Stability of neutrino parameters and self-complementarity relation with varying SUSY breaking scale
Konsam Sashikanta Singh Subhankar Roy Ngangkham Nimai Singh
 
PRD










 Physics Department, Gauhati University, Guwahati 781014, India





 Physics Department, Manipur University, Imphal 795003



             





The scale at which supersymmetry (SUSY) breaks (m_s) is still unknown. The present article, following a top-down approach, endeavors to study the effect of varying m_s on the radiative stability of the observational parameters associated with the neutrino mixing. These parameters get additional contributions in the minimal supersymmetric model (MSSM). A variation in m_s will influence the bounds for which the Standard Model (SM) and MSSM work and hence will account for the different radiative contributions received from both sectors respectively, while running the  renormalization group  equations (RGE). The present work establishes the invariance of the self complementarity relation among the three mixing angles, θ_13+θ_12≈θ_23 against the radiative evolution. A similar result concerning the mass ratio, m_2: m_1 is also found to be valid. In addition to varying m_s, the work incorporates a range of different seesaw (SS) scales and tries to see how the latter affects the parameters.   
 





11.10.Hi, 11.30.Pb, 12.10.Dm, 12.10.Kt, 14.60.Pq


Stability of neutrino parameters and self-complementarity relation with varying SUSY breaking scale
    N. Nimai Singh[nimai03@yahoo.com]
    December 30, 2023
====================================================================================================







§ INTRODUCTION


The physics of neutrino is going through a revolutionary period. From various recent experiments, a small but nonzero value of the reactor angle, θ_13 is confirmed<cit.>. In addition to this, the Dirac CP phase, δ is also observed <cit.>. Recent experiments on neutrino oscillation, 0νββ, and the cosmological observations have revealed precise and important results on the observational parameters like the three mixing angles (θ_13, θ_12, θ_23 ), two mass-squared differences (Δ m_21^2,Δ m_31^2) and possible upper bound on the sum of neutrino masses (Σ m_i) etc. <cit.>. But still we are unable to understand the absolute value of neutrino masses, nature of neutrino mass hierarchy, or its type: Dirac/Majorana etc. The realization that neutrinos are massive in contrast to its old popular assumption that it is massless (according to the SM) is one of the strong signatures that the SM of particle physics has to be extended beyond its present horizon.

Most of the current studies on physics beyond the SM (BSM) relies on the possible existence of supersymmetry (SUSY). But there are other models of BSM physics which does not incorporate the idea of SUSY <cit.>. It is hypothesized that SUSY existed at the early stage of big bang. But with the expansion of our Universe SUSY gets broken and reduced to our present day SM. At what scale that breaking occurs is still an unknown but an important parameter. The general idea is that there are two possible energy scales for the SUSY breaking (m_s): low and high. The low m_s scale <cit.> is expected to be about a few TeV or so as suggested by the grand unified theory (GUT), whereas the high SUSY breaking scale is expected to be somewhere around 10^12 GeV <cit.>. 


One significant finding from the recent LHC experiment which sounds a little disappointing towards the possibility of SUSY is that the experiment, which was operated at an energy scale of 13 TeV, has not provided any evidence of the existence of SUSY particles so far <cit.>. In SUSY inspired neutrino physics, it is predicted that SUSY plays an important role over the neutrino masses and other observational parameters <cit.>. The gauge coupling and Yukawa coupling constants suffer different radiative contributions from the MSSM and SM sectors. Similar to this, we expect that the neutrino observational parameters are also subjected to such kind of effects. 

One of the reasons why the variation in m_s is expected to bring changes to various observational parameters is owing to the changes in the effective range of both MSSM and SM. When we increase the m_s scale, the effective range of SM increases, whereas that for MSSM decreases and vice versa. It will change the amount of radiative correction that each parameter receives from the SM and MSSM, respectively. In our previous work <cit.>, we show the variation of the unification point of the gauge couplings  with varying m_s scale. Such behavior is likely to be seen for the neutrino oscillation parameters too. In this regard, it is important to study the possible effects of varying m_s on the radiative evolution of the neutrinos and hence ,to determine (or narrow down) the possible range of m_s scale.



