INTRODUCTION From the theoretical point of view, neutrinoless double beta decay (0 νββ ) can be writtenas a dimension-9 operator: O νββ = c Λ LNV ¯ u ¯ udd ¯ e ¯ e. (1)Here, Λ LNV is the scale of lepton number violation (LNV). Many beyond the standardmodel contributions to this operator have been discussed in the literature, for a review see[1]. Contributions to the decay rate of 0 νββ decay can be classified as (i) neutrino massmechanism; (ii) long-range [2] and (iii) short-range contributions [3]. Particularly interesting is the possibility that all beyond-standard-model particles, ap-pearing in the ultra-violet completions of this operator, are heavy. This corresponds to theshort-range part of the 0 νββ decay amplitude. In this case, with the current sensitivities of0 νββ decay experiments [6, 7] of the order of roughly O (10 − ) yr, one probes massscales in the range Λ LNV ∼ (1 −
3) TeV - exactly the range of energy explored at the LHC.A list of all possible decompositions of eq. (1) has been found in [8]. Models fall into twoclasses, called topology-I (T-I) and topology-II (T-II), see fig. (1). In this figures outsidelines correspond to the six fermions appearing in eq. (1) , while the internal particles can bescalars, 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-Ias 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 ).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]. Thissignal is generated from the T-I diagram with right-handed neutrinos. The doubly chargedscalar can be searched via vector-boson-fusion, see for example [14, 15]. This corresponds tothe T-II diagram mentioned above. VBF gives the same final state (lljj), but has differentkinematics. We mention in passing that also di-lepton searches can be used to put boundson LR models [16].Both ATLAS and CMS have published results for run-I of the LHC. CMS [17] observedan excess in the electron sample around m eejj ≃ but no excess in the muon sample.CMS interprets the excess as a statistical fluctuation. ATLAS used 20 . 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 0 νββ decay, implies Majorananeutrino masses must be non-zero in all possible models contributing to eq.(1) [4, 5].
14 events with an estimated background of 4 events [17], roughly equal to 2 . σ c.l. v v v Topology I Topology IIv v v v FIG. 1:
Tree-level topologies for the d = 9 0 νββ decay operator. External lines are fermions;internal lines can be fermions (solid) or scalars/vectors. For T-II there are in total 4 possibilitiesclassified as: SSS, VVV, SSV and VVS. Only SSS and VVS can contribute significantly to νββ 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 LHCand their connection to 0 νββ decay has been studied in [19, 20]. In this paper we will studyfuture LHC constraints on topology-II models. We will concentrate on the case where thenon-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 thesesearches, both collaborations give instead upper limits on σ × BR as a function of resonancemass. While dijet data of course can not be used to establish the existence of LNV, non-observation of new resonances in dijet searches at the LHC can be used to obtain limits on0 νββ decay [25]. In our analysis, presented below, we will also estimate the reach of futureLHC 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 andCMS. Lower limits on the masses of first generation LQs from pair production in the √ s = 8TeV data are now roughly of the order of 1 TeV [26, 27]. ATLAS has published first limitsfrom √ s = 13 TeV data with only 3 . / fb, which already give very similar limits [28] despitethe smaller statistics. Searches for singly produced LQs, published by CMS [29], give morestringent limits, albeit only for large values of the LQ coupling to quarks and leptons. Alsothese limits and results of future searches can be used to constrain short-range contributionsto 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-IIcontributions to 0 νββ decay. We give the Lagrangian and necessary definitions for threeexample models. These models cover the optimistic/pessimistic cases for 0 νββ decay. Insection III, we present our numerical results. We then close with a short summary anddiscussion. 3 ediator ( Q em , Q colour ) S S ′ S ′′ ud )(¯ ud )(¯ e ¯ e ) (+1, or ) (+1, or ) ( − , )2 (¯ ud )(¯ u ¯ e )(¯ ed ) (+1, or ) ( − / , ) ( − / , )3 (¯ u ¯ u )( dd )(¯ e ¯ e ) (+4/3, or ) (+2/3, or ) ( − , )4 (¯ u ¯ u )(¯ ed )(¯ ed ) (+4/3, or ) ( − / , ) ( − / , )5 (¯ u ¯ e )(¯ u ¯ e )( dd ) ( − / , ) ( − / , ) (+2/3, or )TABLE I: List of decompositions for topology II from [8]. Only the electric and colour charges ofthe internal bosons are given here. All listed possibilities give short-range contributions. For thecolour charges in some cases there exist two possible assignments.
