Predictions from scoto-seesaw with A 4 modular symmetry

This paper’s novelty lies in introducing a hybrid scoto-seesaw model rooted in A 4 discrete modular symmetry leading to several interesting phenomenological implications. The scoto-seesaw framework leads to generation of one mass square difference (∆ m 2atm ) using the type-I seesaw mechanism at the tree level. Additionally, the scotogenic contribution is vital in obtaining the other mass square difference ( ∆ m 2sol ) at the loop level, thus providing a clear interpretation of the two different mass square differences. The non-trivial transformation of Yukawa couplings under the A 4 modular symmetry helps to explore neutrino phenomenology with a specific flavor structure of the mass matrix. In addition to predictions for neutrino mass ordering, mixing angles and CP phases, this setup leads to precise predictions for (cid:80) m i as well as | m ee | . In particular, the model predicts (cid:80) m i ∈ (0 . 073 , 0 . 097) eV and | m ee | ∈ (3 . 15 , 6 . 66) × 10 − 3 eV range; within reach of upcoming experiments. Furthermore, our model is also promising for addressing lepton flavor violations, i.e., ℓ α → ℓ β γ , ℓ α → 3 ℓ β and µ − e conversion rates while staying within the realm of current experimental limits.


INTRODUCTION
The Standard Model (SM) falls short to provide complete understanding of the properties of neutrinos.Instead of being strictly massless as predicted in the SM, neutrinos have been observed to possess extremely small but non-zero masses through neutrino oscillation data [1].This phenomenon has solidified the evidence for neutrino mixing, indicating that at least two neutrinos have non-zero masses [2][3][4][5].Theoretically and experimentally, neutrinos lack right-handed counterparts in the SM, making it unlikely for them to acquire masses through the Higgs mechanism like other charged fermions.However, the presence of a dimension-five Weinberg operator [6][7][8] can offer a viable means for neutrino mass generation.Nevertheless, the origin and flavor structure of this operator remains debatable.Hence, it is crucial to explore scenarios beyond the standard model (BSM) to account for neutrinos' non-zero masses.Numerous models have been proposed in the literature to explain the experimental data from various neutrino oscillation experiments.One popular mechanism is the seesaw mechanism [9][10][11], and other models include radiative mass generation [12,13], extra dimensions [14][15][16][17][18], etc., among others.A common feature in many BSM scenarios that can generate non-zero neutrino masses is the existence of sterile neutrinos, which are gauge singlets under the SM and are often referred to as right-handed neutrinos.They are connected to the standard active neutrinos through Yukawa interactions.These sterile neutrinos' masses and interaction strengths can vary over several orders of magnitude, leading to a wide range of observable phenomena.
In this paper, we introduce a scotogenic extension of the basic canonical seesaw mechanism, which offers a straightforward explanation for the two distinct oscillation scales and their corresponding messengers that have been observed.Initially, we focus solely on generating the atmospheric scale through the exchange of singlet right-handed (RH) neutrinos, as found in the minimal type-I seesaw mechanism.At this point, two neutrinos remain without mass, resulting in the absence of solar neutrino oscillations.However, we subsequently demonstrate that this degeneracy in mass can be resolved through predictable scotogenic-type radiative corrections leading to a hybrid seesaw called "scoto-seesaw" which is the first of its kind because of the implementation of modular symmetry.
The structure of this paper is as follows.In Sec. 2, we outline the theoretical framework of the scoto-seesaw mechanism with discrete A 4 modular flavor symmetry and its appealing features resulting in simple mass structure for the charged leptons and neutral leptons including light active neutrinos and other two types of sterile neutrinos.Then we discuss the light neutrino masses and mixing in this framework.In Sec. 3, a numerical correlational study between observables of the neutrino sector and model input parameters is established.We also present a brief discussion on lepton flavor violation in Sec. 4, and in Sec. 5, we conclude our results.

