Δ L ≥ 4 lepton number violating processes

We discuss the experimental prospects for observing processes which violate lepton number ( Δ L ) in four units (or more). First, we reconsider neutrinoless quadruple beta decay, deriving a model independent and very conservative lower limit on its half-life of the order of 10 41 ys for 150 Nd. This renders quadruple beta decay unobservable for any feasible experiment. We then turn to a more general discussion of different possible low-energy processes with values of Δ L ≥ 4 . A simple operator analysis leads to rather pessimistic conclusions about the observability at low-energy experiments in all cases we study. However, the situation looks much brighter for accelerator experiments. For two example models with Δ L ¼ 4 and another one with Δ L ¼ 5 , we show how the LHC or a hypothetical future pp collider, such as the FCC, could probe multilepton number violating operators at the TeV scale.


I. INTRODUCTION
So far, no lepton (L) or baryon (B) number violating process has been observed experimentally. However, there are good reasons to believe that neither of these quantities are actually conserved. In fact, even within the standard model (SM), nonperturbative effects such as the sphaleron [1] violate both B and L. More phenomenologically, also the observed baryon asymmetry of the Universe points to the existence of B violation at some point in its early history.
From the viewpoint of standard model effective theory, one can build nonrenormalizable operators which violate B and L [2,3]. The lowest dimensional operator of this kind, the Weinberg operator, appears at dimension 5 (d ¼ 5) and violates L by two units. This operator generates Majorana neutrino mass terms which can be experimentally probed by the observation of neutrinoless double beta decay, 0νββ: ðA; ZÞ → ðA; Z AE 2Þ þ 2e ∓ (for recent reviews see for example [4,5]). Next, at d ¼ 6, one finds ΔB ¼ ΔL ¼ 1 operators. These operators cause proton decay in modes such as the famous p → e þ π 0 . This and other two-body nucleon decays are well known to arise in a variety of models, most notably in grand unified theories-see [6,7] and references contained therein.
The gauge structure of the SM and its field content is such that ΔL ¼ 2n þ ΔB for all nonrenormalizable operators (n being an integer). Thus, for example, even ΔL is associated to even ΔB, so no proton decay mode with ΔL ¼ 2 can exist. However, starting at d ¼ 9 one finds ΔL ¼ 3 operators, associated to ΔB ¼ 1. Also, operators relevant for ΔB ¼ 2 processes, such as neutron-antineutron oscillations, appear first at d ¼ 9. One would naively assume that the rates for ΔL ≥ 3 (or also ΔB ≥ 2) processes are necessarily much smaller than those corresponding to the lower dimensional nonrenormalizable operators. However, this may not be the case, and in fact it is possible that ΔB, ΔL ¼ 1 processes are forbidden altogether. This is exactly what happens in the standard model, since sphalerons are ΔB ¼ ΔL ¼ 3 transitions (thus, sphalerons cannot destroy protons). 1 There is also the possibility that beyond the SM there exists some (so far unknown) symmetry such that lepton number and/or baryon number can only be created or destroyed in larger multiples. For ΔL ¼ 3 this has been recently discussed in [11]; see also [12]. In that case, for example, standard proton decay modes are absent and one is left with p → e þνν , π 0 e þνν , e − ννπ þ π þ and, more interestingly, e þ e þ e þ π − π − . As noted above, these processes are induced by d ¼ 9 or higher operators, which implies that the proton decay rate is suppressed by many powers of the new physics scale Λ. Consequently, one can have Λ ∼ TeV, making it possible for colliders to probe this hypothesis [11].
In this paper, we discuss ΔL ¼ 4 processes (addressing also the possibility of having ΔL ≥ 5), and analyze whether any of these processes can possibly be observed in the foreseeable future. Specific example models are constructed where, due to the presence of some symmetry, operators involving fewer leptons are forbidden.
We start by noting that all ΔL ¼ 4 operators must have dimension greater or equal to 10 (see Table I for examples). Note that at d ¼ 10 there is just one unique ΔL ¼ 4 operator. 2 At d ¼ 12 there are already eight different operators (including derivatives), so we show only some examples for d ≥ 12 [13]. Some operators require more than one generation of fermions, such as the operator at d ¼ 10. Adding two derivatives to this operator, a very similar operator can be realized with just one generation of leptons at d ¼ 12. We discuss these two operators in more detail in Secs. II and IV. The second example at d ¼ 12 simply exchanges one Higgs boson from the operator at d ¼ 10 for two quarksQū c , equivalent to a Yukawa interaction. In this way, higher dimensional operators with fewer Higgs fields can be constructed. This is important, as is seen in Secs. II and III, for experiments at low energies. Also at d ¼ 12 we can find an operator involving only fermions (third example in Table I). However, for this particular operator the final state in any process always involves neutrinos, making it impossible to tag the lepton number experimentally.
We also show in the Table II examples of ΔL ¼ 4 operators with d ¼ 15, both of which violate baryon number. The first one has two L fields; thus the SUð2Þ L contractions again lead to final states involving a neutrino. We are more interested in the other operator, with four charged leptons. In fact, its realization/decomposition necessarily involves new colored fields which can be searched at the LHC, as we point out in Sec. IV.
