Modular flavored dark matter
Alexander Baur Mu-Chun Chen V. Knapp-Perez Saul Ramos-Sanchez





 




UCI-TR-2024-12


TUM-HEP 1520/24

















Alexander Baur^a,b[alexander.baur@tum.de], 
Mu–Chun Chen^c,[muchunc@uci.edu], 
V. Knapp–Pérez^c,[vknapppe@uci.edu],

Saúl Ramos–Sánchez^a,[ramos@fisica.unam.mx],


^a Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México


^b Physik Department, Technische Universität München,
James-Franck-Straße 1, 85748 Garching, Germany


^c Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575 USA









§ ABSTRACT 
Discrete flavor symmetries have been an appealing approach for explaining the observed flavor structure, which is not justified in the SM. Typically, these models require a so-called flavon field in order to give rise to the flavor structure upon the breaking of the flavor symmetry by the VEV of the flavon. Generally, in order to obtain the desired vacuum alignment, a flavon potential that includes additional so-called driving fields is required. On the other hand, allowing the flavor symmetry to be modular leads to a structure where the couplings are all holomorphic functions that depend only on a complex modulus, thus greatly reducing the number of parameters in the model. We show that these elements can be combined to simultaneously explain the flavor structure and DM relic abundance. We present a modular model with flavon vacuum alignment that allows for realistic flavor predictions while providing a successful fermionic DM candidate.












§ INTRODUCTION



The origin of the mass hierarchies and mixings among the three generations of fermions is unexplained in the SM. One possible  solution to address this so-called flavor puzzle
is the introduction of traditional flavor symmetries that allow for transformations among fermions of different flavors. 
These symmetries are independent of moduli and are known to provide explanations for the flavor structure of both the lepton and quark
sectors <cit.>.
To achieve these results, models endowed with traditional flavor symmetries require the introduction
of two kinds of extra SM singlet scalars: i) flavons, whose VEVs are responsible for the non-trivial flavor structure of fermions,
and ii) driving fields <cit.>
that help to shape a suitable potential for obtaining the desired VEV pattern. The traditional flavor
symmetry framework based on non-Abelian discrete symmetries has also proven helpful for DM, where the flavor
symmetry plays the role of a stabilizer symmetry <cit.>. Furthermore, (not necessarily discrete)
flavor symmetries can be successfully combined to explain both DM and flavor anomalies
from SM decays <cit.>.

Another promising approach to explain the flavor parameters without introducing many scalars is provided by so-called modular flavor symmetries <cit.>. This approach has produced
good fits for leptons <cit.>
and quarks <cit.>.
In this scenario, instead of depending on flavon VEVs, Yukawa couplings are replaced
by multiplets of modular forms depending on the half-period ratio τ, which can be
considered a complex modulus. The basic problem then reduces to explaining why
τ stabilizes at its best-fit value ⟨τ⟩, 
for which being close to the symmetry-enhanced points τ = ,  ^2π/3,  ∞ can be advantageous <cit.>. Although this scheme has the potential to avoid the need for any
additional scalar field, the presence of flavons in addition to the modulus can be useful too (see, e.g., Model 1 of <cit.> and <cit.>). Similarly to the traditional flavor case, modular flavor symmetries can also
serve as stabilizer symmetries for DM candidates <cit.>

Modular flavor symmetries arise naturally in top-down constructions, such as magnetized extra dimensions <cit.> or heterotic string orbifolds <cit.>, where they combine with traditional flavor symmetries, building an eclectic flavor group <cit.>. Remarkably, these top-down approaches provide a natural scheme where not only realistic predictions arise <cit.>,
but also the modulus τ can be stabilized close to symmetry-enhanced points <cit.>.


Motivated by these observations, we propose a new supersymmetric model combined with
modular flavor symmetries, which simultaneously accomplishes the following:

   
  * Addressing the flavor puzzle, specifically the origin of the lepton masses and mixing parameters;
   
  * Achieving the vacuum alignment for the flavons;
   
  * Providing a suitable DM candidate with the correct observed DM abundance,
         Ω_DM = 0.265(7) <cit.>.


In order to tackle these issues, we propose a simple supersymmetric model based on a Γ_3≅ A_4 modular flavor symmetry. The model resembles Model 1 of <cit.>, where the neutrino masses arise from a Weinberg operator and the charged-lepton Yukawa couplings are given by the VEV of a flavon. It also resembles the proposal of <cit.>, which studies a DM candidate in a non-modular A_4 model without fitting flavor parameters. In our model, the flavon potential is fixed by the flavor symmetry together with a 1_R2 symmetry, which determines the couplings between the driving field and flavon superfields. The model gives a very good fit to the leptonic flavor parameters, with a low value of χ^2.
Finally, we identify a Dirac fermion composed by the Weyl fermionic parts of both the driving field and flavon superfields; we perform a parameter scan for a correct DM abundance.
The goal of this model is to present a “proof of principle” that driving fields
in modular supersymmetric flavor models can account for both the flavon VEV ⟨ϕ⟩ and DM.