MODEL FRAMEWORK
This section is curated to discuss the formalism involved in describing the model framework, which is based on modular A 4 symmetry in the supersymmetric context.To ensure the minimal charge assignment of superfields, we have considered zero modular weight for two Higgs doublets H u and H d as well as for charged leptons L ℓ , ℓ c R (ℓ = e, µ, τ ).Higgs doublets are trivial singlets under A 4 symmetry.The charged leptons are singlet representations of A 4 , while L ℓ and ℓ c R are assigned as (1, 1 ′ , 1 ′′ ) and (1, 1 ′′ , 1 ′ ) respectively.This setup leads the charged lepton mass matrix to be diagonal, which can be obtained after applying the A 4 product rule (see App. B).Therefore, leptonic mixing matrix can arise solely from the diagonalization of the neutrino mass matrix.To produce the two different mass scales of neutrino oscillation, we have considered the scoto-seesaw setup [104][105][106][107][108], where neutrino mass will be generated at tree level from type-I seesaw and from scoto-loop [109].For the type-I seesaw mechanism, we have incorporated two new superfields N R 1 and N R 2 with modular weight, k = 4, and they transform as 1 and 1 ′ under A 4 respectively.In the case of the scoto-loop scenario, we have introduced additional superfields f and η with weight k = 5 and k = 3, respectively, and they both transform as trivial singlet (1) of A 4 .We have added another scalar superfield η ′ to cancel the gauge anomaly1 .The charge assignment of the superfields and their weights, as well as relevant modular Yukawas, are summarized in Tab.I.

Scalars Yukawa couplings
Fields  The weight and representation of superfields are chosen so that only the superfields f and η can appear at loop level.This property holds because Yukawas can have only even weight under A 4 modular symmetry.To ensure that there is no mixing between the tree and loop level BSM superfields, we have assigned even and odd weights to the tree and loop level BSM superfields, respectively.The above-discussed condition is basically that H u and η have the same hypercharge that will allow H u to appear at loop level and η to appear at that tree level, but the assignment of even and odd weight for H u and η respectively restrict this choice.This analogy also applies to tree level and loop level fermions N R 1 , N R 2 and f , respectively.The leptonic mass matrices are restricted due to specific charge assignment and weight, as discussed below.

Charged lepton and neutrino mass matrices
The charge assignment of leptons (L ℓ , ℓ c R ) as given in Tab.I, allows us to have the following superpotential mentioned below: The above superpotential leads to a diagonal matrix after the application of the A 4 product rule (see App. B).
Here, v d is VEV of H d and m e , m µ , and m τ are the observed charged lepton masses.
To generate the neutrino masses, we have adopted a scoto-seesaw scenario [59].Hence, we will have contributions from both tree and loop levels 2 in the neutrino sector as shown in Fig. 1, where we have taken superpartners to be very heavy.Considering the charge assignments and weights of superfields given in Tab.I, the most general superpotential allowed by A 4 modular symmetry can 2 To generate neutrino masses through the scotogenic mechanism at loop level one needs mass splitting between real and imaginary components of η (η R and η I ).In SUSY framework the term (η † H u ) 2 which provides mass splitting is not allowed.But one can generate this mass splitting through effective interaction.(see App. A).
be expressed as: For the minimal choice of the parameters in our model, we have considered all α's to be the same i.e. α T and κ 1 , κ 2 >> κ 12 .Hence, the above superpotential modifies as: At the tree level, matrices M D and M R are as follows: where, v u is the VEV of H u .Thus, considering the type-I seesaw formula, one can obtain the light neutrino mass matrix at the leading order as Using the expressions of M D and M R from ( 5), we will have where, 1 ′′ .The neutrino mass term can also be generated at one loop level through the scotogenic process due to the presence of the inert doublet η and the fermion f in the loop.The effective superpotential allowed by A 4 modular symmetry is given by, For the minimal choice, we have taken all β's to be the same i.e. β L , then the above superpotential can be written as: where, β L and κ S are the free parameters.The neutrino masses generated effectively at the one loop level can be expressed as follows: where, and the couplings h i , h j are defined as follows: whereas is the loop function given by: Hence, the neutrino mass matrix at the loop level evolves as follows: Using Eqs. ( 7) and ( 13), we can write the total contribution of neutrino mass matrix, which is given as Thus, scoto-seesaw mechanism provides not just neutrino masses but also explains the two different observed mass square differences.The presence of A 4 modular symmetry has also interesting phenomenological implications which we will explore next.

