Stability of the Next-to-Tribimaximal Mixings under Radiative Corrections with the Variation of the SUSY Breaking Scale in MSSM

. We analyze the radiative stability of the next-to-tribimaximal mixings (NTBM) with the variation of the SUSY breaking scale ( m S ) in MSSM, for both normal ordering (NO) and inverted ordering (IO) at the ﬁ xed input value of the seesaw scale M R = 10 15 GeV and two di ﬀ erent values of tan β . All the neutrino oscillation parameters receive varying radiative corrections irrespective of the m S values at the electroweak scale, which are all within the 3 σ range of the latest global ﬁ t data at a low value of tan β . NO is found to be more stable than IO for all four di ﬀ erent NTBM mixing patterns.


Introduction
Neutrino oscillations have been very well established by measuring the neutrino-mixing parameters θ 12 , θ 23 , θ 13 , Δ m 2  21 , and Δm 2 31 [1].One of the promising candidates for explaining the observed Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix is the tribimaximal (TBM) [2] which is ruled out due to the discovery of the nonzero value of θ 13 [3][4][5][6][7][8][9].Hence, in order to accommodate θ 13 ≠ 0, the PMNS matrix is reproduced using the next-to-TBM (NTBM) [10,11] which predicts the correlations among the phase and mixing angles.Existence of the PMNS-mixing matrix, which is the analogue of the CKM matrix in the quark sector, is the consequence of diagonalisation of the neutrino mass matrix.The PMNS-mixing matrix contains three mixing angles θ 12 , θ 23 , and θ 13 and a phase δ CP responsible for CP violation.Two additional phases which do not influence neutrino oscillations are added if we consider neutrinos as Majorana fermions.The measurement of a nonzero θ 13 using reactor neutrinos in 2012 has opened the possibility to measure CP violation in the lepton sector.
The present work is a continuation of our previous work [12] on neutrino masses and mixings with varying SUSY breaking scale m S under RGEs [12][13][14][15][16][17][18][19].We study both normal-and inverted-ordering neutrino mass models.We adopt the bottom-up approach for running gauge and Yukawa couplings from low to high energy scales and the top-down approach for running neutrino parameters from high to low energy scales, along with gauge and Yukawa couplings.
Following the discovery of a nonzero θ 13 , the originally proposed TBM (tribimaximal) mixing pattern, which initially assumed θ 13 to be zero, was no longer considered a valid description.Consequently, extensive research efforts were directed towards exploring various TBM variants capable of accommodating a nonzero θ 13 while accurately reproducing the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix.Among these variants, the next-to-TBM-(NTBM-) mixing scheme has gained prominence. NTBM mixing is characterized by a two-parameter family and has played a crucial role in making precise predictions and establishing correlations among the mixing angles θ ij and the Dirac CP phase δ CP .Multiple model-independent analyses have delved into these correlations, with references available in [10,11,[20][21][22].It is worth emphasizing that the original TBM-mixing pattern, denoted as V TBM , explicitly predicts θ 13 to be exactly zero.However, thorough global fit studies, as summarized in Table 1, have unequivocally demonstrated the existence of a nonzero θ 13 .Consequently, a strong incentive exists to investigate deviations from the TBM-mixing pattern.Phenomenologically, small deviations from the TBM pattern can be easily parameterized by multiplication with a unitary rotation matrix.For example, the TBM 1 -mixing scheme, given in equation (2), can be seen as a modification to the TBM-mixing pattern by applying a 23rotation matrix R 23 from the right to V TBM .In this sense, the rotation matrix R 23 κ 1 , κ 2 could also be interpreted as a perturbation to the exact TBM-mixing pattern.The physical observables-namely, the three mixing angles and the Dirac CP phase δ CP -are correlated via the two parameters κ 1 and κ 2 , which leads us to a well-defined phenomenology.
There are four allowed NTBM patterns [23] depending on the position (left or right) of the multiplication by a unitary rotation matrix to tribimaximal (TBM) mixing.The V TBM is given by The four allowed NTBM patterns are [23] where R 23 , R 13 , and R 12 are the rotation matrices defined as where κ 1 and κ 2 are free parameters within the ranges 0 ≤ κ 1 ≤ π and 0 ≤ κ 2 < 2π, respectively.U 1 , U 2 , U 2 , and U 3 are distinct NTBM-mixing patterns for the TBM 1 , TB M 2 , TBM 2 , and TBM 3 scenarios, respectively.In the present work, we calculate the values of mixing angles and δ CP given by these four different mixing patterns, and these are found to be valid for certain values of κ 1 and κ 2 .We check the stability against radiative corrections by varying the SUSY breaking scale m S within the range of 2-14 TeV, considering two different values of tan β = 30, 50.The paper is organized as follows.NTBM is discussed in Section 2. Analysis for RGEs is discussed in Section 3. Numerical analysis and results are given in Section 4. Section 5 concludes the paper.