As far as we know, the only ΔL ¼ 4 process treated in some detail in the literature is neutrinoless quadruple beta decay (0ν4β), having been discussed for the first time in [15]. A simple power counting suggests that the decay rate associated to 0ν4β is extremely small if all particles which mediate this process are heavy [16]. However, the analysis in [15,16] leaves open the possibility of having an observable 0ν4β decay rate, if several of the mediator particles are very light [here light implies masses of the order of the nuclear Fermi scale, i.e., Oð0.1Þ GeV]. Thus, in Sec. II we calculate a very conservative lower limit on the 0ν4β decay lifetime, based on collider searches for charged bosons. We find that this minimum lifetime is around 20 orders of magnitude larger than the current experimental limit [17], rendering 0ν4β virtually unobservable.
In Sec. III we extend the discussion to other ΔL ¼ 4; 5; 6; … low-energy processes. In all cases our simple rate estimate is far below experimental sensibilities. The most optimistic case, dinucleon decay of the form ðA; ZÞ → ðA − 2; Z − 2Þ þ 2π − þ 4e þ , is expected to be at least some 8 orders of magnitude beyond the current Super-Kamiokande sensitivity. Thus it seems impossible for low-energy experiments to test lepton number violation in 4 or more units.
However, the prospects of observing ΔL ≥ 4 processes at colliders are good, as we discuss in Sec. IV. 3 Indeed, because operator dimensionality becomes irrelevant at energies comparable to the new physics scale, the LHC is in a good TABLE II. List of lowest dimensional operators, after electroweak symmetry breaking, involving charged lepton number violation in ΔL units. For ΔL > 4, more than one generation of leptons and/or quarks is necessary (the alternative would be to add derivatives).

Minimum operator dimension
Operators with smallest dimension We use the word operator to denote a combination of fields, independently of the number of contractions. In the case of L i L j L k L l HHHH (subscripts denote the lepton flavors), there are two contractions: the square of the Weinberg operator, , and another one O 0 ijkl . Nevertheless, it is not necessary to include in the Lagrangian this last one because it is related to the O contraction with the ijkl indices permuted. Furthermore, note that there are several identities among the entries of the tensor O ijkl , so a total of six couplings/numbers encode all possible L i L j L k L l HHHH interactions (for example, if three or more lepton indices are the same, the operator is identically 0). 3 See also [18], where the authors studied Higgs decays into four same-sign leptons in the minimal left-right symmetric model. position to probe the creation or destruction of groups of four or more charged leptons. We construct two example models for ΔL ¼ 4 (and one for ΔL ¼ 5) and calculate production cross sections for the different particles in these models at pp colliders (the LHC and a hypothetical ffiffi ffi s p ¼ 100 TeV collider). From these cross sections we derive lepton number violating event rates and estimate the scales up to which pp colliders can test such kinds of models. We then end the paper with a summary.

II. MINIMUM LIFETIME OF NEUTRINOLESS QUADRUPLE BETA DECAY
The possibility of observing ΔL ¼ 4 processes via neutrinoless quadruple beta decay (0ν4β) was put forward in [15]. Three candidate nuclei for the decay ðA; ZÞ → ðA; Z þ 4Þ þ 4e − were identified: by far the most promising one is 150 Nd, which could decay into 150 Gd plus four electrons with a total kinetic energy of Q ¼ 2.08 MeV. 4 With 36.6g of the isotope, the NEMO-3 detector is well suited to measure this decay provided that it happens at a reasonable rate. Recently, based on an exposure of 0.19 kg · y of 150 Nd, the collaboration reported a lower limit of ð1.1-3.2Þ × 10 21 y on the half-life for this particular process, at a 90% confidence level [17]. The range in this number is explained mostly by the fact that, so far, no reliable theoretical calculation of the single electron spectra has been done for this process. Nevertheless, as can be seen, the result is fairly insensitive to such details.
Given the nonobservation of 0ν2β decay events so far, one has to wonder if 0ν4β decays will ever be observed. If the only contribution to the latter process is the conversion of a pair of neutrons into protons and two electrons, twice repeated (see Fig. 1), then one would expect the approximate relation between the lifetimes of double and quadruple beta decay without neutrinos. 5 In this expression q ∼ 100 MeV stands for the typical momentum transfer in a nucleus. Using the experimental lower limit on τ 0ν2β for various nuclei of the order of 10 26 years [20,21], we can extract the lower bound τ 0ν4β ≳ 10 88 years.
However, this estimate assumes that the main contribution to neutrinoless quadruple beta decay comes from two virtual double beta decays. This does not need to be the case; indeed, as previously mentioned, it is possible to forbid entirely ΔL ¼ 2 processes and still have those with ΔL ¼ 4.
In the following we argue that an important constraint on neutrinoless quadruple beta decay can be derived using data from collider searches of charged bosons. However, before proceeding, let us consider first what is the expected value of the 0ν4β decay rate from simple power counting (see also [15,16]). We start by noting that there are 12 fermions involved, so after electroweak symmetry breaking the relevant operator has dimension 18, Here, κ is just some unspecified dimensionless coefficient.
