A mechanism relating the fermionic mass hierarchy to the flavor mixing
Jin-Lei Yang Hai-Bin Zhang Tai-Fu Feng









Department of Physics, Hebei University, Baoding, 071002, China^1

Key Laboratory of High-precision Computation and Application of Quantum Field Theory of Hebei Province, Baoding, 071002, China^2

Research Center for Computational Physics of Hebei Province, Baoding, 071002, China^3



Considering the hierarchical structure of fermionic masses and the fermionic flavor mixing puzzles in the Standard Model, we propose to relate them by the see-saw mechanism, i.e. only the third generation of quarks and charged leptons achieve the masses at the tree level, the first two generations achieves masses through the mixings with the third generation, and the neutrinos achieve tiny Majorana masses by the so-called Type-I see-saw mechanism. This new picture at the fermion sector can explain simultaneously the flavor mixing puzzle and mass hierarchy puzzle in the SM. In addition, a flavor-dependent model (FDM) is proposed to realize the new mechanism, and observing the top quark rare decay processes t→ ch, t→ uh and the lepton flavor violation processes μ→3e, τ→3e, μ→3μ is effective to test the proposed FDM.






A mechanism relating the fermionic mass hierarchy to the flavor mixing
    Jin-Lei Yang^1,2,3[jlyang@hbu.edu.cn],Hai-Bin Zhang^1,2,3[hbzhang@hbu.edu.cn],Tai-Fu Feng^1,2,3[fengtf@hbu.edu.cn]
    April 29, 2024
======================================================================================================================



The Standard Model (SM) achieves great success in describing the weak, strong, and electromagnetic gauge interactions of fundamental particles. However, the Yukawa couplings in the SM are still enigmatic so far because the large hierarchical structure of masses across the three families

    m_t/m_c≈104, m_c/m_u≈773, m_b/m_s≈51, m_s/m_d≈20, m_τ/m_μ≈17, m_μ/m_e≈207,

while they share common SM gauge group quantum numbers. In addition, the mixing pattern of quarks described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix <cit.> is not predicted from first principles in the SM, and the nonzero neutrino masses and mixings observed at the neutrino oscillation experiments <cit.> make the so-called flavor puzzle in the SM more acutely. It indicates the existing of a more fundamental theory beyond the SM to well explain the large hierarchical structure of fermionic masses, flavor mixings and nonzero neutrino masses. There are some attempts to explain the fermionic mass hierarchies and flavor mixings. For example, the authors of Refs. <cit.> try to explain the fermionic flavor mixings by imposing the family symmetries, some proposals to explain the fermionic mass hierarchies can be found in Refs. <cit.>. These studies illustrate that explaining the observed fermionic mass spectrum and mixings is one of the most enigmatic questions in particle physics, which may help to understand the flavor nature and seek possible new physics (NP).

Weinberg made important efforts to the fermionic mass spectrum <cit.>, flavor problem in the SM and the nonzero neutrino masses. He was the first to propose the Type-I see-saw mechanism to give the tiny neutrino masses naturally <cit.> which is one of the most popular mechanisms so far to arise the Majorana neutrino masses. In 2020 <cit.>, Weinberg proposed a class of models in which only the third generation of quarks and leptons achieve the masses at the tree level, while the masses for the second and first generations are produced by one-loop and two-loop radiative corrections respectively, although he pointed that these models are not realistic for some reasons. Adopting Weinberg's ideas of see-saw mechanism and only the third generation of quarks, leptons achieve the masses at the tree level, we propose to apply the see-saw mechanism to the mass matrices of quarks and leptons, in which the second and first generations achieve masses through the mixings with the third generation, i.e. the mass matrices of quarks and leptons can be written as

    m_q=([        0   m_q,12   m_q,13; m_q,12^*        0   m_q,23; m_q,13^* m_q,23^*   m_q,33 ]),m_e=([        0   m_e,12   m_e,13; m_e,12^*        0   m_e,23; m_e,13^* m_e,23^*   m_e,33 ]),m_ν=([     0 M_D^T;   M_D   M_R ]),

