Neutrinoless Double Beta Decay and Heavy Sterile Neutrinos
Abstract
The experimental rate of neutrinoless double beta decay can be saturated by the exchange of virtual sterile neutrinos, that mix with the ordinary neutrinos and are heavier than 200 MeV. Interestingly, this hypothesis is subject only to marginal experimental constraints, because of the new nuclear matrix elements. This possibility is analyzed in the context of the Type I seesaw model, performing also exploratory investigations of the implications for heavy neutrino mass spectra, rare decays of mesons as well as neutrinodecay search, LHC, and lepton flavor violation. The heavy sterile neutrinos can saturate the rate only when their masses are below some 10 TeV, but in this case, the suppression of the lightneutrino masses has to be more than the ratio of the electroweak scale and the heavyneutrino scale; i.e., more suppressed than the naive seesaw expectation. We classify the cases when this condition holds true in the minimal version of the seesaw model, showing its compatibility (1) with neutrinoless double beta rate being dominated by heavy neutrinos and (2) with any light neutrino mass spectra. The absence of excessive finetunings and the radiative stability of light neutrino mass matrices, together with a saturating sterile neutrino contribution, imply an upper bound on the heavy neutrino masses of about 10 GeV. We extend our analysis to the Extended seesaw scenario, where the light and the heavy sterile neutrino contributions are completely decoupled, allowing the sterile neutrinos to saturate the present experimental bound on neutrinoless double beta decay. In the models analyzed, the rate of this process is not strictly connected with the values of the light neutrino masses, and a fast transition rate is compatible with neutrinos lighter than 100 meV.
Contents

