# Effects of Variations of SUSY Breaking Scale on Neutrino Parameters at Low Energy Scale under Radiative Corrections

Authors: Kh. Helensana Devi; K. Sashikanta Singh; N. Nimai Singh
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

The paper addresses the effects of the variations of the SUSY breaking scalems in the range (2-14) TeV on the three neutrino masses and mixings, in running the renormalization group equations (RGEs) for different input values of high energy seesaw scale MR, and in both normal and inverted hierarchical neutrino mass models. The present investigation is a continuation of the earlier works based on the variation of ms scale. Two approaches are adopted one after another—bottom-up approach for running gauge and Yukawa couplings from low to high energy scale, followed by the top-down approach from high to low energy scale for running neutrino parameters defined at high energy scale, along with gauge and Yukawa couplings. A self-complementarity relation among three mixing angles is also employed in the analysis and it is found to be stable under radiative correction. Significant effect due to radiative corrections on neutrino parameters with the variation of SUSY breaking scale ms is observed. For comparison of the results, variation of tanβ for different MR is also considered.


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## Body

## 1. Introduction

Neutrino physics has registered significant progress in recent years with the measurements of nonzeroθ13 [1–3] and the Dirac CP phase [4, 5], thus indicating a possibility for a sizable CP violation in neutrino sector. The T2K team [5] has concluded with 3σ confidence level that the Dirac Phase δCP lies somewhere between -3.41 and -0.03 for normal hierarchy (NH) and between -2.54 and -0.32 for inverted hierarchy (IH). The interval includes the CP-conserving value of –π=−22/7=−3.14in case of NH, so that the CP conservation is disfavored only at the modest 2σ confidence level. Next generation of neutrino detectors, such as Hyper-Kamiokande in Japan, DUNE in USA, and JUNO in China, may be able to get 5σ confidence level for confirmation of CP violation in neutrino sector. Neutrino oscillations [6–8] have been well studied with the precise measurements of neutrino oscillation mass parameters and mixing angles. But till date, there are still some unsettled questions in neutrino physics such as the correct mass hierarchical order whether normal or inverted, absolute neutrino mass scale, nature of neutrinos whether Dirac or Majorana type, the exact high energy scale of seesaw mechanism, and the supersymmetric breaking scale if all it exists, to mention a few. The information related to the absolute neutrino masses has been continuously updating with the recent Planck data on cosmological upper bound [9, 10] on the sum of three absolute neutrino masses Σmi<0.12 eV, neutrinoless double beta decay [11, 12] results with the upper limit on the effective Majorana neutrino mass mββ<0.036−0.156 eV from the KamLAND-Zen experiment [13] in Japan, and KATRIN [14] result on direct kinematic measurement with the upper bound mνe<0.8 eV. Neutrino mass model, if any, is bound to be consistent with these upper bounds on absolute neutrino masses. On theoretical front, the presence of supersymmetry (SUSY) [15–17] enables us to ensure the stability of hierarchy between the weak and GUT scales with the possible cancellation of quadratic term in radiative corrections to the Higgs boson mass. It is needed to have a precise unification point of three gauge couplings at high GUT scale around 2×1016 GeV [18–20]. It can also provide a natural mechanism for understanding the electroweak symmetry breaking (EWSB) [21, 22] and Higgs physics. Minimal Supersymmetric Standard Model (MSSM) [23] is thus a straightforward extension of the Standard Model (SM) with minimum number of new parameters. All the particles in the same supersymmetric multiplet would have the same mass if the supersymmetry is an exact symmetry. So far, there is no clear evidence for the presence of supersymmetric particles in the ongoing Large Hadron Collider (LHC), and LHC has almost reached its maximum energy of about 14 TeV [24, 25]. Third run of LHC reaches 13.6 TeV slightly higher than that of 13 TeV of the second run [26]. While the existence of supersymmetric particles has been continuously ruling out in LHC, the supersymmetric breaking scale (ms) still remains as an unknown parameter. There are speculations that SUSY particles may have a wide spectrum and are not confined to a single energy scale. For simplicity, one can assume a single scale [27, 28] for all supersymmetric particles and this kind of assumption is valid as long as the mZ or mt<<ms [29, 30]. We assume that the ms scale may lie somewhere in between 2 TeV and 14 TeV within the scope of LHC. The effects of the variations of SUSY breaking scale on the unification of gauge couplings and also Yukawa couplings in MSSM and SUSY GUT models have already been addressed using the two-loop RGEs for gauge and Yukawa couplings within the minimal supersymmetric SU5 model [18], while ignoring for simplicity the threshold effects of the heavy particles, which could be as large as a few percentages. It has already been reported that for gauge couplings, the unification point increases with the increase in the SUSY breaking scale, but for Yukawa couplings the unification points decrease with the increase in SUSY breaking in the reverse order compared to the gauge couplings [18]. This finding has certain implications in other important issues such as running of the renormalization group equations (RGEs) [31–33] for neutrino masses and mixings from high energy seesaw scale to low energy electroweak scale. In this direction, a preliminary analysis with normal hierarchical model has already been carried out on the stability of neutrino parameters and self-complementarity relation [34] with varying SUSY breaking scale ms.The present investigation is a continuation of our previous work on neutrino masses and mixings with varying SUSY breaking scale in the running of RGEs [18, 22, 32, 35–38]. We shall address both normal hierarchical and inverted hierarchical neutrino mass models in both approaches—in the first place, the bottom-up approach for running gauge and Yukawa couplings from low to high energy scale; and in the second place, the top-down approach for running neutrino parameters defined at high energy scale, along with gauge and Yukawa couplings, from high to low energy scale. The present work is confined to the question of stability of neutrino mass models for both normal and inverted hierarchy with the variation of ms scale and other input parameters tanβ and MR scale. Another important applications of RGE analysis such as low energy magnification of neutrino mixings in quark-lepton unification hypothesis at high energy scale, radiative correction to validate the tribimaximal mixings and golden ratio mixing at high scale, and radiative origin of reactor angle and solar neutrino mass-squared parameter at low energy scale are not addressed in the present investigations.The paper is organised as follows. In Section2, we give a brief discussion of gauge and Yukawa coupling RGEs mainly on bottom-up and top-down runnings. In Section 3, we present the numerical analysis and results. In Section 4, we study the effects of variations on neutrino parameters for different values of tanβ. Summary and Discussion are presented in Section 5. We give relevant RGEs for gauge, Yukawa, and quartic Higgs couplings in two-loops for both the SM and MSSM in Appendix A and RGEs of neutrino parameters in Appendix B.


