Quark and lepton flavor model with leptoquarks in a modular $A_4$ symmetry

We propose a quark-lepton model via leptoquarks and modular $A_4$ symmetry. Since the neutrino mass is induced at one-loop level mediated by down quarks as well as leptoquarks, we have to explain lepton and quark masses and mixings with a single modulus $\tau$. Here, we find predictions for lepton and quark sectors with unified modulus $\tau$, and show several constraints originating from leptoquarks.

Considering above issues, one finds that Yukawa flavor structure is also very important to explain them. Recently, powerful symmetries to restrict the number of parameters in Yukawa couplings, so called "modular flavor symmetries", were proposed by authors in refs. [21,22], in which they have applied modular originated non-Abelian discrete flavor symmetries to quark and lepton sectors. One remarkable advantage of applying this symmetries is that dimensionless couplings of model can be transformed into non-trivial representations under those symmetries, and all the dimensionless values are uniquely fixed once modulus is determined in fundamental region. We then do not need the scalar fields to obtain a predictive mass matrix. Along the line of this idea, a vast reference has recently appeared in the litera-  [76][77][78], larger groups [79], multiple modular symmetries [80], and double covering of A 4 [81,82], S 4 [83,84], and the other types of groups [85][86][87][88][89][90] in which masses, mixing, and CP phases for the quark and/or lepton have been predicted 2 . Moreover, a systematic approach to understand the origin of CP transformations has been discussed in Ref. [99], and CP/flavor violation in models with modular symmetry was discussed in Refs. [100][101][102][103], and a possible correction from Kähler potential was discussed in Ref. [104]. Furthermore, systematic analysis of the fixed points (stabilizers) has been discussed in Ref. [105]. A very recent paper of Ref. In this paper, we focus on the quark and lepton masses and mixings based on a LQ model in ref. [11], introducing modular A 4 symmetry to reduce free parameters of Yukawa couplings. Since the quark sector connects to the lepton sector via LQ, charge assignments for quarks(leptons) directly affect the leptons(quarks). In this sense, it would be a good motivation towards unification of quark and lepton flavor in A 4 modular symmetry.
This paper is organized as follows. In Sec. II, we review our model of quark and lepton.
In Sec. III, we have numerical analysis and show several results for normal and inverted hierarchies. We conclude in Sec. IV. In appendix, we summarize several features of modular A 4 symmetry.

II. MODEL
In this section, we review our model. It is known that introducing proper leptoquarks lead us to a radiative seesaw model without any additional symmetries such as Z 2 . Here, where k I is the number of modular weight. we introduce two types of leptoquarks η and ∆ based on Ref. [11]. The color-triplet η has SU (2) L doublet with 1/6 hypercharge, and the color-antitriplet ∆ has SU (2) L triplet with 1/3 hypercharge, where these new bosons and their charges are summarized in Table I.
Then, the valid Lagrangian to induce the quark and lepton mass matrices is given by The Lagrangian for the mixing between the quark and lepton and nontrivial potential are given by where the subscript of the fields represents the electric charge, and w + and z are absorbed by the longitudinal component of the W + and Z bosons, respectively. Due to the µ term in Eq. (II.4), the charged components with 1/3 and 2/3 electric charges mix each other. Here, we parametrize their mixing matrices and mass eigenstates as follows: where their masses are denoted as m A i and m B i respectively. The interactions in terms of the mass eigenstates can be written as The next task is to determine the matrices of y u , y d , h, f, g via modular A 4 symmetry. In the quark sector, we assign Q L to be 3 and −2,ū R to be {1, 1 , 1 } and −4, andd R to be {1, 1 , 1 } and 0 under A 4 and −k, respectively. This assignment is the same as the one in ref. [42], and it is already known that allowed region [27]. Thus, we will work on the same τ region of the lepton sector in our numerical analysis. The up-type quark mass matrix is written as: are complex parameters, and a u , a c and a t can be used to fit the masses of up quarks. The explicit forms of f i and f i are summarized in Appendix. Then m u is diagonalized by two On the other hand, the down-type quark mass matrix is given as: where and a d , a s and a b can be used to fit the masses of down quarks, and Y Finally, we get the observable mixing matrix V CKM as follows: (II.14) A. Lepton sector Now let us move on the lepton sector. We assign L L to be 3 and −2 andē R to be {1, 1 , 1 } and 0 under A 4 and −k, respectively. Here, both of the leptoquark scalars are assigned to be true A 4 singlets with −2 modular weight. The assignments of A 4 and −k are also summarized in Tables I and II. Under these assignments, we can write down the concrete matrices as follows: The charged-lepton sector after spontaneous symmetry breaking is given by The active neutrino mass matrix m ν is given at one-loop level through the following interactions: where F ≡ V † d R f and G ≡ V T d L g and d is mass eigenstate. Then, the neutrino masss matrix in Fig. 1 is given at one-loop level as follows: Here, we define a modified neutrino mass matrix asm ν ≡ m ν /s 2a 1 . Then, we rewrite this diagonalization in terms of the modified formD ν ≡ V T νm ν V ν . Thus, we fix s 2a 1 by and ∆m 2 atm is the atmospheric neutrino mass-squared difference. Here, NH and IH respectively stand for the normal hierarchy and the inverted hierarchy. Subsequently, the solar neutrino mass-squared difference is depicted in terms of s 2a 1 as follows: This should be within the experimental value, where we adopt NuFit 5.0 [107] to our numerical analysis later. The neutrinoless double beta decay is also given by m ee = s 2a 1 |D ν 1 cos 2 θ 12 cos 2 θ 13 +D ν 2 sin 2 θ 12 cos 2 θ 13 e iα 2 +D ν 3 sin 2 θ 13 e i(α 3 −2δ CP ) |, (II. 24) which may be able to observed by KamLAND-Zen in future [108]. The observed mixing matrix of lepton sector [109] is given by