where q=u,d, m_q,33 and m_e,33 are real, M_D, M_R are the 3×3 Dirac and Majorana mass matrices respectively (the nonzero neutrino masses are obtained by the so-called Type I see-saw mechanism). Then considering the measured fermionic masses, under the approximation |m_f,12|, |m_f,13|, |m_f,23|≪ |m_f,33| (f=u, d, e) one can obtain

    m_f,33=m_f_3-m_f_1-m_f_2,
          |m_f,23|=[(m_f_1+m_f_2)m_f,33-|m_f,13|^2]^1/2,
          |m_f,12|=|m_f,13||m_f,23|/|m_f,33|cos(θ_f,12+θ_f,23-θ_f,13)
                  +{[|m_f,13||m_f,23|/|m_f,33|^2cos(θ_f,12+θ_f,23-θ_f,13)]^2+m_f_1m_f_2/|m_f,33|^2}^1/2|m_f,33|,

where θ_f,ij (ij=12, 13, 23) are defined as m_f,ij=|m_f,ij|e^iθ_f,ij, and m_f_k (k=1, 2, 3) is k-generation fermion f mass. Eq. (<ref>) shows that the fermionic flavor mixings are related to the fermionic mass hierarchies after applying the see-saw mechanism to the fermionic mass matrices. In this case, the measured quark masses, lepton masses and their flavor mixings (CKM matrix at the quark sector and PMNS matrix at the lepton sector) will impose strict constraints on the parameters in Eq. (<ref>). Due to the neutrino masses are not fixed experimentally, we focus on the flavor mixings at the quark sector firstly[The analysis about the mass structure and flavor mixings at the lepton sector are similar to the one at the quark sector, and the free parameters in the matrices m_e, m_ν defined in Eq. (<ref>) are constrained less strictly than the quark sector because the nonzero neutrino masses are obtained by the Type I see-saw mechanism.].


As mentioned above, the mixing pattern of quarks is described by the CKM matrix, and quarks' flavor mixing is related to the quarks' mass hierarchies by the new definitions of fermionic mass matrices in Eq. (<ref>). Taking the measured quark masses as input, the remaining free parameters in the matrix m_q are |m_q,13|, θ_q,ij (q=u, d), and they are required to fit the measured CKM matrix <cit.>

    |V_ CKM|=([              0.97435±0.00016              0.22500±0.00067              0.00369±0.00011;              0.22486±0.00067              0.97349±0.00016    0.04182^+0.00085_-0.00074;    0.00857^+0.00020_-0.00018    0.04110^+0.00083_-0.00072 0.999118^+0.000031_-0.000036 ]).



In order to see whether the mass matrices defined in Eq. (<ref>) can fit the measured CKM matrix, we take the pole quark masses m_t=174.3  GeV, m_c=1.67  GeV, m_u=2.16  MeV, m_b=4.78  GeV, m_s=93.4  MeV, m_d=4.67  MeV as inputs and scan the free parameters |m_q,13|, θ_q,ij, (ij=12, 13, 23) in the following range

    |m_u,13|=(0.0∼1.0)  GeV, |m_d,13|=(0.0∼0.1)  GeV, θ_q,ij=(-π∼π),

and keep

    |V_ CKM,12|=(0.224∼0.226), |V_ CKM,21|=(0.22386∼0.22586),
          |V_ CKM,23|=(0.04082∼0.04282), |V_ CKM,32|=(0.0401∼0.0421),
          |V_ CKM,13|=(0.00359∼0.00379), |V_ CKM,31|=(0.00847∼0.00867),

in the scanning. The allowed ranges of |m_d,13|-|m_u,13|, θ_d,12-θ_u,12, θ_d,23-θ_u,23 are plotted in Fig. <ref> (a), (b), (c) respectively. The picture shows that the mass matrices proposed in this work can well coincide with the measured quark masses and CKM matrix. And to fit the measured CKM matrix elements, the values of |m_d,13|, θ_d,23 are related to the chosen values of |m_u,13|, θ_u,23 respectively, and |θ_d,23|-|θ_u,23|≈±(0.04∼0.06)π as shown in Fig. <ref> (c).