## 2. Renormalization Group Equations (RGEs)

We study the radiative corrections to neutrino oscillation parameters using the Renormalisation Group Equations (RGEs) [18, 31, 39] with and without SUSY in two different steps using the low energy observational input values, bottom-up running from low to high energy scale for gauge and Yukawa couplings, and top-down running from high to low energy scale for neutrino mass parameters and mixing angles, along with gauge and Yukawa couplings which are already evaluated at high energy scale MR.


### 2.1. Bottom-Up Running

In the bottom-up running of the RGEs, we divide it into three regions,mZ<μ<mt, mt<μ<ms, and ms<μ<MR. We use the recent experimental data [8, 40] as initial input values at low energy scale, given in Table 1.Table 1
Latest experimental data for fermion masses, gauge coupling constants, and Weinberg mixing angle.
Mass (GeV)Coupling constantsWeinberg mixing anglemZmZ=91.1876αem−1mZ=127.952±0.009sin2θWmZ=0.23121±0.00017mtmt=172.76αsmZ=0.1179±0.009mbmb=4.18mτmτ=1.77The values of gauge couplings,α2 for SU2L and α1 for U1Y, are calculated by using sin2θWmZ=αemmZ/α2mZ and matching condition,
(1)1αemmZ=531α1mZ+1α2mZ.We can also express the gauge couplingsαi’s [18] in terms of normalized couplings gi’s as gi=4παi, where i=1,2,3 denote electromagnetic, weak, and strong couplings, respectively. RGEs at one-loop level [41] is used for evolution of the three gauge coupling constants from mZ scale to mt scale, as given below
(2)1αiμ=1αimZ−bi2πlnμmZ,where mZ≤μ≤mt and bi=5.30,−0.50,−4.00 for non-SUSY case. For fermion masses to define at mt scale, we use QED-QCD rescaling factor η [42], mbmt=mbmb/ηb, and mτmt=mτmτ/ητ, where ηb=1.53 and ητ=1.015. We then convert them to Yukawa couplings, htmt=mtmt/v0, hbmt=mbmb/v0ηb, and hτmτ=mτmτ/v0ητ, where v0=174GeV is the vacuum expectation value (VEV) of SM Higgs field. The calculated numerical values for fermion masses, Yukawa, and gauge couplings at mt scale are given in Table 2.Table 2
Numerical values for fermion masses, Yukawa, and gauge couplings atmt scale.
Fermions massesYukawa couplingsGauge couplingsmtmt=172.76GeVhtmt = 0.9928g1mt=0.4635mbmt=2.73GeVhbmt=0.0157g2mt=0.6511mτmt=1.75GeVhτmt=0.0100g3mt=1.1890We study the effect of variation of SUSY breaking scale (ms) on gauge and Yukawa couplings for running from mt to the MR scale using RGEs, which are given in Appendix A. At ms scale, the following matching conditions are applied at the transition point from SM (mt<μ<ms) to MSSM (ms<μ<MR) as
(3)giSUSY=giSMhtSUSY=htSMsinβ=htSM×1+tan2βtanβ,hbSUSY=hbSMcosβ=hbSM×1+tan2β,hτSUSY=hτSMcosβ=hτSM×1+tan2β.Third generation Yukawa coupling constants are highly affected by input value oftanβ as shown in Equation (3). A detailed numerical analysis shows that high scale value of ht always decreases with input value on SUSY breaking scale ms in the range (2-14) TeV for low and high tanβ=25,35,40,45,and55. However, high scale values of hb and hτ behave in different patterns. In fact hb at high scale MR increases with the increase of ms scale for low value of tanβ=25, but it decreases with ms scale for higher values of tanβ=40−55. For hτ, it has a similar trend with hb but with a little difference in the range of tanβ. In fact hτ at high scale increases with the increase of ms scale for both low and moderate values of tanβ=25,40, but it again decreases with ms for higher value of tanβ=55. For specific case used in the present calculation at input value of tanβ=40, both ht and hb decrease with the increase of ms scale, but hτ increases with the increase of ms scale. This analysis is reflected in Tables 3–6.Table 3
Values of Yukawa and gauge couplings evaluated attR=ln1013GeV=29.93 for tanβ=40, for different choices of ms scale.
ms(TeV)hthbhτg1g2g320.65090.31390.37210.60860.69140.774340.63180.31020.37410.60430.68610.770060.62360.30860.37500.60230.68350.767980.61610.30710.37600.60020.68100.7658100.61260.30630.37650.59920.67970.7648120.60930.30560.37700.59810.67850.7638140.60610.30500.37750.59710.67720.7628Table 4
Values of Yukawa and gauge couplings evaluated attR=ln1014GeV=32.23 for tanβ=40, for different choices of ms scale.
ms(TeV)hthbhτg1g2g320.62890.29770.36500.63160.69820.755440.61040.29460.36740.62680.69150.751460.60220.29310.36840.62450.68890.749580.59470.29170.36950.62220.68630.7475100.59130.29100.37000.62110.68500.7466120.58790.29040.37050.61990.68370.7456140.58480.28980.37100.61880.68240.7447Table 5
Values of Yukawa and gauge couplingstR=ln1015GeV=34.53 for tanβ=40, for different choices of ms scale.
ms (TeV)hthbhτg1g2g320.60840.28290.35790.65740.70260.737840.58910.27980.36010.65210.69710.734160.58090.27840.36130.64940.69440.732380.57350.27710.36240.64680.69170.7305100.57010.27660.36290.64550.69030.7296120.56680.27600.36350.64430.68900.7287140.56370.27540.36400.64300.68770.7278Table 6
Values of Yukawa and gauge couplings evaluated attR=ln1016GeV=36.84 for tanβ=40, for different choices of ms scale.
ms (TeV)hthbhτg1g2g320.58540.26760.34940.68930.70890.720040.56610.26470.35180.68310.70320.716660.55800.26340.35290.68010.70040.714980.56800.26870.36220.67700.66640.7131100.54730.26180.35470.67570.69630.7124120.54410.26130.35530.67420.69490.7116140.54100.26080.35590.67270.69360.7107The output for Yukawa and gauge couplings atMR scale are given in Table 3 for MR=1013GeV, Table 4 for MR=1014GeV, Table 5 for MR=1015GeV, and Table 6 for MR=1016GeV, respectively, for common value of tanβ=40. These values are needed for the next top-down running as input values at high energy scale.