RESULTS
In order to perform the numerical analysis, we have utilized the global fit neutrino oscillation data at 3σ interval from [110,111] as follows.
NO : ∆m Here, we numerically diagonalize the neutrino mass matrix Eq. ( 14) through the relation 3 ), where M = M ν M † ν and U is a unitary matrix, from which the neutrino mixing angles can be extracted using the standard relations: The numerical analysis has been done by performing a random scan over the input parameters space given in Tab.II, taking GeV.After imposing the observed 3σ limits of solar and atmospheric mass squared differences and further constrained by the mixing angles, the typical range of modulus τ is found to be 0.2 ≲ |Re[τ ]| ≲ 0.5 and 0.85 ≲ Im[τ ] ≲ 1.3, satisfying only the normal ordering (NO).In contrast to the above, the present scenario cannot incorporate the inverted ordering (IO) of neutrino mass as briefly discussed in subsection 3.4.Further, we illustrate the results achieved by performing the numerical scan in the following subsections.
We intend to initiate our analysis by embarking focus on the correlations among modular sectors.
In support of the above, the left panel of Fig. 2     preference for Re[τ ] < 0 is evident when considering the implications of δ CP accordance with latest global-fit data [110,111].Note that the δ CP values in Fig. 5 show a reflection symmetry across δ CP = π value, which is a consequence of mirror symmetry exhibited by Re[τ ] across its value at zero as shown in Fig. 3.

Prediction for Neutrinoless Double Beta (0νββ) Decay
In the context of the neutrinoless double beta decay process, the main contribution arising from the model to this process is due to the exchange of light neutrinos.The half life is proportional to the square of the effective mass |m ee | 2 , where |m ee | is given below in Eq. (17).
In our model, we have a very precise prediction for 0νββ as shown in the left panel of Fig. 6 in green color.The purple region represents the parameter space allowed by the latest global-fit data [110,111] up to 3σ level, whereas constraint on the light neutrino mass arising from the Planck (TT, TE, EE + lowE + lensing + BAO) dataset, which has set an upper bound on the sum of neutrino masses m i < 0.12 eV [113] has been shown by the vertical gray band.Not only the neutrino masses are tightly constrained, but additionally, the Majorana phases are not free but highly correlated with each other, as shown in the right panel of Fig. 6.We also noted how Majorana  and the atmospheric mass squared difference (∆m 2 atm ).The green shaded region corresponds to the 3σ confidence level for ∆m 2 atm , while the horizontal gray band denotes the upper limit on the sum of neutrino masses, m i < 0.12 eV [113].It is worth noting that the constraint imposed by ∆m 2 atm leads to a stringent bound on the sum of neutrino masses, confined within the range of (0.073 − 0.097) eV, thus abiding by the established upper limit of 0.12 eV.However, it is imperative to emphasize that the Euclid mission, launched in July 2023, is expected to further constraint the sum of neutrino masses [118].The right panel of Fig. 7 complements these revelations by suggesting a plausible correlation of the mixing angles θ 23 and θ 12 with the sum of neutrino masses m i .

Inverted Ordering (IO) Case
Following a modular setup in a scoto-seesaw scenario, it becomes evident that our framework encounters challenges in explaining the inverted ordering (IO) of neutrino mass.This limitation becomes apparent by looking at Fig. 8.We have shown a robust correlation between two mass square differences of neutrino oscillation, namely ∆m 2 sol and ∆m 2 atm in Fig. 8, while imposing 3σ constraint of all three mixing angles.The green shaded region represents the 3σ allowed values of ∆m 2 sol and ∆m 2 atm .Unfortunately, we can not satisfy both the mass square difference constraints simultaneously, as depicted in Fig. 8. Consequently, our model does not favor IO of neutrino mass.

LEPTON FLAVOR VIOLATIONS
In this section we explore our model's prediction for various lepton flavor violating processes.substantial resources to enhance their sensitivity and improve the current limit on the branching ratio BR, denoted as BR(µ → eγ).The present limit, established by the MEG collaboration [119] is less than 4.2 × 10 −13 , which could reach to the sensitivity of 6 × 10 −14 by the upgraded MEG i.e., MEG-II experiment [120].In the existing theoretical framework, the process of lepton flavor-violating decay µ → eγ takes place at the one loop level through standard Yukawa interactions whose associated Feynman diagrams can be found in Fig. 9.The BR for the rare decay µ → eγ is described in [121].
where, G F ≈ 10 −5 GeV −2 is the Fermi constant, α being the electromagnetic fine structure constant and A 1 is the dipole contribution, expressed as Here, Y 1 , Y 1 ′′ being the modular Yukawa couplings, x = with m + η being the mass of η + which we keep fixed at 400 GeV throughout our computation and G 1 (x) is the loop function In Fig. 10, we have depicted how the BR of µ → eγ varies with respect to the mass of the fermion f .It is evident that as the value of β L increases for a specific M f value, the BR also exhibits an enhancement.

