Contents

UMD-PP-018-11

Natural Seesaw and Leptogenesis from Hybrid of High-Scale Type I and TeV-Scale Inverse

Kaustubh Agashe, Peizhi Du, Majid Ekhterachian, Chee Sheng Fong, Sungwoo Hong, Luca Vecchi

Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, U. S. A.

Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Santo André, 09210-580 SP, Brazil

Department of Physics, LEPP, Cornell University, Ithaca NY 14853, U. S. A.

Theoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne, Switzerland


email addresses: kagashe@umd.edu, pdu@umd.edu, ekhtera@umd.edu, sheng.fong@ufabc.edu.br, sh768@cornell.edu, luca.vecchi@epfl.ch

Abstract

We develop an extension of the basic inverse seesaw model which addresses simultaneously two of its drawbacks, namely, the lack of explanation of the tiny Majorana mass term for the TeV-scale singlet fermions and the difficulty in achieving successful leptogenesis. Firstly, we investigate systematically leptogenesis within the inverse (and the related linear) seesaw models and show that a successful scenario requires either small Yukawa couplings, implying loss of experimental signals, and/or quasi-degeneracy among singlets mass of different generations, suggesting extra structure must be invoked. Then we move to the analysis of our new framework, which we refer to as hybrid seesaw. This combines the TeV degrees of freedom of the inverse seesaw with those of a high-scale ( TeV) seesaw module in such a way as to retain the main features of both pictures: naturally small neutrino masses, successful leptogenesis, and accessible experimental signatures. We show how the required structure can arise from a more fundamental theory with a gauge symmetry or from warped extra dimensions/composite Higgs. We provide a detailed derivation of all the analytical formulae necessary to analyze leptogenesis in this new framework, and discuss the entire gamut of possibilities our scenario encompasses—including scenarios with singlet masses in the enlarged range GeV. The idea of hybrid seesaw was proposed by us in arXiv:1804.06847; here, we substantially elaborate upon and extend earlier results.

Contents

1 Introduction

Viewing the Standard Model (SM) as an effective field theory, Majorana neutrino masses dominantly arise from the unique dimension-five Weinberg operator:

(1)

where and are respectively the SM lepton and Higgs doublets with vacuum expectation value (VEV) . Within our conventions, the new degrees of freedom responsible for generating the operator in eq. (1) are assumed to be characterized by a mass scale and a leading coupling to the SM lepton (and the Higgs) (couplings among new states are measured by other couplings in general). We next elaborate on the “” parameter.

The operator in eq. (1) violates by two units. Such a violation may be induced directly from , as in ordinary type I seesaw scenarios [1]. In all those cases we conventionally say breaking is maximal and set to mean that no further parameter is necessary to generate neutrino masses. On the other hand, in all UV completions in which does not have spurious charge , eq. (1) will have to be proportional to some additional -breaking parameter . In particular, when GeV and such -breaking parameter is forced to be very small, i.e. . Note that while in the former case, setting merely means effectively we did not need the parameter, in the latter case encodes the required breaking. The new parameter in the second scenario could be a ratio of mass scales or couplings within the new sector, or simply be controlled by a new -breaking interaction to the SM. We will refer to these models as scenarios with small breaking.

UV completions of the Weinberg operator have other important physical implications. The necessary source of breaking indicates for example that the UV dynamics responsible for generating eq. (1) may also have the possibility to realize baryogenesis through leptogenesis [2]. Furthermore, the parameters control the possible collider signatures of the new particles involved, suggesting that models with TeV and certainly represent the most promising ones experimentally.

Combining these considerations, we find that small neutrino masses may be obtained in three qualitatively different ways, depending on whether TeV/ or or is the small parameter suppressing :

  • high-scale scenarios TeV in which , are not necessarily small, such as the popular high scale seesaw model [1];

  • scenarios with small couplings and unsuppressed and TeV/, like in low scale seasaw models;

  • scenarios with small breaking, , where and TeV/ may be unsuppressed. The inverse seesaw [3] or linear seesaw [4] belong to this latter class.