II. GENERAL SETUP
In this section we will first recall the general setup of the topology-II contributions to0 νββ 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 ofpossible scalar models given in [8], allow us to cover both the most optimistic and the mostpessimistic 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 decompo-sition of eq. (1) for topology-II. These are listed in table I. Note that in some cases there ismore than one possibility for colour. There are six scalar states in these decompositions: (i)charged scalars, S + and S −− ; (ii) diquarks, S / DQ and S / DQ ; and (iii) leptoquarks, S − / LQ and S − / LQ .Depending on the chirality of the outer fermions, the diquarks could come either fromelectro-weak (EW) singlets or triplets, while the leptoquarks could either be members ofsinglets or doublets. We have examples for each in the three selected models below. Thesingly charged scalar S + necessarily has to be a member of an SU (2) L doublet: S , , / . Hereand everywhere else in this paper the subscripts give the transformation properties underthe SM group in the order SU (3) C × SU (2) L × U (1) Y . Finally, S −− could either come froman EW singlet or a triplet.Considering the full SM group, overall [8] gives 27 different combinations (“models”) forthe five decompositions shown in table I. All of these generate Majorana neutrino masses,from tree-level masses for decompositions with S , , − to 4-loop neutrino masses for thediagram containing S , , − / − S , , − / − S ¯6 , , / [4]. Our three examples correspond to two2-loop and one 1-loop model, see below. This is motivated by the fact that for 2-loopneutrino mass models one can expect that the short-range part of the amplitude for 0 νββ u de + de + S DQ S LQ S LQ S DQ uu uu g g g g g µ νββ : LHC : g S LQ S LQ ¯ u ¯ udd e − e − S DQg g µ ( a ) ( b )( c ) FIG. 2: Quark-level Feynman diagrams for (a) same-sign dilepton plus jets ( lljj ) signal, (b) dijetsignal at the LHC and (c) neutrinoless double beta decay for the example model-1 containing adiquark and a leptquark scalar state. 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 0 νββ decay operator taken from [8]. These examples correspond to T-II-2 BL
1. T-II-4, BL Our first example model contains two new particles: A scalar diquarks and a leptoquark.In the context of 0 νββ decay, diquark contributions were first discussed in [30]. We definescalar diquarks as particles coupling to a pair of same-type quarks. We choose the exampleT-II-4, BL S DQ = S , , / = S / DQ S / DQ S − / DQ − S / DQ , S LQ = S , , / = S / LQ S − / LQ . The interaction Lagrangian of the model is given by: L (1) DQLQ = L SM + g ¯ Qτ · ˆ S DQ · Q c + g ¯ Lτ · S † LQ · d R + µ S † LQ τ · ˆ S DQ · S † LQ + h . c . (2)Here we introduced the notation ˆ S DQ = S (6) DQ,a ( T ¯ ) aIJ , with I, J = 1 − a = 1 − g and g are dimensionless Yukawas and µ has dimension of mass. The symmetric 3 × T and T ¯ can be found in ref. [8].Note that eq. (2) violates lepton number by two units.The inverse half-life for 0 νββ for the diagram of figure 2, is given by [8]: T − / = G | ǫ DQ M DQ | , (3)where G is a phase space integral and ǫ DQ is defined by ǫ DQ = 2 m p G F g g µm DQ m LQ , (4)and the nuclear matrix element is: M DQ = 148 M − M . (5)Here M , are defined in [3], numerical values for Xe can be found in [1].