### 2.2. Top-Down Running

In this running, we use the values of Yukawa and guage couplings which are found atMR scale as initial inputs given in Tables 3–6. In this work, the high energy seesaw scale is the starting point for running the RGEs and it ends at electroweak scale. We give the sum of three neutrino masses Σmi in the range, 0.114 eV -0.121 eV for NH case and 0.1056 eV -0.1072 eV for IH case. The input values of neutrino masses and three mixing angles are indeed arbitrary in order to study the stability of neutrino mass model under RGEs running at low scale, with variations of other free parameters such as tanβ, SUSY breaking scale ms, and high energy scale MR. We select our input values in Tables 7–8 with the aim to produce low energy values of neutrino oscillation parameters consistent with observational data. In our work, we focus on the high energy seesaw scale at MR=1014GeV; but for comparison, it has been supplemented by other values of high energy scales (MR=1013GeV, 1015GeV, and 1016GeV). For input values of mixing angles, we impose a phenomenological relation of lepton mixing angles known as self-complementarity relation (SC), θ23=qθ12+θ13 [43–50]. In this work, we take q=1.1 for fine tuning and we study the stability of SC relation against RGEs. It is observed that this SC relation is nearly stable under radiative corrections. The SC relation for lepton mixing angles is phenomenologically analogous to a relation for quark and lepton mixing angles known as Quark-Lepton Complementarity (QLC) relations, θsun+θC=π/4 [43, 44, 46, 47] between the leptonic 1-2 mixing angle θsun and the Cabibbo angle θC. We also impose the following conditions on input values of neutrino masses.Table 7
Input set of neutrino parameters at high energy scaleMR for NH case. θ23 is used from SC relation, θ23=q×θ13+θ12 with q=1.1. This is common for all cases of ms scale.
Input parametersSeesaw scale (tanβ=40)1016 GeV1015 GeV1014 GeV1013 GeVm1(eV)0.02620.02580.02710.0274m2(eV)-0.0263-0.0259-0.0272-0.0275m3(eV)-0.0615-0.0645-0.0643-0.0664Σmi0.1140.1160.1180.121θ12/032.4633.9533.0132.61θ13/07.397.567.647.70ψ1/0=ψ2/0=ψ/0180180180180δCP/0240240240240Table 8
Input set of neutrino parameters at high energy scaleMR for IH case (m3≠0). θ23 is used from SC relation, θ23=q×θ13+θ12 with q=1.1. This is common for all cases of ms scale.
Input parametersSeesaw scale (tanβ=40)1016 GeV1015 GeV1014 GeV1013 GeVm1(eV)0.05150.05010.05110.0523m2(eV)-0.0516-0.0502-0.0512-0.0524m3(eV)-0.0025-0.0021-0.0022-0.0025Σmi0.10560.10240.10450.1072θ12/031.9432.3931.9932.17θ13/08.538.298.358.40ψ1/0=ψ2/0=ψ/0180180180180δCP/0240240240240The sum of the three neutrino masses should satisfy the latest Planck cosmological dataΣmi<0.12 eV,
(i)
The neutrino mass model should be nearly quasidegenerate at least inm1 and m2 in order to get high value of θ13, and m3 should be nonzeroOne can also express the neutrino mass eigenvalues after absorbing the Majorana CP phases asdiag.m1eiψ1,m2eiψ2,m3. As discussed in ref. [33], from the presence of a term m1eiψ1+m2eiψ2/Δm212 in the evolution of θ12, a nonzero value of the difference ψ1−ψ2 of the Majorana phases damps the RG equation. The damping becomes maximal if this difference equals π, which corresponds to an opposite CP parity of the two nearly degenerate mass eigenstates m1 and m2. A similar term m1eiψ1+m3/Δm312 is also present in the evolution equation of θ13, and this implies opposite CP parity between m1 and m3, though they are not so degenerate. Under this consideration, we make our input choice of CP parity as diag.m1,−m2,−m3 in the present work. The RGEs for evolution of θ12 is also directly proportional to a term A21=m2+m1/m2−m1 which is highly sensitive for the nearly degenerate masses between m1 and m2 [22, 33]. Any possible singularity in the running of RGEs may be avoided with the choice of opposite CP parity between m1 and m2 for nearly degenerate case. In this work, we take two Majorana phases ψ1 and ψ2 at 1800, which are constrained to be equal for simplicity ψ1=ψ2=ψ, and the Dirac CP phase δCP angle at 2400. Our main aim is to study the neutrino oscillation parameters against varying ms for different MR scale.Using all the necessary mathematical frameworks, we analyze the radiative nature of neutrino parameters like neutrino masses, mixings, and CP phases in the top-down approach with the variations ofms scale at different MR scale. The respective RGEs which are given in Appendix B. The input sets are given in Tables 7 and 8.


