The Type-I Seesaw family
Salvador Centelles Chuliá Antonio Herrero-Brocal Avelino Vicente
















The Type-I Seesaw family
 




Salvador Centelles Chuliá^a, Antonio Herrero-Brocal^b, Avelino Vicente^b,c



 ^(a) 

 
 ^(b) 

 
 ^(c) 



mailto:chulia@mpi-hd.mpg.dechulia@mpi-hd.mpg.de,
mailto:antonio.herrero@ific.uv.esantonio.herrero@ific.uv.es,
mailto:avelino.vicente@ific.uv.esavelino.vicente@ific.uv.es





We provide a comprehensive analysis of the Type-I Seesaw family of neutrino mass models, including the conventional type-I seesaw and its low-scale variants, namely the linear and inverse seesaws. We establish that all these models essentially correspond to a particular form of the type-I seesaw in the context of explicit lepton number violation. We then focus into the more interesting scenario of spontaneous lepton number violation, systematically categorizing all inequivalent minimal models. Furthermore, we identify and flesh out specific models that feature a rich majoron phenomenology and discuss some scenarios which, despite having heavy mediators and being invisible in processes such as μ→ e γ, predict sizable rates for decays including the majoron in the final state.





§ INTRODUCTION



The discovery of neutrino oscillations <cit.> highlights the need for Beyond Standard Model (BSM) physics. While neutrinos are massless within the Standard Model (SM) framework, their observed masses not only point towards a novel mechanism beyond the traditional Higgs mechanism but also require a robust explanation for their relatively small scale compared to the electroweak scale. Among various models proposed in the literature, the type-I seesaw mechanism <cit.> stands out for its simplicity and elegance. In this framework, the smallness of neutrino mass is inversely related to a new, higher mass scale M, represented by the Majorana mass of newly introduced neutral fermions.

While elegant from a theoretical standpoint, the type-I seesaw model inherently implies that any phenomenological effects are suppressed by the small seesaw expansion parameter ε^2 = 𝒪(Λ_EW^2 / M^2 ) = 𝒪(m_ν / M ) ≪ 1, where Λ_EW∼ 100 GeV is the electroweak scale. However, this constraint is relaxed in genuine low-scale variants of the model, such as the linear <cit.> and inverse <cit.> seesaws. In these models,

phenomenological effects are not neutrino-mass suppressed, potentially leading to discernible traces in charged Lepton Flavor Violation (cLFV) <cit.>, through direct production of mediators at colliders <cit.>, via non-standard neutrino propagation effects <cit.> or other low-energy probes <cit.>. For a short review on low-scale neutrino mass models see for example <cit.>.

The existence of Majorana masses for neutrinos inherently implies the violation of lepton number symmetry. This violation can occur either explicitly or spontaneously. When the symmetry is global, the latter leads to the presence of a Nambu-Goldstone boson, known as the majoron (J) <cit.>. In this work we analyze the Type-I Seesaw family [From now on, we use capital letters for the family and lowercase for the specific model.], composed by the standard type-I seesaw model and its many low-scale variants. We classify the members of the family and explore novel neutrino mass models that feature rich phenomenology. Cosmological imprints of the majoron such as Δ N_eff <cit.> could provide a complementary approach to the low-scale seesaw signatures discussed above. Here, we will focus on models which feature interactions between the majoron and charged leptons as their main phenomenological signature, opening new avenues for detecting and studying the effects of lepton number violation. 

The paper is organized as follows. We start with a pedagogical introduction of the Type-I seesaw family and the explicit lepton number breaking scenario in Secs. <ref> and <ref>. In Sec. <ref> we focus on the more interesting case of spontaneous symmetry breaking (SSB), where we aim at classifying and analyzing all inequivalent minimal models of the Type-I seesaw family. We also point out those models in which the majoron phenomenology is not neutrino-mass suppressed and flesh them out in Sec. <ref>. Finally, we conclude with a summary in Sec. <ref>. Additional technical details are given in two Appendices.





§ THE TYPE-I SEESAW FAMILY



Let us start by defining the Type-I Seesaw family. A model belongs to the Type-I Seesaw family if its neutral fermion mass matrix can be written as


    ℳ =[     0   M_D; M_D^T   M_F ]  ,


in the basis (ν_i, F_j), where ν_i (i=1,2,3) are the usual 3 SM neutrinos while F_j (j=1,…,n_F) are n_F heavy BSM neutral fermions. M_D is a general 3 × n_F matrix and M_F is an n_F × n_F symmetric matrix. Furthermore, we demand the following condition in order to consider the model as part of the Type-I Seesaw family: 

0.8
The hierarchy (M_D   M_F^-1)_ij≪ 1   ∀  i, j is satisfied. This allows one to expand the relevant physical quantities in powers of ε = Ø (M_D   M_F^-1). These are nothing but the seesaw limit and expansion, respectively.


Which implies

0.8
A light Majorana mass term for the SM neutrinos, M_ν≪Λ_EW, is generated at tree level.


Under this assumption we can compute a general formula for M_ν. By rotating the fields into the mass eigenstates, the symmetric matrix ℳ is brought into diagonal form by means of a Takagi decomposition as


    U^T ℳ  U = ℳ  ,


where U is a unitary matrix and ℳ = diag(m_1,m_2,…, m_3+n_F) is the neutral fermion mass matrix in the mass basis. The matrix U can be expressed as


    U=[ U_l   0;   0 U_h ] [   √(𝕀_3 - P P^†)                P;             -P^† √(𝕀_n_F - P^† P) ]≡ U_2   U_1   .


Here, U_l, U_h and U_1,2 are 3× 3, n_F× n_F and (3+n_F) × (3+n_F) unitary matrices, respectively, P is a 3 × n_F matrix and we denote a general n × n identity matrix as 𝕀_n. This factorization of the unitary matrix U allows one to easily identify the role played by each factor. U_1 brings the neutral fermion mass matrix into a block-diagonal form, while U_2 finally diagonalizes, independently, the light and heavy sectors of the matrix. For instance, the light sector is diagonalized as 


    U_l^T M_ν  U_l = diag(m_1,m_2, m_3)   ,


with m_1,m_2, m_3 the active neutrino masses. Let us now focus on the block-diagonalization of the mass matrix. By expanding P in powers of ε,


    P=∑_i=1^∞ P_i   ,


with P_i ∼ε^i, at leading order in ε one finds


    √(𝕀_3 - P P^†)   = 𝕀_3 + Ø(ε^2)   , 
    √(𝕀_n_F - P^† P)   = 𝕀_n_F + Ø(ε^2)   ,


and


    P = P_1 + Ø(ε^2) = M_D^*  (M_F^-1)^† + Ø(ε^2)     .


Using these results, we find at leading order in ε


    U_1^T ℳ  U_1 ≈[ - M_D   M_F^-1  M_D^T                     0;                     0                   M_F ]  ,


and, therefore, 


    M_ν = - M_D   M_F^-1  M_D^T + Ø(ε^2)   ,


the well-known seesaw formula <cit.>. We can now identify some known models belonging to this family based on the structure of the M_F and M_D matrices, namely, the type-I seesaw, the inverse seesaw and the linear seesaw. We will focus on the minimal realizations of the inverse and linear cases, which require two different neutral fermions, N and S. For simplicity, we will consider the same number of generations for both of them, n_N = n_S = n_F / 2. In this case, the most general matrices are given by


    M_D = [ m_D m_L ]  ,         M_F= [   μ_N   m_R; m_R^T   μ_S ]  ,


where m_D, m_L and m_R are two 3 × n_N and one n_N × n_N general matrices, respectively, and μ_S and μ_N two n_N × n_N symmetric matrices. In order to distinguish between models we must express M_ν in Eq. (<ref>) in terms of the blocks of M_F and M_D. In order to compute M_F^-1, we will consider separately the cases m_R ≠ 0 and m_R = 0 and assume that m_R is invertible in the former case. [This is usually assumed, for instance, in the inverse and linear seesaws. Similar results can be obtained by assuming μ_N or μ_S to be invertible.] One obtains 


    M_F^-1 = {[ [ - ( m_R^T )^-1 μ_S              𝕀_n_S;              𝕀_n_N        -m_R^-1 μ_N ] [     m^-1        0;        0 (m^T)^-1 ]                                                                                                           , if m_R ≠ 0  ,;                                                                                                                                                                                                                                                    ;                                                                                          [ μ_N^-1      0;      0 μ_S^-1 ]                                                                                                           , if m_R = 0  . ].


where we have defined m = m_R - μ_N (m_R^T )^-1μ_S. Thus, the light neutrino mass matrix is given by












    M_ν = {[ ( m_D (m_R^T)^-1μ_S - m_L  )   m^-1  m_D^T + (m_L m_R^-1μ_N - m_D  )   (m^T)^-1  m_L^T + Ø(ε^2),;                                                                                    if m_R ≠ 0  ,;                                                                                                 ;                                                 -m_D  μ_N^-1 m_D^T - m_L  μ_S^-1 m_L^T + Ø(ε^2),;                                                                                    if m_R = 0  . ].