2. T-II-5, BL As a second example we discuss another model with a scalar diquark. However, thisdiquark couples only to down-type quarks. This model was first discussed in [32]. It corre-sponds to the example T-II-5, BL d = 9 0 νββ decay operator [8]. Also this model generates neutrino masses at 2-loop order as discussedin [4].This particular case introduces a singlet diquark S / DQ = S ¯6 , , / and a singlet leptoquark S / LQ = S ¯3 , , / . With these new fields, the Lagrangian contains the interactions: L (2) DQLQ = L SM + g ¯ d cR · ˆ S / DQ · d R + g ¯ Lτ · Q c · S / † LQ + µ S / † LQ · ˆ S / DQ · S / † LQ + h . c . Here, as before, by definition ˆ S / DQ = S / DQ,a ( T ) aIJ .The inverse half-life for the short-range 0 νββ decay in this model has the same form aseq. (3) (with some obvious replacements). In particular, it depends in the same combinationof nuclear matrix elements. 6 . T-II-2, BL Finally, we will discuss a model with a singly charged scalar. We choose the exampleT-II-2, BL S , , / = S S ! , S LQ = S , , / = S (2 / LQ S ( − / LQ ! , S / LQ = S ¯3 , , / . (6)With these new fields, the relevant Lagrangian is: L S LQ = L SM + g ¯ Q · S , , / · d R + g ¯ Qτ · L c · S / † LQ + g d R Lτ · S LQ + µ S † , , / · S LQ · S / LQ + h . c . The inverse half-life for 0 νββ (short-range part of the amplitude) can be written as: T − / = G | ǫ S M S | , (7)where ǫ S is given by ǫ S = 2 m p G F g g g µm S m LQ , (8)and the matrix element is given by: M S = − M . (9)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 currentand future 0 νββ experiments and compare them with the sensitivity of dijet, leptoquarkand dilepton plus jets searches at LHC at √ s = 13 TeV. For definiteness we assume twovalues 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 havecompared our results with the literature [35] and found good agreement with published val-ues, whenever available. Plots for the cross sections can be found in our previous work onT-I contributions for 0 νββ decay [20].From the cross sections we then estimate the future LHC sensitivity as follows. Forthe LNV signal (lljj) we first take a simple fit [20] to the background of existing data ofthe CMS analysis [36] based on 3.6 fb − at √ s = 8 TeV. We checked this fit against theCMS analysis [17] based on 19.7 fb − of data at √ s = 8 TeV, published later, and found7ood overall agreement. In the CMS analysis [17] the main background can be traced to t ¯ t events. We then do a simple estimate which considers that the t ¯ t production cross sectionis very roughly about a factor 3 higher at √ s = 13 TeV than at √ s = 8 TeV. Thus, wescale the original fit to √ s = 8 TeV data with a simple constant and scale the backgroundfunction from L = 3 . − to future expected luminosities of L = 20 / fb and 300 / fb. Forthe estimation of the future dijet background we use the fit of the SM dijet distributionfitted to Monte Carlo simulation given in [37]. For both, dijet and lljj analysis we thenestimate backgrounds as dicussed above and define the sensitivity reach as either the simplesquare root of the background (times two for 95 % c.l.) or 5 signal events, whichever islarger. For future LQ searches at the LHC, we calculate LQ pair production cross sectionsas a function of LQ mass. We simply define the reach of the LQ search then as the mass forwhich there are less then 10 signal events in 20 / fb (300 / fb) at the LHC (before cuts). Thisresults in the simple estimate of m LQ > ∼ . m LQ > ∼ . T νββ / ( Xe ) ≥ . × yr from theKamLAND-Zen collaboration [7]. Several experimental proposals aim at half-life sensitiv-ities of the order of 10 yr. We will use the estimated sensitivity of the nEXO proposal[40, 41] of T νββ / ( Xe ) ≃ × yr for our calculation of the future limits. We con-vert half-life limits into limits on masses and couplings, using the equations discussed inthe 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 QCDcorrections 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 (4 / DQ , S (1 / DQ , S ( − / DQ ,contribute to the dijet cross section. However, the dominant contribution to the dijet crosssection comes from the diquark scalar S (4 / DQ . The Feynman diagram is shown in fig. 2. Wehave assumed for simplicity that the Yukawa couplings g and g are different from zero forthe first quark and lepton generations only. As is shown in fig. 2, the scalar diquark S (4 / 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 area function of the leptoquark mass m LQ and the (unknown) parameters µ and g .In Fig. 3 we show a comparison between 0 νββ decay and dijet, LQ and dilepton plusjets searches at LHC in the plane m DQ vs m LQ , for two fixed choice of g = g (bottom: g = g L , top: g = 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 / µ = m DQ (top). The vertical black line corresponds to future limits from dijet searchesat the LHC, the horizontal purple line is for leptoquark searches and the triangular red curve For the mass mechanism this limit corresponds to h m ν i < ∼ . .
