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

Devi, Kh. Helensana; Singh, K. Sashikanta; Singh, N. Nimai

doi:https://doi.org/10.1155/2023/2365316

Advances in High Energy Physics

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Abstract Introduction Results Discussion and Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 2365316 | https://doi.org/10.1155/2023/2365316

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

Kh. Helensana Devi,¹K. Sashikanta Singh,¹and N. Nimai Singh^1,2

Academic Editor: Theocharis Kosmas

Received01 Mar 2023

Revised22 Sept 2023

Accepted03 Nov 2023

Published24 Nov 2023

Abstract

We analyze the radiative stability of the next-to-tribimaximal mixings (NTBM) with the variation of the SUSY breaking scale () in MSSM, for both normal ordering (NO) and inverted ordering (IO) at the fixed input value of the seesaw scale GeV and two different values of . All the neutrino oscillation parameters receive varying radiative corrections irrespective of the values at the electroweak scale, which are all within the range of the latest global fit data at a low value of . NO is found to be more stable than IO for all four different NTBM mixing patterns.

1. Introduction

Neutrino oscillations have been very well established by measuring the neutrino-mixing parameters , , , , and [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 [3–9]. Hence, in order to accommodate , 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 , , and and a phase 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 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 under RGEs [12–19]. 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 , the originally proposed TBM (tribimaximal) mixing pattern, which initially assumed 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 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 and the Dirac CP phase . Multiple model-independent analyses have delved into these correlations, with references available in [10, 11, 20–22]. It is worth emphasizing that the original TBM-mixing pattern, denoted as , explicitly predicts to be exactly zero. However, thorough global fit studies, as summarized in Table 1, have unequivocally demonstrated the existence of a nonzero . 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 -mixing scheme, given in equation (2), can be seen as a modification to the TBM-mixing pattern by applying a 23-rotation matrix from the right to . In this sense, the rotation matrix 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 —are correlated via the two parameters and , 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 is given by

The four allowed NTBM patterns are [23] where , , and are the rotation matrices defined as where and are free parameters within the ranges and , respectively. , , , and are distinct NTBM-mixing patterns for the , , , and scenarios, respectively. In the present work, we calculate the values of mixing angles and given by these four different mixing patterns, and these are found to be valid for certain values of and . We check the stability against radiative corrections by varying the SUSY breaking scale within the range of 2–14 TeV, considering two different values of . 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.

2. Numerical Predictions in Next-to-TBM (NTBM)

NTBM is defined by multiplying 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: and , are already excluded as they predict zero . The four NTBM patterns provide formulas for the mixing angles and in terms of free parameters and , which characterize the two-parameter family of NTBM patterns [23]. The four patterns are given as follows.

2.1. Pattern

2.2. Pattern

2.3. Pattern

2.4. Pattern

We impose the conditions (NO) and (IO) to constrain the two free parameters and for all NTBM scenarios. The best-estimated numerical values of and for , , , and are provided in Table 2 for specific choices of and in each case, both for NO and IO. We use values of and that fall within the allowed regions depicted in Figures 1 and 2 for both NO and IO, respectively

3. 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, , , and . 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, for , and for , by using and matching condition,

The normalised couplings [18], , where ’s are the gauge couplings and denote electromagnetic, weak, and strong couplings, respectively. One-loop gauge coupling RGEs [28] for the evolution from the scale to scale, are given below: where and for the non-SUSY case. Using the QED-QCD rescaling factor [29], fermion masses at the scale are given by and , where and . The Yukawa couplings at the scale are given by , , and , where 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 scale are given in Table 4.

The evolution of gauge and Yukawa couplings for running from to the scale using RGEs is given in Appendix A. The following matching conditions are applied at the transition point from SM () to MSSM () at the scale, as

In the present work, we have observed the following trends at input values of and . At , both and decrease as the scale increases, while increases with the increment in the scale due to its dependence on , 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 . On the other hand, at , all gauge and Yukawa couplings decrease with increasing , as shown in Table 6.

3.2. Top-Down Running

We use the top-down running approach to study the stability for four patterns using RGEs against varying at a fixed value of the seesaw scale and . In this running, we use the values of the Yukawa and gauge couplings which were earlier estimated at the 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 are all within the favorable range given by latest cosmological bound [30, 31]. We take the two Majorana phases and to be 0 and 180, respectively. We constraint the value of the Dirac CP phase 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 top-down approach with the variations of the scale at a fixed value of for all allowed mixing patterns. In this work, we study the stability of , , , and 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 and are considered which satisfy the condition for NO and for IO. The input set at a high energy scale is given in Table 2.