Eq. (<ref>) constitutes a general result, valid for any model of the Type-I Seesaw family. Note that there is a continuous equivalence between both cases in this equation, i.e. taking the limit m_R → 0 in the case m_R ≠ 0 yields the same result as the exact case m_R=0, as expected.













The different models in the Type-I Seesaw family correspond to different hierarchies among the blocks in the M_D and M_F matrices. In principle, each block is independent, resulting in a unique hierarchy for every pair of blocks. This gives rise to many hierarchies and models, some of which are very popular while others are less well-known. For instance, one finds the usual type-I seesaw whenever there is no hierarchy among the blocks in M_F. Alternatively, if μ_N,μ_S ≪ m_R, Eq. (<ref>) simplifies to


    M_ν =       m_D  ( m_R^T )^-1μ_S m_R^-1 m_D^T + m_L   m_R^-1μ_N ( m_R^T)^-1  m_L^T 
    
        -    m_D ( m_R^T )^-1  m_L^T    - m_L   m_R^-1 m_D^T + Ø(ε^2)   ,


which could either lead to an inverse (if m_L ≪ m_D and μ_S/m_R≫m_L/m_D) or to a linear seesaw (if m_L ≪ m_D and μ_S/m_R≪m_L/m_D). In general, many possibilities exist. In the usual case of one high-energy scale in M_F and one or two low-energy scales (one in M_F and, possibly, one in M_D), all scenarios are summarized in Table <ref>. We refer to Appendix <ref> for a comprehensive discussion considering all possible hierarchies.






We have just seen that different internal hierarchies lead to different mass generation mechanisms in the context of the Type-I Seesaw family. However, one might wonder if these distinct mechanisms represent genuinely different models or if there is some kind of underlying model below them; i.e. if we can find one Lagrangian describing all these models. We will demonstrate that the latter is true in the case of the explicit breaking of lepton number symmetry, U(1)_L, but not when the breaking is spontaneous.




§ EXPLICIT LEPTON NUMBER VIOLATION



We start with a pedagogical Section showing that all models with
explicit lepton number violation can be seen as equivalent to a type-I
seesaw with specific numbers of fermion singlet generations and matrix
textures. Let us again consider n_N generations of the N and S fermion singlets, with the Lagrangian


    -ℒ = y_N  L̅H̃ N + y_S  L̅H̃ S + m_R  N̅^c S + μ_N/2 N̅^c N + μ_S/2 S̅^c S +   .


Here H̃ = i σ_2 H^∗, y_N and y_S are two 3 × n_N Yukawa matrices, m_R is an n_N × n_N matrix with dimensions of mass and μ_N and μ_S are two n_N × n_N symmetric matrices, both with dimensions of mass. The Lagrangian in Eq. (<ref>) violates lepton number explicitly. It can be expanded as


    -ℒ =     y_N  L̅H̃ N + y_N^† N̅H̃^† L + y_S  L̅H̃ S + y_S^† S̅H̃^† L + m_R  N̅^c S + m_R^† S̅ N^c 
       + μ_N/2 N̅^c N + μ_N^*/2 N̅ N^c + μ_S/2 S̅^c S + μ_S^*/2 S̅ S^c   ,


Let us now combine the N and S singlet fermions into the multiplet
F, defined by


    F = ( [ N S ])   .















Now we can make use of the identity ψ̅^c χ^c = χ̅ψ, valid for any two fermions ψ and χ, to find
S̅^c N = N̅^c S. This allows us to write Eq. (<ref>) as


    -ℒ =   Y  L̅H̃ F + M_F/2 F̅^c F +   ,


with the dictionary


    Y    = ( [ y_N y_S ])   , 
    
      M_F    = ( [   μ_N   m_R; m_R^T   μ_S ])   .


The Lagrangian of Eq. (<ref>) is that of a type-I seesaw
with n_F = 2   n_N generations of F singlets and the specific matrix textures given by Eqs. (<ref>) and (<ref>). This proves the equivalence of all models in the type-I family to a specific texture of the standard type-I seesaw. This is not surprising: once we allow for a source of explicit breaking of lepton number in the Lagrangian, the singlets N and S cannot be distinguished and the model becomes a type-I seesaw. As we will show in the next Section, this is no longer true if the breaking of lepton number is spontaneous.





§ SPONTANEOUS LEPTON NUMBER VIOLATION




We now turn towards the more interesting case of spontaneous lepton number violation. Our goal is to analyze the minimal realizations of the Type-I Seesaw family with spontaneously broken global U(1)_L. In defining minimal, we consider models featuring the fields listed in Tab. <ref>. In addition to the SM doublets H and L, with lepton numbers q_H=0 and q_L=1, respectively, we will allow for the presence of the new fermions N and S, as well as a second scalar doublet χ and the scalar singlet σ. The number of generations of N and S will again be denoted by n_N = n_S and left unspecified. The lepton number of N will be fixed to q_N=1, so that the Yukawa term L̅H̃ N is allowed and N can be identified with the usual right-handed neutrino. We also impose that χ couples to S through the Yukawa term L̅χ̃ S, which fixes its lepton number to q_χ = q_S-1, where q_S is the lepton number of S. Finally, the lepton number of σ will be left as a free charge, denoted by q_σ. All neutral scalars will be assumed to get non-zero vacuum expectation values (VEVs),


    ⟨ H^0 ⟩ = v_H/√(2)  ,   ⟨χ⟩ = v_χ/√(2)  ,   ⟨σ⟩ = v_σ/√(2)  .


Therefore, H and χ will be responsible for the breaking of the electroweak symmetry, whereas χ and σ will break U(1)_L if they have non-vanishing lepton numbers. The breaking of lepton number implies the existence of a physical Goldstone boson, the majoron, J <cit.>. Its presence has important consequences for the different energy scales in our setup. Typically, the hierarchy v_χ≪ v_σ is phenomenologically required, as otherwise the majoron would have a sizeable doublet component, allowing the decay Z → J J, which is strongly constrained by LEP <cit.>. Moreover, the additional hierarchy v_χ≪ v_H ensures that the real component of H^0 will be SM-like, as demanded by current LHC data. We note that this is a natural hierarchy, since m_ν∝ v_χ in many models. Finally, we must impose M_D ≪ M_F to guarantee the validity of the seesaw approximation, as required for the model to belong to the Type-I Seesaw family. While we remain agnostic about the hierarchies among Yukawa couplings, we assume that these will not be stark enough to overcome the VEV hierarchies, i.e. the hierarchies in the mass matrix are driven by the VEVs.



Some additional comments are in order:



  * Models that do not include all the fields in Tab. <ref>, but only a subset of them, will also be considered in our analysis. 


  * Models with q_S=1 allow for a significant simplification. First of all, q_H=q_χ=0 and the second scalar doublet χ is actually redundant, since its role can be played by the usual Higgs doublet H. Moreover, q_N=q_S, which implies that the multiplet F = ( [ N S ]) transforms consistently under U(1)_L.


  * We will limit our analysis to models with a single majoron.


It proves convenient to classify the models in the Type-I Seesaw family according to the texture of their neutral fermion mass matrix ℳ. The different possibilities are given by:

[colback=lightblue, title=Class 1]

    [     0   m_D   m_L; m_D^T   μ_N   m_R; m_L^T m_R^T   μ_S ]       [     0   m_D   m_L; m_D^T     0   m_R; m_L^T m_R^T     0 ]       [     0   m_D     0; m_D^T   μ_N   m_R;     0 m_R^T   μ_S ]
 


[colback=lightblue, title=Class 2]

    [     0   m_D   m_L; m_D^T     0   m_R; m_L^T m_R^T   μ_S ]       [     0   m_D     0; m_D^T     0   m_R;     0 m_R^T   μ_S;       ]



[colback=lightblue, title=Class 3]

    [     0   m_D   m_L; m_D^T   μ_N   m_R; m_L^T m_R^T     0 ]





This classification into different classes of matrices is not arbitrary. As will be shown below, the conditions allowing for the more complex matrices (left) can be relaxed to give the simpler models (right). The hierarchies among the mass terms will determine the type of model (type-I, inverse, linear or hybrid). Tab. <ref> shows the covariance of the mass terms in the gauge singlet sector. These results are useful to extract the possible lepton number charges of S and σ that lead to each mass matrix. 

Let us now present all the different minimal realizations of the Type-I Family with SSB. For each realization, we will show the Lagrangian and the mass mechanism. We will also discuss the majoron resulting in each realization and comment on its 1-loop coupling to charged leptons. This coupling is a crucial feature to distinguish among models and was computed analytically in a generic scenario in <cit.>. For the sake of completeness, we summarize the main results of this reference in Appendix <ref>. To the best of our knowledge, among the models in the Type-I Seesaw family, this coupling is only known for some realizations of the conventional type-I seesaw (model C1a) <cit.> and the inverse seesaw (model C1bχ) <cit.>, while other models are studied here for the first time. [Although not exactly the same model as C3, this coupling was obtained for a related version with explicit lepton number breaking in <cit.>.] In particular, we will comment on the neutrino mass suppression (or lack thereof) of the majoron coupling to charged leptons. We will regard the majoron couplings to charged leptons as neutrino mass suppressed when they are proportional to the same VEV ratios as those found in the neutrino mass formula. For instance, in the usual type-I seesaw with three generations of right-handed neutrinos, one finds that the coupling of the majoron to charged leptons scales as g_Jee∝ m_D   m_D^† /v_σ <cit.>. This is neutrino mass suppressed since m_D   m_D^† /v_σ∼ v_H^2/v_σ∼ m_ν. [A neutrino mass suppressed majoron coupling may, in principle, be sizable. As pointed out in <cit.>, it is possible to use the matrix structure of m_D to suppress m_ν as -m_D m^-1_R m^T_D while keeping m_D   m^†_D/v_σ large. However, such a cancellation would require some fine-tuning, so we will ignore this fact.] Models with a majoron coupling to charged leptons that is not neutrino mass suppressed will be regarded as enhanced. A summary of all the results that follow are given in Tab. <ref>.






 §.§ Class 1




Let us start with Class 1 and consider the first matrix in Eq. (<ref>),


    [     0   m_D   m_L; m_D^T   μ_N   m_R; m_L^T m_R^T   μ_S ]  .


As already explained and discussed explicitly below, the other mass matrices (and hence models) in this class can be recovered from this one by just removing some of the fields in our general setup. The presence of the mass term N̅^c N forces |q_σ| = 2. In combination with the term S̅^c S, this implies q_S = ± 1. In this case, N̅^c S is generated, either from a σN̅^c S Yukawa term (once σ gets a VEV) or as a bare Lagrangian term. This leads to two inequivalent realizations, which we now discuss separately.




  §.§.§ Class 1a: q_S = 1  ⇒  q_χ = 0, q_σ= -2




Since in this case q_S = q_N and q_H = q_χ, the Yukawa Lagrangian can be written as


    -Ł =    L̅H̃( y_N   N  +y_S   S ) +σ( λ N̅^c S + 1/2λ_N  N̅^c N  + 1/2λ_S  S̅^c S )+ 
    
    =     Y  L̅H̃ F+ 1/2 σ F̅^c  Λ  F +   .


Here we have identified χ = H, defined Y as in Eq. (<ref>) and introduced 


    Λ = ( [ λ_N   λ; λ^T λ_S ])   .


It is clear that this scenario, which we denote as C1a, is a conventional type-I seesaw, as discussed in Sec. <ref>. The neutrino mass matrix will be given by m_ν∼Y^2 v_H^2/Λ v_σ∼Λ_EW^2/Λ_ H, with v_σ∼Λ_ H≫Λ_EW a large seesaw scale. The majoron phenomenology will therefore be neutrino mass suppressed in this model, as discussed extensively in the literature <cit.>. 




  §.§.§ Class 1b: q_S = -1  ⇒  q_χ = -2, q_σ = -2




With these charges, m_R can exist as a bare mass and the Lagrangian is given by


    -Ł = y_N  L̅H̃ N  +y_S  L̅χ̃ S + m_R N̅^c S + 1/2 λ_N  σ N̅^c N  + 1/2 λ_S  σ^* S̅^c S +   .


Once the electroweak and lepton number symmetries are spontaneously broken we obtain,


    m_D = v_H/√(2)  y_N   ,    m_L = v_χ/√(2)  y_S    ,   μ_N = v_σ/√(2) λ_N   ,   μ_S= v_σ/√(2) λ_S   .


The VEV hierarchy v_χ≪ v_H, required for the Higgs to be SM-like, implies


    m_L ≪ m_D   ,       μ_N ∼μ_S   .


To keep the doublet component of the majoron negligible, we must also impose the hierarchy v_χ≪ v_σ. Depending on the hierarchies between m_L, m_R and μ_S, N one finds either a type-I seesaw, a linear seesaw, an inverse seesaw or a hybrid scenario. In any case, this model (C1b) and the ones in this class will lead to a majoron phenomenology suppressed by neutrino masses.

We now consider the other two matrices in Class 1. By removing the singlet σ we find the second matrix


    -Ł   = y_N  L̅H̃ N  +y_S  L̅χ̃ S + m_R N̅^c S +   →  [     0   m_D   m_L; m_D^T     0   m_R; m_L^T m_R^T     0 ]  .


This model, as it is, features a purely doublet massless majoron and is thus not realistic. However, this can be solved by re-adding the scalar singlet σ with charge q_σ = -1. This allows for the term χ^† H σσ, which would lead to singlet-doublet mixing after symmetry breaking, hence suppressing the doublet component of the majoron. This would be the most straightforward version of the classic linear seesaw with SSB and was studied extensively in <cit.>. We denote this model as C1bσ.

If we instead keep the singlet scalar σ and remove the scalar doublet χ we obtain the model C1bχ, which is the spontaneous version of the inverse seesaw <cit.>. In this case one recovers the third mass matrix in Class 1,


    -Ł   = y_N  L̅H̃ N + m_R N̅^c S +1/2 λ_N  σ N̅^c N  +1/2 λ_S  σ^* S̅^c S +   →  [     0   m_D     0; m_D^T   μ_N   m_R;     0 m_R^T   μ_S ]  .





 §.§ Class 2




We start with the mass matrix

 
    [     0   m_D   m_L; m_D^T     0   m_R; m_L^T m_R^T   μ_S ]  ,


and again look at the U(1)_L covariance of the gauge singlet terms in Tab. <ref>. Since the term N̅^c N must be forbidden in this scenario, q_σ≠± 2. Again, two inequivalent scenarios arise.




  §.§.§ Class 2a: q_S = -1/3 ⇒  q_χ = -4/3, q_σ = -2/3




In this case, the Yukawa Lagrangian can be written as


    -Ł = y_N  L̅H̃ N  +y_S  L̅χ̃ S + λ σN̅^c S +1/2λ_S  σ^*  S̅^c S   +   ,


with


    m_D = v_H/√(2)  y_N   ,    m_L = v_χ/√(2)  y_S    ,    m_R = v_σ/√(2) λ  ,   μ_S= v_σ/√(2) λ_S   .


Again, we must impose the VEV hierarchy v_χ≪ v_H ≪ v_σ, which leads to 


    m_L ≪  m_D        m_R ∼μ_S ≫Λ_EW .


The contribution of m_L is actually subdominant and this model would be a type-I seesaw, with m_ν∼ v_H^2 / v_σ. As such, the same conclusion holds when removing the doublet, in which case we find model C2aχ, with the second mass matrix


    -Ł = y_N  L̅H̃ N  + λ σN̅^c S +1/2λ_S  σ^*  S̅^c S   +   →  [     0   m_D     0; m_D^T     0   m_R;     0 m_R^T   μ_S ]  .


Both options will result in a majoron phenomenology suppressed by neutrino masses.




  §.§.§ Class 2b: q_S = 0  ⇒  q_χ =-1, q_σ = -1




With this charge assignment, the Yukawa Lagrangian is given by


    -Ł = y_N  L̅H̃ N  +y_S  L̅χ̃ S + λ σN̅^c S +1/2μ_S S̅^c S   +   .


As usual, several mass parameters are generated after SSB,


    m_D = v_H/√(2)  y_N   ,    m_L = v_χ/√(2)  y_S    ,    m_R = v_σ/√(2) λ  ,


and, due to the VEV hierarchy v_χ≪ v_H ≪ v_σ, one finds


    m_L ≪ m_D ≪ m_R  .


Like in model C1b, described in Sec. <ref>, the hierarchy between m_L and μ_S determines if the model is an inverse seesaw, a linear seesaw or a hybrid. In this case, removing the singlet scalar is not an option, as it would render some sterile states massless, but we can recover the second matrix by removing the doublet,


    -Ł = y_N  L̅H̃ N  + λ σN̅^c S +1/2μ_S S̅^c S   +   →  [     0   m_D     0; m_D^T     0   m_R;     0 m_R^T   μ_S ]  .


The resulting model, denoted as C2bχ, is a pure inverse seesaw and was analyzed in <cit.>. Both models have enhanced majoron phenomenology, as we will flesh out in Sec. <ref>.




 §.§ Class 3




Finally, we reach Class 3. Again, we consider the mass matrix

 
    [     0   m_D   m_L; m_D^T   μ_N   m_R; m_L^T m_R^T     0 ]  .


By looking at the covariance under U(1)_L of the singlet sector (see Tab. <ref>) we can determine the allowed lepton number charges of the states in the model. The term S̅^c S must be forbidden in this model, which implies that q_S ≠ 0 and |q_σ| ≠ 2 q_S. Thus, the only solution is given by


    q_S = -3  ⇒  q_χ = -4   ,    q_σ = -2


Hence, the Yukawa Lagrangian can be written as


    -Ł = y_N  L̅H̃ N  +y_S  L̅χ̃ S + λ σ^*  N̅^c S +1/2λ_N  σ N̅^c N   +   .


After symmetry breaking,


    m_D = v_H/√(2)  y_N   ,    m_L = v_χ/√(2)  y_S    ,    m_R = v_σ/√(2) λ  ,   μ_N = v_σ/√(2) λ_N   .


Finally, the usual VEV hierarchy v_χ≪ v_H ≪ v_σ implies in this case


    m_L ≪  m_D  ≪  m_R ∼μ_N   ,


where once again the last condition comes from the requirement that the sterile states have to be heavy, in the seesaw spirit. This is a linear seesaw with enhanced majoron phenomenology, which will be fleshed out in Sec. <ref>. To the best of our knowledge, this model has not been discussed before. A similar model was analyzed in <cit.>, where the lepton number breaking is explicit and not spontaneous, leading to a massive majoron.




 §.§ Related models




In addition to the models described previously, we highlight two additional ones that, while not meeting our predefined criteria, remain both potentially viable and interesting. Removing the scalar doublet from the Class 3 mass matrix yields


    [     0   m_D     0; m_D^T   μ_N   m_R;     0 m_R^T     0 ] .


This configuration results in neutrinos that are massless at the tree level. Nonetheless, neutrino masses emerge at the 1-loop level, and thus we do not consider it a part of the type-I seesaw family. This texture of the mass matrix was explored in <cit.> and is referred to as the Extended Seesaw.

Another model, not covered in our analysis, emerges from assigning the lepton number charges as q_S = 0  ⇒  q_χ = -1, q_σ = -2, which leads to the mass matrix


    [     0   m_D   m_L; m_D^T   μ_N     0; m_L^T     0   μ_S ] .


This charge assignment introduces an accidental U(1)_H × U(1)_χ× U(1)_σ symmetry in the scalar potential, resulting in two massless Goldstone bosons, with one being of doublet nature. It is possible to implement a realistic version of this model by deviating from our minimality conditions and incorporating additional singlet scalars. However, it is again beyond the scope of our analysis.






§ PHENOMENOLOGY OF SELECTED MODELS



The classification in the previous Section shows that the Type-I Seesaw family is composed by many distinct models, with potentially distinct phenomenological predictions. In particular, the relevance of the majoron coupling to charged leptons, a crucial feature that can be used to distinguish among models, has been highlighted. Reference <cit.> recently computed general analytical expressions for the 1-loop coupling of the majoron to a pair of charged leptons, valid in any model with a clear hierarchy of energy scales, as required for the seesaw expansion to be consistent. In the context of the Type-I Seesaw family of models, one just has to compute the four diagrams shown in Fig. <ref>. [Other 1-loop diagrams represent corrections to a possible tree-level coupling and can be neglected.] In these diagrams, S_k, P_k and η^+ represent, respectively, the neutral scalars, pseudoscalars and charged scalar in our setup. As already explained, full expressions of the resulting majoron coupling to charged leptons are given in <cit.>. We collect the most relevant results of this reference and apply them to the Type-I Seesaw family in Appendix <ref>.



We will now explore the majoron phenomenology of three specifically chosen models that feature an enhanced interaction of the majoron with charged leptons. As seen in Table <ref>, the majoron interactions in the models not covered in this discussion are suppressed by neutrino masses.




 §.§ Hybrid mechanism with enhanced majoron LFV






Let us start by analyzing model C2b. The lepton number charges of the fields in the model can be read off from Tables <ref> and <ref>, but we give them explicitly in Table <ref> for the sake of clarity. The model is defined by the Yukawa interactions given in Sec. <ref>,


    -Ł = Y L̅ H e_R + y_N  L̅H̃ N  +y_S  L̅χ̃ S + λ σN̅^c S +1/2 μ_S S̅^c S   +   ,


which leads to the neutral fermion mass matrix


    ℳ = [     0   m_D   m_L; m_D^T     0   m_R; m_L^T m_R^T   μ_S ]  ,


with m_D = y_N   v_H/√(2), m_L = y_S   v_χ/√(2) and m_R = λ  v_σ/√(2). By assuming the usual hierarchies, μ_S,   v_χ≪ v_H≪ v_σ, the neutrino mass matrix is found to be


    M_ν≈v_H^2/v_σ^2[y_N  ( λ^T )^-1μ_S λ^-1 y_N^T]   - v_H   v_χ/v_σ[ ( y_N ( λ^T )^-1  y_S^T )   + tr.]   ,


where tr. denotes the transposed of the previous matrix. The first term is the well-known neutrino mass matrix from an inverse seesaw, while the second is the one from a linear seesaw. Thus we deem this model as a hybrid mechanism. For the sake of generality we will be agnostic regarding the hierarchy μ_S / v_χ. λ can be taken to be diagonal and real without loss of generality and, given the hierarchy μ_S ≪λ  v_σ,  λ  v_σ represents the physical masses of the quasi-Dirac pairs of heavy neutral leptons.

In the scalar potential, apart from the usual self-conjugate terms, the following terms are also present:


    V ⊃λ_Hχ 2  (H χ^†) (χ H^†)+ κ  H χ^†σ +   .


The CP-even scalar sector consists of three states: S_1 ≡ h, S_2 and S_3. The lightest of them, h, is identified with the SM-like state discovered at the LHC with a mass m_ S_1≈ 125 GeV. Regarding the CP-odd scalars, there are two massless Goldstone bosons, P_1 ≡ G_Z (absorbed by the Z boson) and P_2 ≡ J (the physical majoron), as well as the massive P_3. Finally one of the two charged scalar states, G_W, is the usual EW Goldstone boson absorbed by the W boson and there is a new massive charged state, η^+. Their masses are given as follows


    
  * CP-even scalars: m_h^2∼ v_H^2, m_ S_2^2 ∼κ  v_H  v_σ/v_χ and m_ S_3^2 ∼ v_σ^2
    
  * CP-odd scalars: m_G_Z = m_J = 0 and m_ P_3^2 = - κ  v_σ v^2/√(2) v_H v_χ𝒩^2
    
  * Charged scalars: m_G_W=0 and m_η^+^2 = - λ_Hχ 2/2 v^2 - κ/√(2)  v_σ v^2/v_χ v_H


where we have defined the normalization factor


    𝒩^2 = 1 + v_H^2   v_χ^2/v_σ^2   v^2  .


Here v is the SM VEV, defined by v^2 ≡ v_H^2 + v_χ ^2. The mass-dimension trilinear κ is a free parameter of the model with a large impact on the scalar spectrum. When κ≫v_χ/v_H v_σ, the BSM states S_2, P_3 and η^+ become heavy and, since we assume v_σ to be a high-energy scale, decouple from the spectrum. Otherwise, our setup leads to a scalar spectrum with new states below v_σ, within the reach of current experiments. As expected, we find a SM-like Higgs and the Goldstone bosons associated to the SSB of the electroweak symmetry, in addition to the one from the SSB of U(1)_L. In the basis ( H^a, χ^a, σ^a), where the a superscript denotes the imaginary component of the corresponding field, the majoron J is given as 


    J    = 1/𝒩(- v_H v_χ^2/v_σ v^2,  v_H^2 v_χ/v_σ v^2,   1 )   ,

where 𝒩 is the normalization factor of Eq. (<ref>).

For the massive charged scalar, in the basis (H^+,  χ^+), we find


    η^+    = v_H/v( - v_χ/v_H,    1 )   .


We can now obtain the majoron coupling to charged leptons. First of all, note that the majoron has a component in the H^a direction, so a tree-level coupling is generated, highly suppressed by the mixing:


    ℒ_ℓℓ J^ = i v_χ^2 v_H/ 𝒩  v^3 v_σ  J ℓ̅ M_ℓ γ_5  ℓ∼v_χ^2   m_ℓ/v^2 v_σ  J ℓ̅ γ_5  ℓ≲v_χ  m_ℓ  m_ν/v^3  J ℓ̅ γ_5  ℓ  .