Introduction 2

Light and heavy neutrino exchange 4

Parameters of the amplitude 4

Role of the nuclear matrix elements 5


Type I seesaw and the nature of the transition 7

Notation 8

Naive expectations for in Type I seesaw model 9

Departing from the naive expectations for neutrino masses 12

A dominant role of heavy neutrino exchange in 15


Going beyond Type I seesaw 24

Extended seesaw 25

Extended seesaw and transition 28

Constraining parameter plane 30


Summary and discussion 32

Appendices 34

References 39
1 Introduction
The study of neutrinoless double beta decay () transition has a special relevance for testing the physics beyond the standard model. On the experimental side, there is a very lively situation [1, 2, 3, 4, 5, 6] and even an experimental claim, that the transition has been already measured at present [7]; but even postponing a judgment on these findings, it is remarkable that there are realistic prospects for orderofmagnitude improvements in the search for the lifetime [5, 6, 8, 9, 10, 11, 12, 13, 14, 15]. On the theoretical side, accepting that neutrinos have masses, see [16, 17, 18, 19, 20, 21, 22, 23, 24] and [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], the investigation of becomes the most natural option; moreover, the observation of leptonnumber violating processes would be a cogent manifestation of incompleteness of the standard model, and could be even considered as a step toward the understanding of the origin of the matter.^{1}^{1}1In fact, the process can correctly be described as a nuclear transition in which some basic constituent of the ordinary matter–i.e., two electrons–are created. Furthermore, once recognized the existence of transitions among different leptonic flavors (again, neutrino oscillations) one should conclude that any global symmetry of the standard model excepting at most is broken; thus, the observation of a lepton number violating process as would imply that also the baryon number is at some level violated.
However, once we enter the theoretical discussion, one has to stress immediately an evident but essential point, that the meaning of depends on the model. To naive eyes, the fact that no neutrino is emitted in this transition leads one to wonder what is the link of with neutrinos. The most popular theoretical answer is that, the hypothesis that neutrinos have Majorana mass [44] suggests that the exchange of virtual, light neutrinos is a plausible mechanism for the occurrence of [45]. An additional (and more recent) theoretical argument is that, listing the effective operators that obey the standard model gauge symmetry, Majorana neutrino masses arise already as dimensionfive operators [46]; thus any further contribution to (or, say, to proton decay) will be due to the higherdimensional operators, and, as such, are expected to be suppressed. However, there is an implicit assumption underlying this approach: namely, that the new physics is at very high scale. This assumption may or may not hold. In fact, the possibility that is mostly due to mechanisms different from the conventional one (light neutrino exchange) has been proposed since long [47] and it is actively discussed, see e.g., [48, 49, 50, 51, 52, 53, 54, 55, 56, 57].
Among the simplest renormalizable extensions of the standard model, that are able to account for neutrino masses, the addition of heavy sterile neutrinos–i.e., so called Type I seesaw [58, 59, 60, 61]–is largely considered as the minimal option. These new particles, if lighter than a some 10 of TeV (see below) can act as new sources of lepton number violation, or in other terms, as potential additional contributors to . The main goal of the present paper is a systematic study of this possibility, considered occasionally in the past [56]. Note incidentally that the possibility that the heavy neutrinos are not ultraheavy, and thus can be potentially tested experimentally, is the first one that has been considered in [58, 59].^{2}^{2}2A theoretical objection to the hypothesis of Type I seesaw, defined introducing the heavy sterile neutrinos as pure gauge singlets as in [60], is that the mass of righthanded neutrinos is not related to any gauge symmetry, differently from the masses of all other known particles. But, as pointed out originally [68] and stressed recently [55], even if such a gauge symmetry is introduced (most plausibly through a SU(2) group) similar considerations hold: the mechanism of is not necessarily light neutrino exchange. The phenomenology of this type of models is however different and to some extent richer than the one we will describe.
The outline of the paper is the following: In Sect. 2, we review the basics of light as well as heavy neutrino exchange in process. The most important result of this section is given in Sect. 2.2; using the updated nuclear matrix element of reference [62], the bound on the activesterile mixing coming from transition is reexamined. The improvement in the uncertainty of the nuclear matrix element leads the bound to be one order of magnitude tighter than the existing one [63]. On the face of this analysis, the bounds coming from other potentially relevant experiments, see [64] for a review, have become relatively less significant. Also see [65] for a specific realization, where the bound on sterile neutrino mass and mixing has been obtained from , astrophysical and cosmological informations.
Following this, we provide the detailed analysis on the nature of transition for the simplest extensions of the standard model with heavy sterile neutrinos. In Sect. 3, we first concentrate on the usual Type I seesaw, and later in Sect. 4, we extend the discussion to the other seesaw scenarios as well, namely Extended seesaw [66, 67].