The possible reason behind the suppression of SUSY motivated effects at the LHC experiments may be due to the low luminosity of the beam. By the end of 2012 LHC's integrated luminosity, running at a center-of-mass energy √(s)= 8TeV, is already over 20fb^-1 <cit.>. The present integrated luminosity of the LHC for √(s)= 13TeV is 35.9fb^-1 for CMS <cit.> and 36.1fb^-1 for ATLAS <cit.>. Some predicted the required integrated luminosity for observing SUSY related events to be 3000 fb^-1 <cit.>, which is approximately 85 times  greater than the present luminosity. Nevertheless, this still gives us a hope for the possible existence of m_s < 13 TeV.  If seesaw (SS) mechanism is the only cause behind the generation of small neutrino masses, then it appears that the right handed neutrino mass scale must lie somewhere within the  range of (10^10-10^16)GeV <cit.>. In our analysis we shall vary the SS scale starting from 10^10-10^15 GeV.  


One sees that the numerical range of three mixing angles within 1σ<cit.> appears as in the following:

    θ_13= 8.44^0^+0.16_-0.17,  θ_12=34.5^0^+1.1_1.0  and  θ_23=41.0^0^+1.1_-1.1.

We see that there may lie a self-complementarity (SC) among these parameters in terms of the following relation

    θ_23=q×(θ_13+θ_12),
 

where, the parameter, q is either unity or 𝒪(1). The SC is an important phenomenological relation <cit.> similar to the quark-lepton complementarity relations <cit.>. The possible existence of such relations among the parameters are expected to be the signatures of a certain flavor symmetry working in the background. The present analysis attempts not to deal with the possible origin of such a kind of an SC relation rather, it insists on the existence of such a relation even at higher energy scale. Our work starts with an assumption that this SC relation holds good at the SS scale. Through our analysis, we will show that this relation remains invariant against the radiative evolution for varying the m_s and SS scale. We emphasize that similar to the works in the literature which focus only on the renormalization group invariant parameters <cit.>, the SC relation can also serve as an RGE invariant relation. 


The present investigation is a continuation of our previous work <cit.>, where we studied the radiative evolution of the three gauge, third generation Yukawa and quartic Higgs couplings following a bottom-up approach, with varying the SUSY breaking scale m_s. It was observed that the unification scales for both the gauge couplings and Yukawa couplings vary but in the opposite trend and tend to attain a fixed value with increasing m_s. There, we vary m_s starting from 500 GeV to 7 TeV. However, in the present work, we follow the top down approach starting from the seesaw scale up to the electroweak scale. We fix, tanβ=58.6, which is relevant in the context of our previous work <cit.>.

 
This paper is organized in the following order. In Sec. II, we give a brief discussion of the neutrinos RGEs. In Sec. III, we study the possible radiative effects on the neutrino parameters at the weak scale. In Sec. IV, we present the numerical analysis. In Sec. V, we  summarize our work, and in the Appendix, we give the RGEs for the gauge, Yukawa and quartic Higgs couplings in two loops for both the SM and MSSM.





 


 














§ RGES FOR NEUTRINO PARAMETERS

Renormalization group approach is a tool for studying physics at a different energy scale, which are otherwise impossible to reach with the current technology, and then to compare it with the available low energy data. Radiative analysis of neutrino parameters requires the RGEs of gauge couplings, Yukawa couplings and the quartic Higgs couplings. The radiative properties of these couplings have been studied extensively in different models and these three gauge couplings are expected to be unified at an energy scale approximately at 2× 10^16GeV <cit.>. The RGEs for the gauge couplings, Yukawa couplings, and quartic Higgs coupling are given in the Appendix. We use 2-loops RGEs for both the SM and MSSM. 


The RGE analysis of the neutrino parameters can be done in two possible ways viz: i) by a run and diagonalize method: where the whole neutrino mass matrix is allowed to evolve using their appropriate RGEs and then the corresponding neutrino parameters can be achieved at the desirable energy scale (μ) by diagonalizing the neutrino mass matrix, ii) by using the RGEs of the corresponding neutrino parameters separately as defined by the equations.(<ref>) to (<ref>). In both the cases, the RGEs of all the neutrino parameters and the RGEs  of various coupling parameters are required to be solved simultaneously.  In this work, we adopt the later stand. 

The input parameters for the gauge, Yukawa, and quartic Higgs couplings at the seesaw scale, given in Table <ref>, are taken form our previous work <cit.>.  In the present analysis, we choose our starting energy scale to be the SS scale.  We consider a different possible SS scale starting from 10^10GeV to 10^15GeV, and  we run down all the observational neutrino parameters from SS scale up to the electroweak scale (m_Z=91.18 GeV) via m_s, which also varies in our analysis. 


The radiative properties of neutrinos has been studied extensively in various models <cit.>. The standard two loops RGEs for the neutrino masses, mixings, and CP phases are shown below. For the three neutrino mixing angles <cit.>, the RGEs are


    θ̇_12   =    -Cy_τ^2/32π^2sin 2θ_12 s^2_23|m_1e^iψ_1+m_2e^iψ_2|^2/Δ m^2_21,	
    θ̇_13   =    -Cy_τ^2/32π^2sin 2 sin 2θ_23m_3/Δ m^2_31(1+ξ)
          ×[m_1cos(ψ_1-δ)-(1+ξ)m_2cos(ψ_2-δ)-ξ m_3cosδ], 
    θ̇_23   =    -Cy_τ^2/32π^2sin 2θ_231/Δ m_31^2[c_12^2|m_2e^iψ_2+m_3|^2+s^2_12|m_1e^1ψ_2+m_3|^2/(1+ξ)],

where, Δ m^2_21=m_2^2-m_1^2 and Δ m^2_31=m_3^2-m_1^2, ξ=Δ m^2_21/Δ m^2_31.



The RGEs for the three phases are,

(for Dirac phase)

    δ̇ = Cy_τ^2/32π^2δ^(-1)/θ_13+Cy_τ^2/8π^2δ^0,

where, 

    δ^(-1)   =   sin 2θ_12sin 2θ_23m_3/Δ m^2_31(1+ξ)×
          [m_1sin(ψ_1δ)-(1+ξ)m_2sin(ψ_2-δ)+ξ m_3sinδ], 
    δ^(0)   =   m_1m_2s^2_23sin(ψ_1-ψ_2)/δ m^2_21
           +m_3s^2_12[m_1cos 2θ_23sinψ_1/Δ m^2_31(1+ξ)+m_2c_23^2sin(2δ-ψ_2)/Δ m^2_31] 
           +m_3c^2_12[m_1c^2_23sin(2δ-ψ_1)/Δ m^2_31(1+ξ)+m_2cos 2θ_23^2sin(ψ_2)/Δ m^2_31],


(for Majorana phases)

    ψ̇_1    =   Cy_τ^2/8π^2{m_3cos 2θ_23m_1s^2_12sinψ_1+(1+ξ)m_2c^2_12sinψ_2/Δ m^2_31(1+ξ). 
          .+m_1m_2c_12^2s_23^2sin(ψ_1-ψ_2)/Δ^2_21} ,
    ψ̇_2    =   Cy_τ^2/8π^2{m_3cos 2θ_23m_1s^2_12sinψ_1+(1+ξ)m_2c^2_12sinψ_2/Δ m^2_31(1+ξ). 
          .+m_1m_2s_12^2s_23^2sin(ψ_1-ψ_2)/Δ^2_21}.


The RGEs for the neutrino mass eigenvalues are


    ṁ_̇1̇   =   1/16π^2[α+Cy_τ^2(2s_12^2s_23^2+F_1)]m_1, 
    ṁ_̇2̇   =   1/16π^2[α+Cy_τ^2(2c_12^2s_23^2+F_2)]m_2, 
    ṁ_̇3̇   =   1/16π^2[α+2Cy_τ^2c_13^2c_23]m_3,


where, 

    F_1    =    -s_13sin 2θ_12sin 2θ_23cosδ +2s_13^2c^2_12c_23^2, 
     
    F_2    =    s_13sin 2θ_12sin 2θ_23cosδ +2s_13^2s^2_12s_23^2,



    α   = -6/5g_1^2-6g_2^2+6y_t^2 
    
      C    = 1
     for MSSM



    α   = -3g_2^2+2y_τ^2+6y_t^2+6y_b^2 
    
      C    = 1/2 for SM


  


With all the necessary mathematical frameworks in hand, we can now study the radiative nature of neutrino masses, mixings and CP phases using the top-down running approach together with the MSSM unification conditions.




In the first step all the parameters are allowed to run down from the seesaw scale to the SUSY breaking scale using their respective MSSM RGEs, and from the SUSY breaking scale further down to the electroweak scale using their SM RGEs. At the transition point from MSSM to SM, we apply appropriate matching conditions as shown below,


    g_i(M^-_SUSY)    =    g_i(M^+_SUSY), 
    λ_t(M^-_SUSY)    =   λ_t(M^+_SUSY)sinβ, 
    λ_b(M^-_SUSY)    =   λ_b(M^+_SUSY)cosβ ,
    λ_τ(M^-_SUSY)    =   λ_τ(M^+_SUSY)cosβ,


where, tanβ=v_u/v_d such that v_u=v sinβ, v_d=vcosβ and v=246 GeV is the vacuum expectation value (VEV) of the Higgs field. In our analysis, we choose a single SUSY spectrum for simplicity and study the radiative stability of the neutrino parameters at the weak scale for varying m_s.




§ RADIATIVE EFFECTS ON THE NEUTRINO OSCILLATION PARAMETERS AND THE CP PHASES

The radiative effects on the neutrino parameters for a strict  normal  or inverted hierarchy is small. If the neutrinos masses have a quasidegenerate spectrum, then the RG evolution between the lowest seesaw scale and electroweak energy scale can have sizable effects <cit.> on the neutrino oscillation parameters. The RG effects may even account for the difference between the mixings in the quark and the lepton sectors <cit.>.





In MSSM, both the atmospheric (θ_23) and solar mixing angle (θ_12) increase with the decrease in energy as predicted by eq.(<ref>) and eq.(<ref>). Out of the three mixing angles, the solar mixing angle is prone to the largest RG effects because of the presence of small Δ m_31^2 in the denominator, whereas θ_13 is subjected to the smallest RG effect.






In the top-down approach, all the three mass eigenvalues behave in a similar fashion, and they all decrease with the decrease in energy scale. Because of the comparatively larger value of α with respect to y_t, y_b, and y_τ, the RG running effect on the mass eigenvalues is less. But, due to the same factor α, there is appreciable running in the RGEs of the mass eigenvalues in the SM case. The running of the mass eigenvalues in the MSSM is defined by a common scaling factor, except for the case of a large tanβ where it deviates considerably.

For nearly degenerate neutrino masses and a large tanβ, the radiative influence of CP phases over other parameters becomes important. All of the phases (both Majorana and Dirac) undergo radiative correction. For different sets of the input phases, the RG effects on the neutrino oscillation parameters may differ. In the context, when the two Majorana phases are equal<cit.>, the evolutions of the parameters are highly suppressed since the leading terms in the RGEs of the phases become zero [See eq.(<ref>)and eq. (<ref>)].



§ NUMERICAL ANALYSIS AND THE RESULTS


The RGEs are differential equations and demand the input values for the parameters to be sought out, at the very outset. In our case, the starting point is the SS scale, and finally, we end up at the EW scale. From SS scale upto m_s scale the RGEs follow  certain pattern (eq.(<ref>)) and reverts to another form in the region from m_s upto EW scale (eq.(<ref>)). Both the SS scale and m_s are unknown to us. Our present analysis although tries to visualize the effect on the neutrino observational parameters for varying m_s, yet gives emphasis on the choice of the SS scale also. We fix the m_s values in between 1 TeV to 13 TeV. In addition, the SS scale is also assigned certain fixed values between 10^10 GeV to 10^15 GeV.    

The parameters, g_1, g_2, g_3, y_t, y_b, y_τ and λ are specified as per Table.<ref>. In the present analysis, we have got nine free parameters: m_1, m_2, m_3, θ_13, θ_23, θ_12, δ, ψ_1 and ψ_2. As stated earlier, the present study presumes the existence of the SC relation (see eq. (<ref>)) at the SS scale. By virtue of this relation, we assign initial input values only to θ_13 and θ_12. Further simplifications are made regarding the initial choice of ψ_1 and ψ_2, which are constrained to be equal,(ψ_1)_0=(ψ_2)_0, for all subsequent calculations (the notations (...)_0 represent the initial input value of the parameter within the bracket). In that way, we assign input values only to six neutrino observational parameters. To simplify, we summarize our strategy in the following way,