In table 1, we summarize how the most common realizations of the above three classes of UV completions of eq. (1) compare with respect to the generation of small neutrino masses, the realization of successful leptogenesis, and the possibility of featuring interesting signatures at colliders. Needless to say, this table reflects our own perspective on the topic, as well as our biases as model-builders. For example, in the table and the remainder of the paper we will often use the term natural. To help the reader appreciate this terminology we hereby attempt to provide an operative definition of this concept, which we may call Dirac naturalness: dimensionless couplings (like or the new physics and SM self-couplings) are of natural size if they are not too far from one, say of ; mass scales () are of natural size if they are either generated via dimensional transmutation of natural couplings or are related to a more fundamental dynamical scale (for e.g. the TeV, the GUT or the Planck scales) by factors of order unity; in the absence of a symmetry reason, the differences among masses and among couplings should be of the same order as the respective masses and couplings themselves (i.e. anarchic masses and couplings). Our naturalness criteria is more restrictive than t’ Hooft’s technical naturalness, which only calls for stability under quantum corrections.

We can now proceed to explain table 1. In High scale type-I seesaw models the new physics is in the form of heavy Majorana right-handed neutrinos with coupling to the SM leptons. Once both a mass for and the coupling are turned on, is broken collectively by and hence the model belongs to class (I). In these cases leptogenesis is realized naturally [2, 6]. Unfortunately, with a high scale GeV there are no detectable LHC or low-energy experimental signals. Small neutrino masses can be obtained quite elegantly. However, the required mass must be a few orders of magnitude smaller than the known fundamental scales (say the Planck or GUT scale of or GeV, respectively), at an intermediate value that is not fully understood. In view of our definitions above, we may view such a scale as almost natural.

UV seesaw model Natural ? Signals? Leptogenesis?
High scale type I Almost No Yes
TeV scale type I Not really Possible Possible
TeV scale inverse/linear Not really Yes Possible
Hybrid (this work and [5]) Possible Yes Yes



Table 1: Our comparison of various UV completions of eq. (1): see explanation in text.

In TeV scale type-I seesaw the symmetry is again maximally broken (). However, here the small neutrino masses ( eV) are obtained for TeV with tiny couplings to the SM, .111Unless we invoke some special textures [7], which we do not consider here. This model thus belongs to our class (II). The smallness of is considered a tuning and hence neutrino mass is not natural according to our definition. The small also makes the direct production of the exotic unlikely. To make this scenario more visible one may consider extensions with additional gauge symmetries ( or LR models) so that may be produced via the associated gauge couplings, which can be sizable, giving collider signals with same-sign dileptons due to Majorana nature of (see references in [8]). Overall, this model scores a “possible” in the experimental signals entry. Finally, leptogenesis is not natural unless one imposes quasi-degeneracy among singlets of different generations to resonantly enhance the CP violation [9]. This cannot be achieved without additional ingredients for example in the form of flavor symmetry. Hence, leptogenesis scores a “possible” here as well.

TeV scale inverse seesaw (ISS) [3] and linear seesaw (LSS) [4] have, besides the right-handed neutrinos (here denoted by ) which couple to the SM lepton and Higgs with Yukawa , additional fermion singlets with left-handed chirality . The latter are introduced such that the singlets can form a Dirac mass term TeV preserving . Such a symmetry is broken by a small Majorana mass term for the singlets in the ISS, or by a small lepton number breaking Yukawa coupling of to the SM in the LSS. In the language introduced in eq. (1) this means

(2)

respectively, and these models belong to class (III). Since TeV is not set by any fundamental scale, and similarly the coupling must be very small, the observed neutrino mass is not obtained naturally according to our criteria. Yet, an attractive feature here is that experimental signals from singlets can arise from sizable Yukawa coupling — including a contribution to the rare process as well as direct production of singlets themselves (see for example [10] and references therein). The model therefore scores a clear “yes” in the signal column.222Even though we get accessibility to the singlets, it is true that it is difficult to directly probe the very small -term, i.e. lepton-number breaking.