Numerical Predictions in Next-to-TBM (NTBM)
NTBM is defined by multiplying V TBM by a unitary rotation matrix on either the left or the right.There are six possible NTBM patterns, but only four patterns given in equations ( 2)-( 5) are allowed since two patterns: U P = V TBM R 12 and U P = R 23 V TBM , are already excluded as they predict zero θ 13 .The four NTBM patterns provide formulas for the mixing angles and δ CP in terms of free parameters κ 1 and κ 2 , which characterize the two-parameter family of NTBM patterns [23].The four patterns are given as follows.
tan θ 23 = We impose the conditions sin δ CP > 0 (NO) and sin δ CP < 0 (IO) to constrain the two free parameters κ 1 and κ 2 for all NTBM scenarios.The best-estimated numerical values of θ ij and δ CP for TBM 1 , TBM 2 , TBM 2 , and TBM 3 are provided in Table 2 for specific choices of κ 1 and κ 2 in each case, both for NO and IO.We use values of κ 1 and κ 2 that fall within the allowed regions depicted in Figures 1  and 2 for both NO and IO, respectively

Analysis for RGEs
Numerical analysis of Renormalization Group Equations (RGEs) [18,24,25] is conducted in two successive steps: first, bottom-up running, and second, top-down running.Two-loop Renormalization Group Equations (RGEs) for gauge and Yukawa couplings are provided in Appendix A for both the Standard Model (SM) and the Minimal Supersymmetric Standard Model (MSSM).The RGEs for neutrino oscillation parameters are presented in Appendix B.
3.1.Bottom-Up Running.Bottom-up running is used to extract the values of gauge and Yukawa couplings at a high energy scale using RGEs which can be divided into three regions, m Z < μ < m t , m t < μ < m S , and m S < μ < M R .We use recent experimental data [1,26] as initial input values at the low energy scale, which are given in Table 3.
We calculate the values of gauge couplings, α 2 for SU 2 L , and α 1 for U 1 Y , by using sin 2 θ W m Z = α em m Z /α 2 m Z and matching condition, 4 Advances in High Energy Physics The normalised couplings [18], g i = 4πα i , where α i 's are the gauge couplings and i = 1, 2, 3 denote electromagnetic, weak, and strong couplings, respectively.One-loop gauge coupling RGEs [28] for the evolution from the m Z scale to m t scale, are given below:  4.
The evolution of gauge and Yukawa couplings for running from m t to the M R scale using RGEs is given in 5 Advances in High Energy Physics Appendix A. The following matching conditions are applied at the transition point from SM (m t < μ < m S ) to MSSM (m S < μ < M R ) at the m S scale, as In the present work, we have observed the following trends at input values of tan β = 30 and tan β = 50.At tan β = 30, both h t and h b decrease as the m S scale increases, while h τ increases with the increment in the m S scale due to its dependence on tan β, as demonstrated in equation (13).These trends are illustrated in Table 5, which will serve as input values for the subsequent top-down running at the high energy scale M R .On the other hand, at tan β = 50, all gauge and Yukawa couplings decrease with increasing m S , as shown in Table 6.

Top-Down
Running.We use the top-down running approach to study the stability for four patterns using RGEs against varying m S at a fixed value of the seesaw scale M R = 10 15 GeV and tan β = 30, 50.In this running, we use the values of the Yukawa and gauge couplings which were earlier estimated at the M R scale, as initial inputs.We consider simple mass relations for both NO and IO in order to minimize the number of input-free parameters, and both their sums of the three neutrino mass eigenvalues Σm i are all within the favorable range given by latest cosmological bound [30,31].We take the two Majorana phases ψ 1 and ψ 2 to be 0 and 180, respectively.We constraint the value of the Dirac CP phase δ CP at 180 °.Using all the necessary mathematical frameworks, we analyze the radiative nature of neutrino parameters like neutrino masses, mixings, and CP phases, using the topdown approach with the variations of the m S scale at a fixed value of tan β for all allowed mixing patterns.In this work, we study the stability of TBM 1 , TBM 2 , TBM 2 , and TBM 3 considering the current experimental data given in Table 1.We assume a relation among the three neutrino mass eigenstates for all the four NTBM-mixing patterns.The values of the free parameters κ 1 and κ 2 are considered which satisfy the condition sin δ CP > 0 for NO and sin δ CP < 0 for IO.The input set at a high energy scale is given in Table 2.

Numerical Analysis and Results
Here, we analyze the impact of varying m S while keeping the values of M R = 10 15 GeV and tan β fixed at 30 and 50, respectively.We consider the effects on neutrino oscillation parameters for both NO and IO, presenting numerical data in Tables 7-10 and graphical representations in Figures 3 and 4.
At tan β = 30, for NO, using the input set which is given in Table 2, it is found that all the neutrino oscillation parameters are in favor with the latest data which are within the 3σ range.All the three mixing angles decrease with increasing m S , but Δm 2 ij increases with increasing m S .δ CP almost maintains stability against the variation of m S .
At tan β = 30, for IO, using the input set which is given in Table 2, it is found that all the neutrino oscillation parameters are in favor with the latest data which are within the 3σ  7 and 9 and their graphical representations in Figures 3 and 5 that NO maintains more stability than the IO.At tan β = 50 and M R = 10 15 GeV, the low energy values of the three mixing angles and δ CP remain stable with the variaion of m S , for both the NO and IO scenarios.However, the low energy values of the Δm 2  21 with the variation of m S are outside the range provided by the global fit data for IO, but for NO, both the low energy values of the two-mass squared differences are within the global fit data, indicating slight preference of NO to IO.These results are presented in Tables 8 and 10 and are graphically represented in Figures 4 and 6.    12 Advances in High Energy Physics If we consider the position (left or right) of the multiplication by a unitary rotation matrix to tribimaximal (TBM) mixing, assuming these unitary rotation matrices are originated from charged lepton mass matrices, TBM 2 and TBM 3 are acceptable as compared to TBM 1 and TBM 2 , as the position of the unitary matrices should be on the left side of the TBM matrix.For example, U 3 = R 12 V TBM .Further, if we also consider the graphical analysis of κ 1 and κ 2 for obtaining the best fit in both NO and IO, TBM 3 seems to be better than the other three types of NTBM, as the ranges of κ 1 and κ 2 in TBM 3 are common for both NO and IO as shown in Figures 1 and 2 for four different NTBM patterns.For the other remaining three cases of NTBM, the ranges of κ 1 and κ 2 are different in NO and IO.Considering the above two points, our analysis shows that TBM 3 is the best candidate.

Discussion and Conclusions
To summarize, we impose the following conditions to obtain the best fit pattern models among the NTBM.
The input of the sum of three neutrino masses should satisfy the latest PLANCK cosmological data Σ m i < 0 12 eV.
(i) We apply the conditions sin δ CP > 0 (NO) and sin δ CP < 0 (IO) in order to constraint the two free parameters κ 1 and κ 2 , respectively, for all the NTBM scenarios (ii) We take the values of κ 1 and κ 2 which give the latest values of three mixing angles given by the latest global fit data (iii) We take different values of κ 1 and κ 2 which lie within the allowed regions as depicted in Figures 1  and 2 for four different NTBM patterns for both NO and IO We study the stability for four different NTBM patterns at a fixed value of M R = 10 15 GeV and two different values of tan β (30, 50) for both NO and IO.
Case A tan β = 30 : for NO, we have studied the stability for four patterns of NTBM using RGEs against the variation of m S .There is a mile decrease of the mixing angles θ ij and δ CP with the increase of m S (2 TeV-14 TeV).These are found to lie within 3σ ranges of observational data.Δ m 2 ij and δ CP increase with increasing m S .The low energy values of Δm 2  21 are found to lie within 2σ whereas those of Δm 2  31 lie within 3σ except for low values of m S .Similarly, for IO, the low energy values of Δm 2 31 and δ CP decrease with the increase of m S (2 TeV-14 TeV) which are found to lie within 3σ ranges.Δm 2  21 is found to lie within 3 σ ranges, which increases with increasing m S .θ ij (except θ 12 ) increases slightly with the increase of m S .
In short, it is found that NTBM-mixing patterns maintain stability under radiative corrections with the variation of m S for a normal ordering case at the fixed value of seesaw scale M R .All the neutrino oscillation parameters receive varying radiative corrections irrespective of the m S values at the electroweak scale, which are all within the 3σ range of the latest global fit data.NO maintains more stabilty as compared to IO with increasing m S .All the four patterns of NTBM are found to be stable with the variation of m S under radiative corrections in MSSM for both NO sin δ > 0 and IO sin δ < 0 .If we consider the graphical analysis for κ 1 and κ 2 both for NO and IO as depicted in Figures 1  and 2, TBM 3 is the best candidate, since it the most consistent one among the four NTBM cases.
Case B (tan β = 50): for both NO and IO, the low energy values of all the three mixing angles with the variation of m S are within the 3σ range.They remain stable with the variation of m S .For IO, the low energy values of the Δm 2  21 with the variation of m S fall outside the range of the global fit data.However, for NO, both the low energy values of the two-mass squared differences fall within the 3σ range of the global fit data.This indicates the slight preference for NO to IO in our numerical analysis.Additionally, all of the low energy values of the neutrino oscillation parameters undergo distinct radiative corrections.The graphical representations can be seen in Figures 4 and 6, accompanied by the numerical data presented in Tables 8 and 10.

