Calculable neutrino Dirac mass matrix and one-loop $\bar θ$ in the minimal left-right symmetric model
Gang Li Ding-Yi Luo Xiang Zhao


















ligang65@mail.sysu.edu.cn



luody25@mail2.sysu.edu.cn



zhaox88@mail2.sysu.edu.cn



Calculable neutrino Dirac mass matrix and one-loop θ̅ in the minimal left-right symmetric model
    Xiang Zhao
    April 29, 2024
===============================================================================================





§ INTRODUCTION



The standard model (SM) of particle physics has achieved great success. However, the origin of neutrino masses and the strong CP problem remains unsolved, serving as compelling motivations for physics beyond the SM (BSM). These two problems might have intrinsic connections, even though they appear in the weak and strong sectors at low energies. 

If neutrinos are Majorana fermions, they could acquire Majorana masses in the seesaw mechanism <cit.>, making them naturally small. In the type-I <cit.> and type-II <cit.> seesaw mechanisms,  right-handed neutrinos and scalar triplet are introduced, respectively. In the minimal left-right symmetric model (MLRSM) <cit.>, 
the neutrino masses can receive contributions from both type-I and type-II seesaw mechanisms M_ν = M_L - M_D^T M_N^-1 M_D (cf. Eq. (<ref>)). 
In case of parity or charge conjugation as the left-right symmetry <cit.>, dubbed case 𝒫 or 𝒞, respectively, the MLRSM is highly predictive, which has been extensively studied <cit.>. Moreover, it was found that in the MLRSM, one can calculate the neutrino Dirac mass matrix M_D in terms of the light and heavy neutrino masses and mixings <cit.>.
As a contrast, the expression of neutrino Dirac mass matrix M_D in Casas-Ibarra parameterization <cit.> in type-I seesaw models is still dependent on an arbitrary complex orthogonal matrix.


The strong CP problem is about the extremely small parameter θ̅≲ 10^-10 <cit.> that violates CP in the strong sector of the SM. The most popular solution to the strong CP problem is the Peccei-Quinn mechanism <cit.>, which leads to the existence of the axion <cit.> and has thus drawn a lot of theoretical attention <cit.> as well as experimental interest <cit.>. Additionally, the strong CP problem can also be addressed by imposing discrete symmetries <cit.>. Parity solutions to the strong CP problem in the left-right symmetric models were considered in Refs. <cit.>, and have been further studied recently <cit.>. In both SM and BSM scenarios, we can separate θ̅= θ +   (M_u M_d), where θ is the coefficient of GG̃ term in the Lagrangian, and (M_u M_d) is included since the up-type and down-type quark mass matrices M_u and M_d are in general non-Hermitian <cit.>. In the MLRSM of case 𝒫, θ vanishes at tree level and θ̅ is equal to (M_u M_d). It has been shown that θ̅≃sinαtan (2β) m_t/(2m_b) <cit.>, where α and β are defined in Eq. (<ref>), m_t and m_b denote the masses of top and bottom quarks, respectively. Thus in order to satisfy the constraint from measurements of neutron electric dipole moments <cit.> on θ̅≲ 10^-10, sinαtan (2β) → 0 is required. 

However, even if the quark mass matrices are (nearly) Hermitian, 
leptonic CP violation would induce θ̅ at one-loop level,
which might exceed the aforementioned bound as pointed out in Ref. <cit.>. 
Instead of being a problem, Senjanovic et al. <cit.> demonstrated that the one-loop θ̅ in the MLRSM implies an upper bound on the masses of sterile neutrinos, which is complementary to the direct searches at the Large Hadron Collider <cit.>. 
As obtained in Ref. <cit.>, θ̅_loop is proportional to (M_N^† M_N [M_D, M_ℓ]), where M_ℓ denotes the charged lepton mass matrix, and the neutrino Dirac mass matrix M_D is determined by the light and heavy neutrino masses and mixings.  However, it was shown that <cit.> θ̅_loop might vanish in the type-I seesaw dominance scenario for specific benchmark choices of the right-handed neutrino mixing matrix V_R, which hindered the attempt to search for sterile neutrinos contributing to θ̅ with neutrinoless double beta () decay <cit.>.

In this work, we propose a parameterization of right-handed neutrino mixing V_R in the MLRSM of case 𝒫 and construct the heavy neutrino mass matrix M_N, for which the Hermiticity of the neutrino Dirac mass matrix M_D is maintained. We then evaluate the one-loop θ̅ for the general seesaw relation for explicit examples of V_R,
and obtain non-vanishing θ̅_loop as a function of the sterile neutrino masses. By using the bound |θ̅_loop| ≲ 10^-10, we can then obtain the upper limits of the sterile neutrino masses. 

The remainder of the paper is organized as follows. In the next section, we provide a brief introduction of the MLRSM of case 𝒫. Sec. <ref> delves into the calculation of neutrino Dirac mass matrix M_D in the Senjanovic-Tello method, and the parameterization of V_R and M_N.  In Sec. <ref>, we evaluate the one-loop θ̅ for explicit examples of V_R. We conclude in Sec. <ref>.




§ MINIMAL LEFT-RIGHT SYMMETRIC MODEL



The MLRSM is based on the gauge group SU(3)_c × SU(2)_L × SU(2)_R × U(1)_B-L, which was proposed to explain the origin of neutrino masses <cit.>. Three right-handed neutrinos ν_R and scalar triplets Δ_L,R are introduced

    ℓ_L,R   = [ ν; e ]_L,R,Δ_L,R = 
    [  δ_L,R^+/√(2)      δ_L,R^++;       δ_L,R^0 -δ^+_L,R/√(2) ] ,

where the flavor indices of leptons are omitted.
Besides, the scalar bi-doublet Φ exists, which is written as 

    Φ = [ϕ_1, iσ_2 ϕ_2^*] ,  ϕ_i = [ ϕ_i^0; ϕ_i^- ] ,   i=1,2 ,

where σ_2 is the second Pauli matrix.
If parity is taken as the left-right symmetry, i.e., case 𝒫, we have

    Δ_L ↔Δ_R ,   Φ↔Φ^† .

The leptonic Yukawa interactions are

    ℒ   = -ℓ̅_L ( Y_1 Φ - Y_2 σ_2 Φ^* σ_2 ) ℓ_R
          -12( ℓ_L^T C Y_L iσ_2 Δ_L ℓ_L + ℓ_L^T C Y_R iσ_2 Δ_R ℓ_R )
          +  h.c. ,

where
C = iγ^0 γ^2 is the charge conjugation matrix,
h.c. denotes the Hermitian conjugate terms. The left-right symmetry is spontaneously broken once the right-handed triplet Δ_R develops a vacuum expectation value (vev), v_R =⟨δ_R^0⟩.
After the electroweak symmetry breaking, Φ develops vevs

    ⟨Φ⟩ = v  diag(c_β,-s_β e^-iα)

with c_β≡cosβ, s_β≡sinβ and v≃ 174. Then the left-handed triplet Δ_L would get the vev v_L, which is generally complex <cit.> and proportional to v^2/v_R <cit.>. 
Defining N_L = ν_R^c, 
one obtains the neutrino mass terms

    ℒ_ν = -12 (ν̅_L^c, N̅_L^c) M_n 
        [ ν_L; N_L ]
        +  h.c. ,

where the full neutrino mass matrix is defined as

    M_n ≡[   M_L M_D^T;   M_D   M_N ] .

The neutrino Majorana and Dirac neutrino mass matrices are

    M_L    = v_Lv_R U_e^T M_N^* U_e^*  ,
    
    M_N    = Y_R^* v_R ,
    
    M_D    = -v ( Y_1 c_β + Y_2 s_β e^-iα) .

After block diagonalizing the neutrino mass matrix, we can obtain the light neutrino masses

    M_ν = M_L - M_D^T 1M_N M_D ,

which is a general seesaw relation including contributions from both type-I and type-I mechanisms. If v_L is negligibly small, it is reduced to the type-I seesaw dominance scenario. 

As shown in Ref. <cit.>, in the MLRSM of case 𝒫 we have

    M_D - U_e M_D^† U_e ∝ s_α t_2β ,

where s_α≡sinα, t_2β≡tan(2β), U_e is the matrix that diagonalizes the charged lepton mass matrix. Thus, in the limit s_α t_2β→ 0, M_D is Hermitian and U_e  = ±1.




§ CALCULABLE NEUTRINO DIRAC MASS MATRIX





 §.§ Senjanovic-Tello method


It has been shown by Senjanovic and Tello <cit.>, the neutrino Dirac mass matrix M_D can be determined with the light and heavy neutrino masses and mixings in the limit s_α t_2β→ 0. In the following, we will briefly introduce the general method they proposed in Ref. <cit.>.