Unfortunately, as we have shown recently in ref. [5] (focusing on the case of strong washout) leptogenesis is not achieved naturally in the ISS (i.e. and ),333Some of these results have been obtained by others (for example, in ref. [11] and more recently, in ref. [13]). according to our naturalness criteria. We will elaborate more on this in section 3. This includes the effects of quasi-degenerate mass among different generation singlets: while such possibility provides a sizable improvement of the final asymmetry, it can barely accommodate the observed value. We also discuss the possibility of weak washout and demonstrate various subtleties, which have not been discussed in the previous literature—though it does not change qualitatively our earlier conclusions. We further extend our conclusions to LSS (i.e. and ) and show that leptogenesis is not natural there either. Another result of the present paper is that even turning on both still requires very small couplings to achieve a successful leptogenesis (see also for example refs. [12, 13]). Our conclusion is that in this scenario leptogenesis scores a “possible”.

Finally, table 1 includes the Hybrid seesaw first presented in ref. [5]. This was designed to overcome simultaneously the two limitations of the ISS: unnaturally small term and difficulty in leptogenesis. The essential idea of hybrid seesaw is to introduce, on top of the ISS module, namely a Dirac pair of fermions with TeV and unsuppressed coupling to leptons , a high scale type I seesaw module, namely heavy ( TeV) Majorana singlets , see figure 1. The theory has no bare -term, but the two modules suitably mix via a IR-scale mass term arising from a scalar vacuum expectation value. In this manner, integrating out the heavy Majorana singlet generates an effective TeV for 444The basic idea of this model is along the lines of ref. [14], but those authors considered TeV instead. For this reason Leptogenesis is not as successful as in our picture, and is not naturally small according to our criteria.

(3)

Taking now TeV we can explain why , and therefore the smallness of neutrino masses.

Figure 1: Schematic representation of physics (seesaw and genesis) of the hybrid model.

The structure of hybrid seesaw, and in particular the characteristic mixing between the low and high scales modules, arises elegantly from warped extra-dimensions (dual to composite Higgs models) [15] as shown by some of us previously [16]. In this sense, the hybrid seesaw could be taken as a “toy” version of the warped/composite one. Alternatively, the peculiar coupling structure in figure 1 can be enforced in weakly-coupled 4D models via a gauge symmetry as we will see in appendix E. Because in the 5D completions are both related to the same fundamental scale (the TeV), that arises dynamically, and simultaneously can be effectively reduced dynamically compared to the Planck scale [16], then it is clear that neutrino masses can be fully natural in the hybrid picture once UV-completed (i.e., strictly speaking, going beyond the hybrid model on which we will focus here). This explains the score “possible” in the appropriate entry in the table (i.e., why it’s not quite an actual “yes”).

How about leptogenesis in the hybrid seesaw model? We have just seen that neutrino masses are suppressed by , suggesting that this is a scenario with small -breaking (class (III) above). However, this is not the complete story. As we will see in detail below (see also [5]), the high scale module violates maximally a global carried by . Because number violation is large at scales , leptogenesis can naturally proceed through the decay of to (analogously to type I seesaw in class (I)), followed by the asymmetry in being transferred to the SM leptons. Hence, the hybrid model also turn out to score a “yes” in natural leptogenesis.

In particular, in ref. [5] we emphasized that high scale leptogenesis with anarchic couplings can be realized for GeV. In this paper, we will study this scenario in more detail and also explore the lower scale GeV where leptogenesis can be realized albeit with hierarchical Yukawa couplings (among different generations of ). Such a relaxation of the lower bound on the heavy singlet mass, compared to the ordinary type I seesaw, might be especially relevant for resolving the SUSY gravitino problem. Overall, due to the hybrid structure, the allowed mass window gets enlarged compared to the usual case, GeV.

Regarding possible experimental signals in the hybrid seesaw, besides signals associated with TeV scale fermions as in the conventional ISS (as mentioned above), the model generally predicts new TeV scale scalars potentially within the reach of present and future colliders as we have shown in ref. [5]. Certain realizations, like the gauge model presented in appendix E, also contain light states that may contribute to and might thus be probed by CMB-Stage-IV [17]. Hence, we put a “yes” in the experimental signals. Remarkably, this model has the ability to realize the most attractive features of the high and low scale modules simultaneously.


Our paper is organized as follows. We begin in the next section with an overview of scenario with small lepton-number breaking (), i.e. the ISS and LSS models, and discuss the constraint from the non-observation of . A thorough analysis of leptogenesis in these models is given in section 3 (see also appendices A and D). Section 4 outlines our hybrid seesaw solution of the problems of the original ISS model. Explicit UV completions of the scenario are presented in appendices E (gauge model) and F (warped/composite model). This is followed by a detailed discussion of leptogenesis in the hybrid model. In section 5 (and appendices B and C) we will provide a systematic derivation of the necessary analytic formalism, which we believe clarifies many of underlying physics. This formalism is then used in section 6 to identify what parameter choices give the right baryon asymmetry, including some interesting benchmark points. We finally conclude in section 7, providing some directions for future work.

2 Scenarios with small breaking

We begin with a review of what we will refer to as small breaking models, that according to our earlier definition have [see eq. (1)]. These are characterized by an effective theory with exotic particles not far from the TeV scale and unsuppressed couplings to the SM, say of order . This guarantees that these scenarios have testable consequences at colliders. Because all new degrees of freedom are heavy, the SM neutrinos are Majorana particles. To ensure that small neutrino masses are generated, these scenarios must possess an approximate lepton number broken by a small dimensionless parameter. The most minimal incarnations of this scenario has been called inverse seesaw and linear seesaw. We will focus on these mostly for simplicity sake.

Let us add to the SM two Weyl fermions and , singlet under the SM, carrying lepton number , respectively. In principle we can combine the pair of Weyl fermions into a Dirac fermion with (or ) playing the role of the left (right) chiralities, but we will not do it here for later convenience. The only invariant couplings, besides the kinetic terms, are:

(4)

with the SM lepton and Higgs doublets, respectively. Gauge contractions are understood, and the flavor indices for (possibly carried by as well) are not displayed here for brevity. We will include the flavor indices in later parts whenever they are relevant. We will take as a reference value. Possible higher dimensional operators (denoted by in eq. (4)) are assumed to be negligible because they are suppressed by a large scale . We will assume is of the order of the Planck scale for definiteness.

In the theory eq. (4) the active neutrinos remain exactly massless. In order to obtain a realistic theory with tiny neutrino masses without adding additional light degrees of freedom, we introduce small sources of breaking. At the renormalizable level there exist only three -breaking couplings:555One may also add . However, after a field redefinition one realizes this is equivalent to a correction to the couplings we show.

(5)

The assumption that the breaking terms are small reads , . The terms correspond to small Majorana masses for the fields . Conventionally, the ISS model is defined by while the LSS model by . Generally, in both of these models, is taken to be zero as well.

The new couplings appearing in eq. (5) can all be assigned a spurionic lepton number, namely and . Because the accidental charges of are the same, in generic UV completions the new couplings in eq. (5) may in fact arise from a unique fundamental coupling with . In that case, a natural consequence of naive dimensional analysis is that, at the order of magnitude level,

(6)

Of course it is possible to build a UV dynamics in such a way that this relation is violated. Yet, the scaling in eq. (6) is what one expects to emerge from truly generic UV theories. More generally, setting one of the couplings in eq. (5) to zero is not always a radiatively stable assumption. For example, inspecting 1-loop diagrams we find that starting with a non-vanishing one generates (from log-divergent piece)

(7)

On the other hand, no renormalization effects are induced by because these correspond to a soft-breaking of . That is, and only self-renormalize and do not radiatively generate other terms.

Majorana masses for the active neutrinos, that have , must be linear in the couplings of eq. (5) to leading order in the small -breaking. This can be readily verified by integrating out at tree-level to obtain, in the leading approximation:

(8)

where GeV and

(9)

Note that does not enter because its charge forces it to appear in front of as complex conjugate, which is not possible at tree-level. With the relation eq. (6) the two contributions in eq. (8) are naturally of the same order. The parameter introduced in eq. (1) may now be identified as

(10)

Lacking a UV description of breaking, it is fair to say that the smallness of is merely an assumption in our effective field theory eqs. (4) and (5). While this model does not truly explain the size of the SM neutrino masses, it provides an interesting laboratory to investigate the phenomenology of scenarios with small breaking. A distinctive feature of these models is the presence of signatures in colliders (see for example [10] and references therein). For TeV and sizable it is in fact possible to produce the pseudo-Dirac fermions at the LHC via mixing with SM neutrinos and observe its subsequent resonant decay. Unfortunately, we will not be able to measure the tiny couplings and hence unambiguously connect the exotic particles to a mechanism for neutrino mass generation. The reason is that in the typical benchmark models from eq. (8) one derives from eq. (9) that , that is certainly out of reach of current and future colliders.

Besides direct production of , there can be indirect signatures in rare processes, like and the electron EDM. At leading order in , the branching ratio can be written as [18]

(11)

where is the fine structure constant, GeV the SM Higgs VEV, and we have neglected corrections of order . The current experimental bound is [19]. For anarchic couplings and masses this translates into . However, the bound can be significantly relaxed by using flavor symmetries. One very efficient way to achieve this is to assume that the Lagrangian eq. (4) has a global symmetry under which the three generations of transform [20].666One may use gauge symmetries to enforce this possibility. This assumption forces , as well as the SM lepton Yukawa coupling, to be diagonal in flavor space and therefore to vanish. The symmetry is then weakly broken by the -violating couplings in eq. (5) to ensure large mixing angles in the PMNS matrix. As a result, we find a huge suppression with respect to the result eq. (11), i.e. . Similarly, one can verify that all CP-odd phases can be removed from , and the first new physics contribution to the EDMs is suppressed by at least . We thus see that the non-observation of rare processes does not represent a robust constraint on this scenario. The most model-independent constraints on come from ElectroWeak (EW) precision tests and are of order TeV (see for example [10] and references therein).

3 Leptogenesis with small breaking

In this section we present analytic estimations of the baryon asymmetry from thermal leptogenesis in TeV scale models with small breaking. We show the results for two specific models: the inverse seesaw and linear seesaw models, as well as combinations of the two. Our qualitative conclusions are however more general and may extend to a broader class of models with small lepton number violation. In section 3.1 we determine the size of the CP parameter, the washout factor and the final baryon asymmetry. Our results will demonstrate that TeV scale models with anarchic (i.e., roughly of same order but not degenerate) couplings and mass parameters tend to predict too small baryon asymmetry. An intuitive interpretation of the parametric dependence of these results is shown in section 3.2 based on the (generalized) Nanopoulos-Weinberg theorem. Finally, in section 3.3, we identify a few possibilities that can give rise to successful (sub-)TeV scale leptogenesis with small lepton-number breaking. Conclusions similar to ours are obtained in the numerical analysis of ref. [13].

3.1 Leptogenesis in TeV scale inverse and linear seesaw

In this section, instead of studying thermal leptogenesis in the most general model [eq. (4) and eq. (5)], we illustrate the main results in two limiting cases, namely the ISS and LSS models. The Lagrangians we consider are

(12)
(13)

where denotes SM lepton flavor index and are the generation indices for . Without loss of generality, we work in the basis where is diagonal and real. For both models, we demand the singlet neutrinos come in two generations (), which is the minimum number of generations required to achieve the realistic neutrino mass matrix. Qualitative results in such two-generation model will not differ much from three-generation one. In the rest of this section, we demand that and define777Since we mostly assume couplings and masses are anarchic in this section, we will simply use variables without generation or flavor indices to show the parametric dependence.

(14)

These are the natural choice of parameters for both seesaw models to obtain the SM neutrino masses and testable collider signals. The smallness of neutrino mass is controlled by the smallness of or [see eq. (9)].

To be concrete here we will present the case of the ISS model. Similar conclusions can be drawn for the LSS model, as we emphasize at the end of section 3.1 and a more quantitative analysis is shown in the appendix A. Starting from eq. (12), we can write

(15)

where we define as the diagonal (off-diagonal) parts of matrix. In general, the matrix is complex. However, since we assume all the phases of each element are order one, and yet we will be doing order of magnitude parametric estimation, including those will make at most changes, but will not modify the parametrics of our estimations. For the sake of simplicity, then we simply treat all elements as real numbers. Assuming , we can diagonalize the mass matrix to first order in and . Defining four Majorana states () with real masses we have

(16)

To first order in and , their masses and couplings are given as (ref. [21])

(17)

where

(18)

From eq. (3.1), we see that and form pseudo-Dirac pairs with small Majorana mass splitting. The mass splitting between a pseudo-Dirac pair is only controlled by diagonal while both and modify the Yukawa couplings. Taking the limit , one can easily find that , and , , as expected for pure Dirac states.

3.1.1 CP asymmetry

Now we are ready to calculate the CP asymmetry from the decay of each Majorana state . After summing over SM lepton flavor , we get:888Assuming anarchy of Yukawa couplings , the lepton asymmetry produced will be distributed among all the lepton flavors in roughly equal proportion. For simplicity, we ignore the small differences in the various flavor asymmetries and sum over . When couplings are hierarchical flavor effects [22] could play a more relevant role, and we will briefly mention about it in section 3.3.

(19)

where comprises a contribution from vertex corrections [23]

(20)

as well as a self energy correction to the decay [11]

(21)

Here is the decay width of .

Let’s take a close look at and in eq. (19):

(22)

Given that the pseudo-Dirac pairs are almost degenerate in mass, the number density of two states are approximately the same. As a result (see appendix D), it is appropriate to consider and as the effective CP asymmetry for each generation. Due to the pseudo-Dirac nature, one finds that

(23)

This means that when we consider the sum of and , parts involving and in will cancel against the corresponding parts with and in to first order. Also, if we consider the generic parameter region of the ISS, i.e.,

(24)

and no hierarchies in mass or couplings among singlet generations and SM flavors, we would get and

(25)

Therefore, the terms involving and dominate in , giving [see eq. (3.1.1)]

(26)

where we have dropped the family indices for and to show only the parametric dependence. Similarly, can be obtained by changing index and in eq. (26), resulting in the same parametric dependence.

For completeness, we also show the parametric dependence of :

(27)

We only use instead of individual or in our study of leptogenesis. However, they are relevant for the argument in appendix D.

If we assume and enforce eV via eq. (9), eq. (26) becomes (see also ref. [11])

(28)

As we will see shortly [eq. (29)], should be to generate the observed baryon asymmetry via leptogenesis and eq. (28) falls short by three orders of magnitude. From eq. (28), it seems that one can obtain a larger value by reducing Yukawa couplings . However, this approach will not allow us to obtain sufficient baryon asymmetry once we, as required, include the washout effects. We will discuss this in the following section.