Our paper is organized as follows. In <Ref>, we review the basics of modular symmetries and its application to solving the flavor puzzle. In <Ref>, we define our model. In <Ref>, we present the numerical fit for the lepton flavor parameters. In <Ref>, we analyze the relevant terms for DM production. We also argue why we need freeze-in (as opposed to the more traditional freeze-out) mechanism for our model to work. We then present a parameter scan for the available parameter space for our DM candidate.
Finally, in <Ref> we summarize our results and future directions for further constraints.




§ MODULAR SYMMETRY





 §.§ Modular groups and modular forms



The modular group Γ:=2,  is given by

    Γ = {[ a b; c d ] |   a,b,c,d ∈ &  ad -bc = 1 }  ,

and can be generated by

    S  = [  0  1; -1  0 ]     and     
    T  = [ 1 1; 0 1 ] ,

which satisfy the general presentation of Γ, ⟨ S,T | S^4 = (ST)^3 = , S^2T=T S^2⟩.
The principle congruence subgroups of level N of Γ are defined as

    Γ(N)  := {γ∈Γ | γ= N } ,

which are infinite normal subgroups of Γ with finite index. We can also define the
inhomogeneous modular group Γ := Γ / {±}≅PSL(2,) and
its subgroups Γ(N) with Γ(2) := Γ(2)/{±} and
Γ(N) := Γ(N) for N>2 (as - does not belong to Γ(N)).
An element γ of a modular group acts on the half-period ratio,  modulus τ, as

    γ τ = a τ + b/cτ + d ,    where  τ∈ℋ

and ℋ is the upper complex half-plane,

    ℋ := {τ∈C  |  τ > 0 }  .

Modular forms f(τ) of positive modular weight k and level N are complex functions of τ, holomorphic in ℋ, and transform as

    f(τ)  γ⟶  f(γ τ )  =  (cτ + d)^k f(τ) ,     γ∈Γ(N) .

In this work, we restrict ourselves to even modular weights, k∈2N, although it is known that modular weights can be odd <cit.> or fractional <cit.> in certain scenarios.
Interestingly, modular forms with fixed weight k and level N build finite-dimensional vector spaces,
which close under the action of Γ. It follows then that they must build representations of a finite
modular group that, for even modular weights, result from the quotient

    Γ_N := Γ / Γ(N) .

Then, under a finite modular transformation γ∈Γ, modular forms of weight k are n-plets
(which are called vector-valued modular forms <cit.>)
Y(τ):=(f_1(τ),f_2(τ),…,f_n(τ))^T of Γ_N, transforming as

    Y(τ)  γ⟶  Y(γ τ )  =  (cτ+d)^k ρ(γ) Y(τ) ,

where ρ(γ)∈Γ_N is a representation of γ.





  §.§.§ Finite modular group Γ_3≅ A_4 



As our model is based on Γ_3≅ A_4, let us discuss some general features of this group and its modular forms.
Γ_3 is defined by the presentation

    Γ_3 = ⟨ S,T  |  S^2  = (ST)^3  =  T^3  = ⟩ .

It has order 12 and the irreducible representations (in the complex basis) are given in <Ref>.

Besides 1^a⊗1^b=1^c with c=a+b 3 and 1^a⊗3=3, where a,b,c=0,1,2 count the number of primes,
we have the nontrivial product rule 3⊗3=1⊕1'⊕1”⊕3_S⊕3_A,
where S and A stand respectively for symmetric and antisymmetric. Considering two triplets ρ = ( ρ_1 , ρ_2 , ρ_3 )^T
and ψ = (ψ_1 , ψ_2 , ψ_3 )^T, in our conventions the Clebsch-Gordan coefficients of ρ⊗ψ are

    (ρ⊗ψ )_1   = ρ_1ψ_1 + ρ_2ψ_3 + ρ_3ψ_2 ,     
      (ρ⊗ψ )_1'   = ρ_1ψ_2 + ρ_2ψ_1 + ρ_3ψ_3 ,
    
      (ρ⊗ψ )_1”   = ρ_1ψ_3 + ρ_2ψ_2 + ρ_3ψ_1 , 
    
      (ρ⊗ψ )_3_S   = 1/√(3)[ 2 ρ_1ψ_1 - ρ_2ψ_3 - ρ_3ψ_2; 2 ρ_3ψ_3 - ρ_1ψ_2 - ρ_2ψ_1; 2 ρ_2ψ_2 - ρ_3ψ_1 - ρ_1ψ_3 ],   
      (ρ⊗ψ )_3_A   = [ ρ_2 ψ_3 - ρ_3 ψ_2; ρ_1 ψ_2 - ρ_2 ψ_1; ρ_3 ψ_1 - ρ_1 ψ_3;                   ].