The analysis above illustrate that applying the see-saw mechanism to the quarks' mass matrices and the first two generations achieve masses through the mixings with the third generation can well fit the measured quark masses and CKM matrix. The fermionic flavor mixings are related to the fermionic mass hierarchies in this new mechanism which can explain simultaneously the flavor mixings puzzle and mass hierarchy puzzle in the SM. This mechanism can be realized simply by setting the Yukawa couplings Y_q,11=Y_q,22=0 in the SM, but it is trivial to set the terms allowed by the gauge symmetries to zero. On the other hand, the mass matrices defined in Eq. (<ref>) can be realized in a NP model by extending the SM with an extra U(1)_F local gauge group where the corresponding U(1)_F charges are related to the particles' flavor, i.e. the gauge group of such flavor-dependent U(1)_F model (FDM) is SU(3)_C⊗ SU(2)_L⊗ U(1)_Y⊗ U(1)_F. The gauge charges of fermions corresponding to SU(2)_L⊗ U(1)_Y⊗ U(1)_F read

    L_1∼(2,Y_L,z),L_2∼(2,Y_L,-z),L_3∼(2,Y_L,0),
          R_1∼(1,Y_R,-z),R_2∼(1,Y_R,z),R_3∼(1,Y_R,0),
          ν_R_1∼(1,0,-z),ν_R_2∼(1,0,z),

where the nonzero z denotes the U(1)_F charge, L_i (L=l, q) denote the i-generation left-handed fermion doublets in the SM, R_i (R=u_R, d_R, e_R) denotes the i-generation right-handed fermion singlets in the SM, ν_R_i (i=1,2) is the right-handed neutrinos[The third generation of right-handed neutrino ν_R_3 has zero U(1)_F, U(1)_Y charges which is trivial, hence only two generations of right-handed neutrinos are introduced.], Y_L, Y_R are the U(1)_Y charges corresponding to doublets L and singlets R in the SM respectively. The chiral anomaly cancellation can be guaranteed for the fermionic charges defined in Eq. (<ref>). The scalar sector is extended by introducing two new doublets and one new singlet

    Φ_1∼(2,1/2,z),Φ_2∼(2,1/2,-z),Φ_3∼(2,1/2,0),χ∼(1,0,2z),

with Φ_3 corresponding to the SM Higgs doublet. The scalar potential in the FDM reads

    V=-M_Φ_1^2 Φ_1^†Φ_1-M_Φ_2^2 Φ_2^†Φ_2-M_Φ_3^2 Φ_3^†Φ_3-M_χ^2χ^*χ+λ_χ (χ^*χ)^2+λ_1 (Φ_1^†Φ_1)^2
              +λ_2 (Φ_2^†Φ_2)^2+λ_3 (Φ_3^†Φ_3)^2+λ'_4 (Φ_1^†Φ_1)(Φ_2^†Φ_2)+λ_4” (Φ_1^†Φ_2)(Φ_2^†Φ_1)
              +λ_5' (Φ_1^†Φ_1)(Φ_3^†Φ_3)+λ_5” (Φ_1^†Φ_3)(Φ_3^†Φ_1)+λ_6' (Φ_2^†Φ_2)(Φ_3^†Φ_3)+λ_6” (Φ_2^†Φ_3)(Φ_3^†Φ_2)
              +λ_7 (Φ_1^†Φ_1)(χ^*χ)+λ_8 (Φ_2^†Φ_2)(χ^*χ)+λ_9 (Φ_3^†Φ_3)(χ^*χ)+[λ_10 (Φ_3^†Φ_1)(Φ_3^†Φ_2)
              +κ(Φ_1^†Φ_2)χ+h.c.],

where

    Φ_1=([                 ϕ_1^+; 1/√(2)(i A_1+S_1+v_1) ]),Φ_2=([                 ϕ_2^+; 1/√(2)(i A_2+S_2+v_2) ]),Φ_3=([                 ϕ_3^+; 1/√(2)(i A_3+S_3+v_3) ]),
          χ=1/√(2)(i A_χ+S_χ+v_χ),

v_i (i=1, 2, 3), v_χ are the vacuum expectation values (VEVs) of Φ_i, χ respectively, (v_1^2+v_2^2+v_3^2)^1/2= v≈ 246  GeV and v_1, v_2< v_3≪ v_χ. Based on the scalar potential in Eq. (<ref>), we can obtain four constraint equations from the minimization