14) eV, depending on nuclear matrixelements [38] ([39]). m DQ H TeV L m L Q H T e V L m DQ H TeV L m L Q H T e V L m DQ H TeV L m L Q H T e V L m DQ H TeV L m L Q H T e V L FIG. 3: Expected future sensitivities for the LHC at √ s = 13 TeV, L = 20 / fb (left) and L =300 / fb (right), compared with current and future double beta decay experiments for the diquarkmodel described in the Lagrangian eq. (2). The vertical black line corresponds to future limitscoming from dijet searches at the LHC, the horizontal purple line from leptoquark searches andthe triangular red curve covers the region for like sign leptons plus two jets search. We use theparameters g = g = g L (bottom) and g = g = 0 . µ is taken as µ = m DQ (bottom)and µ = m DQ (top). The gray region corresponds to the current lower limit for the 0 νββ decayhalf-life of Xe, the blue one corresponds to the estimated future sensitivity of T / = 6 × ys of the nEXO proposal. The dashed line marks the kinematic limit for the lljj search, where m DQ = 2 × m LQ . For more details see text. covers the region probed by the lljj search. The dashed line shows the kinematic limit forthe lljj signal, where m DQ = 2 × m LQ . For masses m DQ < × m LQ , one of the LQs goesoff-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 parameterregions of LNV models contributing to 0 νββ decay already with moderate luminosities.The lljj signal depends very sensitively on the choice of µ , while the dijet signal dependsmostly on the value of g . Smaller values of µ reduce the branching ratio for the lljj finalstate, reducing its reach. However, in this case the branching ratio for the dijet final statesincreases, making the dijet search more powerful, as the figure shows. We stress again, thatwhile dijet searches can be used to exclude parameter regions of LNV models contributing9 - - - Π m DQ H TeV L g - - - Π m DQ H TeV L g FIG. 4: Regions in parameter space of the diquark model described in the Lagrangian (2), whichcan be probed by dijet (black curves) and like sign leptons plus two jets (red curves) searches atLHC at √ s = 13 TeV and L = 300 fb − . We use the parameters m LQ = 1 . g = g L , µ = m DQ (left) and µ = m DQ (right). The gray region is the current lower limit in 0 νββ decayhalf-life, the blue one the estimated future sensitivity of T / = 6 × ys. For more details seetext. to 0 νββ decay, to establish a direct relation between 0 νββ and LHC, a positive result fromthe 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 orderof 8 − g ≥ . µ We have chosen the value of µ = m DQ /
6, because, as the figure on thebottom right shows, negative results from LHC LQ and dijet searches would rule out partial0 νββ decay half-lives in this model below the current experimental limit for µ = m DQ / g = g = g L . For µ ≤ m DQ /
50 negative searches from the LHC would rule outpartial 0 νββ decay half-lives below the future bound of T / = 6 × ys.0 νββ decay depends on the mean of the couplings and masses, see eq. (4). Thus, ingeneral LHC and 0 νββ decay probe complementary parts of parameter space. This can alsobe seen in fig. (3): For large values of µ and/or large values of g and g there is always aregion in parameter space for large values of the DQ mass, where double beta decay is moresensitive than the LHC.In Fig. 4 we show the comparison between the 0 νββ decay and dijet and dilepton plusjets searches at LHC in the plane g − m DQ . The LQ mass was chosen as m LQ = 1 . g = g L , µ = m DQ (left) and µ = m DQ (right).Grey and blue regions show again the sensitivity of 0 νββ decay current and future. Thesolid 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, forwhich there are less than 5 signal events expected in L = 300 fb − .For these choices of parameters, dijet searches can probe larger masses, but the lljj search probes smaller values of the coupling g . Again, for larger choices of µ the branchingratio for the lljj final state is larger and the lljj search becomes more sensitive. Negative10 - - - Π m S H TeV L g - - - Π m S H TeV L g FIG. 5: Future limits for the LHC at √ s = 13 TeV and L = 300 