## 2.1. Bottom-Up Running

In the bottom-up running of the RGEs, we divide it into three regions,mZ<μ<mt, mt<μ<ms, and ms<μ<MR. We use the recent experimental data [8, 40] as initial input values at low energy scale, given in Table 1.Table 1
Latest experimental data for fermion masses, gauge coupling constants, and Weinberg mixing angle.
Mass (GeV)Coupling constantsWeinberg mixing anglemZmZ=91.1876αem−1mZ=127.952±0.009sin2θWmZ=0.23121±0.00017mtmt=172.76αsmZ=0.1179±0.009mbmb=4.18mτmτ=1.77The values of gauge couplings,α2 for SU2L and α1 for U1Y, are calculated by using sin2θWmZ=αemmZ/α2mZ and matching condition,
(1)1αemmZ=531α1mZ+1α2mZ.We can also express the gauge couplingsαi’s [18] in terms of normalized couplings gi’s as gi=4παi, where i=1,2,3 denote electromagnetic, weak, and strong couplings, respectively. RGEs at one-loop level [41] is used for evolution of the three gauge coupling constants from mZ scale to mt scale, as given below
(2)1αiμ=1αimZ−bi2πlnμmZ,where mZ≤μ≤mt and bi=5.30,−0.50,−4.00 for non-SUSY case. For fermion masses to define at mt scale, we use QED-QCD rescaling factor η [42], mbmt=mbmb/ηb, and mτmt=mτmτ/ητ, where ηb=1.53 and ητ=1.015. We then convert them to Yukawa couplings, htmt=mtmt/v0, hbmt=mbmb/v0ηb, and hτmτ=mτmτ/v0ητ, where v0=174GeV is the vacuum expectation value (VEV) of SM Higgs field. The calculated numerical values for fermion masses, Yukawa, and gauge couplings at mt scale are given in Table 2.Table 2
Numerical values for fermion masses, Yukawa, and gauge couplings atmt scale.
Fermions massesYukawa couplingsGauge couplingsmtmt=172.76GeVhtmt = 0.9928g1mt=0.4635mbmt=2.73GeVhbmt=0.0157g2mt=0.6511mτmt=1.75GeVhτmt=0.0100g3mt=1.1890We study the effect of variation of SUSY breaking scale (ms) on gauge and Yukawa couplings for running from mt to the MR scale using RGEs, which are given in Appendix A. At ms scale, the following matching conditions are applied at the transition point from SM (mt<μ<ms) to MSSM (ms<μ<MR) as
(3)giSUSY=giSMhtSUSY=htSMsinβ=htSM×1+tan2βtanβ,hbSUSY=hbSMcosβ=hbSM×1+tan2β,hτSUSY=hτSMcosβ=hτSM×1+tan2β.Third generation Yukawa coupling constants are highly affected by input value oftanβ as shown in Equation (3). A detailed numerical analysis shows that high scale value of ht always decreases with input value on SUSY breaking scale ms in the range (2-14) TeV for low and high tanβ=25,35,40,45,and55. However, high scale values of hb and hτ behave in different patterns. In fact hb at high scale MR increases with the increase of ms scale for low value of tanβ=25, but it decreases with ms scale for higher values of tanβ=40−55. For hτ, it has a similar trend with hb but with a little difference in the range of tanβ. In fact hτ at high scale increases with the increase of ms scale for both low and moderate values of tanβ=25,40, but it again decreases with ms for higher value of tanβ=55. For specific case used in the present calculation at input value of tanβ=40, both ht and hb decrease with the increase of ms scale, but hτ increases with the increase of ms scale. This analysis is reflected in Tables 3–6.Table 3
Values of Yukawa and gauge couplings evaluated attR=ln1013GeV=29.93 for tanβ=40, for different choices of ms scale.
ms(TeV)hthbhτg1g2g320.65090.31390.37210.60860.69140.774340.63180.31020.37410.60430.68610.770060.62360.30860.37500.60230.68350.767980.61610.30710.37600.60020.68100.7658100.61260.30630.37650.59920.67970.7648120.60930.30560.37700.59810.67850.7638140.60610.30500.37750.59710.67720.7628Table 4
Values of Yukawa and gauge couplings evaluated attR=ln1014GeV=32.23 for tanβ=40, for different choices of ms scale.
ms(TeV)hthbhτg1g2g320.62890.29770.36500.63160.69820.755440.61040.29460.36740.62680.69150.751460.60220.29310.36840.62450.68890.749580.59470.29170.36950.62220.68630.7475100.59130.29100.37000.62110.68500.7466120.58790.29040.37050.61990.68370.7456140.58480.28980.37100.61880.68240.7447Table 5
Values of Yukawa and gauge couplingstR=ln1015GeV=34.53 for tanβ=40, for different choices of ms scale.
ms (TeV)hthbhτg1g2g320.60840.28290.35790.65740.70260.737840.58910.27980.36010.65210.69710.734160.58090.27840.36130.64940.69440.732380.57350.27710.36240.64680.69170.7305100.57010.27660.36290.64550.69030.7296120.56680.27600.36350.64430.68900.7287140.56370.27540.36400.64300.68770.7278Table 6
Values of Yukawa and gauge couplings evaluated attR=ln1016GeV=36.84 for tanβ=40, for different choices of ms scale.
ms (TeV)hthbhτg1g2g320.58540.26760.34940.68930.70890.720040.56610.26470.35180.68310.70320.716660.55800.26340.35290.68010.70040.714980.56800.26870.36220.67700.66640.7131100.54730.26180.35470.67570.69630.7124120.54410.26130.35530.67420.69490.7116140.54100.26080.35590.67270.69360.7107The output for Yukawa and gauge couplings atMR scale are given in Table 3 for MR=1013GeV, Table 4 for MR=1014GeV, Table 5 for MR=1015GeV, and Table 6 for MR=1016GeV, respectively, for common value of tanβ=40. These values are needed for the next top-down running as input values at high energy scale.