On the other hand, in the final state there are four electrons and a nucleus which stays essentially at rest, so the 0ν4β decay rate depends on the ð3 × 4 − 1Þth power of the available kinetic energy Q. With these considerations, inserting some numerical factors for the multibody kinematics, we obtain the formula Clearly, this is a very rough estimate for the lifetime of the process, with an uncertainty of a few orders of magnitude. Nevertheless, given the largeness of the numbers involved, it is good enough for the following discussion.
To obtain the lowest possible τ 0ν4β , ideally one would need large, i.e., order Oð1Þ, couplings (κ ∼ 1) and light mediator masses (Λ i ≪ TeV). Note that for very light particles, their mass becomes irrelevant when compared to the typical momentum transfer in the nucleus. Hence the best case scenario is Λ min i ∼ q. One can see easily that, in the limit where all Λ i are of the order of 1 GeV or lower, it becomes conceivable to have τ 0ν4β ð 150 NdÞ smaller than 10 26 years.
FIG. 1. Quadruple beta decay induced by two virtual double beta decays. This contribution to the decay rate is necessarily very small, given the current limits on the 0ν2β decay lifetime of various nuclei. 4 The isotope 126 50 Sn may decay into 126 54 Xe þ 4e − , with Q ¼ 3.13 MeV [19]. However, it can also undergo single beta decay, with a half-life of 2.3 × 10 5 years. 5 This naive estimate is based on a simple dimensional analysis, assuming that both processes (involving potential different parent nuclei) have the same kinetic energy Q. This coarse assumption inevitably introduces a large error in the result which, nevertheless, is of no material significance in the face of such large lifetimes.
However, we now show that the 0ν4β decay diagram contains at least four propagators of charged bosons (scalar or vector). The lightest particle, that can play this role, is the W boson; hence the most optimistic scenario achievable is This corresponds to roughly one 0ν4β decay per year for a mass of ∼10 17 kg of neodymium; hence the observation of this process would be extremely challenging even in the most optimistic scenario. Independent on any concrete model, four or more charged boson propagators are needed for the following reason. Conservation of fermion number implies that the 12 external fermions which make up the 0ν4β operator must be arranged in six currents J i . Out of the six possible pairings (ūū,ūd,ūē, dd, dē andēē) none is electrically neutral. Thus all six J i currents must exchange charge through scalar or vector bosons. (It is also conceivable that this charge exchange occurs through some intermediary vertex; yet this scenario does not lead to a minimal number of internal propagators.) It is straightforward to see that the most economical setup is the one where pairs of currents J i with opposite electric charged are connected by a single charged boson, in which case only three such propagators are needed. However, the only fermion billinears with opposite charges are dd and dē, and clearly one does not obtain the 0ν4β operator with three copies of these fermions. Hence, a minimum of four charged bosons are needed and the lower limit in Eq. (3) applies.
The six possible currents need to be coupled to scalars or vectors. These are charged or doubly charged particles, leptoquarks or diquarks. If one were to construct a loop model for 0ν4β decay and introduce also some new exotic fermions, more exotic scalars/vectors could, in principle, also appear. All of these states, however, necessarily couple to standard model fermions and thus can be searched at accelerators such as LEP and LHC. Again, considering the large numbers involved in Eqs. (3) and (4), a very rough argument suffices for our purpose. Thus, we only quote that LEP data rules out any electrically charged boson, decaying to SM fermions, below roughly 100 GeV [22].
The lower bound in Eq. (4) on the neutrinoless quadruple beta decay lifetime is therefore unavoidable. However, it is also very conservative. If one were to construct lower bounds individually for the six possible currents, (much) larger limits could be derived for the different individual cases. Instead, we now try to see whether it is actually possible to approach this bound, by considering some particularly promising models.
We have established already that 0ν4β decay is necessarily suppressed by the heavy mass of at least four charged bosons and, in order not to further reduce the decay rate, one should avoid colored particles in internal lines. 6 It is easy to check that, out of the 135 distinct ways of partitioning the 0ν4β operator in fermion billinears/currents J i , only ðūdÞ ðūdÞ ðūdÞ ðūdÞ ðēēÞ ðēēÞ makes it possible to have all internal bosons colorless. It is equally simple to arrive at the conclusion that, for this particular partition of the 0ν4β operator, the number of neutral and colorless internal bosons will be minimal (just 4) if and only if the corresponding diagram can be split into two halves, each with fermion billinears ðūdÞ ðūdÞ ðēēÞ, connected by a neutral boson. A particularly interesting way of building such a diagram is the one shown in Fig. 2, where only SM fields plus a neutral scalar are used [15]. In the central part of the diagram, it is clear that four neutrinos are created from nothing; hence the neutrino-scalar interaction is critical for the violation of lepton number. Under the full standard model symmetry, and setting aside a complication which we mention later, this ΔL ¼ 4 operator must be of the form LLLLHHHH (or perhaps higher dimensional). Crucially, this effective interaction does not need to be suppressed by the smallness of neutrino mass; indeed, the operator LLHH might even by absent. One way of generating the LLLLHHHH interaction is by adding to the standard model an extra SUð2Þ L scalar triplet Δ with one unit of hypercharge, as well as a real scalar singlet σ with no gauge interactions. The most general Lagrangian with these two new fields violates lepton number in two units, given the simultaneous existence of the couplings LLΔ and HHΔ Ã . 7 It is possible to construct a model which violates lepton number in units of 4 only, and to do so we retain just the following interactions: This Lagrangian is invariant under the Z 4 ðLÞ lepton- It is straightforward to arrange scalar potential parameters such that only the Higgs doublet acquires a nonzero vacuum expectation value, in which case there is no spontaneous breaking of the Z 4 symmetry. As such, the Weinberg operator is not generated, but LLLLHHHH is [see Fig. 3]. Assuming without loss of generality that the coupling κ is real, σ mixes only with ReðΔ 0 Þ, resulting in two mass eigenstates with masses m 1;2 and a mixing angle θ which can be controlled at will. In particular, it is possible to (a) make one of these states very light, (b) have maximal mixing sin θ, and at the same time (c) keep all other new scalars [ImðΔ 0 Þ, Δ þ and Δ þþ ] arbitrarily heavy. Neutrino masses can be accounted for by introducing right-handed neutrinos, or by the breaking of this Z 4 symmetry, for example by introducing a very small HHΔ Ã coupling. Likewise, the 0ν2β decay rate can be 0 (absence of HHΔ Ã ) or very small (small ΔL ¼ 2 coupling). The complication mentioned earlier is that there is no 1 Λ 2 ν e;L ν e;L ν e;L ν e;L local operator: given that the spinor ν e;L has two components only, the Pauli exclusion principle forbids point interactions with more than two ν e;L . Instead, the four-neutrino interaction must be of the form where Λ is a mass associated to the particle(s) mediating the effective interaction. We used the fact that the distance between the two interaction points x and y is inversely proportional to Λ. In practice, this means that there is an extra suppression factor ðmomentum=ΛÞ 2 due to Pauli blocking on top of the expected 1=Λ 2 suppression for this 4-fermion interaction, 8 and that the relevant operator is ∂∂LLLLHHHH instead of LLLLHHHH.
As pointed out in [15], the uncertainty in the invisible decay width of the Z boson as measured indirectly at LEP [24][25][26][27] constrains the mass of the scalar boson in Fig. 2 to be heavier than roughly Λ ∼ 20 GeV. In this latter process, the momentum p is of the order of the Z mass; hence p ∼ Λ and there is no Pauli blocking. However, the blocking is present in the 0ν4β decay diagram with 4 W's, since p ∼ 100 MeV is much smaller than the mass Λ of the neutral scalar. As a consequence, the 0ν4β decay lifetime is even larger than what has been assumed previously [15,16]; a simple order of magnitude calculation yields τ 4W 0ν4β ∼ 10 69 years, which is very far from the limit given in Eq. (4).
A possible way of avoiding this particular suppression factor is by introducing right-handed neutrinos ν R and W R gauge bosons such that lepton number is violated by a 4fermion interaction 1 Λ 2 ν e;L ν e;L ν e;R ν e;R . This effective operator can be generated with the scalars Δ and σ mentioned earlier, keeping in mind that there is now a ν e;R ν e;R σ interaction. Nevertheless, given the multi-TeV LHC mass limits on the new gauge bosons [28,29], the two diagrams imply a similar 0ν4β decay lifetime, Finally, we make some comments about the possible realizations of the 0ν4β decay operator in Eq. (2) with standard model fields (see also [30]). Due to Pauli's exclusion principle, fermion fields evaluated at the same space-time point anticommute, so for a generic 4-spinor FIG. 3. One possible way of generating a sizable four-neutrino interaction which is not suppressed by m ν =hHi, introducing a scalar triplet Δ and singlet σ. If more than two neutrinos have the same flavor, the amplitude of the process is suppressed by a factor ðmomentum=mediator massÞ 2 . 8 This kind of amplitude suppression can equally appear in operators with no fermions. For example, consider a global SUð2Þ symmetry under which the scalars ϕ ¼ ðϕ 1 ; ϕ 2 Þ T and ϕ 0 ¼ ðϕ 0 1 ; ϕ 0 2 Þ T transform as doublets, and S is a scalar singlet. The local operator ϕðxÞϕðxÞSðxÞSðxÞ is identically 0, but ϕðxÞϕðyÞSðxÞSðyÞ is not; hence it is possible to collide two S's and create a ϕ 1 plus a ϕ 2 . This is done via two diagrams in which ϕ 0 1 and ϕ 0 2 are exchanged in the t and u channels. In the limit where the mass Λ of ϕ 0 1;2 is much larger than the momentum transferred p, the total amplitude varies with 1=Λ 2 × ðp=ΛÞ 2 , and not 1=Λ 2 .
we can only have operators of the form Ψ n ½Á Á Á with n ≤ 4 (if there are no derivatives). 9 Furthermore, if we expand the Dirac indices of such an operator with n ¼ 4, the only nonzero term must be proportional to Ψ ↑ L Ψ ↓ L Ψ ↑ R Ψ ↓ R . This means that the Pauli exclusion principle severely restricts local 0ν4β operators to the unique form and higher dimensional 0νð2mÞβ decay operators, with m > 2, are forbidden entirely unless they have derivatives. Note that quarks have three colors; hence a similar issue arises for operators with more than six quarks of the same charge and chirality.
