Consequences of Neutrinoless Double Decays Dominated by Short Range Interactions

We investigate some consequences if neutrinoless double beta decays of nuclei are dominated by short range interactions. To illustrate our results, we assume that such decays proceed mainly through short range interactions involving two-W-bosons exchanges and confine ourselves to only include new scalars without new gauge interactions for SM fermions. For the neutrino mass problem we propose to solve it by adopting that the active light neutrinos have predominantly Dirac masses and the small Majorana masses induced by the new scalars render them pseudo(quasi)-Dirac particles. This particular aspect of neutrinos may be detectable in the next generation of neutrino oscillation experiments and/or neutrino telescopes. If so this opens a new connection between neutrinoless double beta decays and neutrino physics. We also noted the new physics signals such as high charged scalar states that can be explored in hadron colliders. In particular, we find that a high energy e^- e^- collider will be very useful in testing the origin of lepton number violation which complements neutrinoless double decays studies.


I. INTRODUCTION
In this paper we investigate the possibility that neutrinoless double beta decays (0νββs) of nuclei are dominated by short range physics not involving a heavy sterile right-handed neutrino but due to some other new physics beyond the Standard Model (SM). This is in sharp contrast to the usual assumption that 0νββ is due to the exchanges of light Majorana neutrinos, which constitute a long range exchange force between decaying nucleons in the nucleus. The half-life of the decaying nucleus is directly proportional to the masses of the exchange or virtual neutrinos. They are identified as the active neutrinos of the SM, which are known to be massive but light, i.e. less than 1 eV, due to the observed neutrino oscillations. We refer this as the three-Majorana-neutrino paradigm (3MNP). This is an economical and elegant framework for 0νββ as it involves only physics in the SM with the violation of lepton number encoded in the Majorana masses of the active neutrinos. For an up to date review, see [1,2]. However, since the origin of the active neutrinos masses and their nature is an open question, it behoves us to examine alternatives to the above paradigm and to study consequences that they will lead to, in particular to search for new pathways that are not evident in the 3MNP.
The scenario we are interested in assumes that 0νββsproceed predominately by short range physics beyond the SM. This can happen if all the active neutrinos have physical masses too small to induce 0νββ even if they were Majorana particles. A less stringent possibility is that the Majorana phases are such that they cancel in the effective ν e Majorana mass. This amounts to the first element M ν of the active neutrino mass matrix in the weak basis is vanishing, i.e. (m L ) ee ≃ 0. A third possibility is that the three active neutrinos participating in oscillations have dominantly Dirac masses. If any of the above scenarios takes place and 0νββ is observed in the next generations of experiments then it is likely that some new short range physics is operative. Short range physics contributions to 0νββshave been discussed in [3,4] concentrating in how they affect the hadronic physics. They have been parametrized by effective operators of dimension 7 and 9 [5,6]. Since dimension 7 operators will involve a light neutrino exchange, they will fall out of our assumptions. That leaves dimension 9 as the lowest dimensional operator we need to consider. The theory space for new physics that can generate these operators is large. To reduce that we make a conservative assumption that all SM fermions do not carry additional quantum numbers than dictated by the SM gauge symmetry. This is supported by LHC having not seen any new gauge bosons and numerous low energy precision measurements that set stringent limits on their masses and couplings. This leaves new scalars and fermions transform nontrivially under the SM gauge symmetry that can carry color and color singlets as new degrees of freedom to be studied. Here, we shall concentrate on scalars and leave new fermions for a future work.
We organize our paper as follows. In Sec. 2, we take two-W-boson exchange as the lowest state for the dimension 9 operator. Then new physics for 0νββ will proceed via W W → ee.
Tree level new physics for this will involve colorless scalars. We then compare the constraints given by 0νββ with that from the LHC and future colliders on these new objects. Since the interactions involve must violate lepton number by 2 units, one has to check that they do not generate (m L ) ee at a large enough value so as to invalid our short range dominance proposition. This is independent of the black box theorem [7], which generates a Majorana mass for ν e at the 4-loop level. Quantitatively, this yields (m L ) ee 10 −28 eV [8]. This value is inconsequential for 0νββ if they were to be discovered in the current or next generation experiments. In Sec. 3, we investigate the issue of neutrino masses if 0νββ is driven by the short range interactions proposed. Our conclusions are given in Sec. 4.