## 2.2. Top-Down Running

In this running, we use the values of Yukawa and guage couplings which are found atMR scale as initial inputs given in Tables 3–6. In this work, the high energy seesaw scale is the starting point for running the RGEs and it ends at electroweak scale. We give the sum of three neutrino masses Σmi in the range, 0.114 eV -0.121 eV for NH case and 0.1056 eV -0.1072 eV for IH case. The input values of neutrino masses and three mixing angles are indeed arbitrary in order to study the stability of neutrino mass model under RGEs running at low scale, with variations of other free parameters such as tanβ, SUSY breaking scale ms, and high energy scale MR. We select our input values in Tables 7–8 with the aim to produce low energy values of neutrino oscillation parameters consistent with observational data. In our work, we focus on the high energy seesaw scale at MR=1014GeV; but for comparison, it has been supplemented by other values of high energy scales (MR=1013GeV, 1015GeV, and 1016GeV). For input values of mixing angles, we impose a phenomenological relation of lepton mixing angles known as self-complementarity relation (SC), θ23=qθ12+θ13 [43–50]. In this work, we take q=1.1 for fine tuning and we study the stability of SC relation against RGEs. It is observed that this SC relation is nearly stable under radiative corrections. The SC relation for lepton mixing angles is phenomenologically analogous to a relation for quark and lepton mixing angles known as Quark-Lepton Complementarity (QLC) relations, θsun+θC=π/4 [43, 44, 46, 47] between the leptonic 1-2 mixing angle θsun and the Cabibbo angle θC. We also impose the following conditions on input values of neutrino masses.Table 7
Input set of neutrino parameters at high energy scaleMR for NH case. θ23 is used from SC relation, θ23=q×θ13+θ12 with q=1.1. This is common for all cases of ms scale.
Input parametersSeesaw scale (tanβ=40)1016 GeV1015 GeV1014 GeV1013 GeVm1(eV)0.02620.02580.02710.0274m2(eV)-0.0263-0.0259-0.0272-0.0275m3(eV)-0.0615-0.0645-0.0643-0.0664Σmi0.1140.1160.1180.121θ12/032.4633.9533.0132.61θ13/07.397.567.647.70ψ1/0=ψ2/0=ψ/0180180180180δCP/0240240240240Table 8
Input set of neutrino parameters at high energy scaleMR for IH case (m3≠0). θ23 is used from SC relation, θ23=q×θ13+θ12 with q=1.1. This is common for all cases of ms scale.
Input parametersSeesaw scale (tanβ=40)1016 GeV1015 GeV1014 GeV1013 GeVm1(eV)0.05150.05010.05110.0523m2(eV)-0.0516-0.0502-0.0512-0.0524m3(eV)-0.0025-0.0021-0.0022-0.0025Σmi0.10560.10240.10450.1072θ12/031.9432.3931.9932.17θ13/08.538.298.358.40ψ1/0=ψ2/0=ψ/0180180180180δCP/0240240240240The sum of the three neutrino masses should satisfy the latest Planck cosmological dataΣmi<0.12 eV,
(i)
The neutrino mass model should be nearly quasidegenerate at least inm1 and m2 in order to get high value of θ13, and m3 should be nonzeroOne can also express the neutrino mass eigenvalues after absorbing the Majorana CP phases asdiag.m1eiψ1,m2eiψ2,m3. As discussed in ref. [33], from the presence of a term m1eiψ1+m2eiψ2/Δm212 in the evolution of θ12, a nonzero value of the difference ψ1−ψ2 of the Majorana phases damps the RG equation. The damping becomes maximal if this difference equals π, which corresponds to an opposite CP parity of the two nearly degenerate mass eigenstates m1 and m2. A similar term m1eiψ1+m3/Δm312 is also present in the evolution equation of θ13, and this implies opposite CP parity between m1 and m3, though they are not so degenerate. Under this consideration, we make our input choice of CP parity as diag.m1,−m2,−m3 in the present work. The RGEs for evolution of θ12 is also directly proportional to a term A21=m2+m1/m2−m1 which is highly sensitive for the nearly degenerate masses between m1 and m2 [22, 33]. Any possible singularity in the running of RGEs may be avoided with the choice of opposite CP parity between m1 and m2 for nearly degenerate case. In this work, we take two Majorana phases ψ1 and ψ2 at 1800, which are constrained to be equal for simplicity ψ1=ψ2=ψ, and the Dirac CP phase δCP angle at 2400. Our main aim is to study the neutrino oscillation parameters against varying ms for different MR scale.Using all the necessary mathematical frameworks, we analyze the radiative nature of neutrino parameters like neutrino masses, mixings, and CP phases in the top-down approach with the variations ofms scale at different MR scale. The respective RGEs which are given in Appendix B. The input sets are given in Tables 7 and 8.


## 3. Numerical Analysis and Results

The effects of the variation ofms on the outputs of neutrino mass parameters and mixing angles are given in Tables 9–12, along with the graphical representations in Figure 1 for normal hierarchical (NH) model, and in Tables 13–16 and Figure 2 for inverted hierarchical (IH) case. In each case, we also present the results for variation of high energy seesaw scale (1013 -1016) GeV. Similar patterns with the variations of seesaw scale are observed in all the Figures 1 and 2.Table 9
Effects on the output of neutrino parameters at low energy scale, on varyingms for NH case (tanβ=40, MR=1013 GeV).
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi (eV)22.5044.83145.3732.698.40235.260.09542.5695.96145.4232.728.47235.360.09462.5936.66645.4532.738.52235.410.09482.6146.96245.4732.748.54235.430.094102.6217.28845.4832.758.56235.450.093122.6247.52645.5032.758.58235.470.093142.6207.77945.5232.768.60235.490.092Table 10
Effects on the output of neutrino parameters at low energy scale, on varyingms for NH case tanβ=40,MR=1014GeV.
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi (eV)22.3364.77145.7532.098.35235.080.09242.4185.85045.7932.118.41235.160.09362.4286.54145.8233.138.46235.220.09282.4356.85945.8433.148.48235.240.091102.4407.18045.8633.158.51235.270.091122.4417.49045.8833.168.53235.290.090142.4547.65445.8933.168.55235.300.090Table 11
Effects on the output of neutrino parameters at low energy scale, on varyingms for NH case tanβ=40,MR=1015GeV.
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi(eV)22.3734.37146.7134.038.24235.350.09042.4615.35046.7634.058.31235.430.09062.4795.96446.7934.078.35235.480.09082.4926.26446.8134.088.38235.510.090102.4986.63346.8334.098.41235.540.089122.4986.83246.8434.098.43235.560.089142.4997.02846.8634.108.45235.570.088Table 12
Effects on the output of neutrino parameters at low energy scale, on varyingms for NH case (tanβ=40, MR=1016 GeV).
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi (eV)22.3544.43544.8332.548.08234.980.09342.4275.45944.8732.568.14235.070.09462.4446.10844.8932.578.18235.130.09482.4726.45744.9132.588.20235.160.094102.4766.73444.9232.598.22235.180.093122.4757.03344.9432.608.24235.210.093142.4867.18244.9532.608.26235.220.093Figure 1
Effects on the low energy output results inθij, Δmij2, and δCP with variation of ms for IH case at tanβ=40. Four different choices of MR scale are presented.Table 13
Effects on the output of neutrino parameters at low energy scale, on varyingms for IH case (tanβ=40, MR=1013 GeV).
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi(eV)22.585.9644.7032.218.41239.940.11942.556.6444.7132.228.41239.930.11762.537.1044.7232.228.41239.930.11682.527.3244.7332.238.41239.920.115102.527.5244.7332.238.41239.920.114122.517.7344.7432.238.41239.920.114142.507.8644.7432.238.41239.920.113Table 14
Effects on the output of neutrino parameters at low energy scale, on varyingms for IH case (tanβ=40, MR=1014 GeV).
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi(eV)22.465.6844.4432.048.35239.950.11442.446.3044.4632.058.35239.940.11262.426.7544.4732.058.35239.930.11082.417.0544.4832.058.35239.930.106102.407.2544.4832.068.35239.930.109122.407.3744.4932.068.35239.930.108142.397.4944.4932.068.35239.920.108Table 15
Effects on the output of neutrino parameters at low energy scale, on varyingms for IH case (tanβ=40, MR=1015 GeV).
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi(eV)22.375.5644.8232.448.29239.950.11142.346.1144.8432.458.29239.940.10962.336.5544.8532.458.29239.940.10782.326.7344.8532.458.29239.930.107102.316.9144.8632.468.29239.930.106122.307.1544.8732.468.29239.930.105142.307.2344.8732.468.29239.930.105Table 16
Effects on the output of neutrino parameters at low energy scale, on varyingms for IH case (tanβ=40, MR=1016 GeV). Four different choices of MR scale are presented.
ms scale (TeV)Δm31210−3eV2Δm21210−5eV2θ23/0θ12/0θ13/0δCP/0Σmi (eV)22.515.2244.5931.988.54239.940.11842.495.7344.6031.988.54239.930.11662.476.1444.6131.998.54239.930.11582.476.2644.6231.998.54239.920.114102.466.4644.6231.998.54239.920.114122.456.6844.6332.008.54239.920.113142.456.7444.6332.008.54239.920.112Figure 2
Effects on the low energy output results inθij, Δmij2, and δCP with variation of ms for NH case at tanβ=40. Four different choices of MR scale are presented.The neutrino oscillation parameters are found to be almost stable with the variation ofms at low energy scale except Δm212 which is found to be very sensitive with the change of ms scale. One difference between NH and IH model is that in NH case, all the low energy parameters Δm312, Δm212, θ23, θ12, θ13, and δCP are found to increase with the increase of ms scale, but in IH case, Δm212, θ23, and θ12 increase whereas Δm312 and δCP decrease with ms. The low energy values of Σmi are below the latest Planck data Σmi<0.12 eV, where the values for NH are smaller than those of IH case.