4. Numerical Analysis and Results

Here, we analyze the impact of varying while keeping the values of GeV and 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 , 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 range. All the three mixing angles decrease with increasing , but increases with increasing . almost maintains stability against the variation of .

At , 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 range. All the three mixing angles and maintain more stability as compared to . Both and increase but decreases against the variation of . increases whereas decreases with increasing , and maintains more stability as compared to at higher . It is observed from Tables 7 and 9 and their graphical representations in Figures 3 and 5 that NO maintains more stability than the IO.

At and GeV, the low energy values of the three mixing angles and remain stable with the variaion of , for both the NO and IO scenarios. However, the low energy values of the with the variation of 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.

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, and are acceptable as compared to and , as the position of the unitary matrices should be on the left side of the TBM matrix. For example, . Further, if we also consider the graphical analysis of and for obtaining the best fit in both NO and IO, seems to be better than the other three types of NTBM, as the ranges of and in 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 and are different in NO and IO. Considering the above two points, our analysis shows that is the best candidate.

5. 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 eV. (i)We apply the conditions (NO) and (IO) in order to constraint the two free parameters and , respectively, for all the NTBM scenarios(ii)We take the values of and which give the latest values of three mixing angles given by the latest global fit data(iii)We take different values of and 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 GeV and two different values of (30, 50) for both NO and IO.

Case A : for NO, we have studied the stability for four patterns of NTBM using RGEs against the variation of . There is a mile decrease of the mixing angles and with the increase of (2 TeV–14 TeV). These are found to lie within ranges of observational data. and increase with increasing . The low energy values of are found to lie within whereas those of lie within except for low values of .

Similarly, for IO, the low energy values of and decrease with the increase of (2 TeV–14 TeV) which are found to lie within ranges. is found to lie within ranges, which increases with increasing . (except ) increases slightly with the increase of .

In short, it is found that NTBM-mixing patterns maintain stability under radiative corrections with the variation of for a normal ordering case at the fixed value of seesaw scale . All the neutrino oscillation parameters receive varying radiative corrections irrespective of the values at the electroweak scale, which are all within the range of the latest global fit data. NO maintains more stabilty as compared to IO with increasing . All the four patterns of NTBM are found to be stable with the variation of under radiative corrections in MSSM for both NO and IO . If we consider the graphical analysis for and both for NO and IO as depicted in Figures 1 and 2, is the best candidate, since it the most consistent one among the four NTBM cases.

Case B (): for both NO and IO, the low energy values of all the three mixing angles with the variation of are within the range. They remain stable with the variation of . For IO, the low energy values of the with the variation of 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 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 where and are function coefficients in MSSM, and, for the non-supersymmetric case, we have

A.1. Two-Loop RGEs for Yukawa Couplings and Quartic Higgs Coupling [24]

For MSSM, where

For the non-supersymmetric case, where and is the Higgs self-coupling, GeV is the Higgs mass [32], and GeV is the vacuum expectation value.

The beta function coefficients for non-SUSY case are given as follows: , and

B. RGEs for Three Neutrino-Mixing Angles and Phases [33]: Neglecting Higher Order of

where , , and .

B.1. RGEs for the Three Phases [33]

For the Dirac phase , where

For the Majorana phase [33],

For the Majorana phase ,

B.2. RGEs for Neutrino Mass Eigenvalues [33]

where

For the MSSM case,

For the SM case, and is the Higgs self-coupling in the SM.

Data Availability

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

One of the authors (KHD) would like to thank SCOAP³ for the financial support. This paper has previously been presented as an arXiv [34] (10.48550/arXiv.2302.11852). The editable file of this manuscript is updated on the zenodo website https://zenodo.org/records/10068005?token=eyJhbGciOiJIUzUxMiJ9.eyJpZCI6IjMwNWEzODFlLTllNmQtNDAzZC1hOGVmLTM3OWFlODcwZGYwMiIsImRhdGEiOnt9LCJyYW5kb20iOiJmNDhjN2QxMzU2MzgwNDQ1NWI4NDBiMzYxYjMxNGE3MiJ9.vqifuZt0X5NWkVv_UoWYmbanadokRELpky5nSA6sAp-aP5qtdNxfX0If82B3_NRM8bZMh9q-77VfcyWeRUJB2g. Here, out of 32 figures, 24 editable figure files are also given.

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Copyright

Copyright © 2023 Kh. Helensana Devi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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