The lowest-weight modular forms of Γ_3 furnish a triplet Y=(Y_1,Y_2,Y_3)^T
of weight k_Y=2, whose components are given by <cit.>

    Y_1(τ )     = /2π[ η'(τ/3)/η(τ/3) + η'(τ +1 /3)/η(τ +1 /3) + η'(τ +2/3)/η(τ +2 /3) - 27 η'(3τ )/η (3 τ )]  , 
    
    Y_2 (τ )     =   -/π[ η'(τ/3)/η(τ/3) + ω^2η'(τ +1 /3)/η(τ +1 /3) + ωη'(τ +2/3)/η(τ +2 /3) ] , 
    
    Y_3 (τ )     =   -/π[ η'(τ/3)/η(τ/3) + ωη'(τ +1 /3)/η(τ +1 /3) + ω^2 η'(τ +2/3)/η(τ +2 /3) ] ,

where η( τ ) is the so-called Dedekind η function

    η ( τ )  =  q^1/24∏_n = 1^∞(1 - q^n )
          with     q := ^2πτ .

Higher-weight modular forms can be constructed
from the tensor products of the weight 2 modular forms given in <Ref>.





 §.§ Modular supersymmetric theories



We consider models with 𝒩=1 global SUSY, defined by the Lagrange density

    ℒ = ∫^2 θ ^2θ̅  K(Φ, Φ) + (∫^2 θ  W (Φ ) + h.c.) ,

where K (Φ, Φ ) is the Kähler potential, W (Φ) is the superpotential, and
Φ denotes collectively all matter superfields φ^i of the theory and the modulus τ. 


Under an element of the modular symmetry γ∈Γ, τ transforms according to <Ref>, and matter superfields
are assumed to transform as

    φ^i   γ  (cτ + d)^-k_iρ_i(γ) φ^i ,

where k_i are also called modular weights of the field φ^i, which transform as Γ_N multiplets. Modular weights k_i are not restricted to be positive integers because
φ^i are not modular forms. Analogous to <Ref>, the matrix ρ_i(γ) is a representation of the
finite modular flavor group Γ_N.

For simplicity, we assume a minimal Kähler potential[In principle, there could be further
terms in the Kähler potential with an impact on the flavor predictions <cit.>, which are ignored here.]
of the form

    K (Φ, Φ )  =  -log( -τ + τ̅ ) + ∑_i(-τ + τ̅ )^-k_i |φ^i |^2  .

Making use of <Ref>, we see that K(Φ, Φ) transforms
under a modular transformation γ∈Γ as

    K (Φ, Φ )  γ   K (Φ, Φ ) + log (cτ +d ) + log (cτ̅ +d ) .

Thus, realizing that the Kähler potential is left invariant up to a global supersymmetric Kähler transformation,
in order for the Lagrange density of <Ref> to be modular invariant, we need the superpotential
to be invariant under modular transformations, i.e.

    W (Φ )  γ  W (Φ )  .

The superpotential W (Φ) has the general form

    W (Φ )  = μ_ij(τ )φ^iφ^j + Y_ijk(τ)φ^iφ^jφ^k + G_ijkℓ(τ)φ^iφ^jφ^kφ^ℓ  ,

where μ_ij(τ ), Y_ijk(τ ) and G_ijk(τ ) are modular forms of level N.
Because of <Ref>, each term of <Ref> must be modular invariant.
Let us illustrate how we can achieve this by taking the trilinear coupling
Y_ijk(τ)φ^iφ^jφ^k. The Yukawa coupling Y_ijk transforms
under a modular transformation γ∈Γ as

    Y_ijk(τ )  γ  (cτ + d)^k_Yρ_Y (γ) Y_ijk(τ )  ,

where k_Y is the even integer modular weight of the modular form Y_ijk(τ). Then, for Y_ijk(τ)φ^iφ^jφ^k
to be invariant and using the superfield transformations of <Ref>,
we must demand that k_Y = k_i+k_j + k_k and that the product
ρ_Y⊗ρ_i⊗ρ_j⊗ρ_k contains an invariant singlet.


Since we shall be concerned with SUSY breaking, let us briefly discuss the soft-SUSY breaking terms in the Lagrange density.
They are given by

    ℒ_soft =  -1/2( M_aλ̂^aλ̂^a + h.c)
                                  - m̃_i^2φ̅̂̅^iφ̂^i
                                  - (A_ijkφ̂^iφ̂^jφ̂^k + B_ijφ̂^iφ̂^j + h.c.)  ,

where M_a are the gaugino masses, λ̂^a are the canonically normalized gaugino fields, and m̃_i are the soft-masses.
We use a notation, where φ̂^i stands for both the canonically normalized chiral superfield and its scalar component <cit.>. We do not assume any specific source of SUSY breaking, and thus the parameters in <Ref> are free, in principle. However, in our model B-terms are forbidden at tree level and the A-terms do not play a role in DM production at tree-level, see <Ref>.




§ DARK MODULAR FLAVON MODEL




We propose a model of leptons, governed by the modular flavor group
Γ_3≅ A_4. In our model, the lepton doublets L transform as a flavor triplet and charged-lepton singlets E^c_i transform as three distinct singlets under Γ_3. The particle content of our model is summarized in <Ref>. 


In our model, neutrino masses arise from the Weinberg operator

    W_ν = 1/Λ( H_u L H_u L Y(τ ) )_1 ,

where Λ is the neutrino-mass scale and Y(τ) is the modular-form
triplet Y=(Y_1,Y_2,Y_3)^T of weight 2 given by <Ref>. 
Note that there exists no other modular multiplet of weight 2 in Γ_3≅ A_4. Hence, under our assumptions, the neutrino sector obtained from <Ref> is highly predictive as it only depends on the parameter τ, the VEV v_u of H_u, and Λ.

The charged-lepton superpotential at leading order is

    W_CL = α_1/Λ_ϕ E_1^c H_d (L ϕ_3)_1+ α_2/Λ_ϕ E_2^c H_d (L ϕ_3)_1'+α_3/Λ_ϕ E_3^c H_d (L ϕ_3)_1” ,

where α_i, i=1,2,3, are dimensionless parameters, Λ_ϕ denotes the flavor breaking scale, and the subindices refer to the respective Γ_3 singlet components of tensored matter states in the parentheses. The charged-lepton mass matrix can be determined by the Γ_3 triplet flavon VEV ⟨ϕ_3⟩ as in Model 1 of <cit.>. However, we have taken a different value of ⟨ϕ_3⟩, which eventually leads to a better fit.


The flavon superpotential is given by

    W_ϕ   = Λ_ϕβ_1  ζ_3 ϕ_3 + β_2ζ_3 ϕ_3 ϕ_3    +  β_3/Λ_ϕζ_3 ϕ_3 ϕ_3 ϕ_3 + Λ_ϕβ_4 ζ_1”ϕ_1' + β_5/Λ_ϕζ_1”ϕ_1'ϕ_3ϕ_3 
        + β_6/Λ_ϕζ_3 ϕ_3 ϕ_1'ϕ_1' ,

where β_i, i=1,… ,6, are dimensionless couplings. This superpotential gives rise to the desired VEV pattern with driving superfields ζ_r, r = 1”, 3, and an extra flavon ϕ_1', where the subindices label the respective A_4 representations. To fix the flavon superpotential, we impose a symmetry 1_R2, similarly to <cit.>. As usual, the 1_R R-symmetry forbids the renormalizable terms in the superpotential that violate lepton and/or baryon numbers.
We remark that the flavon superpotential <Ref> is important for both finding the correct vacuum alignment for the flavons, as discussed below, and for identifying a viable DM candidate as described in <Ref>.


The flavon VEV is then attained by demanding that SUSY remains unbroken at a first stage, i.e. we require vanishing F-terms.
Recall that the F-term scalar potential in a global supersymmetric theory is given schematically by

    V  =  F^iK_iF̅^  ,

where

    F^i  = -∂ W/∂Φ^i ,
        F̅^ = F^j*    and    
    K^i = ( K_i)^-1 = (-τ + τ)^k_iδ_ij̅  .


Recall that a superfield Φ can be expanded in its components as <cit.>

    Φ   = Φ (x) - θσ^μθ̅∂_μΦ (x) - 1/4θ^2 θ̅^2∂^2 Φ  (x) + √(2)θψ_Φ (x)  
       + /√(2)θ^2∂_μψ_Φ (x)σ^μθ̅+θ^2 F_Φ (x) ,

where we have used the notation that Φ represents both the superfield and its scalar component, ψ_Φ is the fermionic component and F_Φ is the F-term.
For SUSY to be preserved, we must have ⟨ F_Φ⟩ = 0. We assume that the only possible sources of SUSY breaking are given either by τ or a hidden sector. Thus, we demand ⟨ F_φ^i⟩ = 0, for all i, where φ^i represents the matter fields in our model (cf. <Ref>).
We solve these F-term equations at the VEV's of the flavons ϕ_3 and ϕ_1' and Higgs fields,

    ⟨ϕ_3 ⟩  =: v_3 ( 1, a, b)^T,     ⟨ϕ_1'⟩  =: v_1' ,     ⟨ H_u ⟩  =: v_u     and    ⟨ H_d ⟩  =: v_d  .

where we assume a,b,v_3,v_1'∈R. All F-term equations are trivially satisfied except the ones corresponding to the driving fields given by 

    ⟨ F_ζ_3i⟩ =  0     and    ⟨ F_ζ_1”⟩ =  0 ,

where ζ_3i, i=1,2,3, are the three components of the triplet. Thus, we obtain the following relations

    β_2     =  c_2(a,b)  β_1 Λ_ϕ/v_3 ,          β_3     =  c_3(a,b) β_1Λ_ϕ^2 /v_3^2 ,
    β_5     =  c_5(a,b) β_4Λ_ϕ^2 /v_3^2 ,    β_6     =  c_6(a,b) β_1Λ_ϕ^2 /v_1'^2 ,

where c_i(a,b), for i=2,3,5,6, are coefficients that depend only on the values of a,b. The numerical values of a,b dictate the charged lepton mass matrix. Hence, they are determined by the fit to the flavor parameters that we do in <Ref>. Through the flavor parameter fit, we have found values of ⟨ϕ_3⟩, ⟨ϕ_1'⟩ that satisfy simultaneously <Ref>, thus yielding vacuum alignment.

Finally, we assume that the flavon VEV scale from <Ref> is below Λ_ϕ, such that

    v_3   =  v_1' =  0.1 Λ_ϕ .

Furthermore, we identify the DM candidate as a Dirac fermion built as a combination of the Weyl components of ζ_i and ϕ_i with the scalar component of the flavon ϕ_i serving as a mediator.
The parameters in <Ref> shall play an important role in finding the current DM abundance since they set the couplings in <Ref>.




§ FLAVOR FIT



Having defined our model of modular flavored dark matter, we next assess its capability to reproduce the experimentally observed charged-lepton masses, neutrino squared mass differences, and the mixing parameters of the PMNS matrix
while providing predictions for yet undetermined observables, such as the three absolute neutrino masses, the Dirac phase, and the two Majorana phases.

The explicit neutrino mass matrix can be determined from <Ref>. By calculating the tensor products to obtain the symmetry invariant part,
we find that the neutrino mass matrix is predicted to be

    M_ν = v_u^2/Λ[ 2 Y_1  -Y_3  -Y_2;  -Y_3 2 Y_2  -Y_1;  -Y_2  -Y_1 2 Y_3 ].

The charged-lepton mass matrix M_CL arises from <Ref>. Substituting the flavon VEVs as defined in <Ref> and calculating the tensor products,
we arrive at

    M_CL = v_d v_3/Λ_ϕ[   α_1 α_1 b α_1 a; α_2 a   α_2 α_2 b; α_3 b α_3 a   α_3 ] =  0.1 v_d[   α_1 α_1 b α_1 a; α_2 a   α_2 α_2 b; α_3 b α_3 a   α_3 ]  ,

where we have used <Ref>. The values of v_u and v_d are determined by the Higgs VEV, v = √(v_u^2 + v_d^2) =246 GeV, and tanβ = v_u/v_d, which we assume to be tanβ = 60. The neutrino mass scale is determined by Λ, while we choose α_3 to set the mass scale of charged leptons. By using the standard procedure (see e.g. ref. <cit.>), one arrives at the lepton masses and the PMNS mixing matrix.
For our model, the resulting 12 flavor observables depend on 6 real dimensionless parameters τ, τ, a, b, α_1/α_3, and α_2/α_3, as well as two dimensionful overall mass scales v_u^2/Λ and 0.1 v_d α_3.





To show that the model can accommodate the observed flavor structure of the SM lepton sector,
we scan its parameter space and compare the resulting flavor observables to experimental data,
with the best-fit values shown in <Ref>.
As an approximate measurement of the goodness of our fit, we introduce a χ^2 function

    χ^2  = ∑_iχ_i^2 ,

consisting of a quadratic sum of one-dimensional chi-square projections for each observable.
Here, we assume that the uncertainties of the fitted observables are independent of each other
and do not account for the small correlations among experimental errors of sin^2θ_23 and other quantities.
For the mixing angles and the neutrino squared mass differences, we determine the value of χ_i^2
directly from the one-dimensional projections determined by the global analysis NuFIT v5.3 <cit.>,
which are available on their website. This is necessary
to account for the strong non-Gaussianities in the uncertainties of the mixing parameters in the PMNS matrix.
For these observables, we refrain from considering corrections from renormalization group running,
given that their contribution is expected to be small compared to the size of the experimental errors.
For the charged-lepton masses, we determine the value of χ_i^2 by

    χ_i  = μ_i,exp. - μ_i,modelσ_i ,

where μ_i,model denotes the resulting value for the ith observable of the model,
while μ_i,exp. and σ_i refer to its experimentally observed central value
and the size of the 1σ uncertainty interval given in <Ref>, respectively.
The total value of χ^2 for all considered observables may then be interpreted to indicate an
agreement with the experimental data at a √(χ^2) σ confidence level (C.L.).


To scan the parameter space of the model and minimize the χ^2 function, we use the dedicated code
 <cit.>. We find that the model is
in agreement with current experimental observations.
The best-fit point in the parameter space of our model is at

    τ   = -0.0119 +   1.005  ,            a   =-0.392 ,     b   =0.380 ,     0.1 α_3 v_d     = 0.130 GeV
    α_1/α_3   = -22.0 ,    α_2/α_3   =4.78 10^-3 ,    v_u^2/Λ   =0.0221 eV ,

where we obtain χ^2=0.08, meaning that all resulting observables are within their experimental 1σ interval, cf. also <Ref>.
In <Ref>, we present the regions in moduli space that yield results with χ^2≤ 25.





For the specific values given in <Ref>, the relations among the couplings of the flavon superpotential of <Ref> read

    β_2 = 4.27  β_1  ,     β_3 = 83.5 β_1 ,     β_5 = 142 β_4 ,     β_6 = 121 β_1 .

Any values of β_1 and β_4 then solve the F-term equations of the driving fields, cf. <Ref>, and ensure the specific vacuum alignment of <Ref>. 
The resulting observables at the best-fit point given in <Ref> lie well within the 1σ intervals of the experimental data shown in <Ref>, and read

    m_e/m_μ   = 0.00473 ,     m_μ/m_τ   = 0.0451 ,      y_τ    = 0.795  , 
    sin^2θ_12   = 0.306 ,              sin^2θ_13   = 0.02231 ,      sin^2θ_23   = 0.568 ,
    δ_^ℓ/π   = 1.52  rad  ,    η_1/π   = 1.41   rad,    η_2/π   = 0.351   rad  ,
    
    m_1    = 49 meV ,     m_2    = 50 meV ,     m_3    = 0.75 meV .

Moreover, the resulting sum of neutrino masses, the neutrino mass observable in ^3H beta decay, and the effective neutrino mass for neutrinoless double beta decay, are

    ∑ m_i = 100 meV ,     
    m_β = 50 meV ,     and    
    m_ββ = 48 meV ,

which are consistent with their latest experimental bounds
∑ m_i < 120 meV <cit.>, m_β < 800 meV <cit.>, and m_ββ < 156 meV <cit.>.
It is to be noted that our predicted value of the effective neutrino mass m_ββ is challenged by experimental bounds determined with certain nuclear matrix element estimates <cit.>, as illustrated in <Ref>.


We remark that the model can be consistent with both octants of θ_23, while only being compatible with Dirac -violating phases in the range of 1.36 < δ_^ℓ < 1.55 at a 3σ C.L., as shown in <Ref>.
Moreover, the inverted-ordering neutrino masses are predicted to lie within the narrow ranges

    48 meV < m_1 < 50 meV ,   
    49 meV < m_2 < 51 meV ,   
    0.72 meV < m_3 < 0.78 meV ,

at a 3σ C.L. The numerical analysis suggests that the model prefers a neutrino spectrum with inverted ordering.
For a normal-ordering spectrum we only obtain a match with experimental data just barely in the 3σ interval with χ^2≈ 7.









§ DM ABUNDANCE



Let us start by identifying the DM candidate. Since the flavon field only couples to charged leptons,
the phenomenological implications for DM in our model are determined by the charged-lepton Yukawa interactions and the flavon potential.
The interactions between the SM charged leptons and the flavon scalar ϕ_3 are given by

    ℒ_CL    =  -∑_i=1^3v_d/2α_i ψ_E_i^C(ψ_Eϕ_3)_1 +h.c 
        ⟶  -∑_i=1^3v_d/2α_i  ψ_E_i^C(ψ_E(ϕ_3  +v_3 (1,a,b)^T))_1 +h.c. ,

where ψ_E denotes the fermionic part of the charged-lepton component of L. In the last line, we develop the flavon ϕ_3 in the vacuum with its VEV given by v_3 (1,a,b)^T, see <Ref>. From the leading flavon superpotential terms in <Ref>,[We ignore the terms suppressed by Λ_ϕ.] we obtain

    ℒ_ϕ    =  -1/2Λ_ϕβ_1 ( ψ_ζ_3ψ_ϕ_3)_1 -1/2β_2 (ψ_ζ_3ψ_ϕ_3ϕ_3)_1 -1/2Λ_ϕβ_4 (ψ_ζ_1”ψ_ϕ_1')_1+h.c.
        →  -1/2Λ_ϕβ_1 ( ψ_ζ_3ψ_ϕ_3)_1 -1/2β_2 (ψ_ζ_3ψ_ϕ_3(ϕ_3  +v_3 (1,a,b)^T))_1
             -1/2Λ_ϕβ_4 (ψ_ζ_1”ψ_ϕ_1')_1+h.c.
        =  -1/2Λ_ϕβ_1 ( ψ_ζ_3ψ_ϕ_3)_1 -4.27/2β_1 (ψ_ζ_3ψ_ϕ_3(ϕ_3  +v_3 (1,a,b)^T))_1
             -1/2Λ_ϕβ_4 (ψ_ζ_1”ψ_ϕ_1')_1+h.c. ,

where ϕ_3 is again expanded around its VEV in the second line and we use the relations of <Ref> in the last line.

The DM candidate is the lowest mass-state Dirac fermion built as a linear combination of the Weyl components of the driving fields and flavon fields, ψ_ζ_i, ψ_ϕ_i, whereas the mediator is a linear combination of the flavon scalars. The particle content relevant for DM production is outlined in <Ref>. After the Higgs and the flavons acquire VEVs as given in <Ref>,[The reheating temperature is chosen to be T_R =150 GeV such that DM production begins after EW symmetry breaking. Recall that the EW symmetry breaking temperature of crossover is around 160 GeV with a width of 5 GeV <cit.>. We acknowledge that higher reheating temperatures are also possible, in which case, the production of DM happens before EW symmetry breaking. We leave this possibility for future work.] the symmetry group of <Ref> gets broken down to 1_EM×1_R.



Interactions in <Ref> allow for processes as shown in the diagrams in <Ref>. 
Furthermore, we also consider the scalar potential

    V = ∂ W/∂φ_iK^i∂ W^*/∂φ_^* + ∑_i=1,2,3 | m_ϕϕ_3,i |^2 + | m_ϕϕ_1'|^2 ,

where we have added the soft-masses  m_ϕ for the flavon scalars. We assume that  m_ϕ is the same for the four scalars and should be of order of the SUSY breaking scale. It has been shown in <cit.> that for modular A_4 supersymmetric models, the SUSY breaking scale is constrained to be above 6 TeV.
Therefore, we choose values of order m_ϕ =20-1000 TeV. Furthermore, the A-term in <Ref> does not play a significant role in DM production at tree-level since the production of DM through flavon scalar annihilations are suppressed due to the low reheating temperature we consider; therefore, we also ignore the A-term in our analysis.




It turns out that the correct DM relic abundance can be obtained with a freeze-in scenario <cit.> in this model (cf. <cit.> where a freeze-out production mechanism is utilized). This can be seen as follows. First, note that the effective scalar flavon mass m_ϕ is obtained by diagonalizing the mass matrix obtained from <Ref>. If we assume that m_ϕ≫   m_E_i,m_ψ_ϕζ, where m_E_i, m_ψ_ϕζ=m_DM∼Λ_ϕβ_1 represent the mass of the charged leptons and the DM respectively, then the mediator mass (m_ϕ∼m_ϕ + m_ψ_ϕζ) is much larger than the DM and the charged-lepton masses.  Therefore, the diagram  <Ref> reduces to the effective 4-fermion operator  as indicated in <Ref> with a coupling of ∼β_2 α_m/m_ϕ^2. In a freeze-out mechanism, if the rate ⟨σ v ⟩ at which DM is annihilated decreases, then the amount of  DM relic abundance increases. Since we must require β_2 ≪ 4 π to retain perturbativity, and we expect α_1 = -6.97, as explained at the end <Ref>, then

    ⟨σ v ⟩∼|β_2 α_m/m_ϕ^2|^2 ≪|88/m_ϕ^2|^2 .


Using  5.3.41 <cit.>, we find that for the chosen values of Λ_ϕ and m_ϕ, too much DM is produced. Since increasing m_ϕ or lowering β_2 would only decrease ⟨σ v ⟩, the DM abundance can not be decreased to the observed DM abundance in a freeze-out scenario.[If we had chosen high enough reheating temperatures, then the A-terms of <Ref> would have dominated the production of DM. In this case, the connection between DM and flavor parameters is weakened.] On the other hand, for a freeze-in scenario, we have the opposite behavior. Specifically, if ⟨σ v ⟩ decreases, the amount of produced DM also decreases. So, we can choose smaller β_2 or larger m_ϕ values to obtain the observed relic abundance of DM in the Universe.



We now proceed to present the predictions of our model for the DM abundance after performing a parameter scan. We use  5.3.41 <cit.> for the DM abundance computation and FeynRules 2.0 <cit.> to create the  <cit.> model files. As mentioned earlier, we assume tanβ = 60 and a low reheating temperature of T_R =150 GeV.

From our discussion we see that we have 4 free parameters to determine the DM in our model:


  * the scalar flavon soft-mass m_ϕ,

  * the flavor breaking scale Λ_ϕ,

  * the coupling β_1 in <Ref>, and

  * the coupling β_4 in <Ref>.

We fix 10^-6≤β_1, β_4 ≤ 10^-4 and  20 TeV ≤m_ϕ,  Λ_ϕ≤ 1000 TeV.
These bounds for the couplings β_1, β_4 respect the perturbativity of all the couplings β_i (cf. <Ref>).



In <Ref> we show the prediction for DM relic abundance as a function of β_1 and β_4, for Λ_ϕ > m_ϕ (<Ref>), m_ϕ = Λ_ϕ (<Ref>), and m_ϕ < Λ_ϕ (<Ref>). The unshaded region indicates the excluded parameter space where too much DM is produced. We see that all plots in <Ref> exhibit similar behavior. It is possible to have a coupling up to β_4, β_1 = 10^-4, but not at the same time. This is consistent with the fact that freeze-in normally requires small couplings <cit.>. Furthermore, the abundance increases as we increase either β_1 or β_4, which is consistent with the fact that  the DM relic abundance increases with ⟨σ v ⟩.




<Ref> shows the predicted DM relic abundance as a function of m_ϕ and Λ_ϕ for β_1 > β_4 (<Ref>), β_1 = β_4 (<Ref>), and β_1 < β_4 (<Ref>). The unshaded space represents the excluded parameter space. We observe a similar behavior for the three plots in <Ref>. The DM abundance decreases if either m_ϕ or Λ_ϕ grows. Furthermore, a soft-mass of m_ϕ= 20 TeV and a simultaneous flavon breaking-scale Λ_ϕ= 20 TeV is excluded in all cases for the chosen values of other parameters.