In this sense, quadruple beta decay (with or without neutrinos) is a borderline case between allowed and excluded local processes. An interesting consequence is that there are only three 0ν4β operators of minimal dimension (¼ 18), ∼QQ u c u c QQd cdcLL e c e cHH : ð14Þ

III. OTHER LOW-ENERGY PROCESSES WITH ΔL ≥ 4 INVOLVING CHARGED LEPTONS
We now move on to a brief discussion of other lepton number violating processes, involving low energies, with four or more charged leptons. Rough estimates for their rates are given in the following. We stress that for an experimental proof of L violation, final states should not contain neutrinos.
In Table II we give the lowest dimension at which a given ΔL ≠ 0 operator can appear, together with some examples.
(Note that, since we are interested here in low-energy processes, neither Higgs nor gauge bosons can appear as final states.) The table starts with the ΔL ¼ 1 operators which induce the standard proton decay modes (hence ΔB ¼ 1), and these are followed by the ΔL ¼ 2 operator associated to neutrinoless double beta decay. With larger ΔL, the dimension of the operators keeps rising and at some point one expects that the rates of associated lowenergy processes becomes too small to be observed. We now discuss briefly this point, by focusing on the most promising processes.
It is important to distinguish those cases where baryon number is violated from those scenarios where ΔB ¼ 0. This is simply due to the fact that the available energy in ΔB ≠ 0 processes is fixed by the nucleon mass, of the order of ∼GeV, while kinetic energy of the charged leptons is much smaller (∼MeV) in the ΔB ¼ 0 case.
Let us consider first the latter case, ΔB ¼ 0. This implies immediately that ΔL must be an even number. The relevant processes are then 0νð2nÞβ with n > 2. We discuss only β − decays, since for quadruple beta decays it has been shown already in [15] that the positron emission or electron capture processes are even more hopeless, due to their smaller Q-values. For 0νð2nÞβ with n > 2 the same observation applies.
The process 0ν6β is induced by an operator with 18 fermions; hence the decay width Γ is suppressed at least by a factor Q 17 q 29 =Λ 46 relative to the nucleon Fermi momentum q ∼ 100 MeV. Moreover, we note that at most four electrons can be at a single point x; therefore the operators for these 0νð2nÞβ decays require at least two derivatives, and consequently the decay width is suppressed by four more powers of q=Λ, compared to the simpleminded estimate quoted above. It is also straightforward to check that at least six electrically charged bosons are needed to mediate the process; hence, the same logic as discussed for neutrinoless quadruple beta decay applies.
Finally, kinematically 0ν6β and larger is only allowed for neutron-rich nuclides which are far from the valley of stability; hence these isotopes have a very short half-life. The longest-lived isotope seems to be 134 52 Te, which can decay into 134 58 Ce þ 6e − with a Q value of 2.3 MeV, but also decays by single beta emission with a half-life of 41.8 minutes [19]. Let us now turn to processes where baryons are destroyed and hence, the available energy is much larger. Here, we consider the cases ðΔL; ΔBÞ ¼ ð4; AE2Þ and ðΔL; ΔBÞ ¼ ð5; AE1Þ; violation of lepton or baryon number in greater quantities is associated with even bigger minimum lifetimes. 9 If only one chirality X ¼ R, L is involved, then Ψ n X ½Á Á Á with n ≤ 2 are the only possibilities. For example, an electron-neutrino mass term ν e;L ν e;L is allowed, but interactions of the form ν e;L ν e;L ν e;L ½Á Á Á or ν e;L ν e;L ν e;L ν e;L are not.
The lowest dimensional operator with four charged leptons (after electroweak symmetry breaking) is eeeeuuuuuu=Λ 11 (see Table II). It leads, for example, to diproton decay, ðA; ZÞ → ðA − 2; Z − 2Þ þ 2π − þ 4e þ . In the most optimistic scenario, this operator can be built in such a way that only seven powers of Λ correspond to the mass of mediators with color (Λ C ), while the remaining four powers of Λ are related to the mass of fields with electroweak interactions only (Λ EW ). Hence, using the values Λ EW ≈ 200 GeV and Λ C ≈ 2 TeV. The prefactor 10 −13 takes care of the fact that this is a six-body decay. Super-Kamiokande has searched for other diproton decay modes, imposing limits of the order of 10 32 years on the associated lifetimes [31]. It seems therefore very hard to probe ΔL ¼ 4 processes at low energies, even for those cases where baryon number is violated. Note, however, that due to the much larger energy release, the gap between the experimental sensitivity and the most optimistic expectation is only 8 orders of magnitude, compared to the (minimum of) 20 orders found for quadruple beta decay. For ΔL ¼ 5 (or larger) the decay rates are necessarily even more suppressed. For ΔL ¼ 5, the lowest dimensional operators have d ¼ 21, as shown in Table II; therefore the decay widths are suppressed by 34 powers of ∼m p =Λ when compared to m p . In summary, we conclude that observation of charged lepton number violation in four or more units in low-energy experiments is impossible in the foreseeable future.