From M_D = M_D^†, 
Eq. (<ref>) can be expressed as

    HH^T=v_L/v_R1-
        1/√(M_N) M_ν^* 1/√(M_N) ,

where the Hermitian matrix H is defined as 

    H=1/√(M_N) M_D 1/√(M_N) .

One can then decompose HH^T as 

    H H^T = O s O^T ,

using the fact that HH^T is symmetric. In the above, O is a complex orthogonal matrix and s is the symmetric normal form. The matrices O and s are obtained
from Eqs. (<ref>) (<ref>).
The matrix H itself can be expressed as

    H = O √(s) E O^† .

with E being determined by the Hermitian condition H = H^†.


    √(s) E = E √(s^*) ,  
        E^T = E^* = E^-1 .



Comparing Eq. (<ref>) with Eq. (<ref>), one readily get 

    M_D = √(M_N) O √(s) E O^†√(M_N^*) .

Notice that O, s and E depend on M_ν and M_N, the neutrino Dirac mass matrix, we can calculate M_D  once the light and heavy neutrino masses and mixings are known.


Although the above method is applied to the general seesaw relation in the Hermitian case (cf. Eq. (<ref>)), no general M_D could be obtained since M_N is arbitrary <cit.>.
In terms of the physical masses and neutrino mixing matrices, 

    M_ν = V_L^* m_ν V_L^† ,  
        M_N = V_R m_N V_R^T ,

thus we should have a priori knowledge of V_R and m_N besides the inputs of m_ν and V_L from the measurements of neutrino oscillation <cit.>.

If V_R = V_L is assumed,we could obtain <cit.>

    M_D = V_L m_N √(v_Lv_R - m_νm_N) V_L^† .


While it is straightforward to calculate M_D for a different V_R, the following condition 


    ImTr[v_L/v_R-1/M_NM_ν^*]^n=0 ,   n=1,2,3

makes it more complicated, which results from the Hermiticity of H. The above relation implies that
the phases of light and heavy neutrino mass matrices are not independent <cit.>.

That is to say, for any V_R being assumed,  it is necessary to verify the condition in Eq. (<ref>) with the resulting heavy neutrino mass matrix M_N. Therefore, an appropriate choice of V_R is crucial and non-trivial. 




 §.§ Parameterization of V_RVR and M_NMN



Notice that if v_L is real, the condition in Eq. (<ref>) is reduced to ImTr[M_N^-1 M_ν^*]^n=0. This enables us to obtain possible forms of M_N and V_R, the details of which are given in Appendix <ref>. 

We find that in the MLRSM of case 𝒫 for Hermitian M_D and real v_L, the right-handed neutrino mixing matrix V_R can be parameterized as

    V_R=PV_L√(m_N m_ν)^-1 ,

where P is a Hermitian or anti-Hermitian matrix, 

    P = ± P^† .
 

For convenience, we can further write V_R as

    V_R = P̂ V_L ,   P̂≡ P V_L √(m_N m_ν)^-1 V_L^† .

Note that P has the mass dimension one, while P̂ is dimensionless.
As V_L and V_R are unitary <cit.>, it follows that P̂ must also be a unitary matrix, thereby imposing constraint on P.
If V_R = V_L, P̂ =  1, we readily get the Hermitian matrix P = V_L √(m_N m_ν) V_L^†.

From Eq. (<ref>), one can construct the heavy neutrino mass matrix

    M_N= PM_ν^-1 P^T ,

which satisfies the condition in Eq.(<ref>). 

If V_R = V_L, and m_N = v_R/v_L m_ν, using Eq. (<ref>), we can readily get M_N = v_R/v_L M_ν^*. Thus the above parameterization of M_N is compatible with the type-II seesaw dominance scenario. 





§ ONE-LOOP Θ̅THETA



As pointed out in Ref. <cit.>, θ̅ can be generated from the leptonic CP violation, which contributes to the Higgs potential at one-loop level:

    V ⊃[α_2 (Δ_R^†Δ_R) +  h.c.]  (Φ̃Φ) ,

where the coupling α_2 is complex and Φ̃≡σ_2 Φ^* σ_2.
It is shown that <cit.>

    θ̅_loop≃116π^2m_tm_bTr( Y_R^† Y_R [Y_1, Y_2] )lnM_Plv_R ,

where the Dirac Yukawa couplings Y_R and Y_1,2 are defined in Eq. (<ref>), and M_Pl=1.22× 10^19 denotes the Planck scale. In terms of the mass matrices, we have <cit.>

    θ̅_loop   ≃116π^2m_tm_b1v_R^2 v^2 
         ×Tr( M_N^T M_N^* [M_D, M_ℓ] )lnM_Plv_R .

where the charged lepton mass matrix M_ℓ is diagonal due to U_e = ± 1.
If V_R= V_L,
by using the expressions of M_N and M_D given in Eqs. (<ref>) (<ref>), we can easily verify that θ̅_loop is exactly zero. This also applies when V_R =  1 <cit.>. 



