Confronting the inverse seesaw mechanism with the recent muon g-2 result

Since the heavy neutrinos of the inverse seesaw mechanism mix largely with the standard ones, the charged currents formed with them and the muons have the potential of generating robust and positive contribution to the anomalous magnetic moment of the muon. Ho\-we\-ver, bounds from the non-unitary in the leptonic mixing matrix may restrict so severely the parameters of the mechanism that, depending on the framework under which the mechanism is implemented, may render it unable to explain the recent muon g-2 result. In this paper we show that this happens when we implement the mechanism into the standard model and into two versions of the 3-3-1 models.

It represents a hint of physics beyond standard model (BSM) [3]. The easiest way to generate new contributions to the g-2 of the muon is by means of new interactions involving the leptonic sector (See Ref. [4] for an extended review). Furthermore, the SM cannot address neutrino masses. Therefore, it is appealing to investigate models that can accommodate both problems.
There are several ways to explain neutrino masses, but in this work, we will focus on the inverse seesaw (ISS) mechanism [5][6][7]. It is a genuine TeV scale seesaw mechanism whose signature, heavy neutrinos N 's, is supposed to manifest at TeV scale. This interesting aspect allows us to confront the mechanism with existing TeV scale probes. In light of the recent measurement conducted by the Fermilab g-2 experiment, one might wonder whether one can successfully explain g-2 with the ISS.
The minimum SM extension capable of embedding the ISS has been addressed previously to analyze the corresponding g-2 contributions for sterile fermion states [8]. We consider the contributions for the g-2 of the muon due exclusively to the heavy neutrinos of the ISS mechanism. To do this, we consider models that do not accommodate the recent muon g-2 result (SM and some versions of the 3-3-1 model) and implement the ISS mechanism in them economically, where only the new neutral fermion singlets are added to the models in question. We then try to obtain a region of parameter space that could accommodate the recent measurement of g-2 of the muon and agree with some bound over the mechanism.
Our work is organized in the following way: In Sec. II, we present the main ingredients of the mechanism and develop our approach. In Sec. III, we implement the mechanism into the standard model, the minimal 3-3-1 model and the 3-3-1 model with right-handed neutrinos and explicit the terms that give the main contributions to the g-2 of the muon due to the heavy neutrinos. In sec. IV, we present our numerical results. We finalize presenting our conclusions in Sec. V.

A. Framework and nonunitarity costraint
The mechanism involves nine neutral fermions with specific chirality ν L = (ν e L , ν µ L , ν τ L ), S R = (S e R , S µ R , S τ R ) and S L = (S e L , S µ L , S τ L ) composing the following mass terms where M D , M and µ are generic 3 × 3 mass matrices. We rearrange things so that in the basis ν = (ν L , S C L , S L ) we may write where, In recognizing, we write This mass matrix may be diagonalized by a mixing matrix W given by [9,10] W ≈ such that where we have In terms of the original parameters, we have, Observe that the diagonalization of m light must lead to the eigenvalues m = (m 1 , m 2 , m 3 ), which we assume are associated with the following eigenvectors n L = (n 1 L , n 2 L , n 3 L ).
Now comes the essence of the mechanism. For m D belonging to the electroweak and M to the TeV scale, we just need that lepton number be explicitly violated at sub-keV scale (µ ∼ 0.1keV) in order to have m at sub-eV scale. Of course, we are assuming that n L are the standard neutrinos.
Concerning the heavy neutrinos, we assume that m heavy is diagonalized by U R , and that the eigenvectors are given by N L = N i L , with i = 1...6, whose masses lie around M . Moreover, the mixing among the neutrinos allows us to write the standard flavor neutrinos as the following combinations of the physical neutrinos [11] where F = M D M −1 . In view of the structure of this mixing, all deviation from unitarity is determined by the Hermitian matrix, We point out that it does not depend on the parametrization of the PMNS matrix. This η matrix is known as the nonunitarity parameter and have a crucial role in this paper. Current bounds on nonunitarity effects gives [12] | η bound |<

B. Our approach
In the usual approach, the matrices M and µ are considered as diagonal ones. In this case M D determines the texture of m light .
We take a different approach now. We assume that which leaves µ responsible for the texture of m light . It happens that in our approach the parameter η, defined in Eq. (10), takes the form which, according to the bound in Eq. (11), yields, This is nice because the nonunitarity constraint is favoring µ below keV scale to yield standard neutrino masses at eV scale.
After diagonalizing m light above, we obtain where m ν = diag(m 1 , m 2 , m 3 ). Inverting these matrices, we get, Once we know the absolute values of m 1 , m 2 , m 3 and having U P M N S from the experiments, we obtain the texture of µ.
Now we focus on the pattern of U R . For this we diagonalize m heavy given in Eq. (4) which in our approach: For m N >>| µ ij | the eigenvalues of m heavy are, which means that the six heavy neutrinos are practically mass degenerate with the values lying around m N . In this case, the mixing matrix, U R , that diagonalizes m heavy develops the following approximated pattern In summary, in our approach, µ is responsible for the texture of the standard neutrino mass matrix, while the assumption m N >>| µ ij | infers the pattern of U R . This method eases the assessment of the heavy neutrinos contributions to g-2 significantly consistently.
With this method at hand, we can solidly investigate if the inverse seesaw mechanism may explain g-2 [1].
We remind the reader that we will adopt throughout the most recent result from the Muon g-2 Experiment at Fermilab, which combined with previous Brookhaven National has a significance of 4.2σ [1,3]. See Ref. [4] for an extensive explanation about how the anomaly can be weakened or strengthened.
The implementation of the ISS requires the introduction of new neutral fermions, N i , to the original fermion content of the model in question. In extended gauge theories that feature the presence of new vector gauge bosons, such neutral fermions will appear in the charged currents involving exotic charged gauge bosons represented by W ± . In a general way, we write down these interactions through the following terms After some steps, the contribution of these new interactions to the g-2 of the muon is given by [4], where We apply this method to obtain the contribution to the g-2 stemming exclusively from the new ingredients of the ISS mechanism when implemented in models from starters that do not accommodate g-2.

A. Standard model
To implement the ISS mechanism in a minimal way into the standard model, we need to add six new neutral fermions as singlets field, so the leptonic content becomes, where a = e , µ , τ . The scalar sector is assumed to be composed uniquely by the standard Higgs H With these new fields we can form the following new terms in addition to all standard ones, where ν R = (ν e R , ν µ R , ν τ R ) T and S L = (S e L , S µ L , S τ L ) T . When of the spontaneous breaking of the symmetry, where H = v √ 2 , we obtain the following mass terms, where M D = Y D v √ 2 and ν L = (ν e L , ν µ L , ν τ L ) T . Observe that Eq. (24) recovers Eq. (1) which means that we have the ISS mechanism.
Thus, all the step done above is valid here.
Through mixing, Eq. (9), the ISS mechanism gives rise to the following Lagrangian, where N k are heavy pseudo-Dirac neutrinos. Notice that the product (M D † )[M R † ] −1 U R can be approximated by, It is clear that the six new pseudo-Dirac neutrinos contribute to g-2. Then, from Eq.
(25) and Eq. (19) we can easily identify g ij v1 , and g ij a1 that appear in Eq. (20). We also set λ = m µ /m W and f = = m N /m µ . After all these considerations, we are ready to calculate the contribution from Pseudo-Dirac N 's to g-2 due to the interactions in Eq. (25). [13,14]. In it, all leptons of each family compose a triplet in the following way,

B. Minimal 3-3-1 model
where a = e , µ , τ . Through this feature, we can understand the quantization pattern of electric charges [15]. Moreover, the gauge anomalies are canceled only when three families are considered at once [16]. There are several additional interesting aspects [17][18][19][20][21]. The gauge sector of the model features the standard gauge bosons and five other ones as a new Z , two new single charged gauge bosons W ± and two doubly charged gauge bosons U ±± . Their respective neutral and charged currents are found in [22]. It has been shown in [23] that all these interactions are not enough to accommodate the recent muon g-2 result. Hence, it brings up whether the ISS mechanism can foot the bill while successfully generating small masses.
For the implementation of the mechanism, we add the six neutral fermions ν R and, S L as discussed previously. The minimal set of scalars necessary to break spontaneously the symmetry and generate masses for all massive particles, except neutrinos, are three triplets η , ρ, χ and a sextet of scalar S. With this we form the terms where These mass terms recover the characteristic mass matrix of the ISS mechanism given in Eq. (3). Then all approach developed in Sec. II is applicable here.
In what concern the contributions to the g-2 of the muon due to the heavy neutrinos N 's, they arise from the interactions of these neutrinos with the charged gauge bosons W ± and W ± , M D , M R and U R are exactly those in Sec. II. Our task here is to check if this new interactions give significant contributions to the g-2 of the muon.

C. 3-3-1 model with right-handed neutrinos
In this version of the 3-3-1 models [24][25][26] the third component of the leptonic triplet is occupied by the right-handed neutrinos invoked by the ISS mechanism [27] This version shares the same main features that the minimal one in respect to anomaly cancellation, electric charged quantization and strong-CP problem with the additional one of having natural dark matter candidates [28][29][30]. However, it is a new model with its gauge sector being composed by the standard gauge bosons and five other ones as a new Z , two new single charged gauge bosons W ± and two neutral, but non-hermitian, gauge bosons U 0 , U 0 † .
The interactions of these gauge bosons with the neutral and charged currents is found in [31]. It was checked in [23] that all these interactions are not enough to accommodate the recent muon g-2 result, and again this provides a strong motivation for we check if the implementation of the ISS mechanism would give significant contributions to the g-2 of the muon.
For the implementation of the mechanism, we just need to add three new neutral fermions in the singlet form, S L . In this model the minimal set of scalars necessary to break spontaneously the symmetry and generate masses for all massive particles, except neutrinos, are three triplets of scalars η , ρ, χ. With this minimal particle content we form the following terms that will trigger the ISS mechanism When ρ and χ develop their vev's, ρ = vρ √ 2 and χ = vχ √ 2 , we have the following mass terms, These mass terms recover the characteristic mass matrix of the ISS mechanism given in Eq. (3). Then all approach developed in Sec. II is applicable here with specific care because, now, M D is an antisymmetric matrix and, due to this, we can not take it diagonal. But it is possible to take it degenerated. In this case we have Following this, taking M = m N I, we have for the nonunitary parameter, defined in Eq. (10) In what concern the contributions to the g-2 of the muon due to the heavy neutrinos N 's, they arise from the interactions of these neutrinos with the charged gauge bosons W ± and W ± , to interactions involving charge currents and single charged gauge bosons as represented in Fig. 1 for V being the standard gauge boson W ± or W ± . For previous works addressing g-2 into this model, see Refs. [32,33] FIG. 1: New Feynman diagram contributing to the muon g-2. N are the heavy pseudo-Dirac neutrinos characteristic of the ISS mechanism and V ± = W ± , W ± .
We are now ready to calculate such contributions in all three cases. We do this and present our numerical results in the next section.

IV. NUMERICAL RESULTS
For our numerical calculations, we made use of the code [34], developed in [23]. Then, after configuring this algorithm for each case, we solved numerically the integral in Eq. (20).
Our first result is presented in Figs Note that the nonunitarity bound lies orders of magnitude below the favored region for g-2.
In summary, ISS is not the solution to g-2.
Finally, we show in Figs. 6 and 7 our results for the case of the ISS implemented into the 3-3-1 model with right-handed neutrinos. Looking at the numerical results, we arrive at the same conclusion. The ISS mechanism cannot address g-2 even in the scope of a 3-3-1 model with right-handed neutrinos. We would like to emphasize that our conclusion concerning the inverse seesaw is applicable to the minimal version of the inverse seesaw, in other words, when it is implemented in the standard model, and to the 3-3-1 models studies. Our conclusion is not valid to all possible implementations of the inverse seesaw as one can always add fields in a way to change the contribution to the g-2 or circumvent the non-unitarity bounds, and consequently find different results. Obviously, such constructions are necessarily more complex. A concrete example can be found in [35][36][37].

VI. CONCLUSIONS
We revisited the inverse seesaw mechanism and assessed the possibility to explain g-2 when nonunitarity constraints are considered. Firstly, we added the necessary fields to the Standard Model spectrum to realize inverse seesaw mechanism. Later, we implemented the inverse seesaw in the minimal 3-3-1 model, and in the 3-3-1 model with right-handed neutrinos. We studied the impact of the new neutral singlet fields on the lagrangians relevant for g-2 and nonunitarity studies. In all three cases, we conclusively showed that the ISS mechanism cannot address g-2 due to the stringent nonunitarity constraint that forces the masses of the particles to be orders of magnitude higher than the one required to explain g-2. This conclusion is clear in Fig. 6. In summary, our findings show that the nonunitarity constraint is severe enough to avoid the inverse seesaw mechanism to explain the recent muon g-2 result within these frameworks.