IV. PROBING LEPTON NUMBER VIOLATION AT COLLIDERS
We now turn to a discussion of probing models with multiple lepton number violation at colliders. We do not aim at a full, systematic analysis of all possibilities. Instead, we discuss two simple models with ΔL ¼ 4 and then present one example for a model with ΔL ¼ 5. Models which lead to different ΔL ¼ 4 operators at low energy or models with larger ΔL violation can be easily constructed following the same principles that we use in our examples.
Our first model is inspired by the discussion on neutrinoless quadruple beta decay in Sec. II. In this model, called model I below, we add only two new fields to the standard model. Both are scalars: (i) Δ ¼ S 1;3;1;−2 and (ii) T ¼ S 1;3;0;−2 . Here, the subscripts stand for the transformation properties or charge under the SM gauge group and lepton number, SUð3Þ C × SUð2Þ L × Uð1Þ Y , L (also we use an S for scalars and, later on, an F for 2-component Weyl spinors). The only change with respect to the model discussed in Sec. II is that we have replaced the singlet σ ¼ S 1;1;0;−2 with the Y ¼ 0 SUð2Þ L triplet field T.
As before, we enforce a Z 4 ðLÞ symmetry which ensures that leptons can only be created or destroyed in groups of 4. The Lagrangian of the model is 10 Note that this Lagrangian is also Uð1Þ B invariant. In other words, baryon number is preserved; hence processes such as dinucleon decay are completely absent in this model. Note that Δ is the same field that appears in the type-II seesaw mechanism. However, our symmetry forbids the term HΔH, which in seesaw type II is the source of ΔL ¼ 2. This implies that for m 2 Δ ≥ 0, in our model there is no induced vacuum expectation value for Δ 0 and thus no Majorana neutrino mass term.
We have not written generation indices in Eq. (16). In general Y Δ is a complex symmetric (3,3) matrix. All terms in the Lagrangian, with the exception of those proportional to T 2 or T 4 , conserve lepton number. For the phenomenology discussed below it is important that m 2 T violates ΔL in four units.
The term proportional to λ HHΔT leads to mixing between the neutral and singly charged components in T and Δ after electroweak symmetry breaking (EWSB), as well as to a mass splitting between the CP-even and CP-odd components in Δ 0 . Thus, after EWSB the model has two new neutral CP-even scalars S 0 1;2 and two singly charged scalars S AE 1;2 plus one neutral CP-odd scalar, A 0 , and one doubly charged scalar, Δ AEAE . In our numerical calculations we always diagonalize all mass matrices and consider mass eigenstates correctly. However, in the following discussion we simply use T 0 and Δ 0 for S 0 2 and S 0 1 , respectively. (And similarly for the singly charged states T AE and Δ AE .) This is done only for the clarity of the discussion; it does not affect any of our conclusions. Note that for typical choices of masses m 2 T and m 2 Δ above ð500 GeVÞ 2 mixing between the different states is small unless m 2 T ≃ m 2 Δ . 10 We use the notation Δ ≡ For the numerical calculations shown below, we have implemented the model into SARAH [32,33]. The implementation is then used to generate SPheno code [34,35] for the numerical generation of spectra. The UFO model files generated by SARAH are used for cross section and decay calculations with MadGraph [36].
Both Δ and T can be produced with sizeable rates at colliders. Figure 4 shows cross sections for the most important production modes in pp colliders for two values of ffiffi ffi s p : To the left ffiffi ffi s p ¼ 13 TeV, to the right ffiffi ffi s p ¼ 100 TeV.
The numerically largest cross section is found for Δ AEAE pair production. However, as we discuss below, for the observation of ΔL ¼ 4 processes the interesting production modes are pp → ðT 0 þ T þ Þ and pp → ðT 0 þ T − Þ. Both processes proceed through an off-shell W-diagram; see Fig. 5. The cross section for pp → ðT 0 þ T þ Þ is larger than for pp → ðT 0 þ T − Þ, reflecting the fact that the initial state is positively charged. We first discuss the decays of Δ. The different components of Δ decay according to Δ AEAE → l AE α l AE β , Δ AE → ν α l AE β and Δ 0 → ν α ν β with 100% branching ratio, when summed over α and β. Since cross sections are largest and background lowest for Δ AEAE , the most stringent constraints on m Δ come from searches for Δ AEAE . Both ATLAS [37] and CMS [38] have searched for doubly charged scalars decaying to charged leptons. Limits depend quite strongly on the flavor of the charged leptons. CMS [38] gives limits as low as m Δ AEAE ≃ 535 GeV for a Δ þþ decaying with 100% to pairs of taus, while limits are in the range of (800-820) GeV, if the Δ þþ decays only to electrons or muons. ATLAS [37], on the other hand, has established lower limits on m Δ AEAE of roughly (600-800) GeV, for branching ratios to either electrons or muons in the range of (0.2-1). We therefore use two choices of m Δ in our numerical examples below, namely, m Δ ¼ 0.6 TeV and m Δ ¼ 1 TeV. The former is allowed only for a Δ with coupling mostly to τ's, while the latter is currently unconstrained. Note, however, that with the predicted L ¼ 3=ab for the high luminosity LHC m Δ in excess of m Δ ¼ 1 TeV will be probed, while for a ffiffi ffi s p ¼ 100 TeV collider we estimate from Fig. 4 that up to m Δ ∼ 5 TeV could be tested with L ¼ 3=ab, in agreement with the numbers quoted in [39].