In Type I seesaw, the generic naive expectation (as precisely defined in Sect. 3.2) leads us to believe that the heavy sterile neutrino contribution in process is much smaller than the light neutrino contribution. For one generation of light and and heavy sterile neutrino state, this naive expectation is established by the very basic seesaw structure (see Sect. 3.2.1). Going beyond one generation, it is however possible to reach the opposite extreme; i.e., one can obtain a dominant and even saturating [1] sterile neutrino contribution, which is not inherently linked with the light neutrino contribution. The systematic study of this possibility requires, that the light neutrino contribution should be smaller than the naive expectation suggested by seesaw. This consideration is carried out in Sect. 3.3, where we analyze the vanishing seesaw condition and its perturbation, leading to small neutrino masses. We classify the different cases, where the light neutrino spectra is not necessarily degenerate, and even possibly hierarchical. All these cases can provide a dominant sterile neutrino contribution in process, as discussed in Sect. 3.4. We derive an useful parameterization in Appendix A to study these cases analytically. For completeness we also provide explicit numerical example in Sect. 3.4.6.
Assuming a saturating contribution [1] from heavy sterile neutrino exchange, in Sect. 3.2.2, we provide a naive estimation on the prospect of heavy Majorana neutrino search at LHC [69, 70, 71], as well as in lepton flavor violating process [72]; these prospects turn out to be weak due to the stringent constraints coming from process. The possible issue, like radiative stability of the light neutrino mass matrices, below the naive seesaw expectation, is discussed in Sect. 3.3.3. This, along with the request of a dominant heavy sterile neutrino contribution in process gives an upper bound of about GeV on the heavy sterile neutrino mass scale (Sect 3.4.5) and also an upper bound on the necessary perturbation of the vanishing seesaw condition.
Following the analysis of Type I seesaw, in Sect. 4 we then consider a natural seesaw extension, namely Extended seesaw scenario [66, 67]. We describe the basics of Extended seesaw in Sect. 4.1 and after that quantify the different sterile neutrino contributions in process (Sect. 4.2). As for the Type I seesaw, the sterile neutrino states in this case can also give significant contributions in process. In this particular seesaw scenario, the light neutrino contribution depends on a small lepton number violating parameter, while to the leading order, the activesterile neutrino mixing is independent of that parameter. Due to this particular feature, the sterile neutrino contribution to process is totally independent of the light neutrino contribution. In the next section i.e., Sect. 4.3, we discuss the possibility of obtaining a saturating contribution [1] from the sterile neutrino states, the scope of finding sterile neutrinos at LHC [69, 70, 71], and as well as the possibility of obtaining a rapid lepton flavor violation[72]. Possible issues, such as, the higherdimensional correction to the activesterile mixing angle and transition amplitude, that will arise due to the small lepton number violating scale, has also been discussed. The details of higherdimensional correction to the mass and mixing matrix has been evaluated in the Appendix B. Finally, in Sect. 5, we present the summary of our work.
The analysis presented in this paper clearly shows that the heavy sterile neutrino states can certainly dominate the transition; this possibility has the potential to overcome any conflict [73] between the cosmological bound and the experimental hint on obtained by Klapdor and collaborators [7], or more in general it permits to reconcile a fast (and potentially observable) rate of and small neutrino masses. In addition, for Type I seesaw, the demand of radiatively stable light neutrino mass matrices lowers the mass scale of the heavy sterile neutrino states below 10 GeV. Note that, based on the updated nuclear matrix elements [62], the bound from is now much more stringent than the previous consideration [63, 64]. Further improvement in meson as well as neutrinodecay experiments have a certain potential to provide us with more information on (and possibly a measurement of) the activesterile mixings, which is similar to the conclusion obtained in the MSM model by Shaposhnikov and collaborators [74].
2 Light and heavy neutrino exchange
The phenomenological possibility that some heavy neutrino state contributes to transition amplitude has been remarked since long: see [75] for a model that however contradicts the current understanding of neutrino masses and interactions, [68] for the first modern discussion within gauge theories, [76] for a further earlier contribution.
The relevant discussion is summarized in this section, emphasizing: 1) the large number of free parameters, Sec. 2.1; 2) the role of (and the remarkably large uncertainties in) nuclear matrix elements for heavy neutrino exchange, Sec. 2.2. Subsequently, we will apply the results of this discussion to the specific model of interest for heavy neutrinos, Type I seesaw, where the number of free parameters is smaller.
2.1 Parameters of the amplitude