fb − compared with current andfuture double beta decay experiments. The gray region is the current lower limit in 0 νββ decayhalf-life whereas the blue region represents the parameter region accessible in near future 0 νββ experiments. The colored lines shows sensitivity limits for the LHC for dijet (left) and dileptonplus jets (right) searches for production of three different scalar bosons S +1 (red), S DQ / (purple)and S DQ / (black). These limits were calculated using g = g L and m LQ = 1 . µ = m DQ .For more details see text. results from the dijet searches would exclude large part of the parameter space explorableby future 0 νββ decay experiments. However, for large values of µ there is always a cornerof parameter space for large couplings and DQ masses, where 0 νββ decay is more sensitive.Finally in Fig. 5 we plot a comparison of sensitivities of 0 νββ decay and the dileptonplus jets (Fig. 5 right) and dijet (Fig. 5 left) searches at LHC for the three different modelsdiscussed in section II: T-II-2 BL µ = m DQ / m LQ = 1 . g = g L . The LHC ismost sensitive for the case of the triplet diquark model (T-II-2 BL S (4 / DQ islarger than the one for the diquark S ( − / DQ (purple curve) and the singled charged scalar S (red curve). Fig. 5 shows also current and future limits from 0 νββ decay for the respectivemodels in consideration. The gray area is the currently excluded part of parameter spacefrom non observation of Xe decay with T / > . × yr and the blue one the estimatedfuture sensitivity, as before. The full lines are for the two diquark models (which have thesame nuclear matrix elements, see above). The dashed lines are for the singly charged scalarmodel (T-II-2 BL S ( − / DQ is intermediatebetween the other two.Finally, we briefly comment on other T-II models. As shown in table I, all T-II decompo-sitions contain either a diquark or a charged scalar (in one case two different diquarks). Thethree example models, which we used in the numerical analysis, covers the cases with thelargest and smallest cross sections at the LHC. It also covers the models with the largest and11mallest matrix elements for the 0 νββ decay. Thus, our sensitivity estimate for the futurecovers the extreme cases, both optimistic and pessimistic, and all other models should liesomewhere in between.In case of a discovery in the future at the LHC, one important question to ask is, which ofthe different model possibilities is the one realized in nature. As in the case of T-I [20], thismight be achieved by investigating mass peaks in different variables and by the measurememtof the “charge asymmetry”, i.e. the measurement of the number of events in l − l − jj relativeto l + l + jj . IV. DISCUSSION AND SUMMARY
We have discussed how future LNV and dijet searches at the LHC can be used to constrainscalar 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 scalarshort-range contributions to 0 νββ decay given in [8]. Two of these models contribute to0 νββ decay through short-range diagrams mediated by diquark scalars and one of themby a singly charged scalar. For these models we have shown that the future LNV and dijetsearches at the LHC will provide stringent constraints on the parameter space of the models,complementary to 0 νββ decay experiments. Except for small parts of the parameter regionof these LNV models, a 0 νββ decay signal corresponding to a half life in the range T / < 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 regionmeasurable in 0 νββ decay. We note that, while we have concentrated on three particularexamples, similar constraints will apply to any scalar short-range contributions to 0 νββ .Finally, we mention that the observation of lepton number violation at the LHC and/orin double beta decay will have important consequences for high-scale models of leptogenesis[43, 44]. Acknowledgements
This work was supported by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064 and SEV-2014-0398 (from the
Ministerio de Econom´ıa y Competitividad ), as well asPROMETEOII/2014/084 (from the
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