    M_Φ_1^2=λ_1v_1^2+1/2[(λ_4'+λ_4”) v_2^2+(λ_5'+λ_5”) v_3^2+v_2/v_1v_3^2  Re(λ_10)+√(2)v_2/v_1 v_χ Re(κ)+λ_7v_χ^2],
          M_Φ_2^2=λ_2v_2^2+1/2[(λ_4'+λ_4”) v_1^2+(λ_6'+λ_6”) v_3^2+v_1/v_2v_3^2  Re(λ_10)+√(2)v_1/v_2 v_χ Re(κ)+λ_8v_χ^2],
          M_Φ_3^2=λ_3v_3^2+ Re(λ_10)v_1v_2+1/2[(λ_5'+λ_5”) v_1^2+(λ_6'+λ_6”) v_2^2+λ_9 v_c^2],
          M_χ^2=λ_χ v_χ^2+1/2[λ_7 v_1^2+λ_8 v_2^2+λ_9 v_3^2+√(2)v_1v_2/v_χ Re(κ)].


Firstly, we focus on the Higgs mass spectrum and scalar potential in the FDM which contains four CP-even Higgs, two CP-odd Higgs and two singly charged Higgs. For simplicity, all parameters in Eq. (<ref>) are assumed to be real and v_1=v_2, λ_4'=λ_4”=λ_4/2, λ_5'=λ_5”=λ_5/2, λ_6'=λ_6”=λ_6/2. Scanning the following parameter space

    v_1=(0, 40)  GeV, λ_i=(0, 5)   with  (i=1,...,9,χ), λ_10=(-5, 0),
          v_χ=(4, 10)  TeV, κ=(-3,-0.1)  TeV,

we plot M_H_1 versus v_1 (a), M_H_2 versus v_1 (b), M_H_3 versus κ (c), M_H_4 versus v_χ (d), M_A_1 versus λ_10 (e), M_A_2 versus κ (f), M_H_1^± versus λ_10 (g) and M_H_2^± versus κ (h) in Fig. <ref>.

In the scanning, we keep the next-to-lightest CP-even Higgs mass M_H_2 in the range 124  GeV<M_H_2<126  GeV[The reasons for taking the next-to-lightest CP-even Higgs to be the SM-like Higgs: 1, considering the observed 2.8σ excess at about 95GeV by CMS <cit.> and the observed 2.3σ local excess at about 98GeV by LEP <cit.>, we will explore in our next work whether the additional scalars in the FDM can account for these low-mass excesses; 2, the additional Higgs mass below 125 GeV by far is not excluded experimentally <cit.>; 3, if the low-mass Higgs is excluded completely in future, we verify numerically that the next-to-lightest CP-even Higgs mass can reach hundreds GeV by taking the lightest CP-even Higgs mass around 125 GeV and scanning the parameter space in Eq. (<ref>).] and the scalar potential at the input v_1, v_2, v_3, v_χ is smaller than all the other stationary points to guarantee the stability of vacuum.



The right-handed neutrino singlets obtain the Majorana masses after χ achieving nonzero VEVs. Then the Yukawa coupling in the FDM can be written as

    ℒ_Y=Y_u^33q̅_3 Φ̃_3 u_R_3+Y_d^33q̅_3 Φ_3 d_R_3+Y_u^32q̅_3 Φ̃_1 u_R_2+Y_u^23q̅_2 Φ̃_1 u_R_3+Y_d^32q̅_3 Φ_2 d_R_2
                +Y_d^23q̅_2 Φ_2 d_R_3+Y_u^21q̅_2 Φ̃_3 u_R_1+Y_u^12q̅_1 Φ̃_3 u_R_2+Y_d^21q̅_2 Φ_3 d_R_1+ Y_d^12q̅_1 Φ_3 d_R_2
                +Y_u^31q̅_3 Φ̃_2 u_R_1+Y_u^13q̅_1 Φ̃_2 u_R_3+Y_d^31q̅_3 Φ_1 d_R_1+Y_d^13q̅_1 Φ_1 d_R_3
                +Y_e^33l̅_3 Φ_3 e_R_3+Y_e^32l̅_3 Φ_2 e_R_2+Y_e^23l̅_2 Φ_2 e_R_3+Y_e^21l̅_2 Φ_3 e_R_1+ Y_e^12l̅_1 Φ_3 e_R_2
                +Y_e^31l̅_3 Φ_1 e_R_1+Y_e^13l̅_1 Φ_1 e_R_3+Y_R^11ν̅^c_R_1ν_R_1χ+Y_R^22ν̅^c_R_2ν_R_2χ^*+Y_D^21l̅_2 Φ̃_3 ν_R_1
                +Y_D^12l̅_1 Φ̃_3 ν_R_2+Y_D^31l̅_3 Φ̃_2 ν_R_1+Y_D^32l̅_3 Φ̃_1 ν_R_2+h.c..

The mass matrices in Eq. (<ref>) can be obtained after Φ_1, Φ_2, Φ_3, χ receive nonzero VEVs. The gauge boson masses in the model are

    M_γ=0, M_W=1/2g_2 v, M_Z≈1/2(g_1^2+g_2^2)^1/2 v, M_Z'≈ 2|zg__F| v_χ,

where the higher orders of v/v_χ are neglected for M_Z and M_Z'.


One of the most effective ways to test the FDM is measuring the top quark rare decay processes t→ ch and t→ uh, because the tree level coupling between top quark and H_3 (which corresponds to the SM Higgs doublet) can make significant contributions to these two processes as shown in Eq. (<ref>). The branching ratios can be written as <cit.>

    Br(t→ q_u h)=|ℳ_t q_u h|^2√(((m_t+m_h)^2-m_q_u^2)((m_t-m_h)^2-m_q_u^2))/32π m_t^3Γ^t_ total,

where q_u=u, c, the amplitude ℳ_tq_uh can be read directly from the Yukawa couplings in Eq. (<ref>), and Γ^t_ total=1.42GeV <cit.> is the total decay width of top quark. Then taking the points obtained in Fig. <ref> and Fig. <ref> as inputs, we plot the results of Br(t→ uh)- Br(t→ ch) in Fig. <ref> (a) and Br(t→ uh) versus |m_u,13| in Fig. <ref> (b), where the red and blue lines denote the upper bounds on Br(t→ ch) and Br(t→ uh) from Particle Data Group <cit.> respectively. The picture indicates that the model predicts Br(t→ uh) and Br(t→ ch) can reach about 10^-5 and 10^-4 respectively, which have great opportunities to be observed in the next generation of experiments. In addition, large Br(t→ uh) prefers large |m_u,13| (|m_u,13|≳0.65  GeV) as shown in Fig. <ref> (b) while large |m_u,13| is hard to fit the measured CKM matrix elements.

Due to the introducing of flavor dependent gauge symmetry U(1)_F, Z and new Z' bosons in the FDM can make contributions at the tree level to the lepton flavor violation (LFV) processes μ→3e, τ→3e, μ→3μ, which suffer strict experimental constraints Br(μ→3e)<1× 10^-12 <cit.>, Br(τ→3e)<2.7× 10^-8 <cit.>, Br(τ→3μ)<2.1× 10^-8 <cit.>. In addition, Higgs bosons can also make contributions to these LFV processes at the tree level, we take |m_e,13|=0.01  GeV, θ_e,12=θ_e,13=θ_e,23=0 and the U(1)_F charge z=1 for simplicity. The corresponding amplitude can be written as <cit.>

    ℳ(e_j→ e_i e_ie̅_i)=C_1^Lu̅_e_i(p_2)γ_μ P_L u_e_j(p_1) u_e_i(p_3)γ^μ P_L ν_e_i(p_4)
                +C_1^Ru̅_e_i(p_2)γ_μ P_R u_e_j(p_1) u_e_i(p_3)γ^μ P_R ν_e_i(p_4)
                +[C_2^Lu̅_e_i(p_2)γ_μ P_L u_e_j(p_1) u_e_i(p_3)γ^μ P_R ν_e_i(p_4)
                +C_2^Ru̅_e_i(p_2)γ_μ P_R u_e_j(p_1) u_e_i(p_3)γ^μ P_L ν_e_i(p_4)-(p_2↔ p_3)]
                +[C_3^Lu̅_e_i(p_2) P_L u_e_j(p_1) u_e_i(p_3) P_L ν_e_i(p_4)
                +C_3^Ru̅_e_i(p_2) P_R u_e_j(p_1) u_e_i(p_3) P_R ν_e_i(p_4)-(p_2↔ p_3)],