III. NUMERICAL ANALYSIS
Here, we perform numerical analysis. Before searching for allowed region, we fix some mass parameters as m A 2 = m A 1 and m B 2 = m δ = m B 1 , where we require degenerate masses for the components of η and ∆ to suppress the oblique parameters ∆S and ∆T . Notice here that our theoretical parameters a u,c,t , a d,s,b , a , b , c are used to determined the experimental masses for quarks and charged-leptons. Thus, only the following input parameters are randomly selected in the range of (m A 1 , m B 1 ) ∈ [1 , 100 ] TeV, |g u1,u2,u3 | ∈ 10 −5 , 1.5 , (|a η |, |b η |, |c η |, |a|, |b|, |c|, |b |, |c |) ∈ 10 −5 , 10 . (III.1) Above the range, we have numerical analysis in cases for quark and lepton, where experimental data in the quark sector should be within the range at 3σ. While the one in the lepton sector is discussed in the range within 3σ(yellow dots) and 5σ(red dots) applying χ 2 analysis in Nufit 5.0.

A. NH
For NH case, we show our results of lepton sector in Figs. 2, 3, 4, 5. In Fig. 2, allowed value of τ is shown where yellow(red) points present the values within 3(5)σ. One finds that allowed space is rather localized. Especially, the region at nearby τ ∼ 1.75i would be interesting since it is close to the fixed point that has a remnant Z 3 symmetry. In Fig. 3 In addition to the lepton sector, we will search for our allowed region of quark sector in Figs. 6, 7, 8. Here, the dotted red line at 3σ interval while the black line is best fit value.
And the yellow(red) points correspond to 3(5)σ of the lepton sector where τ is commonly used.
In Fig. 6, we show the CP phase of quark δ in term of (1, 3) component of CKM matrix; |V ub |, and find whole the region is allowed at 3σ interval. In Fig. 7, we show |V ub | and |V td |, and found that there is a weak linearly correlation between them. In Fig. 8, we show |V cb | and |V ub |, and find that there is also a weak linear correlation between them.
Bench mark point for NH: We also give a benchmark point to satisfy the quark and In case of IH, we obtain less allowed parameter points compared to the case of N H since it is more difficult to fit the data. Since there are no points within 3σ region but few points  within 5σ region, we will explain the tendency instead of showing scattering plots. The value of τ is interestingly localized at nearby two fixed points i and 1.74i, each of which has remnant symmetry of Z 2 and Z 3 . In addition to the lepton sector, we discuss our allowed region of quark sector. Even  though the allowed points are not so many, we might say something from our analysis as follows. As for the CP phase of quark δ in term of (1, 3) component of CKM matrix; |V ub |, we found whole the region is allowed at 3σ interval. As for |V ub | and |V td |, we find that there is a weak linearly correlation between them. As for |V cb | and |V ub |, we find that there is also a weak linear correlation between them.
Bench mark point for IH: We give two interesting benchmark points; τ ≈ 1.06i, 1.76i to satisfy the quark and lepton masses and mixings as well as phases in the center and right sides of Tables III and IV The quark mixings are given by τ ≈ 1.06i : Then, we have performed numerical analysis to search for allowed region satisfying experimental measurements for both quark and lepton sector, depending on NH and IH. In case of NH, we have found rather wide allowed space within 3σ interval and obtained tendency of observables for quark and lepton. Especially, we have found allowed region at nearby τ = 1.75i that is close to the fixed point of τ = i∞. Thus, we have also shown a promising bench mark point at around the solution.
In case of IH, we would not found the allowed region within 3σ interval, but found within 5σ interval. Although the number of allowed point is few, we have found all the allowed regions are localized at nearby τ = i, 1.76i, both of which are nearby fixed points. We have shown them as benchmark points. These would be tested near future.
= [y 1 , y 2 , y 3 ] T , transforming as a triplet of A 4 is written in terms of Dedekind eta-function η(τ ) and its derivative: