LHC signals of radiatively-induced neutrino masses and implications for the Zee-Babu model

Contrary to the see-saw models, extended Higgs sectors leading to radiatively-induced neutrino masses do require the extra particles to be at the TeV scale. However, these new states have often exotic decays, to which experimental LHC searches performed so far, focused on scalars decaying into pairs of same-sign leptons, are not sensitive. In this paper we show that their experimental signatures can start to be tested with current LHC data if dedicated multi-region analyses correlating different observables are used. We also provide high-accuracy estimations of the complicated Standard Model backgrounds involved. For the case of the Zee-Babu model, we show that regions not yet constrained by neutrino data and low-energy experiments can be already probed, while most of the parameter space could be excluded at the 95 % C.L. in a high-luminosity phase of the LHC.


I. INTRODUCTION
The Standard Model (SM) gives a natural and simple explanation for exactly massless neutrinos. However, neutrino oscillation data provide an irrefutable evidence of neutrino masses, which are much smaller than the rest of the fermions. Neutrino masses can be accommodated in extensions of the SM involving new particles or parametrized by non-renormalizable operators violating lepton number conservation (LN).
The most straightforward extension is obtained by adding three families of singlet right-handed neutrinos. If they have a large Majorana mass term M , LN is broken, left-handed neutrino masses are generated at tree level and their smallness can naturally be explained by the see-saw formula m ν ∼ v 2 /M , with v the vacuum expectation value (VEV) of the SM Higgs doublet. This is the so-called see-saw model type I [1][2][3][4] which arises in Grand Unification models like the ones based on SO (10).
The simplicity of the model and the fact that it appears naturally in well motivated extensions of the SM makes the see-saw type I as the most appealing explanation of neutrino masses. A more complicated extension, but richer phenomenologically, is obtained by replacing the singlet neutrinos by triplets of fermions without hypercharge (see-saw type III [5,6]). Both, type I and type III see-saws, contain only one new physics scale, the Majorana mass of the new fermions, M , and lead to the same see-saw formula. Therefore, to explain the observed tiny neutrino masses one needs an extremely large M making very difficult to test these mechanisms 1 .
Alternatively, to generate neutrino masses at tree level one can enlarge the SM with a scalar triplet with hypercharge Y = 1, which develops a VEV (see-saw type II [9][10][11][12][13][14]). The model contains two mass scales, the mass of the new particles, M , and a trilinear coupling which breaks explicitly LN, µ. Then, the neutrino masses are m ν ∼ µv 2 /M 2 . If µ ∼ M we are in a situation similar to see-saws type I and III, the scale must be very large and the model difficult to test. However, since µ is protected by symmetry (it is the only coupling in the Lagrangian that breaks LN), it can be naturally small, µ M . Then, M can be at the electroweak scale and the new particles could be produced at the LHC and/or give large effects effects in low energy experiments by virtual exchange (for instance, lepton flavour violating processes).
In the see-saws one explains the smallness of neutrino masses by introducing large mass scales as compared to the electroweak scale. The new heavy particles give huge loop contributions to the SM Higgs mass, which show in all its crudity the hierarchy problem of the SM. This problem can be alleviated in models in which neutrino masses are suppressed by loop factors. For instance, if we enlarge the SM with only scalars, other than triplets with Y = 1, neutrino masses cannot arise at tree level. However, if LN is broken in the scalar potential, sooner or later Majorana neutrino masses will appear as radiative corrections. Typically this mechanism gives a neutrino mass formula like the see-saw type II, but suppressed by loop factors, m ν ∼ 1 (16π 2 ) n µv 2 /M 2 , where n = 1, 2, 3, · · · is the number of loops at which the mass is generated and µ is the coupling of the potential that breaks LN. Moreover, the breaking of LN involves the simultaneous presence of several Yukawa couplings which produces further suppressions in the neutrino mass formula. With all these suppressions, the mass of the new particles, M , can be rather low even if µ is not small as compared with M . This makes these models testable in present and near future experiments.
The simplest of these models is the Zee-Babu model [15,16], which contains only two complex scalar singlets, singly and doubly charged, which we will denote as h and k respectively, and gives neutrino masses at twoloops. The model has a very rich phenomenology that has been widely studied, see for instance [17][18][19][20][21]. The Zee-Babu model is just a representative of a large class of interesting models which give small radiative neutrino masses by extending only the Higgs sector. Archetypes of this class of models are the Zee model [22] for masses generated at one loop (see [23] for one-loop models with leptoquarks), refs. [15,16,24,25] for two-loop masses (see [26] for a model with leptoquarks) and [27,28] for three-loop masses (see also [29] for a recent review and a complete list of references). These models are further motivated by the fact that, contrary to the rest of the SM interactions, departures from the SM can be plausibly hidden in the scalar sector, which is not precisely measured yet. Most of the models we shall be interested contain doubly-charged scalars 2 , which have a very rich and peculiar phenomenology.
Several experimental searches for doubly-charged scalars have been carried out at the LHC [30][31][32][33][34][35]. They all concluded with negative results. However, unlike some widespread sociological feelings, these results should not be discouraging. On the contrary, a critical assessment of these analyses reveals that they are all not sensitive to doubly-charged scalars with decays other than into samesign leptons. Departures from this assumption have been already considered in the literature. Thus, pair-produced doubly-charged scalars decaying into W bosons [36,37] or with both W boson and leptonic decays [38] have been studied. However, no LHC study of doubly-charged scalars with exotic decays, as those arising in models of radiatively-induced neutrino masses (e.g. the Zee-Babu model), which are the ones that must really have TeV masses, has been worked out so far. In fact, the Zee-Babu model contains the coupling µk ++ h − h − , which is essential in the generation of neutrino masses and can lead to the decay k ±± → h ± h ± . The aim of this paper is to make progress in this direction.
With this spirit, in section II we motivate several doubly-charged scalar exotic decay modes that will be subsequently studied in section III. In this section we highlight the most promising LHC observables and signal regions defined out of them to test doubly-charged scalars in a variety of realistic models of neutrino masses. Given the technical difficulties of determining the SM background with good accuracy 3 , and the fact that our proposed analysis can be eventually used to study prospects for many other scenarios, we also provide background estimation for each of the signal categories considered in this work in appendix A. In section III, we investigate the reach of this search for the famous Zee-Babu model [15,16], in (very broad) parameter space regions not yet constrained by neutrino data. Finally, we conclude in section V.

II. EXOTIC DECAYS OF DOUBLY-CHARGED SCALARS
We will restrict ourselves to SM extensions with only uncolored scalars 4 with electric charged, at most, Q = 2. We will denote by k the doubly-charged scalar whose decays we are interested in. In addition, we will call χ, h and S any additional doubly-charged, singly-charged or neutral scalar, respectively. Besides the leptonic decay of k, k ±± → ± ± , new decay modes are typically present. In fact, k must also couple linearly to non-leptonic fields, because otherwise it would not break LN and therefore would not generate (LN violating Majorana) masses for the neutrinos. (Note that Dirac masses can not be induced by purely scalar extensions of the SM.) In particular, the following k decay modes take place in a variety of models: tor with an SU (2) L triplet with Y = 1; see the left panel of figure 1 (the W bosons are hidden in the longitudinal components of the Higgs doublet). Provided the VEV of the neutral component of this triplet is large enough, k decays predominately into gauge bosons. As an example, for a k mass m k ∼ 500 GeV, and assuming the neutrino masses to fulfill m 2 νi = 0.1 2 eV 2 , the triplet VEV has to be only above ∼ 0.0001 GeV [40]. (Note that this value does not spoil the ρ parameter bound on this VEV, few GeV.) Finally, this decay mode appears also naturally in extended composite Higgs models [37], which are further motivated by the gauge-hierarchy problem.
Most importantly, it is the only other possible k decay that occurs in the Zee-Babu; see the center panel of figure 1.
k ±± → h ± h ± , h ± → W ± S. It has been shown to occur in models where both h and S are odd under a Z 2 symmetry, while k is even; see e.g. reference [27]. A more recent example is given by the model of reference [28]; see figure 1 right panel. Contrary to the first one, in this model h is part of an SU (2) L triplet with Y = 1 (instead of a doublet with Y = 1/2) containing an additional doubly-charged scalar χ. Consequently, we can also find the following decay mode: χ ±± → k ±± S, k ±± → ± ± . An interesting aspect of this channel is that, for given χ and S masses, it can sensibly strengthen the bounds on m k coming from current searches for doubly-charged scalars.
Finally, despite being potentially present, we do not consider cascade decays of k via emission of W bosons. As we will comment below, these are very hard to constrain experimentally.