II. 2-W-BOSONS MECHANISM FOR 0νββ
At the quark level, 0νββ can be represented by Fig. 1 with the 2-W-boson mechanism being the leading approximation as depicted. The Lagrangian for the short range interaction can be symbolically written as where i denotes different electron currents (j i ) and quark currents (J i ), ǫ i represents the particle physics involved, and m p stands for the proton mass. Here, Lorentz contractions have to be taken and are not shown, while the quark currents are to be sandwiched between initial and final nuclear states for the full matrix element. The half-life for a given decay may be generically expressed as where G i is the nuclear phase space factor, and M i . The calculations of these two factors are given in [3]. In this paper, we are concerned with models that give rise to ǫ i and it is dimensionless. As a comparison, the long range neutrino exchange is given by ǫ ν = (m L ) ee /(0.01eV).
For the 2-W-boson scattering mechanism one can construct tree level new physics that induces W − * + W − * → e − + e − . Immediately one can recognize that a doubly charged scalar T −− will be involved. A generic T −− exchange mechanism is displayed in Fig. 2. The nature (T ++ , T + , T 0 ). A coupling between T and the lepton, given by y ℓl c T ℓ, can be constructed. This type of Higgs triplet models is popular in type II and radiative seesaw models for neutrino masses, see e.g. [9,10]. The T 0 component must pick up a VEV, v T , in order to get a W − W − T ++ coupling. This in turn generates a tree level neutrino Majorana mass; hence y ℓ v T < 1eV. Thus, the W W T coupling is ∼ gv T , where g is the SU(2) gauge coupling.
Since no doubly charged scalar has been found at the LHC [11,12], a lower bound on the mass is 1.3 TeV by using the signal of same sign dileptons and assuming 100% branching ratio. The rate for 0νββ is given by gy ℓ v T m p /M 2 T . We estimate that ǫ 10 −24 and so this case is uninteresting. We conclude that triplet Higgs with SM quantum numbers (1, 3, 1) in usual notations will not be useful for us.
The electron pairs are in a singlet state of (1, doubly charged scalar singlet Φ(1, 1, 2) can couple to them with the coupling 1/2y Φē c R e R Φ. Now y Φ is unconstrained by neutrino masses. On the other hand, Φ will have no tree level coupling to the two W-bosons. This necessitates the introduction of additional Higgs scalars. We have previously ruled out the scalar (1, 3, 1) and it is easy to see that triplets with |Y | ≥ 2 cannot be used. This leaves the option of higher SU(2) representations. The next lowest representation that can be used is Ψ(1, 4, 3/2) and explicitly the quadruplet can be generated and is igv ψ /2.. The hypercharge assignments also prevents tree level couplings to active neutrinos.
The next ingredient is to provide the mixing between Ψ ++ and Φ ++ . The price to pay is to introduce yet another scalar . Then the gauge invariant term HT ′ ΨΦ † is allowed 1 . We obtain the desired mixing if t 0 is non-vanishing and H = v. We note that both t 0 and ψ 0 must be less than a GeV from precision electroweak measurements. Moreover, without exotic fermions the hypercharges of the new scalars are such that they have no tree level couplings to active neutrinos. Thus, 0νββ is given by Fig. 3.
Our solution is not unique and higher SU(2) representations can be used. Constructing 1 If economy on new states is desired one can construct a soft term such as ΨΨΦ † . Mixing between Ψ and Φ is induced after the spontaneous symmetry breaking (SSB) of Ψ. However, this will require that Ψ be in an odd dimensional SU (2) representation. For our case the lowest one would be a quintuplet [13].
where v is the SM Higgs VEV and λ is a free parameter that controls the strength of lepton number breaking. The physical doubly charged scalars can be obtained by diagonalizing a 2 × 2 mass matrix with off diagonal terms given by Eq. (3). The details depend on the scalar potential and are not essential for this discussion. It suffices to know that the mixing angle α is given by where M Ψ and M Φ are the masses of the respective fields before diagonalisation and we have omitted some unimportant constants. v T is constrained to be less than a few GeV, and the masses in the denominator are of order 0.5 TeV. This is the lower bound from [11] when the branching ratio of a doubly charged scalar into a given same sign dilepton pair is 10 %.
This is more appropriate for us since we expect Φℓℓ ′ to be approximately equal. This gives 9 such decays. If the mass splitting is also of order 100 GeV, we expect α 10 −3 if λ ∼ 1.