## 4. Effects of the Simultaneous Variation ofms and tanβ Values

In this section, we again study how the estimated low energy values of neutrino oscillation parameters behave with the simultaneous variations ofms and tanβ over a wide range. As a representative case, we consider only one seesaw scale at MR=1014 GeV for both NH and IH, using the same high scale input parameters given in Tables 7 and 8, respectively. We consider the range of tanβ=25−55 along with the range of ms=2−14TeV. Significant effects of the variation of tanβ for a given ms value have been observed as shown in Tables 17 and 18 and Figures 3 and 4. The following observations on the low energy neutrino oscillation parameters can be drawn.
(i)
All the three mixing angles are observed to increase with increasing values oftanβ and ms for both NH and IH casesTable 17
Effects on the neutrino oscillation parameters at low energy scale, on varyingms and tanβ for NH case (MR=1014 GeV).
ms (TeV)tanβ=55tanβ=45tanβ=40tanβ=35tanβ=252θ23/046.6345.8545.7544.2543.96446.6445.8845.7944.2944.00646.6445.9145.8244.3144.02846.6445.9245.8444.3244.041046.6545.9345.8644.3544.061246.6545.9445.8844.3644.071446.6645.9545.8944.3644.072θ12/032.9532.7932.0931.8231.38433.2133.1732.8232.2132.01633.2333.1933.1332.4332.14833.2833.2133.1432.5432.181033.2933.2333.1532.6332.281233.3133.2633.1632.6632.291433.3333.2833.1632.6732.312θ13/08.968.428.358.147.9549.028.488.418.208.0069.048.538.468.238.0289.068.548.488.258.04109.098.578.518.288.06129.118.598.538.308.07149.128.618.558.308.082Δm21210−5eV24.184.364.774.804.8345.235.365.855.966.0265.565.766.546.616.7486.086.216.856.917.01106.376.597.187.237.38126.766.827.497.517.69146.887.117.657.717.892Δm31210−3eV22.042.322.332.452.5642.092.362.412.492.6062.142.392.422.522.6382.162.402.432.542.64102.162.412.442.552.65122.172.412.442.562.66142.182.422.452.572.672δCP/0230.15234.35235.08236.72238.394230.49234.49235.16236.81238.436230.49234.58235.22236.84238.458230.68234.62235.24236.86238.4610230.77234.66235.27236.90238.4812230.90234.70235.29236.92238.4914231.00234.73235.30236.92238.50Table 18
Effects on the neutrino oscillation parameters at low energy scale, on varyingms and tanβ for IH case (MR=1014 GeV).
ms(TeV)tanβ=55tanβ=45tanβ=40tanβ=35tanβ=252θ23/044.5844.4644.4444.4144.38444.6244.4944.4644.4244.39644.6444.5044.4744.4344.39844.6644.5144.4844.4444.401044.6744.5244.4844.4444.401244.6844.5244.4944.4444.401444.6944.5344.4944.4544.402θ12/032.07832.05832.04832.02632.001432.08332.06132.05132.02932.008632.08832.06832.05232.03132.016832.09632.07132.05632.03632.0211032.10132.07632.06132.04232.0261232.10832.08132.06432.04432.0311432.11132.08332.06732.04632.0332θ13/08.3788.3628.3518.3398.32948.3798.3638.3518.3428.33268.3818.3648.3528.3458.33488.3838.3668.3538.3478.336108.3858.3698.3548.3488.338128.3888.3718.3558.3518.339148.3908.3728.3568.3538.3402Δm21210−5eV212.487.985.685.554.38413.959.086.306.084.93614.429.436.756.455.13815.1510.057.056.895.441015.3810.257.257.095.541215.7310.537.377.125.741415.9510.747.497.225.812Δm31210−3eV22.322.412.462.472.5142.282.352.442.452.4962.252.292.422.442.4782.232.272.412.432.46102.222.252.402.422.45122.212.242.402.422.43142.202.222.392.412.422δCP/0239.899239.932239.95239.960239.9784239.897239.922239.94239.956239.9766239.896239.918239.93239.953239.9748239.891239.914239.93239.946239.97210239.884239.912239.93239.944239.96912239.881239.911239.93239.942239.96514239.879239.909239.92239.937239.962Figure 3
Effects on the low energy output results inθij, Δmij2, and δCP with variation of ms for NH case at MR=1014 GeV. Five different choices of tanβ are presented.Figure 4
Effects on the low energy output results inθij, Δmij2, and δCP with variation of ms for IH case at MR=1014 GeV. Five different choices of tanβ are presented.For NH case, bothΔm212 and Δm312 decrease with increasing value of tanβ, but increase with increasing ms. For IH case, Δm212 increases but Δm312 decreases with the increase of both tanβ and ms.For NH case,δCP decreases with increasing tanβ but increases with increasing ms, whereas for IH case, δCP decreases with the increase of both tanβ and ms.