III. SEARCH STRATEGY
Both the ATLAS and the CMS collaborations have developed a large amount of searches for doubly-charged scalars. These include analyses of the total collected luminosity at 7 [31,32], 8 [41,42] and 13 [34,35,43] TeV of center of mass energy. However, they are all inspired by the see-saw type II and therefore look for final states

FIG. 2:
Orange: Invariant mass distribution of each pair of two same-sign leptons in events pp → k ++ k −− with k ±± → ± ± . Green: Same as before but with k ±± → h ± h ± , h ± → ± ν. Red: cut imposed by the experimental collaboration [35]. m k and m h have been set to 500 and 100 GeV, respectively. containing pairs of same-sign leptons reconstructing a narrow invariant mass. This requirement is only (and slightly) relaxed in final states with taus. Consequently, doubly-charged scalars with exotic decays can easily be missed. As an example, we compare in figures 2 and 3 the invariant mass distribution of the two same-sign lepton pairs resulting from the decay of doubly-charged scalars into ± ± (orange) and h ± h ± with h ± → ± ν (green). Clearly, the narrow cut removes most of the signal in this latter case. Moreover, the interplay between different variables such as the invariant mass of pairs of leptons or the missing energy is never fully exploited. In addition, other analyses, and in particular searches for Supersymmetry in multi-lepton final states, are also non constraining (even for small doubly-charged scalar masses). We have tested this by means of CheckMATE v2 [44], which implements, among others, searches for gluinos in final states with 2 same-sign leptons or 3 leptons, jets and missing energy [45]. A last analysis that might be sensitive to the signals we are interested in is given by the CMS search for the seesaw type-III in multi-lepton final states [46]. Again, this search is of narrow scope. Among the characteristics that make it not suitable to explore generic doubly-charged scalars we find that it focuses on final states with same-flavour opposite-sign leptons. For models such as the Zee-Babu, where the doubly-charged scalar can produce equally muons or electrons, this requirement kill half of the signal. Furthermore, final states with only two light leptons, which are abundant in the models of interest, are disregarded. In any case, that CMS paper does not provide detailed information (e.g number of background events in each signal category), and so a proper estimation of its (presumably limited) reach to these models can not be precisely stated. To the best of our knowledge, no other multi-lepton analysis with the latest data has been made public yet.
As things stand, new analyses are necessary to fully explore models of radiatively-induced neutrino masses. With the aim of being able to test different scenarios (though still focusing on neutrino models 5 ), we propose a broad scope search containing several signal regions and categories.
They all contain several same-sign leptons. One of the main challenges of this kind of analysis is the correct estimation of the background, which originates mainly from the charge miss-identification of electrons. This requires time-consuming simulations of SM processes with many particles in the final state. Consequently, given the numerous signal regions that we work out in the next section, our background estimation will be valuable for many LHC studies of particles producing same-sign lepton events. 5 Despite being interesting, inclusive searches looking "everywhere" without any theoretical biased, such as the modelindependent analysis of reference [47], are not very efficient in the search for new physics.

A. New signal regions
Prior to the selection of events, the relevant physical objects are constructed in the following way. Electrons (muons) are defined to have p T > 20 (10) GeV and |η | < 2.5 (2.6). Jets are clustered using the anti-k t algorithm with R = 0.4. They are defined by p j T > 20 GeV and |η j | < 2.4. Despite their small phenomenological relevance, we have also fixed the b-tagging efficiency to 0.7 and the τ -tagging efficiency to 0.5. Of major importance is the probability of an electron (positron) to be identified as a positron (electron). Following reference [34] we estimate it by P (|η |, and Throughout the text, we will refer to electrons and muons simply as leptons, while taus will be excluded of this definition. Only events with at least two-same sign leptons are selected. Out of this sample, we define three orthogonal signal regions (SRs), containing two, three and four leptons, respectively. SR 1: inspired by the recent ATLAS analysis of reference [34], it contains events with two leptons. If more than two same-sign leptons are present, only that pair with the highest invariant mass is considered for computing the observables defined below. SR 2: inspired by the recent CMS analysis of reference [35], it contains events with exactly three leptons, with exactly two of opposite sign. SR 3: inspired by the same CMS analysis, it contains events with exactly two positive and two negative leptons.
We further consider the following observables. 1) S T , defined as the scalar sum of the p T of all leptons in the signal region. 2) The invariant mass of each same-sign lepton pair, m ± ± . 3) The transverse mass of each samesign lepton pair, as well as the one of the third lepton in SR 3. We will denote both collectively by m T . Following reference [37], we define the former by  where E miss stands for the missing energy. 4) The stransverse mass, m T 2 , defined as In SR1, L 1 and L 2 are given by the harder and the softer lepton, respectively. In SR2, L 1 stands for the vectorial sum of the two same-sign leptons, while L 2 is given by the third one. In SR3, L 1 represents the vectorial sum of the two positive charged leptons, and L 2 the vectorial sum of the two negative ones. For each SR, and for each observable O = m ± ± , m T , m T 2 , we consider 81 different categories defined by S T > X and O > Y with X, Y = 100, 200, ..., 900 GeV. In accord with reference [35], we consider the following dominant backgrounds: Drell-Yan with m + − > 100 GeV, tt, W Z, W W , ZZ, W W W , W W Z, W ZZ, ZZZ, ttW , ttZ. Background events are generated at NLO in α s with MadGraph v5 [48]. Initial and final state radiation and showering is performed by Pythia v6 [49]. The cross sections of all relevant backgrounds are shown in table I. The uncertainty due to the choice of scale is also shown. Finally, we also provide the number of generated Monte Carlo events.
In order to validate the goodness of our Monte Carlo generation as well as the appropriateness of equations 1 and 2, we recast the analysis of reference [34] and compare the distribution of m e ± e ± > 345, 410, 485, 575, 680, 810 and 1020 GeV provided by the experimental collaboration with that obtained by us. For this goal, we use homemade routines based on MadAnalysis v5 [50] and ROOT [51]. The result is depicted in figure 4.
We also compare the distribution of m ± ± given in reference [35] with ours; it can be seen in figure 5. Clearly, our results are in perfect agreement with those provided by the experimental collaborations. Moreover, we have checked that the contribution of each background in table I to the total SM expectation agrees with that reported by both ATLAS and CMS. Consequently, we have computed the number of expected background events in each of the categories mentioned above. These are listed in tables II, III, IV, V, VI, VII, VIII, IX and X in the appendix A.

B. Applications
Based on this information, we can estimate the reach of current data to several signals mediated by doublycharged scalars. Let us start considering the standard case k ±± → ± ± . This is the only one that has been considered in experimental analyses so far; it will allow us to further validate our approach. We focus on pairproduction of doubly-charged scalars. As a matter of fact, this channel is always present, while associated production is absent in many models; see section II.
We implement the relevant interactions in Feynrules v2 [52]. We generate signal events using MadGraph v5 at LO in α s , using again Pythia v6 as parton shower. We then estimate the number of signal events in each of the categories defined above for L = 35 fb −1 . (We restrict to this value because no experimental analysis with more data is still publicly available.) We subsequently look for those three categories that give the largest sensitivity defined as S/ √ B in SR1, SR2 and SR3, respectively. (Obviously, in the present four-lepton case, the sensitivity is by far driven by the categories with m ± ± m k ±± −100 GeV in SR3.) These three categories are orthogonal, meaning that no single event can belong to more than one of them. We can therefore simultaneously consider these three independent categories to analyze to what extent the signal is compatible with the observed data given the SM predictions of appendix A. To this aim, we adopt the CL s method [53]. The corresponding statistic is computed using MCLimits [54], which takes also into account the systematic uncertainty due to the finite number of generated Monte Carlo events. On top of it, we include a systematic uncertainty in the background normalization of 10 %. For each value of m k = 300, 500, 700, 900 GeV, we estimate the lowest cross section that can be excluded at the 95 % C.L. using this procedure. Exclusions for intermediate masses are obtained by linear interpolation.  works of references [34] and [35]. As things stand, masses as large as m k ∼ 700 GeV are excluded in this channel by using current data. Note however that these are significantly weakened in models in which k is an SU (2) L singlet, whose production cross section is roughly a factor of 2 smaller. One can easily derive approximate prospects for a larger luminosity, by scaling the cross section bound masses goes down one order of magnitude for L = 3 ab −1 . The upper limit on the triplet masses turns out to be in this case ∼ 1.1 TeV. We note, however, that some corrections to this result might be needed, given the limited statistic of some Monte Carlo samples (see table I) for large luminosities.
We repeat this exercise for the different signals commented in section II, to which current analyses, relying on a narrow cut on the invariant mass of any pair of same-sign leptons, are not sensitive at all. The first such a signal appears when k ±± → W ± W ± , what can happen also in the see-saw type II. We still restrict to the pairproduction mode. (Note that the associated production channel is not necessarily present even in these models with this decay; see for example reference [55].) The most sensitive categories are those with m T , S T 100 − 300 GeV in SR1, m T 100 − 300 GeV and S T > 400 − 600 GeV in SR2 and the ones with m ± ± 100 − 300 GeV and S T 600 GeV in SR3. In this case, we have checked that the combination of the three categories with two, three and four leptons improve the sensitivity of LHC data by almost an order of magnitude with respect to that obtained using only the most sensitive category. Even so, the presence of W bosons in the final state makes this channel almost unaccessible with current data; see figure 7. In the long term, instead, masses up to m k ∼ 400 GeV might be probed. Analyses specifically dedicated to this channel in composite Higgs models could improve over this result; see reference [37].
Using this same broad-scope strategy, we analyze the LHC reach for pair-produced doubly-charged scalars decaying into exotic channels (with 100% branching ratio). The results are shown in figures 8, 9 and 10. We see that cascade decays with W bosons in the final state    are still hard to tag. It can be then easily understood that decay chains with emissions of soft W bosons are also very unconstrained. Therefore, a large parameter space of models giving these decays is perfectly allowed by collider data. On the contrary, the pair-production of k ±± k ∓∓ with the subsequent decay of k ±± → h ± h ± is very constrained. The most sensitive signal categories in this respect are those with m T 2 ∼ S T m k /2 in SR1 and those with m T ∼ S T m k /2 in SR2 and SR3.
Likewise, χ ±± → k ±± S with S stable (i.e. missing energy) and k → ± ± is almost as constrained as the standard pair-production of doubly-charged scalars with leptonic decays. Thus, in models in which this production mode is present, constraints on k can be significantly altered with respect to those obtained using standard searches for k alone (see figure 11) because the latter is more copiously produced.

IV. IMPLICATIONS FOR CONCRETE MODELS
In concrete models, the doubly-charged particles can decay into several different channels, and so the bounds on different parameter space points can not be read from the plots above. Instead, the full process of comparing signal, background and data outlined in section III B must be done. Note, however, that the background, which is the most complicated and time-consuming task in this respect, does not need to be computed again. It can be just taken from the tables in appendix A. We illustrate this procedure in the Zee-Babu model. It has been previously considered in several references [15][16][17][18][19][20][21]. All of them have assumed that the exotic decays of the doublycharged scalars were impossible to tag at the LHC. Excluded region for a model containing an SU (2)L singlet k ±± decaying into ± ± as well as a doubly-charged component of a Y = 1 SU (2)L triplet, χ ±± , decaying mostly into κ ±± and a neutral scalar S when kinematically accessible (dashed orange line). The bounds on m k are ∼ 100 GeV above those obtained assuming the presence of this particle alone. We also show the bounds when κ ±± decays mostly into ± τ ± (green solid line), as suggested by the recent proposed model of reference [28]. For consistency with this model, we have in both cases assummed the mass of S to be 200 GeV.