Results for µ → 3e
The three body LFV decay processes ℓ α → ℓ β ℓ β ℓ β can proceed through penguin and box diagrams, which are shown in Fig. 11.Our model supports the µ → 3e decay and the corresponding branching ratio can be expressed as [121,122], where, r µe = mµ me with F RR and F RL arising from the Z-boson contribution which are given as [122]  with In this context, M Z is the mass of the Z boson, g 2 represents the SU (2) L gauge coupling, and θ W stands for the weak mixing angle.The coefficient F Z is given below as It is important to note that these terms are subjected to suppression by the masses of the charged leptons, m e and m µ .The form factor A 1 is dipole contribution and is given in Eq. (19).The other form factor A 2 , given as can be generated by non-dipole contribution, whereas B box , induced by box diagrams, is given as The loop functions G 2 (x), D 1 (x, x), and D 2 (x, x) can be given as [121] G In the limit x → 1, these loop functions have the values The current upper limit of BR(µ → 3e) is 1 × 10 −12 , while the upcoming Mu3e experiment [123] aims to achieve a remarkable sensitivity level of O(10 −16 ), by adopting a phased strategy.Fig. 12 illustrates how the BR of µ → 3e changes as a function of the fermion mass (M f ).Analogous to µ → eγ process, the branching ratio increases with β L for any specific value of M f .FIG. 13: Penguin contributions to µ − e conversion in nuclei.

µ − e conversion in Nuclei
The most stringent limitation on Lepton Flavor Violation (LFV) decays is currently represented by the µ → eγ process.Nonetheless, we anticipate an enhanced level of sensitivity in the future, particularly from the µ − e conversion within the nucleus.Several experiments, such as Mu2e, DeeMe, COMET, and PRISM/PRIME [124][125][126], are currently at their peak aiming to establish an upper limit of 4.3 × 10 −14 (for Titanium nucleus) and aspire to achieve future sensitivity as low as 10 −18 .In the following, we provide a brief overview of the contribution arising from µ − e conversion in the nucleus, as depicted in Fig. 13.The conversion rate for µ − e within the nucleus is presented as follows: LV + g LV + g RV + g RV + g Here, the proton and neutron numbers inside the nucleus are expressed by Z and N , Z eff represents the effective atomic charge and can be found in [127] for different nuclei, F p & Γ capt denote the nuclear matrix element and the total muon capture rate respectively.These parameters can be determined based on the choice of the nucleus and can be found in Ref. [128,129].Other parameters used in the above equation are provided below, where X = L, R and K = V, S: FIG. 14: The left plot represents the variation of the conversion ratio of µ − e for Ti nucleus with the fermion mass M f .However, the right plot elaborates the correlation between the conversion rate and BR for µ → eγ, with vertical and horizontal bands representing their respective upper bounds.
The numerical values of G K coefficients are taken from [129][130][131].Here, g XK(q) being the effective couplings, given as follows where, ) is generated from photon penguins, Q q represents electric charge of the corresponding quark, and Further, g q L and g q R can be can be retrieved from Ref. [122].We compute the conversion rate of µ − e in Titanium ( 4822 Ti) nucleus (relevant details can be found in [129]).The left panel of Fig. 14 projects the conversion rate versus fermion mass M f , and the right panel signifies its correlation with BR(µ → eγ).The horizontal dashed line corresponds to the upper bound [132].

Comments on Muon g − 2
In April 2021, Fermilab made a groundbreaking announcement with its inaugural measurement result [133] concerning the muon's anomalous magnetic dipole moment (referred to as (g−2) µ ).When this result is combined with the findings from BNL [134], it reveals a substantial 4.2σ deviation from the Standard Model prediction [135][136][137].This remarkable deviation has significantly strengthened the confidence of particle physicists in their pursuit of uncovering new physics that extends beyond the Standard Model.In this context, we have identified three potential candidates, denoted as N i , f , and η, in our model which could account for this anomaly.The individual contribution from N i can be found in Ref. [138], revealing that due to the substantial mass of N i , its contribution is exceedingly small.Furthermore, delving deeper into the analysis, the contributions of both f and η can be found from Ref. [139][140][141][142], demonstrating that they contribute negatively to the muon's anomalous magnetic moment.Moreover, our model is rooted in the Supersymmetric context and includes H u and H d , which can be viewed as a two-Higgs doublet model (2HDM) of type-II.However, it is worth noting that the contribution of this specific type of 2HDM does not align with experimental limitations [143][144][145].