where i=1, 2 for j=3, i=1 for j=2, u_e_i denotes the spinor of lepton, ν_e_i denotes the spinor of antilepton, P_L=(1-γ_5)/2, P_R=(1+γ_5)/2, and p_k denotes the momentum of charged lepton with k=1,2,3,4. The coefficients C_1,2,3^L,R from the contributions of Higgs bosons and Z, Z' gauge bosons, can be obtained through the Yukawa couplings in Eq. (<ref>) and the definition of covariant derivative corresponding to SU(2)_L⊗ U(1)_Y⊗ U(1)_F

    D_μ=∂_μ+i g_2 T_a A_μ^a+i g_1 Y B_μ+i g__F F B'_μ+i g__YF Y B'_μ, (a=1,2,3)

respectively. In Eq. (<ref>), g__YF is the gauge coupling constant arises from the gauge kinetic mixing effect, (g_2, g_1,  g__F), (T_a, Y, F), (A_aμ, B_μ,  B'_μ) denote the gauge coupling constants, generators and gauge bosons of groups (SU(2)_L, U(1)_Y, U(1)_F) respectively. Then we can calculate the decay rate <cit.>

    Γ(e_j→ e_i e_ie̅_i)=m_e_j^5/1536π^3[1/2(|C_1^L|^2+|C_1^R|^2)+|C_2^L|^2+|C_2^R|^2+1/8(|C_3^L|^2+|C_3^R|^2)].

The total decay widthes of μ, τ are taken as Γ^μ_ total=2.996× 10^-19GeV, Γ^τ_ total=2.265× 10^-12GeV <cit.>.


Taking the points obtained in Fig. <ref>, Fig. <ref> as inputs, and keeping Br(μ→3e)<1× 10^-12, Br(τ→3e)<2.7× 10^-8, Br(τ→3μ)<2.1× 10^-8 in the computation, we plot the allowed ranges of M_Z'-g_F in Fig. <ref> (a), the results of Br(τ→ 3e)- Br(μ→ 3e) in Fig. <ref> (b), the results of Br(τ→ 3μ)- Br(μ→ 3e) in Fig. <ref> (c).

It is obvious from Fig. <ref> (a) that the experimental upper bounds on the branching ratios of these LFV processes give the constraints M_Z'/g_F≳70  TeV in our chosen parameter space. And Fig. <ref> (b), Fig. <ref> (c) show that Br(μ→3e)≳ 10^-16, Br(τ→3e)≳ 10^-16, Br(τ→3μ)≳ 10^-10 in the FDM, which have great opportunities to be observed in the future <cit.>.



Focusing on the hierarchical structure of fermionic masses puzzle and fermionic flavor mixings puzzle, we propose to relate these two puzzles by the see-saw mechanism, i.e. only the third generation of quarks and charged leptons (the SM neutrinos obtain tiny Majorana masses through the so-called Type-I see-saw mechanism by introducing the right-handed neutrinos) achieve the masses at the tree level, and the first two generations achieve masses through the mixings with the third generation. Taking the pole quark masses as inputs, we found that the mechanism proposed in this work can well fit the measured CKM matrix elements. Finally, a flavor-dependent model (FDM) are proposed in this letter to realize the new mechanism mentioned above. The model extends the SM by an extra U(1)_F local gauge group, two scalar doublets and two scalar singlets, where the new U(1)_F charges are related to the particles' flavor. And we point out that observing the top quark rare decay processes t→ ch, t→ uh and the lepton flavor violation processes μ→3e, τ→3e, μ→3μ is an effective way to test the FDM.









The work has been supported by the National Natural Science Foundation of China (NNSFC) with Grants No. 12075074, No. 12235008, Hebei Natural Science Foundation with Grant No. A2022201017, No. A2023201041, Natural Science Foundation of Guangxi Autonomous Region with Grant No. 2022GXNSFDA035068, the youth top-notch talent support program of the Hebei Province.







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