Appendix
A. RGEs for Gauge Couplings [24] The two-loop RGEs for gauge couplings are given by  [24].For MSSM, where , For the non-supersymmetric case, and λ = m 2 h /v 2 0 is the Higgs self-coupling, m h = 125 78 ± 0 26 GeV is the Higgs mass [32], and v 0 = 174 GeV is the vacuum expectation value.
The beta function coefficients for non-SUSY case are given as follows:

Figure 1 :
Figure 1: Constraints on the two free parameters κ 1 and κ 2 are applied in the context of the NO spectrum.The 3σ-allowed mixing angles are shaded in black, and regions where sin δ CP > 0 are shaded in green, both for the four different mixing patterns of NTBM.
and b i = 5 30,−0 50,−4 00 for the non-SUSY case.Using the QED-QCD rescaling factor η [29], fermion masses at the m t scale are given by m b m t = m b m b /η b and m τ m t = m τ m τ /η τ , where η b = 1 53 and η τ = 1 015.The Yukawa couplings at the m t scale are given by h t m t = m t m t /v 0 , h b m t = m b m b /v 0 η b , and h τ m τ = m τ m τ /v 0 η τ , where v 0 = 174 GeV is the vacuum expectation value (VEV) of the SM Higgs field.The calculated numerical values for fermion masses, Yukawa couplings, and gauge couplings at the m t scale are given in Table

Figure 2 :
Figure 2: Constraints on the two free parameters κ 1 and κ 2 are imposed within the inverted-ordering (IO) spectrum.The 3σ-allowed mixing angles are shaded in black, while regions where sin δ CP < 0 are shaded in green, for the four different mixing patterns of NTBM.

3 TBM1Figure 3 : 3 Figure 4 : 3 Figure 5 :
Figure 3: The variation of m S for different NTBM patterns at M R = 10 15 GeV and tan β = 30 for NO leads to effects on the low-energy output results of θ ij , Δm 2 ij , and δ CP .

2 TBM 3 Figure 6 :
Figure 6: The variation of m S for different NTBM patterns at M R = 10 15 GeV and tan β = 50 for NO leads to effects on the low-energy output results of θ ij , Δm 2 ij , and δ CP .

Table 2 :
The allowed input set of neutrino parameters at the high-energy scale M R = 1015GeV and tan β = 30, 50 encompasses four different patterns.θ 23 , θ 12 , θ 13 , and δ CP are estimated values, while the others are arbitrary allowed input values.In this paper, we consider both the NO and IO.

Table 3 :
Current experimental data for fermion masses, gauge coupling constants, and Weinberg mixing angle.

Table 4 :
Numerical calculated values for fermion masses, Yukawa couplings, and gauge couplings at m t scale.

Table 5 :
Yukawa and gauge couplings were evaluated at M R = 1015GeV for tan β = 30, considering different choices of the m S scale.

Table 6 :
Yukawa and gauge couplings were evaluated at M R = 1015GeV for tan β = 50, considering different choices of the m S scale.
range.All the three mixing angles and δ CP maintain more stability as compared to Δm 2 ij .Both θ 23 and θ 13 increase but θ 12 decreases against the variation of m S .Δm 2 21 increases whereas Δm 2 31 decreases with increasing m S , and Δm 2 31 main-tains more stability as compared to Δm 2 21 at higher m S .It is observed from Tables

Table 7 :
Effects on the output of θ ij , Δm 2 ij , and Σm i at a low-energy scale, on varying m S for four different NTBM patterns (NO) at tan β = 30 and M R = 10 15 GeV.

Table 8 :
Effects on the output of θ ij , Δm 2 ij , and Σm i at a low-energy scale, on varying m S for four different NTBM patterns (NO) at tan β = 50 and M R = 10 15 GeV.

Table 9 :
Effects on the output of θ ij , Δm 2 ij , and Σm i at a low-energy scale, on varying m S for four different NTBM patterns (IO) at tan β = 30 and M R = 10 15 GeV.

Table 10 :
Effects on the output of θ ij , Δm 2 ij , and Σm i at a low-energy scale, on varying m S for four different NTBM patterns (IO) at tan β = 50 and M R = 10 15 GeV.