3.1.2 Washout and baryon asymmetry

The final baryon asymmetry through leptogenesis from decays of can be parametrized as follows

(29)

where is the number density of baryons (anti-baryons) and is the total entropy density of the thermal bath. The pre-factor comes from relativistic number density of normalized to the entropy density . The efficiency factor is always less than unity and parametrizes the effect of washout processes. It is obtained by solving the Boltzmann equations. The efficiency of leptogenesis can be parametrized by the so-called washout factor [6]

(30)

where is the Hubble rate with being the thermal bath (photon) temperature, the number of relativistic degrees of freedom and GeV the Planck mass. In the ISS scenario, due to the approximate lepton number conservation, the washout from inverse decay is actually controlled by [24]999The appearance of may be understood as follows. In the limit , since lepton number is preserved, no process can washout (or produce) the asymmetry. Therefore, the effective washout factor must vanish as . Another (more technical) way to see this is to recall that the washout from the inverse decay can be obtained by the on-shell part of scattering. Due to the near degeneracy, this scattering gets contribution from both s-channel and and importantly, most of their contributions cancel. The surviving piece comes from interference of the two and is proportional to .

(31)

where with or . Also, we dropped generation index for simplicity of notation and we will do so below when there is no chance of confusion. Consistently, this quantity vanishes in the lepton number conserving limit. Notice that we can express eq. (26) as

(32)

where we have taken .

If a few, the washout from inverse decay () is efficient (strong washout regime) and (see appendix B.2.2). In this regime, substituting eqs. (32) and (31) into eq. (29), the baryon asymmetry is estimated to be

(33)

where we have taken . This analytic estimation was first obtained in our earlier paper [5]. Clearly, a TeV scale will result in a too small asymmetry compared to the observed value  [25]. Remarkably, in the strong washout regime, the final baryon asymmetry () for the ISS model with anarchic couplings and masses reduces to the simple formula [eq. (33)] which does not depend on and .

To complete our discussion, we also need to consider the weak washout regime, where . ISS model has a peculiar feature that the production of singlets is controlled by [eq. (30)], whereas the washout is controlled by [eq. (31)]. Assuming no initial abundance of , there are two cases in weak washout region and the corresponding efficiency factors are

(34)

as derived in appendix B.2.1. We emphasize that such parametric dependence of is qualitatively different from that of usual type I seesaw (i.e., ). To the best of our knowledge, this analytic result, especially which of , should appear in , has not been discussed in the literature. If we, on the other hand, assume has been kept in thermal equilibrium with SM particles by interactions other than those due to Yukawa coupling ,101010For instance, if is charged under new gauge symmetries (e.g. ), they can acquire an initial thermal abundance. the efficiency factor is of the order

(35)

Putting everything together, in the weak washout regime, we have

(36)

We see that in all cases the final baryon asymmetry in the weak washout regime is smaller compared to that of strong washout in eq. (33). Therefore, the TeV scale ISS model with anarchic mass and coupling cannot provide successful leptogenesis.

An analogous calculation for the LSS model is shown in appendix A and the parametric dependences of the final baryon asymmetry of the two seesaw models are in fact the same, as we summarize in table 2. Therefore, we conclude that TeV scale ISS and LSS model with anarchic parameters and sizable cannot give rise to successful leptogenesis.

Model CP asymmetry () Efficiency () Baryon asymmetry ()
Inverse seesaw
Linear seesaw
Table 2: Summary of the parametric dependence of CP asymmetry, washout, baryon asymmetry in inverse [see eq. (12)] and linear seesaw [see eq. (13)]. The parameters are defined in eq. (14), wheres below eq. (31).

3.2 Nanopoulos-Weinberg theorem

As discussed in the previous section, the CP asymmetries in TeV scale ISS and LSS models are small because they are respectively 111111One might wonder why the lepton-number violation is captured by in the case of the neutrino mass, and by in the case of CP-violation (and leptogenesis). This may follow from the fact that while the generation of is off-shell phenomenon (i.e. simply integrate out ’s), that of CP-violation and related asymmetry generation occurs near on-shell. Especially, when the genesis goes through the resonance-enhancement, on top of parametric lepton-number violation , it acquires extra kinematic (resonance-)enhancement , yielding the associated net breaking parameter . and , where are the tiny parameters characterizing the small lepton number violation. We will argue in this section that this feature can indeed be anticipated due to Nanopoulos-Weinberg (NW) theorem (ref. [26]) and similar conclusions can be drawn in some variations of ISS or LSS models, or combination of both.