A. The Zee-Babu model
The Zee-Babu model extends the SM scalar sector with two SU (2) L singlets, h and k, with hypercharges Y = 1 and Y = 2, respectively. The relevant Lagrangian for our discussion reads where L SM stands for the SM Lagrangian and L aL , a with a = 1, 2, 3 are the first, second and third generation SM lepton doublets and singlets, respectively, andL L = iσ 2 L c L with σ 2 the second Pauli matrix. Overall, the model depends only on the antisymmetric (resp. symmetric) dimensionless couplings f ab (resp. g ab ), the physical masses of the new scalars, namely m k and m h and the dimensionless parameter κ defined by µ = κ min{m h , m k }. The relevant decay widths read and Naturalness arguments, together with the requirement that no charge-breaking global minimum is developed by the potential, imply that κ 4π. Neutrino oscillation data and low-energy constraints restrict the allowed parameter space. We consider two large regions permitted by current experiments, depending on whether the neutrino mass hierarchy is normal (NH) or inverted (IH) [20,21].
As a result, k decays mainly into leptons (for m k < 2m h ), while h decays into a lepton and a neutrino around 60 % of times and, to a lesser extent (∼ 40 %), into a tau and a neutrino. Thus, for m k > 2m h , the pair-production of doubly charged scalars give rise to two, three, and fourlepton events in 35 %, 30 % and 15 % of the cases, respectively. Being 0 and 1-lepton events weird, our search strategy can then capture most of the signal. The scalar widths are small enough so that the narrow-width approximation holds. Consequently, we proceed as follows. We vary m k and m h in the range 100, 200, ..., 1000 GeV. For each pair, and having fixed all couplings to the values mentioned before, we compute Monte Carlo events and estimate the efficiency ( ) for selecting events in each of the categories described in previous sections. The number of expected signal events in each category for a given luminosity (L) can then be computed as Then, we compare again the three most sensitive categories (one for each SR) with the corresponding SM background. The results are shown in figure 12. The grey triangle at the bottom of the plot is already excluded by neutrino and low-energy data. The green region enclosed by the solid green line is the area that can be excluded using the already collected luminosity L = 70 fb −1 (counting 35 for each experiment, ATLAS and CMS). It is worth noting that, if we take κ > 5, the triangle excluded by neutrino data goes down to 400 GeV. Consequently, current LHC data can already constrain regions not bounded before by other experiments. Likewise, the orange region enclosed by the dashed orange line is the region that can be excluded in a high-luminosity phase of the LHC with L = 3000 fb −1 . Doubly-charged scalar masses as large as 1 TeV could be probed for k ±± → ± ± , but also in exotic decays.

Inverted hierarchy
The IH parameter space of the Zee-Babu model is very constrained by neutrino data, being m k and m h out of the LHC reach for most values of φ and δ. These parameters stand for the physical Majorana and Dirac phases in the PMNS matrix, respectively. (Note that, in the Zee-Babu model, one of the neutrinos is massless.) However, these bounds are significantly weakened if φ ∼ δ ∼ π, and even smaller for large values of κ. For definiteness, we take These values are allowed by current data, even for small values of m k , m h ∼ 100 GeV [21]. We emphasize that there is very small room for variations in this hierarchy of couplings. Moreover, although O(1) modifications in their absolute values might be in principle allowed, the expected scalar branching ratios, and therefore the LHC phenomenology, would remain the same. We proceed as in the NH case and test which region of the m k −m h plane can be probe with current and future LHC data. The result is shown in figure 13. The smaller region for m k < 2m h in comparison with the NH case is due to the smaller k branching ratio into leptons. Again, there are nonpreviously bounded regions that can be excluded with the current LHC data, even for m k > 2m h . Likewise, masses in the TeV region could be tested with future analyses.

V. CONCLUSIONS
We have argued that current LHC analyses can not probe doubly-charged scalars with exotic decays, as those arising in models of radiatively induced neutrino masses. Novel searches, as the ones proposed in this article, com- bining different signal regions and observables, are however sensitive to these particles. Thus, masses as large as 500 GeV can be accessed with current LHC data for doubly-charged scalars k decaying as k ±± → h ± h ± , with h ± → ± ν. These numbers are only slightly smaller than those for doubly-charged scalars decaying into pairs of same-sign leptons. This result has important implications for concrete scenarios, most importantly the Zee-Babu model. In particular, we have shown that parameter space regions of this model not yet constrained by neutrino and low-energy experiments can be tested with current LHC data, while much larger regions could be excluded at the 95 % C.L. in a high-luminosity phase.
Conversely, models in which k decays predominately via the emission of W bosons are by far less constrained.
In any case, our results (most importantly the selection of signal regions and observables, as well as the precise determination of the SM background) can be applied to very different models of neutrino masses. Therefore, we expect that this work will be of interest for many forthcoming studies.