As we recall, several experiments testify that the usual left neutrinos () are subjected to flavor transformations, as expected if they have mass. Considering only the minimal case these neutrinos have Majorana mass; this minimal ansatze amounts to postulate that the particle content of the SU(2)U(1) standard model theory remains the same, while the Lagrangian is endowed with a nonrenormalizable term, which after spontaneous symmetry breaking reads h.c., with
(1) 
In the above, the unitary matrix is the leptonic mixing matrix, , and are respectively the flavor and mass basis; the physical masses of the neutrinos are real and nonnegative; the possible Dirac and Majorana phases are included into .
This hypothesis not only accounts for oscillations, but also has some predictive power for the lepton number violating neutrinoless double beta decay process. Indeed, the eeelement of the mass matrix:
(2) 
contributes to the neutrinoless double beta decay rate (see [77], [78], [79], [80] for recent reviews). Evidently this quantity cannot exceed , and since the differences of neutrino mass squared are measured and the relevant elements of the mixing matrix are sufficiently wellknown, there is a upper bound on as a function of the lightest neutrino mass [40, 81, 82], or equivalently of the other mass scales, such as the mass probed in direct search for neutrino masses , or the sum of the neutrino masses probed in cosmology (see [83] and e.g., [84, 85]). For the lightest neutrino mass scale eV, relevant to the case of present experimental sensitivities, one can approximate the bound as,
(3) 
with an accuracy better than 10%, that moreover improves for normal mass hierarchy. It is interesting that the experimental hint on obtained by Klapdor and collaborators [7], according to [73], challenges the bound obtained in cosmology, though this conclusion depends on which cosmological bound is considered and which nuclear matrix element is used, see below for a more quantitative statement.
Of course, in less minimal models new contributions to are expected and the conflict between the cosmological bound and the experimental hint on [7] can be overcome. This can happen when the usual lefthanded neutrinos contain heavy neutrino components, too,
(4) 
where we have considered heavy neutrinos with masses and small mixing (this condition is discussed later). In fact, the amplitude of is proportional to
(5) 
which includes the contribution due to heavy neutrino exchange (also called direct contribution, or contact term). Here, is the virtuality of the exchanged neutrino. We have , for the time component is of the order of the value of the reaction, few MeV, wheres its space component is much larger and essentially determined by the separation between neutrons, MeV. In the above expression for the amplitude, we considered the case of interest,
(6) 
Indeed, if we have a virtual Majorana neutrino with mass and with momentum , its propagator implies that the amplitude is proportional to:
(7) 
which in the limits of or reduces to and respectively.
2.2 Role of the nuclear matrix elements
A traditional expression for the halflife is:
(8) 
where the parameters of light and heavy neutrinos, as defined in Eq. 5, are presented through the complex dimensionless parameters and , and, conventionally, the reference mass scales are chosen to be the electron and the proton masses and . In the above, and are the nuclear matrix elements corresponding to the light and heavy neutrino exchange in process.
Whenever necessary, we will assume for the nuclear matrix elements the values given in [62], where we read that, in the case of Ge transition, the ‘phase space’ factor is yr and the matrix elements are and . These values imply, for instance, that the result of Klapdor, yr [7] would be consistent with and ; with and ; or with linear combinations of these limiting cases (possibly allowing for an overall sign). The first possibility, where only the neutrino mass is present, would imply
(9) 
namely, a degenerate neutrino spectrum, partly testable with KATRIN experiment and of great interest for cosmological investigations, since it implies eV (see Eq. 3). In the following, we will be especially interested to explore the opposite limit, when the is dominated by the second term, and the neutrino mass spectrum is not necessarily degenerate, and even possibly hierarchical.
The traditional expression in Eq. 8 can be recast into the following equivalent form:
(10) 
where we set and following [55] we defined,
(11) 
Note the resemblance of Eq. 10 with the expression of the amplitude given in Eq. 5. With the values of [62], we get , in remarkable accordance with the rough expectations for described in Sec. 2.1.
An alternative presentation of the lifetime, valid for generic values of the neutrino masses, is
(12) 
This agrees with Eq. 8 if one identifies the following sets of parameters,
(13) 
and, at the same time, the following limits hold:
(14) 
the scale of comparison being (200 MeV). A simple and useful analytical approximation of the general expression of Eq. 12 has been proposed in [86]:
(15) 
The advantage of this formula is that it emphasizes the role of the neutrino propagator given in Eq. 7 and allows one to switch easily from the regimes of light and heavy neutrino exchange (compare with Eq. 10 in the limit of light and heavy neutrinos), even being slightly inaccurate in the region where [57, 63].
Using the last formula and the present experimental bound on lifetime yr [1], we obtain the upper bound on the mixing , which is shown in Fig. 1. When this is compared with the other experimental constraints on the model, compiled by [64], it is quite evident that play the most important role. The details of the figure are as follows,