However, we obtain larger couplings at 1-loop. To compute them we need the couplings for the dominant diagrams: W, Z and η^+, i.e. diagrams (a), (b) and (c) of Fig. <ref>, while diagram (d) turns out to be subdominant. From Eqs. (<ref>) and (<ref>) we obtain, 


    A̅   = 1/2[      0   A̅_L; A̅_L^T   A̅_H ]  ,     A̅_L = iv_χ v_H/√(2)𝒩 v^2 v_σ[ -v_χ y_N  v_H y_S ]  ,     A̅_H = i/√(2)𝒩[   0   λ; λ^T   0 ]  , 
    D̅_R    = 1/v[ - v_χ  y_N  v_H   y_S ]  .


With this, it is trivial to obtain the necessary matrices. For the gauge boson diagrams we find,


    ∑_j ∼ l Γ_βα j^1,0,0   = ∑_j ∼ lΓ̃_βα j^1,0,0≃ - i/2 v_σ (m_D m_D^†)_βα  = - iv^2/4 v_σ( y_N   y_N^†)_βα  , 
    ∑_j ∼ hΔ_βα j^0,1,-1   = ∑_j ∼ hΔ̃_βα j^0,1,-1≃i/ 2v_σ (m_D m_D^†)_βα = iv^2/ 4 v_σ (y_N   y_N^†)_βα  ,


while for the η^+ contribution, 


    ∑_j Γ̃_s p j^1,0,0   ≃∑_j ∼ hΔ_s p j^0,1,-1≃i/2 𝒩 v_σ[ m_R^2     0;     0 m_R^2 ]_sp≃i v_σ/4 𝒩[ λ^2   0;   0 λ^2 ]_sp  ,
    ∑_j Γ̃_s p j^1,1,0   ≃∑_j ∼ hΔ_s p j^0,1,1≃i /2 𝒩 v_σ[ m_R^4     0;     0 m_R^4 ]_sp≃i v_σ^3/8 𝒩[ λ^4   0;   0 λ^4 ]_sp  .


Finally we obtain the leading order contribution for the 1-loop coupling of the majoron to charged leptons,


    ℒ_ℓℓ J =iJ/32π^2 v_σℓ̅[M_ℓ (y_N   y_N^†)  γ_5 +2 M_ℓ (y_N   y_N^†-  y_S  Θ  y_S^†) P_L- 2  (y_N   y_N^† - y_S  Θ  y_S^†)M_ℓ P_R]ℓ  ,


where the matrix Θ is given by


    Θ_sp≡(m_R^2)_s/( (m_R^2)_s - m_η^+^2 )^2( (m_R^2)_s- m_η^+^2 + m_η^+^2 logm_η^+^2/(m_R^2)_s) δ_sp  .


The strength of the majoron interactions showcased in Eq. (<ref>) is not neutrino mass-suppressed. Indeed, we can consider the limit μ_S, v_χ→ 0, which leads to M_ν→ 0 in Eq. (<ref>). Even in this unrealistic scenario with massless neutrinos, Eq. (<ref>) would lead to a lepton flavor violating majoron with potentially observable signatures.



 §.§ Inverse seesaw with enhanced majoron LFV





The C2bχ model can be obtained by removing the doublet χ from the field inventory of model C2b. Therefore, the charges of the fields in this model are those shown in Table <ref>. The Yukawa Lagrangian is given by


    -Ł = Y L̅ H e_R + y_N  L̅H̃ N  + λ σN̅^c S +1/2 μ_S S̅^c S   +   ,


which corresponds to that of the previous model after setting y_S = 0. One obtains the neutral fermion mass matrix


    ℳ = [     0   m_D     0; m_D^T     0   m_R;     0 m_R^T   μ_S ]  ,


with m_D = y_N   v_H/√(2) and m_R = λ  v_σ/√(2). By assuming the VEV hierarchy μ_S ≪ v_H≪ v_σ, the neutrino mass matrix is found to be


    M_ν≈v_H^2/v_σ^2  y_N  ( λ^T )^-1μ_S λ^-1 y_N^T   ,


which is nothing but the well-known neutrino mass formula obtained in the inverse seesaw. The spectrum of the scalar sector of this model can be easily derived by adapting the results from the previous Section. Similarly, the majoron coupling to charged leptons in this model can be obtained simply by
taking the limit m_η^+→∞ (or, equivalently, y_S → 0) in Eq. (<ref>). One finds


    ℒ_ℓℓ J =iJ/32π^2 v_σℓ̅[M_ℓ (y_N   y_N^†)  γ_5 +2 M_ℓ  y_N   y_N^†  P_L- 2   y_N   y_N^†  M_ℓ P_R]ℓ  .


We note once again that the couplings in Eq. (<ref>) do not vanish in the limit μ_S → 0, which implies sizable majoron LFV rates even in the absence of neutrino masses.




 §.§ Linear seesaw with enhanced majoron LFV






Let us now consider model C3 which, as described in Sec. <ref>, also features enhanced rates of majoron LFV processes. The charges of the fields under the electroweak and global U(1)_L symmetries are given explicitly in Table <ref>. The relevant Yukawa Lagrangian is


    -Ł = Y L̅ H e_R + y_N  L̅H̃ N  +y_s  L̅χ̃ S + λ σ^*  N̅^c S +1/2 λ_N  σ N̅^c N   +   ,


which leads to the neutral fermion mass matrix


    ℳ = [     0   m_D   m_L; m_D^T   μ_N   m_R; m_L^T m_R^T     0 ]  ,


with m_D = y_N   v_H/√(2), m_L = y_S   v_χ/√(2), μ_N = λ_N   v_σ/√(2) and m_R = λ  v_σ/√(2). This texture, assuming v_χ≪ v_H ≪ v_σ, leads to a linear seesaw mechanism for neutrino masses, with


    M_ν≈ - m_L   m_R^-1  m_D^T + tr. = -v_χ v_H/v_σ  y_S  λ^-1  y_N^T + tr.


One of the matrices λ and λ_N can be taken to be diagonal without loss of generality by performing adequate rotations of the fields N and S. In the scalar sector, the U(1)_L charges of χ and σ are -4 and -2, respectively. Then, the scalar potential, apart from the usual self-conjugate terms, also includes the terms


    V ⊃λ_Hχ 2 (H χ^†) (χ H^†) + λ_G  χ^† H σσ +   .


Using the same notation for the scalar and pseudoscalar states and assuming the same VEV hierarchy (v_χ≪ v_H ≪ v_σ) as in Sec. <ref>, the mass spectrum of the model is given by:


    
  * CP-even scalars: m_h^2∼ v_H^2, m_ S_2^2 ∼ v_σ^2 v_H/v_χ and m_ S_3^2 ∼ v_σ^2
    
  * CP-odd scalars: m_G_Z = m_J = 0 and m_ P_3^2= -λ_G v_σ^2 v^2/2 v_H v_χ𝒩^2∼ v_σ^2   v/v_χ 
    
  * Charged scalars: m_G_W=0 and m^2_η^+ = -λ_Hχ 2/2v^2 - λ_G/2v_σ^2 v^2/v_H v_χ∼ v_σ^2   v/v_χ 


Here we have introduced the normalization factor

    𝒩^2 = 1 + 4  v_H^2   v_χ^2/v_σ^2   v^2  .


Again, as expected, we find a SM-like Higgs, h, as well as the usual Goldstone bosons, including the majoron due to the spontaneous breaking of global U(1)_L.

The masses of the massive pseudoscalar 𝒫_3 and the charged scalar η^+ lie above the v_σ scale by a factor of order √(v / v_χ)≫ 1 and thus decouple from the theory at lower energies. Therefore, one can easily verify that the diagrams (c) and (d) of Fig. <ref> give subdominant contributions and we have to compute only the gauge boson contributions, diagrams (a) and (b) of the same figure. We start by writing the profile of the majoron J in the basis  ( H^a, χ^a, σ^a),


    J =  1/𝒩( - 2  v_χ^2   v_H/v_σ  v^2,   2  v_H^2   v_χ/v_σ  v^2,   1 )   .


Again, the superscript a denotes the imaginary component, and we have used the normalization factor of Eq. (<ref>).

Given the hierarchies at play, v^2 ≈ v_H^2 and the majoron is mainly a singlet, with a small doublet component suppressed by a factor of order m_ν / v or smaller. In fact, this component leads to a very suppressed tree-level diagonal coupling between the majoron and charged leptons:


    ℒ_ℓℓ J^ =2 i v_χ^2 v_H/𝒩v^3 v_σ J ℓ̅ M_ℓγ_5 ℓ∼m_ν  m_ℓ  v_χ/v^3  J ℓ̅ γ_5  ℓ  .