Finally, in <Ref> we show the correlation between the minimum scalar flavon mass m_ϕ_MIN and DM mass m_DM for fixed values of β_1 > β_4 (<Ref>), β_1 = β_4 (<Ref>), and β_1 < β_4 (<Ref>). By comparing <Ref>, we find that for a given value of the soft-mass m_ϕ, the minimum scalar flavon mass m_ϕ_MIN constrained by the DM relic abundance can be derived. We see that the scalar flavon mass can be as low as ∼63 TeV for all cases illustrated in <Ref>. Moreover, the DM mass can be as low as ∼16 GeV for β_1 = 6× 10^-5 and β_4 =10^-4.





§ CONCLUSION




We constructed a flavor model based on the finite modular group Γ_3≅ A_4 that simultaneously explains the flavor parameters in the lepton sector and accounts for the observed DM relic abundance in the Universe. The 12 lepton flavor parameters are determined by 8 real parameters: the two components of the modulus VEV, the four flavon VEVs, ⟨ϕ_i⟩, and two dimensionful parameters that set the mass scales of charged-lepton and neutrino masses. We identify a DM candidate composed of the fermionic parts of the flavon superfields and the driving superfields. The mediator is the scalar part of the flavon superfield which interacts with the charged-lepton sector of the SM. We obtain a good fit to the lepton flavor parameters (3 charged lepton masses, 3 mixing angles, 1 phase, 2 neutrino squared mass differences) with χ^2 = 0.08 for an inverted-hierarchy neutrino spectrum. The lepton flavor fit fixes the couplings of the charged leptons to the DM mediator as well as the flavon VEVs ⟨ϕ_i⟩.
These VEVs satisfy the F-term equations to retain supersymmetry at high energies, determining thereby the coupling between our DM candidate and the mediator. Interestingly, our model exhibits 4 additional degrees of freedom that are left free and serve to achieve a DM relic abundance which  does not exceed the observed value Ω_CDM = 0.265. These parameters are 2 dimensionless couplings β_1, β_4, the flavor breaking scale Λ_ϕ, and the soft-mass for the flavon m_ϕ. We find that if the mediator mass is assumed to be much larger than the DM and the charged-lepton masses, then the appropriate DM production mechanism is freeze-in rather than freeze-out. We observe that a viable DM relic abundance can be generated in regions of the parameter space constrained by 10^-6≤β_1,β_4 ≤ 10^-4,  20 TeV ≤m_ϕ,  Λ_ϕ≤ 1000 TeV, tanβ = 60, and T_R = 150 GeV.

Although some amount of tuning is necessary in our model to identify the best parameter values, we point out that it is the choice of charges and modular weights that render the right flavon superpotential, which in turn delivers the alignment of the flavon VEVs ⟨ϕ_i⟩. Further, the phenomenologically viable value of the modulus VEV ⟨τ⟩ might be achieved through mechanisms as in <cit.>.

As a feasible outlook of our findings, it would  be interesting to study the possibility of applying this new scenario in top-down models such as <cit.>. Another interesting aspect to explore would be the study of additional constraints on the DM free parameters β_1, β_4, Λ_ϕ and m_ϕ. This could be done by using constraints from electron scattering searches like XENON1T <cit.>, SENSEI <cit.> or DAMIC <cit.>. Furthermore, this model only accounts for the lepton sector, but it should be extended to the quark sector. In this case, we could also use searches from nucleon scattering like LUX <cit.>, DEAP-3600 <cit.>, PandaX-II <cit.>, DarkSide <cit.> or EDELWEISS <cit.>. We leave these intriguing questions for upcoming work.





 §.§ Acknowledgments

The work of M.-C.C. and V.K.-P. was partially supported by the U.S. National Science Foundation under Grant No. PHY-1915005. 
The work of S.R.-S. was partly supported by UNAM-PAPIIT grant IN113223 and Marcos Moshinsky Foundation. This work was also supported by UC-MEXUS-CONACyT grant No. CN-20-38.
V.K.-P.  would like to thank Kevork N. Abazajian, Max Fieg, Xueqi Li, Xiang-Gan Liu, Gopolang Mohlabeng, Michael Ratz and Miša Toman for fruitful discussions. M-C.C.and V.K.-P. would also like to thank Instituto de Física, UNAM, for the hospitality during their visit. A.B. would like to thank the Department of Physics and Astronomy at UCI for the hospitality during his visit. V.K.-P. also thanks the opportunity to use the cluster at Instituto de Física, UNAM. This work utilized the infrastructure for high-performance and high-throughput computing, research data storage and analysis, and scientific software tool integration built, operated, and updated by the Research Cyberinfrastructure Center (RCIC) at the University of California, Irvine (UCI). The RCIC provides cluster-based systems, application software, and scalable storage to directly support the UCI research community. https://rcic.uci.edu








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DMdark matter
SUSYsupersymmetry
VEVvacuum expectation value
EFTeffective field theory
MFVminimal flavor violation
SMStandard Model
EWElectroweak