The upper yellow region is disallowed from neutrinoless double beta decay consideration. Part of this region is as well constrained from different meson decays, neutrino decaysearches as well as other experiments, shown explicitly in the figure. The lower blue region is the allowed one from as well as the various above mentioned experiments.

The middle grey band which has been obtained considering the exchange of a single heavy neutrino with mass in process, corresponds to the uncertainty of the nuclear matrix elements and . For the thick black line in this grey band, we have adopted the parameters of [62], and , i.e., ; for the upper thin black line and , namely, ; while for the lower thin line and , namely, . The upper line agrees numerically with the results of [63] (see also [86]); the large difference with [62] should be attributed to the new short range correlations and improved nucleon form factors.^{3}^{3}3We thank F. Šimkovic for clarifying discussions on this issue. The lower line, instead, is meant to convey a conservative idea of the uncertainties; see [87, 88] for the most stringent upper bound on we could have at present, depending on the size of the nuclear matrix elements. For each of these black lines, the region above the line is disallowed from transition.

The span of values of used in Fig. 1 is much more conservative than the one of [62], quoted above. It corresponds to the range given in the compilation [89], see their Fig. 1. By comparing with a similar compilation of about 10 years ago [82], see their Fig. 2, one understand that the new nuclear physics calculations obtained a reduction of the uncertainty of a factor of two for the regime of light neutrino exchange. This improvement is of enormous importance: the lifetime scales only as the square of the nuclear matrix elements, while in presence of background, the improvement of the bound on the lifetime scales as the square root of the exposure.