For the observation of ΔL ¼ 4 processes, we need to produce the hyperchargeless triplet T. We therefore now turn to a discussion of the decays of T 0 and T AE . First of all, note that all decay rates for these particles are proportional to the coupling λ HHΔT . This is due to the fact that this term is the only one linear in T allowed by Zð4Þ L . Mixing between T 0 and T AE with Δ 0 and Δ AE induces two-body decays for these states into leptonic final states. However, these always involve neutrinos and thus are not useful to establish experimentally lepton number violation. More important are then decays of T 0 and T AE to Δ AEAE and gauge bosons. Figure 5 shows the most important Feynman diagrams. Apart from T þ → W − þ Δ þþ , T þ can decay to W þ þ A 0 , W þ þ Δ 0 , 11 as well as Δ þ þ h and Δ þ þ Z 0 . The branching ratio for T þ → W − þ Δ þþ is always close  11 Recall that we use Δ 0 in this discussion synonymous for the CP-even scalar S 0 1 .
to 25%, if the mixing between T þ and Δ þ is small. Similarly, T 0 can decay to a number of final states involving gauge (and Higgs) bosons. Figure 6 shows the most important branching ratios for T 0 decays as a function of m T . As the plot shows, for small values of m T the twobody decays T 0 → A 0 þ Z 0 (and T 0 → Δ 0 þ h) have the largest branching ratios. However, for large values of m T , the decay mode T 0 → Δ AEAE þ 2W ∓ becomes dominant. T 0 decays to both Δ þþ and Δ −− with equal rates. That both decays have the same rate can be understood as a mass insertion in the decay of T 0 ; see With the cross sections, see Fig. 4, and the decay branching ratios we can then calculate the number of ΔL ¼ 4 events as a function of m T (and m Δ ). We have performed this exercise for the LHC and find that, taking into account the lower limit on m Δ , there is less than 1 event even after 3=ab of data has been taken. However, the prospects look much brighter at a hypothetical ffiffi ffi s p ¼ 100 TeV collider; see  Table II, the smallest dimensional operator generating four charged leptons is 15 dimensional, e 4 u 6 . We therefore choose to implement it in our second example, model II. Any model leading to this operator necessarily involves beyond-the-SM colored fields.
Model II introduces three new states, two fermions O ¼ F 8;1;0;−2 and D c ¼ F¯3 ;1;1=3;1 (together with its vector partnerD c ¼ F 3;1;−1=3;3 ) and one scalar, S d ¼ S 3;1;−1=3;1 . By enforcing once again Z 4 ðLÞ invariance, we obtain this time a Lagrangian with an enlarged accidental symmetry group G SM × Uð1Þ 2B−L . It might not be immediately obvious that this latter group contains Z 4 ðLÞ, but this is nevertheless true. Indeed, the Lorentz and standard model group G SM force all operators with SM fields to be Z 2 ðB − LÞ invariant; hence we may write L ¼ 2n þ B for some integer n. Together with L ¼ 2B, it is then quite easy to see that L and B are forced to be multiples of 4 and 2, respectively. Crucially, unlike in model I, due to the Uð1Þ 2B−L symmetry it is not possible to break lepton number without breaking baryon number as well. As such, one can have small/unobservable dinucleon decays rates, but neutrinoless quadruple beta decay is strictly forbidden.
The Lagrangian contains the following terms, We have calculated the pair production cross sections for the new particles in our model II using again MadGraph [36].
The coupling λ HHΔT was chosen λ HHΔT ¼ 0.1 in this example, while the entries in Y Δ were arbitrarily put to be smaller than Oð0.1Þ. With these choices, decays to purely leptonic final states are negligible and therefore not shown. The results are shown in Fig. 8. Again, the plot to the left is for the LHC; the one on the right is calculated for ffiffi ffi s p ¼ 100 TeV. In all cases one expects that gluon-gluon fusion gives the largest contribution to the cross section; see Fig. 9. The largest cross section is found for the fermionic octet. More than ten events from O-pair production are expected in L ¼ 3=ab for octet masses up to m O ≃ 3 TeV. A ffiffi ffi s p ¼ 100 TeV collider would be able to collect more than ten events for octet masses up to m O ≃ 15.5 (18.5) TeV for L ¼ 3=ab (30=ab). In model II, since O is an electrically neutral state, it decays with equal branching ratios to D c 1=3 þ S d;−1=3 and D c −1=3 þ S d;1=3 . This implies that ΔL ¼ 4 final states 4l þ þ 6j have the same rate as the ΔL ¼ 0 final states 2l þ þ 2l − þ 6j. Thus, the production cross section (event number) of O-pair production gives directly the limit on the scale up to which ΔL ¼ 4 can be tested in model II.