A larger mixing can occur if the masses are accidentally degenerate or λ ∼ 10. The physical states denoted by S ±± 1,2 are related to the weak states Φ ±± and ψ ±± via Without lost of generality, we assume that S 1 is the lighter state. As we shall see later, the mixing is small and S 1 is mostly Φ. The masses are denotes by M 1,2 , respectively.
Referring to Eq.(1), our model gives only one contribution and the current correlation given by It is sensitive to the difference of the inverse mass squared Currently, the half-life of the decay 136 Xe → 136 Ba e − e − [14] gives the most stringent limit [3] Thus, we get This shows the complementarity of 0νββ to direct searches at the LHC. The direct search is sensitive to one states at a time and depends on the decay products of the state in question due to experimental constraints. If y Φ /g ≃ 1, it implies that the branching ratio of decay of the doubly charged scalar into same sign dilepton pair is not negligible compare to that into a pair of same sign W-bosons. This is in fact the preferred search mode at the LHC. This holds true if the mixing is small as we argue. However, if the scalars are more degenerate and the mixing becomes large, then the gauge bosons decays can become more important. This is more challenging experimentally but important to test the physics involved and must not be ignored.

III. NEUTRINO MASS GENERATION
An examination of Fig. 3 will show that the lepton number violating interaction constructed will yield a 2-loop contribution to a Majorana mass to ν e . The Feynmann diagram is given in Fig. 4, where it is depicted in the weak basis. Evaluating the diagram [9] for a given physical scalar S gives The integral I is given by A similar cancelation between S 1 and S 2 takes place as in 0νββ . Assuming M 2 ≫ M 1 ≫ M W , one has that Comparing Eqs. (6), (9) and (11), we see that (m L ) ee is completely determined by the amplitude for 0νββ . There are three suppression factors at play. Firstly, the mixing α is small, 10 −3 , secondly there is the 2-loop factor which is ∼ 10 −5 and finally the smallness of m e . The last one is the biggest suppression because m 2 e /M 2 S ∼ 10 −12 for a TeV scalar. Even with v ψ ∼ 1 GeV, we get M ee 10 −12 to 10 −11 eV. Hence, the active ν e will give a negligible contribution to 0νββ and is consistent with our short range physics dominance hypothesis.
where we have assumed that the Yukawa couplings of Φ to e, µ, τ are the same. Since the neutrinos oscillation data involve the mass squared differences of the neutrinos, it is obvious that we are many orders of magnitude away from explaining the data [17]. The smallest mass squared difference is 7.39 × 10 −5 eV 2 . Pushing some Yukawa couplings to their perturbative limits will not change the above conclusion.
A solution to this conundrum will be to introduce 3 SM singlet right-handed neutrinos N R that can be used to give Dirac masses to the active neutrinos. The physics here can be seen simply by considering the single family case with only ν eL and only one N R . Since we assume that lepton number violation occurs only in the scalar sector, we take N R to have a vanishing tree level Majorana mass. ν e develops a relatively large Dirac mass via the usual Yukawa coupling of y ν ν L N R H. If y ν ∼ 10 −12 , then a Dirac mass m D ∼ 0.1 eV is generated for ν e after the SSB. We treat all Yukawa couplings as free parameters and will not go into a deeper understanding of the hierarchy shown in known Yukawa couplings. Moreover, such a small Yukawa coupling can be implemented in extra dimensional models [18] For typographic simplicity, we drop all subscripts for the lepton states in the following.
The 2 × 2 mass matrix in the ν, N c basis is represented by and m D ≫ m ee as seen previously. We have set the lower left corner value to zero but it can be generated at 3-loop in our model. Since this has a very small value comparing to even m ee we can safely set it to zero. The eigenvalues for Eq. (14) are and the states are almost maximally mixed, i.e.
with the small mixing given by θ = m ee /(4m D ). The mass eigenstates are a pair of Majorana leptons with opposite CP phases with a very small mass splitting, which in our example is proportional to m ee ∼ 10 −11 eV. This is known as quasi or pseudo-Dirac neutrinos [19,20]. 3 Generalization to 3 families of neutrinos is straightforward but nontrivial. Firstly, three righthanded SM singlet neutrinos N αR are introduced and denote their masses by the matrix (m R ) α , where α denotes weak eigenbasis. Explicitly, the weak eigenstates are where the superscript c denotes charge conjugation. The neutrino mass matrix is now a 6 × 6 matrix denoted by where each entry m is a 3 × 3 matrix. In our model (m L ) αβ can be obtained by calculating similar diagrams of Fig. 4. m D is an obvious generalization of the Dirac mass and is the dominant matrix, i.e. all the elements are such that m D ≫ m L ≫ m R ≃ 0. It is more convenient to diagonalize the product M † M, which reads as where the dominance of m D has been employed. This can be diagonalized by [22] U is the usual Pontecorvo-Maki-Nakagawa-Sakata matrix [23,24] and it renders m D † m D diagonal with the eigenvalues m 2 1 , m 2 2 , m 2 3 . Defining ε i = (U † m L U) ii ) and P a diagonal phase matrix e iφ j = ε j /|ε j |. The mass eigenvalues are with i = 1, 2, 3. Clearly, U will not diagonalize m L in general and also m D is diagonalized by U † m D U R as for SM charged leptons. We label the mass eigenstates ν ± j corresponding to right corner given by m R = 0 [21]. m R ≫ v is the celebrated Type I seesaw mechanism. Since most the signatures for psuedo-Dirac neutrino involve detecting mass splittings one cannot distinguish this from our scenario.
the eigenvalues of Eq. (21) and as a result the three active neutrinos are related to the mass eigenstates via As we have argued before the largest element in m L is the τ τ component; thus, we predict the ratios ε 1 : ε 2 : ε 3 ≃ |U 1τ | 2 : |U 2τ | 2 : |U 3τ | 2 ≃ 0.04 : 1 : 1 with the current neutrino oscillation data given by [17].
The neutrinos flavor conversion probability can be expressed as where L is the baseline of the neutrino experiment and E is the neutrino energy. The ν α survival probability is then wherex = L/(2E) and m +2 i is given in Eq. (21). In the limit all ε → 0 this reduces to the standard expressions.
Eq. (24) shows that there are long wavelength oscillations superimposed on the observed ones. In order to be able to observe the effects of ε, the oscillation length is given by The next generation reactor experiment JUNO [25] with a base line of 57 km and neutrino energy in the MeV range is well suited for studying pseudo-Dirac neutrino oscillations with splittings O(10 −4 )eV. Smaller splittings will require astrophysical neutrinos sources and neutrino telescopes. We defer a detail study of this intricate oscillation phenomena to a future study. For some early discussions of the pseudo-Dirac neutrino phenomenology, see [21,26,27].
In conclusion by assuming the short range interactions to be dominated, we have broken the connection between 0νββ and direct neutrino mass measurements using kinematics of weak decays of nuclei such as the Katrin experiment [28] and Project 8 [29]. This is not surprising since the neutrino exchange is no longer assumed. If future experiments do not confirm the expected connections within expected uncertainties, then short range interactions must be taken into account. Interestingly, our study has opened up a new connection between 0νββ and the phenomenology of quasi-Dirac neutrinos. These effects may be probed in future neutrino oscillation experiments and neutrino telescopes and further studies are warranted.

IV. HIGH ENERGY COLLIDER PROBES
To implement the short range dominance in 0νββ , we have introduced a moderate number of new scalars. There are a pair of neutral spin-0 states, ψ 0 and t 0 . The real parts of which are two Higgs scalars with masses in the Tev range. There is also one heavy pseudoscalar from a linear combination of the imaginary parts. The orthogonal combination will be a massless Majoron since lepton number is spontaneously violated in the model. This can serve as a candidate for dark radiation and the phenomenology has been extensively discussed in the literature [30][31][32]. In addition, there are two pairs of singly charged scalars (t ± , ψ ± ) and two pairs of doubly charged scalars (Φ ±± , ψ ±± ) and a triply charged pair ψ ±±± .
Their masses are all expected to be in the TeV range. Of all these the experimentally more spectacular ones are the multiply charged states. They are easily produced with sufficient energy. Their production cross sections are enhanced due to the high charges. The LHC search efforts concentrate mainly on the doubly charged ones using multileptons as signatures which we have already discussed. The triply charged states are more unusual can also be searched for at the LHC [13]. Much of the detail phenomenology is model dependent and in particular is sensitive to the parameters of the scalar potential. Instead we will focus on more model independent signatures without having to spell out the details of the potential.