In order to evaluate θ̅_loop for other choices of V_R, we use the parameterization in Sec. <ref>, and consider V_R = P̂ V_L with the following textures of P̂:

    P̂_1   =i[ 1 0 0; 0 0 1; 0 1 0 ] ,  P̂_2=i[ 0 1 0; 1 0 0; 0 0 1 ] ,  
    P̂_2   =i[     1/2     1/2 -√(2)/2;     1/2     1/2  √(2)/2; -√(2)/2  √(2)/2       0 ] ,

where we have included the factor of i to maintain the Hermiticity of the neutrino Dirac 
mass matrix M_D [In the type-I seesaw dominance scenario, where v_L is negligibly small, M_D in Eq. (<ref>) is anti-Hermitian if we assume V_R = V_L, a fact that appears to have been overlooked. We find that for V_R = i V_L, θ̅_loop also vanishes, which can be verified analytically. 
]. One can directly verify that for these cases the matrix P = P̂ V_L √(m_N m_ν) V_L^† is anti-Hermitian.





We assume that the active neutrino masses m_ν≡diag(m_1,m_2,m_3) are in the normal hierarchy, and that the sterile neutrino masses  m_N ≡diag(m_4,m_5,m_6) are correlated with m_ν:

    m_1 m_4 = m_2 m_5 = m_3 m_6 ,

and choose the parameters as

    m_1 = 10^-3  eV ,   v_L = 1  eV ,   v_R = 15 .




For the cases in Eq. (<ref>), we obtain the magnitude of non-vanishing θ̅_loop as a function of the heaviest sterile neutrino m_ N max = m_4 in Fig. <ref>. 
Since θ̅_loop approximately increases with the sterile neutrino masses (m_ Nmax )^5/2, by requiring |θ̅_loop|≲ 10^-10 [ We obtain that θ̅_loop has definite sign for P̂ being given in Eq. (<ref>).   
], we obtain the upper bound on the sterile neutrino masses, which was highlighted in Ref. <cit.>. For P̂=P̂_1, P̂_2 and P̂_3, we obtain m_ N max≲ 2.5, 6 and 2, respectively.

As a benchmark, we take P̂ = P̂_1 and assume
m_4 =2.86   TeV, m_5=3.32   GeV, and m_6 =57.2   MeV.
The heavy neutrino mass matrix is given by


    M_N=
        ([   -1.95× 10^12-7.05×10^5 i -1.16× 10^12+5.60× 10^10 i  6.38× 10^11+6.47× 10^10 i; -1.16× 10^12+5.60× 10^10 i -6.96× 10^11+6.69× 10^10 i  3.85× 10^11+2.03× 10^10 i;  6.38× 10^11+6.47× 10^10 i  3.85× 10^11+2.03× 10^10 i -2.09× 10^11-4.24× 10^10 i;                            ])  eV .

Using the Senjanovic-Tello method, we obtain the matrices in Eq.(<ref>)

    O=
            ([  -0.1344+0.04691i -0.4861-0.006028i  0.8648+0.003902i; 0.6396-0.0002750i   0.6240+0.01683i   0.4499-0.02296i;  0.7584+0.008545i  -0.6125+0.02193i  -0.2263-0.03073i ]) ,
            
       E=1 , s=
            ([  8.576×10^-7            0            0;            0 4.602×10^-10            0;            0            0 6.867×10^-13 ]) ,

and the neutrino Dirac mass matrix

    M_D=
        ([           545380.  343623.+13636. i -204348.+16763. i;  343623.-13636. i           272102. -116364.+17374. i; -204348.-16763. i -116364.-17374. i           109404.;                   ])  eV .


The resulting value of θ̅_loop is -1.241335×10^-10. While the heaviest sterile neutrino is within reach of direct searches at the Large Hadron Collider <cit.>, the lighter sterile neutrinos could give significant contributions to neutrinoless double beta decay <cit.>.