Comments on Dark Matter
In our model, we have employed scoto-seesaw mechanism [59], wherein neutrino masses are generated through type-I seesaw at tree level and from scotogenic mechanism [109] at one loop level.
The particles running inside the scoto-loop (η and f ) can be potential DM candidate depending upon their masses.Here, the role of Z 2 dark symmetry plays by the modular weight to stabilize the DM candidate.The dark sector particles, i.e., scalar η and fermion f , have odd modular weights while all other particles have even modular weights.Since Yukawa couplings can have only even weight [63], the dark sector particles do not mix with other particles.Therefore, the lightest of scalar η and fermion f will be a stable particle hence, a viable candidate for DM.The underlying dark matter phenomenology, in principle, aligns with the scoto-seesaw mechanism [59], where both the fermion and neutral scalar can be a DM candidate, hence, we do not discuss it in this paper.

CONCLUSION
The primary objective of this paper is the introduction of A 4 modular symmetry in the context of neutrino phenomenology, employing the scoto-seesaw framework to explore its unique implications.
We have achieved the generation of neutrino mass both at the tree level, utilizing right-handed neutrino superfields N R i , and at the one loop level, involving a fermion superfield f and an inert scalar doublet superfield η.Notably, the BSM fermions N R i , f and the inert scalar η are singlets under the A 4 symmetry, characterized by modular weights of 4, 5 and 3, respectively.To maintain the invariance of the superpotential, we have harnessed higher-weight Yukawa couplings, which also are singlets under the A 4 symmetry, with modular weights of 4, 8, and 10, and are expressed in terms of lower-weight modular coupling, denoted as Y where F 1l and F 2l are the loop function given by: The symmetry group A 4 is an even permutation group of four objects.It has 4!/2=12 elements and can be generated by two generators S and T obeying the relations: The group has four irreducible representations: three singlets 1, 1 ′ , 1 ′′ , and a triplet 3. The product rules for the singlets and triplets are: where, 3 S(A) denotes the symmetric (anti-symmetric) combination.In the complex basis where T is a diagonal matrix, we have following representation for S and T , The scalar fields ′ kinetic term is as follows which doesn't change under the modular transformation, and eventually, the overall factor is absorbed by the field redefinition.Thus, the Lagrangian should be invariant under the modular symmetry.

(C13)
Due to the constraint given in Eq. (C11), we see that Y In general, the dimension (d k ) of modular forms of the level 3 and weight k is k + 1 [63,149].The

FIG. 1 :
FIG. 1: Neutrino mass generation from "scoto-seesaw" mechanism.The left diagram corresponds to the tree level seesaw while the right diagram represents the effective one loop scotogenic contribution to neutrino masses.
illustrates a plausible correlation between Im[τ ] and Re[τ ].Furthermore, the middle and right panel of Fig. 2 depicts the plausible ranges for modular Yukawa couplings, as well as their associations with Re[τ ] and Im[τ ].

3. 1 .
Fig.3(region in green color is 3σ allowed values of θ 23 and θ 12 ).Proceeding further, in Fig.3, one can clearly see that in our model, mixing angles have symmetric distribution around the origin for

FIG. 6 :
FIG. 6: The left panel corresponds to the effective neutrino mass |m ee | [eV] as a function of the lightest neutrino mass m lightest [eV] predicting NO for neutrino mass while the right panel shows the correlation between Majorana phases.

FIG. 7 :FIG. 8 :
FIG. 7: The left panel showcases the correlation between the sum of neutrino masses m i with atmospheric mass square difference ∆m 2 atm .While the right panel depicts the correlation of mixing angles θ 23 and θ 12 with m i .

FIG. 12 :
FIG. 12: Here, we have summarized the anticipated BR values for the µ → 3e decay.These values are computed based on the contributions outlined in Fig. 11.The color bar positioned on the right side represents the changing values of the free parameter β L .

TABLE I :
Particle content and modular Yukawa couplings of the model and their charges under SU (2) L × U (1) Y × A 4 , where k I is the number of modular weight.

TABLE II :
Ranges of the parameters used for the numerical scan.