However, larger couplings are found at the 1-loop level. To compute these coupling we must determine A̅ (see Eq. (<ref>)) in this model. One finds


    A̅ = 1/2[      0   A̅_L; A̅_L^T   A̅_H ]  ,     A̅_L = 2iv_χ v_H/√(2)𝒩 v^2 v_σ[ -v_χ y_N  v_H y_S ]  ,     A̅_H = i/√(2)𝒩[  λ_N   -λ; -λ^T    0 ]  ,


and then


    ∑_j ∼ l Γ_βα j^1,0,0   = ∑_j ∼ lΓ̃_βα j^1,0,0≃i/2 v_σ (m_D m_D^†)_βα  =  iv^2/4 v_σ( y_N   y_N^†)_βα  , 
    ∑_j ∼ hΔ_βα j^0,1,-1   = ∑_j ∼ hΔ̃_βα j^0,1,-1≃ -i/ 2v_σ (m_D m_D^† -2 m_D μ_N^†μ_N ( m_R^T)^-1( m_R^*)^-1 m_D^†)_βα
       = -iv^2/ 4 v_σ (y_N   y_N^† -1/2 y_N λ_N^†λ_N ( λ^T)^-1( λ^*)^-1 y_N^†)_βα  .


Finally, with these results at hand, one can write the leading contribution for the coupling of the majoron to a pair of charged leptons as


    ℒ_ℓℓ J = -iJ/32π^2 v_σ ℓ̅ {    M_ℓ  [( y_N  (𝕀_3 -1/3R ) y_N^†) ] γ_5 + M_ℓ  y_N  ( 2  𝕀_3 -5/12R ) y_N^† P_L 
       -  2 y_N  ( 2  𝕀_3 -5/12R^†) y_N^†  M_ℓ P_R} ℓ  ,


where R ≡λ_N^†λ_N ( λ^T)^-1( λ^*)^-1. As in the previous two example models, the strength of the majoron interactions in Eq. (<ref>) is not neutrino mass suppressed. Again, considering the limit v_χ→ 0 leads to vanishing neutrino masses (M_ν→ 0 in Eq. (<ref>)), but leaves Eq. (<ref>) intact.




 §.§ Analysis




We have just examined three models featuring enhanced majoron couplings to charged leptons. To determine whether this enhancement will be phenomenologically significant we will now analyze the current constraints on this coupling and their future improvement prospects. We will also compare them with other typical signals of low-scale seesaws, such as μ→ e γ.




  §.§.§ Current and future constraints


We start by writing down a general majoron interaction Lagrangian with a pair of charged leptons,


    ℒ_ℓℓ J = J ℓ̅( S_L P_L + S_R P_R) ℓ +   =J ℓ̅( S P_L + S^† P_R) ℓ  ,


with S = S_L + S_R^†. There are stringent constraints on both diagonal and off-diagonal majoron couplings to
charged leptons <cit.>. The flavor conserving couplings are constrained by energy loss mechanisms in astrophysical observations <cit.> and yield


    | Im(S_ee^exp) |    < 2.1 × 10^-13  , 
    | Im(S_μμ^exp) |    < 3.1 × 10^-9  .


The presence of non-zero off-diagonal couplings in Eq. (<ref>) allows the non-standard decay μ^+ → e^+   J. In particular, we find 


    Γ(ℓ_α→ℓ_β  J) = m_ℓ_α/32  π | S^βα|^2   ,


where we have defined 


    | S^βα| = ( | S^βα_L |^2 + | S^βα_R |^2 )^1/2  .


The best limits on this process were obtained at TRIUMF <cit.>. Taking into account all
possible chiral structures for the majoron coupling, one can estimate
the limit <cit.>

    ( μ→ e   J ) ≲ 10^-5  ,

which in turn implies


    | S^e μ| < 5.3 × 10^-11  .



Finally, the currently best experimental limits on τ decays
including majorons were set by the Belle II
collaboration <cit.>. They can be used to derive the
bounds


    | S^e τ| < 3.5 × 10^-7  , 
       | S^μτ| < 2.7 × 10^-7  .


Future experiments such as Mu3e <cit.> will improve these constraints. In particular, for a massless majoron, phase-I of the Mu3e experiment is expected to find at 90% CL


    (μ→ e   J )_Mu3e≲ 6  ·  10^-7  .



A typical signal of many BSM models, and in particular low-scale seesaws, is the lepton flavor violating process μ→ e γ. The MEG collaboration reported <cit.>


    (μ→ e  γ)_MEG≲ 4.2  ·  10^-13  .


This bound will be improved by MEG-II <cit.> to


    (μ→ e  γ)_MEG-II≲ 6  ·  10^-14  .





  §.§.§ Phenomenology



We now proceed to analyze the majoron phenomenology of our selected models C2b, C2bχ and C3. In all of them we can estimate


    Γ(ℓ_α→ℓ_β  J) ∼m_ℓ_α^3/v_σ^2  .


On the other hand, the ℓ_α→ℓ_βγ process will be mediated in our general setup by the W boson and the charged scalar η^+ whenever it is present, see Fig. <ref>. While the exact expressions are well-known, see for instance <cit.>, we can again estimate






    Γ(ℓ_α→ℓ_β γ) ∼m_ℓ_α^5/m_R^4


where m_R is the mass of the heavy neutral fermion running in the loop. In all the models considered, this mass will be approximately given by the lepton number breaking scale v_σ times some Yukawa coupling λ. Then, the ratio between the branching ratio of both processes will scale as


    (ℓ_α→ℓ_βγ)/(ℓ_α→ℓ_β J )∼(m_ℓ_α/m_R)^2 λ^-2  ,


and hence one expects (ℓ_α→ℓ_βγ) ≪(ℓ_α→ℓ_β J ), except possibly when λ≪ 1. For instance, for ℓ_α = μ and m_R ∼ 1 TeV,  (ℓ_α→ℓ_βγ) ∼(ℓ_α→ℓ_β J ) would require λ≲ 10^-4. In summary, the key signature in these models is the flavor violating majoron emission in lepton decays. 

The previous analysis is overly simplistic, as it only considers the relevant energy scales and their hierarchy. Therefore, a rigorous numerical exploration that takes into account the various numerical factors arising from the loops and the freedom in the scale of the Yukawa couplings is required to validate it. As we proceed to show, a correct treatment proves that the parameter space where the process μ→ e γ would be observed before μ→ e J is extremely limited, particularly needing small Yukawa couplings, thus moving away from the motivation for low-scale seesaws. In other words, while these models could feature sizeable μ→ e γ decays, as expected in similar low-scale seesaw models, most of the parameter space in which this happens would lead to majoron interactions which are already ruled out by astrophysical observations and cLFV experiments.

Since our analysis holds true irrespective of the charged scalar η^+ mass, we can study the majoron-charged lepton phenomenology of the three models in the limits κ≫ v_σ v_χ/v, leading to Θ→ 0 in Eq. (<ref>), and λ_N → 0 in Eq. (<ref>). In both simplifying scenarios the neutrino mass remains unaffected and the majoron-charged lepton interaction becomes equal for the three models, C2b, C2bχ and C3, [Note that there is a non-physical global sign between the results in  C2b and C2bχ and the one in C3 that appears due to the lepton charge choice.] i.e.


    ℒ_ℓℓ J =iJ/32π^2 v_σℓ̅[M_ℓ (y_N   y_N^†)γ_5 +2 M_ℓ  y_N   y_N^†  P_L- 2  y_N   y_N^† M_ℓ  P_R]ℓ  .


Comparing this expression with the generic Lagrangian in Eq. (<ref>), we identify


    S_L    =  i/64 π^2 v_σ[2 M_ℓ  y_N y_N^† - M_ℓ Tr(y_N y_N^†)]  , 
    
        S_R     = S_L^†  .


For any given Yukawa matrix y_N it is possible to find suitable μ_S, y_S and v_χ that fit neutrino data <cit.>. Out of the astrophysical constraints, we expect the constraint of Eq. (<ref>) to be more stringent than the one of Eq. (<ref>). Indeed, in order for this not to be the case we would need


    S_μμ/S_ee≳ 10^4   ,


but from Eqs. (<ref>)-(<ref>), and taking into account that S = S_L + S_R^†, one finds


    S_μμ/S_ee = m_μ/m_e×-(y_N   y_N^†)_11+(y_N   y_N^†)_22 - (y_N   y_N^†)_33/(y_N   y_N^†)_11-(y_N   y_N^†)_22 - (y_N   y_N^†)_33  ,


which implies that fine-tuned Yukawas couplings are required to cancel the contribution to S_ee. If instead we assume the Yukawa couplings y_N and λ to be of 𝒪(1), we can use Eqs. (<ref>) and (<ref>), as well as the couplings in Eqs. (<ref>)-(<ref>), to derive a lower limit for the lepton number breaking scale v_σ,


    v_σ > 3   m_e/32 π^21/| Im(S_ee^exp) |   ∼ 10^4  TeV  ,


and an estimate for the branching ratios of the flavor violating processes μ→ e J and μ→ e γ obtained for that scale,


    (μ→ e J)    ≈ 10^ -5  , 
    (μ→ e γ)    ≈ 10^-23  .