It is worth noting that due to s_α t_2β∝α_2, the neutrino Dirac mass matrix M_D is not exactly Hermitian. 
A delicate examination of non-Hermitian M_D was recently conducted in Ref. <cit.>. Nevertheless, the correlation between M_D and θ̅  poses a challenge for recursive evaluation. Unless there is accidental cancellation, assuming that M_D is exactly Hermitian, as we have done, is adequate for estimating the upper limit on the sterile neutrino masses.



§ CONCLUSION



In this work, we have proposed a parameterization of right-handed neutrino mixing V_R = P V_L √(m_N m_ν)^-1 with P being a Hermitian or anti-Hermitian matrix in the MLRSM of case 𝒫, and constructed heavy neutrino mass matrix as M_N = P M_ν^-1 P^T. In this parameterization, the Hermiticity of the neutrino Dirac mass matrix M_D is maintained. 
We then evaluate the one-loop θ̅ generated from leptonic CP violation for the general seesaw relation with explicit examples of V_R and obtain non-vanishing θ̅_loop as a function of the sterile neutrino masses. By requiring |θ̅_loop| ≲ 10^-10, we obtain the upper bound on the sterile neutrino masses.

Our parameterization of V_R and M_N is applicable to other phenomenological studies. In particular, it enables us to study the interplay of 0νββ decay and one-loop θ̅ mediated by the sterile neutrinos in the MLRSM of case 𝒫.









§ HERMITCITY OF HH



In Sec. <ref>, we have provided the parameterization of the right-handed neutrino mixing V_R and heavy neutrino mass matrix M_N, which guarantees the Hermiticity of the matrix H hence neutrino Dirac mass matrix M_D. In this appendix, we will give more details.

From Eq. (<ref>), we expand

    [v_L/v_R-1/M_NM_ν^*]^n
    
        =   C_1 + C_21/M_NM_ν^* +...+C_n(1/M_NM_ν^*)^n ,

where all the C_n for n∈N are real numbers. So all we need to check is (M_N^-1M_ν^*)^n. 


First, we assume

    M_N = P M_ν^-1 Q ,

where Q is a matrix with dimension one. 

We observe that if Q = ± P^*, 

    ImTr[(1/M_NM_ν^*)^n]
        
    
        =   ±ImTr[(P^-1*M_νP^-1M_ν^*)^n] .

Defining A = P^-1^* M_ν, we have

    [ (A A^*)^n ]    = [ A^* (A A^*)^n-1 A ]
       =[ (A^* A)^n ] ,

so that 

    ImTr[(1/M_NM_ν^*)^n] 
        = 
        ImTr[(1/M_NM_ν^*)^n*] ,

which implies that

    ImTr[(1/M_NM_ν^*)^n] = 0 .

Therefore, the condition in Eq. (<ref>) is satisfied. 

From Eq. (<ref>), we have

    M_N
         
        =   PV_L m_ν^-1V_L^T Q
    
        =    V_R m_N V_R^T .

To find a possible form of V_R, we define

    F = √(m_ν) X √(m_N) ,

where X is an orthogonal matrix, X X^T = 1,
and obtain

    m_ν=Fm_N^-1 F^T .





Then Eq. (<ref>) becomes

    PV_L (F^T)^-1 m_NF^-1V_L^T Q = V_R m_N V_R^T .

If 

    V_R=PV_L (F^T)^-1 ,  
        V_R^T=F^-1V_L^T Q ,

the relation in Eq. (<ref>) must be satisfied. Hence, Q = P^T = ± P^* with P being a Hermitian or anti-Hermitian matrix.
Without loss of generality, we assume X= 1, and then F=√(m_ν m_N). Other choices of X can also yield appropriate V_R and P, which may also be of interest.




§ ACKNOWLEDGEMENTS

GL would like to thank Jordy de Vries and Juan Carlos Vasquez for fruitful discussions, and Ravi Kuchimanchi for helpful correspondence regarding Ref. <cit.>. DYL is grateful to Ken Kiers for generously sharing the codes of Ref. <cit.> and for providing valuable suggestions and help.
This work is partly supported by the National Natural Science Foundation of China under Grant No. 12347105, Fundamental Research Funds for the Central Universities in Sun Yat-sen University (23qnpy62), and SYSU startup funding. 






apsrev4-1