## 5. Summary and Discussion

To summarize, the present work is a continuation of the earlier investigations [18, 32] on the effect of the variations of SUSY breaking scale ms in the running of RGEs for neutrino masses and mixing parameters from high to low energy scale. Among many other applications of RGEs on neutrino physics such as magnification of neutrino mixings at low energy scale in quark-lepton unification hypothesis at high energy scale, generation of suitable radiative corrections to validate the tribimaximal or golden ratio neutrino mixings at high scale, and radiative origin of reactor mixing angle and solar neutrino mass-squared parameter at low energy; the present work focuses only on the question of the stability of neutrino mass models for both NH and IH, under RGEs analysis with the variations of SUSY breaking scale ms and input value of tanβ.The numerical analysis in the present investigation is confined to the effects on the variations of three important free parameters in the ranges—high energy seesaw scaleMR=1013−1016GeV, the SUSY breaking scale ms=2−14TeV, and tanβ=25−55. As a special representative case, we choose MR=1014GeV for tanβ=40 in both NH and IH models. For simplicity of comparison, the results for other choices of MR are also presented. To study the stability criteria of neutrino mass model, we start with arbitrary high energy scale input values of neutrino masses and mixings which satisfy certain conditions including Planck cosmological bound. The input value for the Dirac CP phase angle is taken at 2400 for all cases and two Majorana phase angles at 1800 for simplicity. In order to avoid any possible singularity in running RGEs for nearly quasidegenerate case, we considered Majorana CP conserving parity in the mass eigenvalues m1,−m2,−m3. Majorana phases are more insensitive as compared to δCP against RGEs analysis and we omit to report these results. We conclude with the following important points of our results.The input value oftanβ sharply affects the evolution pattern of the third generation Yukawa coupling constants (ht, hb, and hτ) with energy scale. It has been observed that ht at MR scale always decreases with the increase of SUSY breaking scale ms for both low and large tanβ values. However, the high energy scale values of hb and hτ are observed to increase with the increase in ms scale for low tanβ values, but decrease with ms for larger tanβ values. For moderate tanβ values, hb decreases with ms again but hτ increases with ms.
(i)
The effect of the variation ofms scale on neutrino oscillation parameters at low energy scale is very mild except Δm212 which is very sensitive with ms and tanβ values. Both low energy scale values of Δm312 and Δm212 increase with the increase in ms scale for NH case. However, Δm212 increases with the increase of ms, but Δm312 decreases with ms for IH case. The low energy scale values of three mixing angles (θ12, θ13, and θ23) have mild increasing trend with the increase of ms scale for both NH and IH cases. The Dirac CP phase δCP at low energy scale increases with the increase in ms for NH but it decreases with ms for IH caseThe simultaneous variations ofms and tanβ on low scale neutrino oscillation parameters have significant effects. It is observed that all the low energy values of mixing angles (θ12, θ13, and θ23) increase with the increase in ms and tanβ values for both NH and IH. Both low energy values of Δm212 and Δm312 decrease with the increase of tanβ, but they increase with ms scale for NH. For IH case, Δm212 increases and Δm312 decreases with the increase of ms and tanβ. The Dirac CP phase δCP decreases with the increase of tanβ but increases with the increase of ms for NH case. For IH case, the low energy value of δCP decreases with the increase of both tanβ and ms.The complementarity relation (SC) is found to satisfy at high and low energy scale under RGEs with the variations of bothtanβ and ms scale.The numerical analysis in the present work shows the stability of both NH and IH neutrino mass models with the variation of SUSY breaking scalems, and also the other two input parameters MR scale tanβ for a wide range of input values. The present analysis can be applied to check the validity at low energy scale of certain mixing patterns such as tribimaximal [51–54] and golden ratio mixing patterns defined at high energy scale [34, 55–58].



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