This rather strong conclusion can be relaxed by allowing smaller Yukawa couplings. Indeed, in the left panel of Fig. <ref> we show the relation between S̃_eμ and the lepton number breaking scale v_σ, while the right panel of this figure shows a comparison between the branching ratios of the two lepton flavor violating processes discussed above. In the numerical scan leading to this figure we have imposed the seesaw condition (M_D · M_F^-1)_ij < 10^-2, the astrophysical constraints of Eqs. (<ref>) and (<ref>), as well as neutrino masses and mixing in agreement with current data <cit.>. In addition, in both panels we have explicitly shown the predictions of the model when the Yukawa couplings are taken to be of 𝒪(1) (orange points) and when this assumption is relaxed and smaller Yukawa couplings are allowed (blue points). First of all,  we observe on the left side of the figure that the current limit on BR(μ→ e J) from TRIUMF (see Eq. (<ref>)) already restricts the parameter space of the model, ruling out some points not excluded by the other constraints (Eq. (<ref>) being the most important one). We also find that the decay μ→ e J can be used to test the model at Mu3e even if the lepton number breaking scale v_σ is as high as ∼ 4 · 10^4 TeV. As already explained, this decay has much better chances of being observed than the more conventional μ→ e γ. This is illustrated on the right side of Fig. <ref>, where we see that only a very small fraction of the points resulting from our numerical scan will be tested by MEG-II, even though this experiment is sensitive to much smaller branching ratios than those tested by Mu3e for μ→ e J. Importantly, and consistently with our previous considerations, the very few points that will be tested by MEG-II involve small Yukawa couplings, hence departing from the main motivation for low-scale seesaw models. 







§ SUMMARY AND DISCUSSION


Models belonging to the Type-I Seesaw family are among the most promising for explaining neutrino masses and their scale. In these models, lepton number is broken explicitly or spontaneously, resulting in a Majorana mass matrix for neutrinos. At first, we focused on the mass matrix, overlooking the origin of the U(1)_L breaking. Using the general seesaw expansion, we derived a comprehensive formula for neutrino masses in the Type-I Seesaw family. This formula reproduces the known results of models such as the type-I, linear, or inverse seesaws, and enables us to identify hybrid models or those with less-known hierarchies.

Regarding the U(1)_L breaking origin, we observed that in the first scenario, with explicit breaking, the differences between models of the family are spurious since lepton number is not a good symmetry, allowing us to describe all models with the same Lagrangian —the one of the conventional type-I seesaw— with just different textures. However, when U(1)_L is spontaneously broken, models become distinguishable, and we can no longer describe them with only one Lagrangian. With this realization, we analytically derived all the different minimal models of the Type-I Seesaw family with SSB of U(1)_L. In this case, a Goldstone boson arises in the spectrum, the majoron, providing a clear signal of these models and allowing us to distinguish between them based on their phenomenology. We systematically classified this in Section <ref>, demonstrating that while in most models the majoron couplings to charged leptons are suppressed by neutrino masses, one also finds some models where this interaction is enhanced. Finally, we considered some example models of the latter type and analyzed their phenomenology, finding a promising signal: although we may not observe the usual flavor violating process μ→ e γ, the exotic μ→ e J decay may be within the reach of near-future experiments, even for relatively high lepton number breaking scales.

Our work explores both established models as well as novel ones. Moreover, it provides a comprehensive framework for working with models belonging to the Type-I Seesaw family. While our focus has been on the minimal realization of the family, the formulas for majoron-charged leptons interactions provided in Appendix <ref> open up possibilities for exploring promising non-minimal realizations. Additionally, these formulas are useful not only for scenarios with exact SSB of U(1)_L, but also for cases with soft symmetry breaking, where the majoron is not strictly massless but rather light. This allows for the application of our framework to versions featuring a massive majoron, of potential interest in cosmology and low-energy experiments.





§ ACKNOWLEDGEMENTS


Work supported by the Spanish grants PID2020-113775GB-I00
(AEI/10.13039/501100011033) and CIPROM/2021/054 (Generalitat
Valenciana). The work of AHB is supported by the grant
No. CIACIF/2021/100 (also funded by Generalitat Valenciana).  AV
acknowledges financial support from MINECO through the Ramón y Cajal
contract RYC2018-025795-I.






§ MODELS IN THE TYPE-I SEESAW FAMILY



Based on the dependence of the neutrino mass matrix on the relevant physical scales, we consider three generic mass generation mechanisms:


    
  * Type-I seesaw: if M_ν∼m_1^2/M, with m ≪ M,
    
  * Inverse seesaw: if M_ν∼m_1^2 m_2/M^2, with m_2 ≪ m_1 ≪ M,
    
  * Linear seesaw: if M_ν∼m_1 m_2/M with m_2 ≪ m_1 ≪ M.


There are many models in the Type-I Seesaw family, due to the many different possible hierarchies among the blocks in the M_D and M_F matrices. They can be classified according to these hierarchies. We find three possible cases.



  §.§.§ Case 1: No hierarchy among the blocks in M_F


If the blocks in the M_F matrix are all of the same order, m_R ∼μ_N ∼μ_S ∼Λ_H, where Λ_H is a high-enery scale, the neutrino mass matrix can be written as


    M_ν = c_1   m_D  Λ_H^-1 m_D^T + c_2   m_D  Λ_H^-1 m_L^T   + c_3   m_L  Λ_H^-1 m_L^T + c_4   m_L  Λ_H^-1 m_D^T + Ø(ε^2)   ,


where c_i, with i=1, … ,4, are constants. It is clear that one finds a type-I seesaw, regardless of the hierarchy between m_D and m_L.



  §.§.§ Case 2: m_R ≫μ_N (m_R^T )^-1μ_S


In this case the neutrino mass matrix can be expressed as


    M_ν =      - m_D  ( m_R^T )^-1μ_S   m_R^-1 m_D^T - m_L   m_R^-1μ_N ( m_R^T)^-1  m_L^T 
        + m_D ( m_R^T )^-1  m_L^T    + m_L   m_R^-1 m_D^T + Ø(ε^2)   .


Then, depending on the relative scale of m_D and m_L and their hierarchies with the blocks in M_F, we can classify the resulting models as follows:



  2.1 If m_L/m_D≫μ_S/m_R   ,   m_L/m_D≪m_R/μ_N  and   m_L ≪ m_D   or    m_D ≪ m_L: linear seesaw

  2.2 If m_L/m_D≪μ_S/m_R   ,   m_L^2/m_D^2≪μ_S/μ_N  and  μ_S ≪ m_R: inverse seesaw (Dirac mass term: m_D)

  2.3 If m_L/m_D≪m_R/μ_N   ,   m_L^2/m_D^2≫μ_S/μ_N  and  μ_N ≪ m_R: inverse seesaw (Dirac mass term: m_L)

  2.4 Otherwise: type-I seesaw




  §.§.§ Case 3: m_R ≪μ_N (m_R^T )^-1μ_S


In this case the neutrino mass matrix can be written as


    M_ν =  m_D  μ_N^-1 m_D^T + m_L  μ_S^-1 m_L^T - m_D  μ_N^-1  m_R  μ_S^-1  m_L^T - m_L  μ_S^-1  m_R^T  μ_N^-1  m_D^T  + Ø(ε^2).


Again, depending on the relative scale of m_D and m_L and their hierarchies with the blocks in M_F, the resulting models can be classified as follows:



  3.1 If m_L/m_D≪m_R/μ_N   ,   m_L/m_D≫μ_S/m_R  and   m_L ≪ m_D   or   m_D ≪ m_L: linear seesaw

  3.2 If m_L/m_D≪m_R/μ_N   ,   m_L/m_D≫μ_S/m_R  and   m_R ≪μ_S    or   m_R ≪μ_N: inverse seesaw (Dirac mass term: m_D)

  3.3 Otherwise: type-I seesaw






§ THE MAJORON COUPLING TO CHARGED LEPTONS



Reference <cit.> provides general analytical expressions for the 1-loop coupling of the majoron to a pair of charged leptons which can, in general, be written as <cit.>


    ℒ_ℓℓ J = J  ℓ̅_β( S_L^βα  P_L + S_R^βα  P_R ) ℓ_α +  = J  ℓ̅_β[ S^βα  P_L + ( S^αβ)^*   P_R ] ℓ_α  ,


where P_L,R = 1/2( 1 ∓γ_5 ) are the usual chiral projectors while ℓ_α,β are the charged leptons, with α,β two generation indices. In the Type-I Seesaw family S is given by


    S^βα = 1/8 π^2( δ^βα Γ_Z^α +  L_W^βα +  L_η^+^βα+  L_S^βα)   .