A well known general mechanism for the pair production of new particles is via the Drell-Yan process. Specifically, we can have The decays of ψ proceed as The final signature is a resonance of a same sign dilepton with a same sign W -boson. In this reaction all the couplings are known with the only model dependence coming in the branching ratio of S 1 → ℓℓ ′ . An equally interesting reaction is followed by the decay of ψ +++ as in Eq. (27) and a same opposite sign dilepton recoiling against it. Notice that none of the new charged states couples directly to quarks; hence, the Drell-Yan mechanism is the best for their production.
High energy lepton colliders will be ideal probes for the new states, in particular if we have a e − e − collider option. Such an advanced lepton collider is expected to operate with the center of mass (cm) energy in the multi-TeV range [33] and an exploratory luminosity of at least 10 36 cm −2 s −1 . Optimistically, one can search for direct production of the doubly charged states as a dilepton resonance via where ℓ, ℓ ′ = e, µ, τ . Sinceȳ Φ ≤ 1, this rate is not be suppressed. If M 1 < √ s with s the cm energy, one will see a peak in the total cross section. For M 1 > √ s, the cross section For the diagonal terms ℓ = ℓ ′ , it is a factor of 2 larger. This cross section is O(40)fb for √ s = 1TeV if we set y ee = y ℓℓ ′ = e.
Similarly, one can consider the case of e − e − → S * −− → W − W − , which will be easier to search for in a lepton collider than a hadron collider. The reaction e − e − → W − W − is the inverse of 0νββ if the latter proceeds via a virtual W exchange. Thus, it provides a model independent test of the two-W-boson mechanism for 0νββ . Here, the W-boson pair is on shell. This can be seen in the righthand diagram of Fig. 1. Now the two electrons are incoming from the top and the two W-bosons are outgoing and decay into two jets each or ℓν pair. This reaction must occur if 0νββ were observed and proceeds via the 2-W exchange.
We refer this as inverse 0νββ and was first discussed in [34]. In our model, this is a s-channel process, hence any one of the W-bosons will have an isotropic scattering angle distribution.
This contrasts with the previous discussions on this reaction which were mainly focussed on of probing heavy Majorana neutrino exchanges [34]. The latter has a characteristic t channel angular distribution that peaks at π/2. It is instructive to note that for a 1 TeV Majorana neutrino N that mixes with ν e with the mixing parameter 10 −3 , the cross section at high s is ∼ 4.2 fb. In passing we also note that similar probe reaction at hadron colliders such as the LHC using two W fusion to two same sign leptons have challenging backgrounds, see e.g. [35]. On the positive side searches at hadron colliders probe two units lepton number violation that are not both electrons. Up till now the only constraints come from rare meson decays such as K → πµµ(µe) and µ − → e + conversion in nuclei.

V. CONCLUSIONS
Lepton number violation (LNV) is a crucial question in particle physics. It is intimately connected to the question of the neutrino mass generation, which remains unknown despite tremendous progress in the experimental front in establishing neutrino oscillations. It is also widely believe that it is violated by a small amount in the SM in terms of the small active neutrino Majorana mass. However, this is far from being established. Observation of 0νββ will then be the explicit demonstration of LNV independent of any model. There is now a world wide effort in improving the current experiment [36]. The usual theoretical discussion begins with the assumption of long range exchanges of light Majorana neutrinos as the dominant mechanism for 0νββ . Here, we make the assumption that 0νββ proceeds mainly through short range interactions involving the two-W-boson exchange. We also confine ourself to no new gauge interactions for the SM fermions. This turns out to be very restrictive and new scalars with high SU(2) representations can induce such decays. The new physics also generates a small Majorana neutrino mass for ν e that is insignificant for 0νββ .
While this is consistent with our hypothesis for 0νββ but inconsistent with the oscillation data. We propose to solve it by assuming that the light neutrinos have predominantly Dirac masses and the small Majorana masses induced by the new scalars render them quasi-Dirac particles. The splitting although small but may be detectable in the next generation of neutrino oscillation experiments and/or neutrino telescope. This is a new connection between 0νββ and neutrino physics which is yet to be studied in detail. Conversely the search for evidence of pseudo-Dirac nature of neutrinos can shed light on the mechanism for 0νββs.
We also noted the new physics signals such as the high charged states that can be explored in hadron colliders. In particular, we find that a high energy e − e − will be very useful in testing the origin of LNV and complements the 0νββ studies.