The limits from which has been derived using the result of [62] and presented in Fig. 1, are significantly tighter than the previous limits on mass and mixing given in [63] (the result of [63] has also been adopted in recent global analysis [64]). Conversely, the impact of other constraints, in particular those from meson decays, neutrinodecay searches and other experiments, becomes relatively less important: See again [64] (and in particular their Fig. 2) where full reference to the original literature is provided.
3 Type I seesaw and the nature of the transition
Type I seesaw is in many regards the simplest extension of the standard model: only heavy sterile neutrino states are added to the spectrum of the SU(3)SU(2)U(1) theory [58, 59, 60, 61], with a primary purpose to account for light neutrino masses in a renormalizable gauge model. However, these heavy states might lead to measurable effects, in particular, for the neutrinoless double beta decay.
In this section we discuss the nature of transition within the Type I seesaw [56, 57], emphasizing the possibility discussed occasionally in the literature that the heavy neutrino exchange contribution plays the main role for . In the present study, we analyze in greater detail the parameter space of Type I seesaw.
Let us describe in detail the outline and scope of this section. First, we recall the basic notations for the model (Sect. 3.1). In Sect. 3.2 we provide a precise formulation of a naive and widespread expectation: within Type I seesaw, the contribution of the heavy neutrino states to the decay is smaller than the one due to light neutrino states. Actually, for one generation this naive estimation works perfectly well (see Sect. 3.2.1) but for more than one generation, it is possible to obtain a large and dominant contribution to from the heavy neutrino states, which is not necessarily inherently linked with the light neutrino contribution. This will be discussed in detail, after the mathematical premise of Sect. 3.3, aimed at outlining the cases when the light neutrino masses are much smaller than suggested by the naive expectations from seesaw. Finally, we discuss in Sect. 3.4 the possible cases when heavy neutrino exchange provide us with a large effect in . We exhibit explicit examples when this happens. We prove that this possibility can be implemented within Type I seesaw, without occurring into limitations on the structure of the light neutrino mass matrix. As an extreme possibility, we show that it is possible to arrange a large contribution from the heavy Majorana neutrino exchange, even when the light neutrino contribution to is negligible. We discuss the possible issues, like radiative stability in Sect. 3.3.3, the possibility of relatively less finetuning in Sect. 3.4.2, and derive bounds on heavy neutrino mass scale and finetuning parameter in Sect. 3.4.5. Finally, in Sect. 3.4.6 we present an explicit numerical example, where the heavy neutrino contribution is the dominant one and the light neutrino contribution is negligibly smaller than the heavy neutrino contribution.
3.1 Notation
In our subsequent discussion of process and its relation with Type I seesaw, we denote the standard model flavor neutrino states by and the heavy Majorana neutrinos by . The Lagrangian describing the mass terms is the following,
(16) 
For three generation of standard model neutrinos and generation of sterile neutrino state , and will be of and dimension. From the above Lagrangian one obtains this following neutral lepton mass matrix,
(17) 
The neutrino flavor state is related to the neutrino mass state by the unitary mixing matrix where,
(18) 
The mixing matrix which diagonalizes the above mentioned neutral lepton mass matrix satisfies the following relation , where is the diagonal neutrino mass matrix. We denote as follows,
(19) 
where and represent the light and heavy neutrino masses respectively. It is convenient to introduce a couple of auxiliary matrices as . The first matrix blockdiagonalizes , namely it satisfies . Subsequently, is further diagonalized by the matrix , that satisfies the relation . Let us denote the blockdiagonalized matrix as,
(20) 
It is possible to operate a systematic expansion of and is powers of [90], thus enforcing the seesaw approximation, . Up to leading order in powers of , we have simply for the heavy neutrino mass matrix, while the light neutrino mass matrix reads,
(21) 
Keeping terms up to 2nd order in , the mixing matrix is,
(22) 
Next, we denote the mixing matrix as follows,
(23) 
where the mixing matrices and diagonalize the light and heavy neutrino mass matrices and respectively: and . From Eq. 22 and Eq. 23, one immediately obtains,
(24) 
To the leading order, the mixing matrix is simply,
(25) 
Finally and quite importantly, we note that the mixing between light and heavy neutrino states is . According to convention of Eq. 4, this mixing matrix is denoted as , namely
(26) 
In the basis where the heavy Majorana neutrino mass matrix is diagonal, , we rewrite the mixing matrix as follows,
(27) 
and the Dirac mass matrix in this basis is simply,
(28) 
We note in passing, that Eq. 4 is actually valid only when ; the deviations that should be expected (due to the unitarity constraints) are formally evident from Eq. 24, and are usually small.
3.2 Naive expectations for in Type I seesaw model
In this section, we aim to define precisely which are the naive expectations from the Type I seesaw model for various interesting measurable quantities. Subsequently, we argue for the interest in exploring alternative possibilities.
3.2.1 Heavy neutrino exchange in the single flavor case
Let us begin the analysis by showing that, a single flavor Type I seesaw implies that the light neutrino exchange in process is never subleading. In other words, it is not possible to attribute the transition to the heavy neutrino exchange with one light and one heavy neutrinos only.
Assume one light and one heavy Majorana neutrino and with masses and respectively. The flavor state is mixed with the mass states as follows,
(29) 
and thus the transition amplitude receives a contribution proportional to,
(30) 
The expression is valid whatever value the mass has. Now, since by hypothesis we are considering Type I seesaw, the leftleft element of the mass matrix given in Eq. 17 is zero. Hence we have, . Thus we conclude that the amplitude of Eq. 30 can be rewritten as:
(31) 
Since , we see that the effect of the heavier state can only reduce the strength of the transition; this becomes negligible, leaving only the contribution due the light neutrino exchange, in the limit when .
It is also useful to note that, upon expanding Eq. 31 in powers of we get the following,
(32) 
As expected from Eq. 5, the new contribution has the form of a contact term–i.e., it is a constant. It is clearly evident from above that for the limit , the second term within the bracket is much smaller than unity.
With this, we conclude that for one generation case, the contribution of the heavy sterile Majorana neutrino state to process is always much smaller than the one of the light Majorana state. This implies that, in order to have a large contribution to within Type I seesaw, we need to consider the multiflavor case. However, such a property is not generic of a multiflavor case, as shown later.
3.2.2 Naive expectations: , colliders and lepton flavor violation
Next, we consider the naive expectations from Type I seesaw. Since this discussion is quite important for the following discussion, we begin by providing a precise definition of what is meant by ‘naive expectations’. Most of these expectations correspond to simple scaling laws, obtained replacing the Dirac mass matrix in Eq. 17 with a single mass scale , and likewise the Majorana mass matrix with a single mass scale . In other words, here we assume that all heavy neutrino masses are of the order of , all light neutrino masses of order of (see Eq. 21) all mixing angles with heavy neutrinos (with and ) are of the order of (see Eqs. 26 and 27); and, when we speak of “seesaw”, we simply mean that we restrict to the case .
Fig. 2 shows the relevant portion of the plane. The three gray bands correspond to the following boundaries: (1) MeV, i.e., heavy sterile neutrinos are assumed to act as pointlike interactions in the nucleus, as discussed in Sect. 2.1, see in particular Eq. 7; (2) GeV, in order to ensure perturbativity of the Yukawa couplings; (3) , namely, to the seesaw in a conventional sense. The plane is divided in various regions (halfplanes) by the three oblique lines, that corresponds to power laws in loglog plot, and are defined as follows:

The leftmost oblique line separating white and blue region corresponds to the condition that the naive formula for neutrino mass gives a small enough result: eV. The inequality eV, excluded experimentally, corresponds to the region that lies below the line. (We will discuss below the cases when this bound can be evaded).