In this equation, each term represents the contribution of one of the Feynman diagrams shown in Fig. <ref>. These contributions, expressed as functions of certain general matrices and loop functions, are provided in <cit.>. Here, we present them for the sake of completeness. They are given by


    Γ_Z^α   =   i   m_ℓ_α/v^2 [ ∑_s=1^3 ( ∑_ j ∼ l Γ̃_ssj^1,0,0+Γ_ssj^1,0,0/6 -∑_j∼ hΔ̃_ssj^0,1,-1+Δ_ssj^0,1,-1/3)  ]  , 
    
    
    
      L_W^βα   = 2   m_ℓ_β/v^2 [  ∑_j ∼ l( Γ_αβ j^1,0,0   */12+ 2/3Γ_βα j^1,0,0) - ∑_j∼ h( Δ̃_αβ j^0,1,-1   */6 +7/12Δ̃_βα j^0,1,-1)]    , 
    
    
    
    
    L_S^βα   = ω/4 ( m_ρ^2 - m_σ^2 )^2{( G   C^†  M_ℓ)_βα( -m_ρ^2 + m_σ^2 +m_ρ^2 logm_ρ^2/m_σ^2) 
       - ( M_ℓ G^†  C  )_βα( -m_ρ^2 + m_σ^2 +m_σ^2 logm_ρ^2/m_σ^2) 
    -2 ( G   M_ℓ  C )( m_ρ^2 - m_σ^2 ) logm_ρ^2/m_σ^2}  , 
    
    
    
    
      L_η^βα   =   m_ℓ_β[ D̅_R^β p(D̅_R^α s)^* (L_η^RR)^*_sp - (D̅_R^α p)^* D̅_R^β s(L_η^RR)_sp]   ,


where in the last equation we have defined


    L_η^RR =     f_7  ∑_j∼ hΔ^0,1,-1_spj+∑_j ( f_8  Δ^0,1,1_spj-F_5,7 Γ̃^1,0,0_spj -F_6,8 Γ̃^1,1,0_spj) 
       + ∑_j∼ l( F_5,7,-1 Γ̃^1,0,0_spj+F_6,8,-2 Γ̃^1,1,0_spj)    , 
    L_η^RR =    L_η^RR ( f_(1,   2,  5,   6,  7,   8 ) ↔ f_(3,   4,  9,   10,  15,   16 ) ,  f_13↔ F_1, -3,   f_14↔ F_2, -4)   ,


and where, following the conventions of <cit.>, ω denotes the coupling between J  P_k  S_k, C represents the coupling between charged leptons and S_k, and G is the coupling between the charged leptons and P_k. The loop functions f and F are also provided in <cit.>. We note that in these expressions there are sums that extend over all states, or only over the light (j ∼ l) or heavy (j ∼ h) ones. The precise definitions of the Γ, Γ̃, Δ, and Δ̃ matrices are also given in this reference. One can particularize them for the Type-I Seesaw family and find the combinations that appear in Eqs. (<ref>)-(<ref>). The sums relevant for the gauge boson contributions are given by


    ∑_ j ∼ l Γ_βα j^1,0,0    = ∑_ j ∼ l Γ̃_αβ j^1,0,0 =1/2A̅_L M_D^† -1/2( M_D   M_F^-1 A̅_H   M_D^†)_βα  , 
    ∑_ j ∼ hΔ̃_βα j^0,1,-1   =∑_ j ∼ hΔ_αβ j^0,1,-1 = 1/2M_D  (M_D^†A̅_L + M_F^†A̅_L^T   M_D^*   ( M_F^† )^-1 + M_F^†A̅_H ) M_F^-1  ( M_F^† )^-1  M_D^†  ,


whereas the sums relevant for the triangle diagram with the η charged scalar are


    ∑_ j Γ̃_s p j^1,0,0    =1/2( M_D^†A̅_L + M_F^† A̅_H  )_s p  , 
    ∑_ j Γ̃_s p j^1,1,0    = 1/2( M_F^†  M_F   M_F^† A̅_H  )_s p  , 
    ∑_ j Δ_ s p j^0,1,1   = 1/2(M_D^†A̅_L M_F^† M_F + M_F^†A̅_L^T   M_D^* M_F  + M_F^†A̅_H   M_F^†  M_F )_sp  ,
    ∑_ j ∼ hΔ_ s p  j^0,1,-1   =1/2[ M_F^-1( (M_F^†)^-1 M_D^†A̅_L M_F^†  +  A̅_L^T   M_D^*  + A̅_H   M_F^†) M_F ]_sp  .


Here we have used the fact that the charged scalar does not couple to the charged leptons and SM neutrinos. The amplitudes in these expressions depend on the couplings entering the loops, namely the majoron coupling to a pair of neutral fermions, A̅, and the charged scalar coupling to a neutral lepton and a charged lepton, D̅_L,R.








These are given in the gauge basis and can be readily computed for all variants in the Type-I Seesaw family. Let us consider the general Lagrangian


    -Ł = y_N  L̅H̃ N  +y_s  L̅χ̃ S + λ σ_NS N̅^c S + 1/2λ_N  σ_N N̅^c N  + 1/2λ_S  σ_S S̅^c S +   ,


which reduces to the models discussed in Sec. <ref> by properly matching the scalar fields to those in each model. Since we introduce only one singlet scalar σ in our field inventory, and this not always participates in all fermion singlet Yukawa terms, σ_NS, σ_N or σ_N may be absent in some models. In some cases λ_i  σ_i may represent a bare mass and, alternatively, some of the σ_NS, σ_N and σ_N singlet may correspond to the same singlet σ or its conjugate. Similarly, the doublet χ may coincide with H in some models. In general, the majoron will be a linear combination of the CP-odd parts of H, χ and σ, and then one can write


    J = α_H H^a +α_χχ^a +α_σσ^a ≡α_H H^a +α_χχ^a +α_σ_NSσ_NS^a + α_σ_Nσ_N^a + α_σ_Sσ_S^a    ,


where the superscript a refers to the CP-odd part of each scalar. The α_i coefficients encode the mixing in the CP-odd sector and can be easily computed in any given model. In some cases, it may occur that σ_NS = σ_N, σ_S = σ_N^* or any other combination. This does not affect the majoron but does affect how our α_i coefficients must be taken. For example, if we have N̅^c S as a bare mass term,  the bilinear N̅^c N coupling to σ and the bilinear S̅^c S coupling to σ^*, then we must take α_σ_NS=0, α_σ_N = α_σ and α_σ_S = -α_σ.
This fact can be expressed by the relation


    α_σ = |α_σ_NS|+|α_σ_N|+ |α_σ_S|/3-δ_0  α_σ_NS-δ_0  α_σ_N-δ_0  α_σ_S  .


After these preliminaries, the majoron coupling to neutral fermions in the gauge basis is given by 


    A̅ = i/2 √(2)[           0   α_H   y_N   α_χ   y_S; α_H   y_N^T   α_σ_N λ_N    α_σ_NS λ; α_χ   y_S^T  α_σ_NS λ^T   α_σ_S λ_S ]≡1/2[      0   A̅_L; A̅_L^T   A̅_H ]  ,


with 


    A̅_L = i/√(2)[ α_H   y_N α_χ   y_S ]  ,       A̅_H = i/√(2)[  α_σ_N λ_N   α_σ_NS λ; α_σ_NS λ^T  α_σ_S λ_S ]  .


The charged scalar η^+ is also an admixture of two different fields,


    η^+ = β_H   H^+ +β_χ χ^+  ,


where the β_i coefficients parameterize the mixing. Then, the couplings of η to a charged lepton and a neutral lepton are given, in the gauge basis, by


    D̅_R    = i/√(2)[ β_H    y_N   β_χ  y_S ]
      
      
        ,


where, again, the absence of a coupling between η, the charged leptons and SM neutrinos has been used. Here, D̅_R is given in the (N    S) basis. With these expressions for the couplings, we have derived the general form of the majoron coupling to charged leptons in the Type-I Seesaw family, enabling us to adapt it for each specific model.

Let us now briefly discuss these results. We start with the matrices entering the W and Z contributions. Note that A̅ is nothing but a Yukawa coupling, so A̅_ij≤ 1. This implies that the dominant terms will be ∼M_D^2/M_FA̅_H and ∼ M_D  A̅_L. For the first one, it is obvious that if we want to achieve a coupling that is not naturally suppressed by the light neutrino masses, we need A̅_H to alter the structure of M_F^-1. Regarding the second one, the reason is phenomenological. The majoron doublet admixture must be suppressed, as demanded by various experimental constraints. Of course, this suppression will be given by the VEV of σ. Then, the suppression will be of the order v/v_σ, v_χ/v_σ, or some combination leading finally to, at least, the same suppression that we found in the first term. One can argue in the same way for the scalar diagrams, thus concluding that we need A̅ to alter the structure of M_F^-1 if we want sizable rates for flavor processes involving the majoron.


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