It is straightforward to use the ideas discussed above to construct also models, which lead necessarily to larger ΔL. We discuss only one example with ΔL ¼ 5. This model III introduces five new states. We need two copies of S d , to which we assign different lepton numbers, S L¼1 d ¼ S 3;1;−1=3;1 and S L¼0 d ¼ S 3;1;−1=3;0 . The vectorlike down quarks also now come in two copies. We have D c L¼2 ¼ F¯3 ;1;1=3;2 ; its vector partner isD c L¼3 ¼ F 3;1;−1=3;3 , and also D c L¼1 ¼ F¯3 ;1;1=3;1 with its vector partnerD c L¼4 ¼ F 3;1;−1=3;4 . Finally, the model also has the fermionic vector octet, O ¼ F 8;1;0;2 and O ¼ F 8;1;0;3 . With these lepton number charges, we then enforce a Z 5 ðLÞ symmetry. Just as with model II, there is a bigger, accidental symmetry group in this model, Uð1Þ 5B−3L . In other words, for each group of five leptons created, three new baryons should appear as well, and for this reason the proton is completely stable in this model III.
ΔL ¼ 5 processes at the LHC can then occur through diagrams such as the example shown in Fig. 10, whereD c L¼3 is pair produced via gluon fusion. Note that the decay chains of bothD c L¼3 and ðD c L¼3 Þ Ã end with the same number of SM fermions: 7. One can assign the source of lepton number violation in this diagram to the mass termD c L¼3 D c L¼2 . If all other particles in the diagram are lighter than D c L¼2 and all couplings the same order, ΔL ¼ 5 and ΔL ¼ 0 final states from these decay chains have similar rates. 12 Nevertheless, note that even if theD c L¼3 D c L¼2 mass term was switched off, lepton number would still be broken; in fact, B  12 If all masses and all couplings are numerically the same, the branching ratio for ΔL ¼ 5 and ΔL ¼ 0 final states becomes equal. and L conservation would be restored only if the vector masses of D c L¼1 and O were absent as well. We can estimate the mass reach of the LHC to test this kind of diagram from the cross sections shown in Fig. 8. We estimate that more than ten events (before cuts) would remain for masses of D c L¼2 below 2.7 TeV in L ¼ 3=ab. At a ffiffi ffi s p ¼ 100 TeV collider more than ten events would occur for m D c below 13.3 (15.5) TeV in L ¼ 3=ab (30=ab). Thus, there is ample parameter space that could be tested in future colliders even for models with ΔL > 4. Many different models of this kind can be readily constructed.

V. SUMMARY
Given the current experimental situation, the total number of leptons L might be a conserved quantity. Standard probes for ΔL ≠ 0 are proton decay (ΔL ¼ 1, 3) and neutrinoless double beta decay (ΔL ¼ 2) experiments. However, neither has found any signal so far. It is therefore possible that L is violated only in larger multiplicities, i.e., three, four or more units.
In this context, we have discussed that the decay of a nucleus into four electrons and no neutrinos will likely never be observed. This and other ΔL ¼ 4 low-energy processes where the leptons are electrically charged must necessarily be mediated by several heavy bosons; hence their amplitudes are severely suppressed, given the current accelerator constraints. We have calculated a very conservative lower limit on the half-life for neutrinoless quadruple beta decay, which is 20 orders of magnitude larger than current experimental sensitivities. The same conclusion is valid if the lepton number is broken in five or more units. The least pessimistic scenario that we have found is the one where baryon number is also violated. In this case two protons in a nucleus could decay into four positrons plus pions, for example. The energy scale in this process is set by the proton mass, but even so, the high dimensionality of the operators involved implies that proton decay experiments would need to increase their exposure by at least 8 orders of magnitude before meaningful constraints could be derived experimentally.
Colliders, on the other hand, can explore the possibility that lepton number is violated in four or more units. The reason behind this observation is rather simple: Even though these processes involve many exotic particles, if the energy of the collider exceeds the mass of those exotics, the suppression associated with the high dimensionality of these ΔL ≥ 4 operators disappears. In this work, we presented two models for ΔL ¼ 4 and one for ΔL ¼ 5, which make use of this idea. We have calculated cross sections for the LHC and a possible future ffiffi ffi s p ¼ 100 TeV collider, estimating the rates for ΔL ¼ 4 (and ΔL ¼ 5) processes. Naturally, for hadron colliders the expectations are highest for models which contain colored particles. In this case the LHC (the ffiffi ffi s p ¼ 100 TeV collider) could probe ΔL ¼ 4 up to scales of roughly 3 TeV (18 TeV). We expect that the high multiplicity of these events associated with ΔL ≥ 4 will make them virtually background free, giving a rather spectacular signal.