The NW theorem states that, in the CP-violating decay process, if the particle can decay only through baryon (lepton) number violating parameters (e.g. Type-I), a nonzero CP asymmetry can be generated starting at third order in baryon (lepton) number violating parameters. In addition, the generalized version of the NW theorem (ref. [27]) says that, if the decaying particle, on the other hand, can decay through both baryon (lepton) number violating and conserving couplings, the CP asymmetry may be generated at second order in baryon (lepton) number violating parameters.

Now we apply both theorems to check our results for the ISS and LSS models. The CP asymmetry in decay width is given in eq. (26) for ISS and eq. (106) for LSS:

(37)

where we sum over almost degenerate states and all final states .

For the ISS, if we assign the lepton number charges , the Yukawa coupling is lepton number conserving and is the only lepton number violating parameter. Then can decay also via number-conserving interactions and, following the extended version of the NW theorem, the CP asymmetry should be . The CP asymmetry in eq. (37) indeed contains , hence .

Similarly for the LSS, we can always assign lepton number such that only one of or violates lepton number. Since can decay through either or , it always follows the extended NW theorem. Therefore, we expect the CP asymmetry is proportional to two powers of and two powers of , which matches the result in eq. (37).

In general, NW theorem forces the CP asymmetry from singlets decay to be or , which is suppressed in models with small lepton number breaking. Adding further lepton number conserving decay channels or new generations of leptons would not alter this result.

3.3 Possible variations to achieve successful leptogenesis

Our discussion so far assumed anarchic couplings and masses and considered either a small or a small , separately. In this subsection we relax these assumptions with the aim of looking for models with small violation that can result in larger final asymmetry compared to eq. (33).

3.3.1 Inverse seesaw with degeneracy among different generations

We first consider the possibility that the singlet masses are quasi-degenerate among different generations:

(38)

with so that our previous formulae in section 3.1 still apply. Although quasi-degeneracy in mass within a pseudo-Dirac pair is naturally obtained due to approximate lepton number, to realize quasi-degeneracy in mass among singlets of different generations in a natural way, an approximate family symmetry is necessary as was done, for example, in the resonant leptogenesis scenario [28]. In scenarios with minimal flavor violation, even if is set to zero at the tree level, generally Yukawa couplings might break the family symmetry, generating at loop level of the size

(39)

In this case, the which parametrically is given by (see eq. (18))

(40)

can be enhanced. Substituting eq. (40) into eq. (26), one has

(41)

When two generations are nearly degenerate, thus, the CP asymmetry is enhanced compared to eq. (26) by a factor of . The washouts are nevertheless unchanged.121212The contribution to has two pieces in the non-degenerate case: . The second term is suppressed compared to the first one and thus we only keep the first term in the previous estimation. In the case we discussed here, where there is degeneracy among different generations, the first term is still unchanged. This is because the first term is controlled by the mass splitting within each generation, which will not be modified by the degeneracy among different generations. The second term, however, is enhanced by : . Now these two terms are comparable due to the assumption in eq. (39) and the parametric dependence of remain the same as in eq. (31). So the final result scales as

(42)

The right size of may be obtained by choosing the right size for . However, we are not completely free to choose its value here. In particular, our analysis is done under the assumption that 131313Implicitly, we also assumed to get a concrete expression. However, a straightforward check can confirm that while CP and washout factor will change (basically replacing with ), the final asymmetry will be the same as the one we show above. and (technical) naturalness indicates that . Combining these two with the constraint from the neutrino mass, i.e. , gives rise to an upper bound on the enhancement factor