The condition that the naive contribution from the heavy neutrino exchange to amplitude: saturates the present experimental bound [1] gives the line close to the diagonal and which separates the pink and blue region of the parameter space. The halfplane below this line defines the region where this contribution is larger than allowed experimentally.

The area below the oblique line separating the pink and yellow region is the region in plane, where the production of the heavy Majorana neutrinos in colliders is not suppressed by a small coupling. The oblique line corresponds to the heavy and light neutrino mixing angle , and the region below this line corresponds to : the muon flavor refers to the possibility to have a same sign dimuon signal [91].
Note that the last region is divided in two parts by the bound that applies to , with , and that translates into [92]. This excludes the rightmost region; the almost vertical line corresponds to the fact that when , the limit is relaxed. For one generation of standard model neutrino and one heavy sterile state , of course the bound would be absent.
If taken rather literally, the naive expectations on Type I seesaw (as defined here) would suggest that the existing bounds on neutrino masses imply that there is no room for large contributions to from heavy neutrinos, or to produce heavy neutrinos at colliders, or to expect a rapid transition.
3.2.3 Alternative possibilities in the multiflavor case
However, as it is known in the literature and as we discuss in great details later, the naive expectation on discussed in the previous section, can be strongly relaxed in certain multiflavor cases.
This offers interesting possibilities for the phenomenology of Type I seesaw and in particular for the investigations of the nature of transition within this model. Indeed, the impression that one receives from Fig. 2 is that, even after evading the constraint from neutrino mass, the constraints from are likely to be relevant in a large portion of the parameter space, which agrees with the conclusion of [70]. We will show in the following that this impression is confirmed by a detailed analysis in a large portion of the parameter space. When we depart from the condition that the mixing angles are all the same, we can decouple the size of the various amplitudes: in fact, the amplitude of depends on , the one of samecharge dimuon signal depends instead on , and finally depends on . This fact, already, provides a large freedom to phenomenological investigations. However, in this work we prefer to proceed systematically, and will be mostly concerned to classify which cases (which matrices) evade the constraint from neutrino mass, showing that at the same time, the heavy neutrino exchange contribution can play a relevant role for the decay process. This can be achieved if light neutrino mass is strongly suppressed than the naive expectation from seesaw.
With these phenomenological motivations in mind, we proceed to investigate in detail the different cases when the neutrino masses are much smaller than suggested by the naive expectations.
However, we would like to sketch out in passing an important theoretical consideration, that will be developed in the following. Consider the case when the treelevel neutrino massmatrix is very suppressed or zero. We expect that the radiative corrections will provide us with nonzero massmatrix; let us say, for definiteness, of the order of , where is some orderone gauge coupling. Thus, the tighter constraint depicted in Fig. 2–the one denoted ‘ mass too large’–can be relaxed, but only by a couple of orders of magnitude. This implies that the sterile (i.e., heavy) neutrino exchange contribution to can have a dominant role only in a limited region of the parameter space, when the sterile neutrinos are not too heavy. Stated differently, the lighter the sterile neutrinos, the less problematic is to reconcile a dominant role of the sterile neutrinos for the transition with the values of the masses of the ordinary neutrinos.
3.3 Departing from the naive expectations for neutrino masses
We are interested to the cases when the naive expectation for the light neutrino mass, , typical of Type I seesaw, does not hold. Thus, we first proceed in Sect. 3.3.1 by a mathematical analysis of the vanishing seesaw condition and then we classify in Sect. 3.3.2 which are the perturbations of this condition that permit us to have neutrino masses much smaller than suggested by this naive expectation.
3.3.1 Solving the condition : Lightneutrino masses=0
We describe here a direct procedure to find the nontrivial solutions to the matricial condition . Consider the three flavor scenario with two matrices and , where is assumed to be invertible. Using a suitable biunitary transformation, we go to the basis where the Dirac mass matrix is diagonal, i.e., . This basis is very convenient but only to solve the condition ; eventually, we have to return to the original flavor basis, where the weak interactions and the charged lepton mass matrix are diagonal. The condition is compatible with an invertible matrix if , that is the trivial solution, but also if one diagonal element in is nonzero; all other cases are excluded.^{4}^{4}4If , and are all nonzero, we immediately find . If, e.g., only and are nonzero, we have , which implies , that is again incompatible with the existence of the inverse of . In mathematical terms, recalling that the characteristic of a matrix is basis invariant, we conclude that has characteristic 0 or 1. Let the nonzero element be the third one, i.e., and : We have to satisfy . This means that the block including , and has zero determinant, i.e.,
(33) 
Rotating this block of into a diagonal form, it has one diagonal element equal to zero. We conclude that, in a given basis, we are dealing with the following Lagrangian including the Dirac and the Majorana masses:
(34) 
where and are nonzero since , while and are free parameters.
Evidently, the characteristic of this matrix is 3: there are three null eigenvalues. More in details, we see that operating a rotation in the and the fields, we can transform it into a mass matrix where the new ‘Dirac’ block is just zero. The physical meaning of these mathematical results is that the condition always implies that the mass matrix of the light neutrinos is zero to all order, and that this condition can be realized in a nontrivial manner only arranging for a ‘large’ mixing (i.e., order ) between the left and the right neutrinos. (This conclusion has been derived previously using a systematic expansion of the neutrino mass matrix [90, 93] see also [94, 95, 96].) From the above proof, it is easy to understand that, up to change of basis, the previous nontrivial solution of the condition is the most general one.
3.3.2 Perturbing the condition : Lightneutrino masses
Here, we perturb both and , maintaining the first one diagonal. From the previous discussion, it is pretty evident that to satisfy the vanishing seesaw condition with an invertible , one should have at most one nonzero diagonal element in . Hence, we want to keep only one large diagonal element in the Dirac mass matrix; while the other two diagonal elements appear due to perturbation: In formulae, we write
(35) 
with and . In the following, we denote by a small parameter, which we will use to explicitly tune the smallness of the light neutrino masses. We will show how it is possible to organize the elements of in powers of in order to maintain a special suppression of the light neutrino mass matrix. To simplify the notation, we refrain from writing explicitly the coefficients of of the mass matrices , and , but we will use the symbol to keep track of this simplification; i.e., to say, in each of the matrix elements of , and , we show only the leading order. In short, in the following formulae we emphasize the necessary suppressions of the matrix elements in powers of the small parameter . We identified 3 main cases:
Case A:
Consider the following Dirac and Majorana mass matrices:
(36) 
(The elements (12) and (13) of could be much smaller without affecting the argument.) The analysis of this case essentially reduces to analysis of the matrices. This case yields one massless and two massive light neutrinos and will be discussed in details later, being a prototypical case.
Cases B:
A similar situation is realized for the following mass matrices:
(37) 
that correspond to the following light neutrino mass matrices:
(38) 
The analysis of these cases is pretty similar to the analysis of the previous one. It is easy to see that, for all of them:

The elements of the neutrino mass matrix are at most of the order of . Thus all neutrino masses are more suppressed than the what naive seesaw formula would suggest.

However, the determinant of the light neutrino mass matrix is . This essentially implies that the lightest neutrino mass is order , i.e., very small.
Both features of Case B are in common with those of Case A, where the lightest neutrino mass is just zero.
Case C:
Finally we consider an interesting case, which is in favor of nonsuppressed lightest neutrino mass, i.e.,
(39) 
Now the elements of the light neutrino mass matrix are at most of the order of (that is the same as before up to the redefinition ) but the determinant of the mass matrix is . Hence, depending on , it is possible to have a lightest neutrino mass, which is not small a priori.
Remark
Before passing to the discussion of the previous mass matrices, we note that the presence of a very light (or just massless) neutrino, common to Cases A and B–but not necessarily to Case C–could be tested experimentally by future cosmological measurements [84, 85]: In fact, in the case of normal (resp., inverted) hierarchy, we would expect that the sum of neutrino mass is (resp., twice) the atmospheric mass scale meV.
3.3.3 Quantification of the finetuning and lower bound on
The common feature of the neutrino mass matrices classified in the above section is that they are smaller than suggested by the naive seesaw formula; using the symbols as in Sects. 3.2.2 and 3.3.2,
(40) 
where is a small parameter.^{5}^{5}5 For case C, this requires a redefinition of . The question arises, whether such a structure is stable under radiative corrections.
It has been remarked in [71] that, for nonsupersymmetric Type I seesaw, the decoupling of two heavy neutrinos with different masses and produces a correction to the neutrino mass matrix of the order of
(41) 
where is a gauge or Higgs coupling, since the renormalization group evolution of the effective operator differs from the evolution of the Yukawa couplings (or Dirac mass), as shown in [97]. This can be seen as a minimum natural size of the coefficient in Eq. 40,
(42) 
unless we want to accept very finetuned mass matrices, an unattractive possibility that we could however consider, if the data should force us to do so: see Sect. 3.4.7 for a discussion.
Besides the possibility to enforce , by an (approximate) global symmetry [71, 98] to put to zero the radiative correction of Eq. 41, there are also other cases when the resulting condition on (Eq. 42) can be, if not avoided, at least relaxed. First, it is known [99] that in supersymmetry, the effective operator and the neutrino mass matrix receive the same radiative correction, so that the conclusion of Eq. 41 does not hold. Second, also in nonsupersymmetric model, heavy neutrinos with masses below the electroweak scale, , will not give logarithmic corrections, but smaller polynomial corrections only:
(43) 
simply because the electroweak physics should decouple.^{6}^{6}6When we enforce lepton number conservation in the model; this agrees with the fact that Eq. 43 vanishes in this limit. These corrections can be attributed to finite diagrams, where the usual tree level operator for neutrino mass is dressed by (or by higgs boson) exchanges (see Fig. 3). The condition of radiative stability of the tree level neutrino mass matrix, bounds from below:
(44) 
The discussions in this section can be summarized as follows,
(45) 
where GeV; as we see in the following, it is the regime where the corrections are smaller, , the one in which we will be mostly interested.
3.4 A dominant role of heavy neutrino exchange in
In this section we discuss how it is possible that heavy neutrino exchange provides us with a dominant contribution to within Type I seesaw models. First, we state precisely in Sect. 3.4.1 what is the role of the heavy neutrino exchange for the amplitude of , and we consider in Sect. 3.4.2 the case when this contribution is large building on the mass matrices discussed in Sect. 3.3. Then, we pursue a detailed investigation of this case for two flavors (Sect. 3.4.3) and as well as for three flavor mass matrices (Sect. 3.4.4). We discuss the compatibility with finetuning issues in Sect. 3.4.5 and conclude in Sect. 3.4.6 with numerical example.
3.4.1 The amplitude of and the role of heavy neutrino exchange
The mixing between the heavy sterile neutrino state and the standard model neutrino state is caused by the Dirac mass matrix (see Eq. 16). As clear from Fig. 4, the amplitude for the process is proportional to the following factor stemming from the vertices and the propagator,
(46) 
where we consider the expressions in leading (second) order in and is the Dirac mass matrix in the basis where the heavy neutrinos are diagonal, see Eq. 28. Reminding that this expression in sandwiched between chiral projectors, its contribution to the amplitude is just,
(47) 
Focussing again on the case , we can expand Eq. 47 as follows:
(48) 
where we have used the diagonalizing relation . The first term in brackets, evidently due to the exchange of light Majorana neutrinos, is the usual one; the second one is the effect of heavy neutrino exchange in which we are interested. For real , it can be written simply (up to the sign) as,
(49) 
a quantity with dimension of an inverse mass. In the following, we will refer often to this as a ‘contact term’, having in mind the nature of the operator that induces the transition. Using Eqs. 21 and 26, it is easy to verify that Eq. 48 coincides with Eq. 5 fully; but the new expression is more convenient for the subsequent theoretical analysis of Type I seesaw model.
3.4.2 The case for a large contact term
Being ready to consider the special class of neutrino mass matrices classified in Sect. 3.3, it is possible to understand the cases when the naive expectations of Type I seesaw discussed in Sect. 3 (and in particular Sect. 3.2.2) do not work. These special cases are of great phenomenological interest, especially for neutrinoless double beta decay, but have also some theoretical interest, in view of the fact that they are based on the simplest renormalizable extension of the standard model, that includes massive neutrinos.
In formal terms, and using the symbols as in Sects. 3.2.2 and 3.3.2, we are considering the possibility that the neutrino mass matrix is smaller than suggested by the seesaw formula and at the same time the contribution of heavy Majorana neutrino states in process,
(51) 
where is a coefficient which depends on the specific particle physics model, and will be determined later in this section for the cases of interest. We are particularly interested in heavy neutrino masses that saturate the experimental bound [1],
(52) 
where the nuclear matrix element and the halflife apply both to Ge (see Sect. 2). It is important to note that by reducing the mass scales, the need of finetuning (i.e., too small ) diminishes; indeed Eq. 40 and 51 are left unchanged by the scaling
(53) 
where implies that will be smaller than GeV, maintaining perturbativity of the Yukawa couplings. (Actually, the fact that decreasing and we need less finetuning can be understood also from an inspection of Fig. 2.) Also note the very mild dependency of the heavy neutrino mass scale in Eq. 52 on the true value of the halflife; if the central value found by Klapdor and collaborator is confirmed, this would change by only 3%, but even if the true lifetime should turns out to be 100 times larger, i.e., yr, the mass in Eq. 52 would just be doubled. The dependence on the matrix elements is also quite mild. Finally, even stretching the perturbativity condition on GeV), from to , the mass would increase only by a factor of 2.3.
3.4.3 Dominating heavyneutrino contribution with two flavors
Here we analyze a prototypical two flavor case. We start from rather specific mass matrices, given in the basis where the Dirac mass matrix is diagonal, and then switch to the flavor basis (where the charged current interactions and the charged lepton masses are diagonal). We show that, in this way, we can obtain information on the structure of the contact term.
Structure of the contact term in the basis where is diagonal
Suppose that we have the matrices