  (Step 1) We vary the initial values of the six neutrino parameters at a fixed m_s scale. To ensure that the initial choice of the parameters beget the numerical values at the EW scale which are consistent within 3σ range, we follow a simple mechanism. To illustrate, let us fix m_s at 5 TeV, SS scale at 10^14 GeV, and assume, (m_2)_0=2.34×10^-2 eV and (δ)_0=90^∘. The remaining parameters, (θ_13)_0, (θ_23)_0, (m_1)_0, (m_3)_0 and (ψ_1)_0 are assigned with certain numerical values, so that the final output at the EW scale lies within 3σ. Next, we vary the parameter, (ψ_1)_0 and see how the remaining parameters, like, (θ_ij)_0 and (m_i)_0 are to be adjusted in order to keep the outcome within the 3σ range. For details, see Table <ref>, Figs. <ref>, and <ref>. We see that, except (m_3)_0 which varies a little, the other input parameters are almost stable against changing (ψ_1)_0. The motivation behind performing this step is to ensure that the final numerical values in concern with the neutrino observational parameters are not too  sensitive to the initial input of the Majorana phase. This observation helps us to choose an arbitrary value for (ψ_1)_0. We take (ψ_1)_0=45^∘ for all subsequent calculations.   
,

  (Step 2) The SUSY breaking scale m_s, is attributed to the following numerical values like, 1, 3, 5,...13 TeV and in accordance with that, we categorize seven sets of input values as, A1, A3, A5...A13 respectively. For example, the set A_5 corresponds to the set of input (θ_ij)_0, (m_i)_0, (δ)_0, and (ψ_1)_0, at m_s=5 TeV. For all the above mentioned sets, we fix (δ)_0=90^∘. Similarly, we assign sets, B1, B3, B5...B13 with (δ)_0=270^∘. This is to be noted that both kinds of sets Aj and Bj are the input values of the neutrino parameters, at the SS scale of 10^14 GeV. There is another 𝒪(1) parameter, q which appears in eq. (<ref>) is tuned between 0.95 to 0.97. For details, see Table. <ref>.



  (Step 3) In this step, keeping a certain input set, say A5 fixed, we vary the m_s scale between 1 TeV  to 13 TeV, and check the stability of the neutrino observational parameters at the EW scale. The details are shown in the Tables. ref90m12 - <ref>.


  (Step 4) We repeat step 3, for different values of the SS scale, such as 10^10, 10^11...10^15 GeV.



We will now discuss the results of our analysis.




 §.§ For varying m_s at fixed SS scale:


We keep track of the the numerical values of the neutrino observational parameters at the EW scale. From Tables.  <ref>-<ref>, one sees that, except, Δ m_21^2, other parameters like θ_13, θ_12, θ_23, and Δ m_31^2 show stability at the face of changing m_s. For all the three mixing angles, the fluctuations are consistent within 3σ bound <cit.>.But for Δ m_31^2, the fluctuations sometimes cross the 3σ bound. Although the input entries corresponding to different neutrino parameters are almost the same for all the sets Aj and Bj, the solar mass squared difference at the EW scale is found quite sensitive towards both the initial input as well as to the m_s scale. To illustrate, one can see that for the input data set, say A5, which results in Δ m_21^2=7.57× 10^-5 eV^2, and this is consistent within 1σ bound, for m_s being set at 5 TeV. If m_s is changed a little, say to 3 TeV and 7 TeV, we see that for the same input data set A5, the Δ m_21^2 become, 9.16× 10^-5 eV^2 and 6.67× 10^-5 eV^2 respectively. This output lies strictly outside the 3σ region. However, if we achieve an acceptable Δ m_21^2, against a higher m_s scale, we can expect a little stability. To exemplify, if for A11, we achieve, Δ m_21^2=7.54× 10^-5 eV^2 (within 1σ bound), against m_s=11 TeV, then changing the m_s to either 9 or 13 TeV, will not take this parameter outside 3σ. In addition, both solar and atmospheric mass squared difference decreases, with the increase in m_s scale. The CP violating phases also vary a little if m_s were changed. With the increase of the latter, δ decreases, whereas, the two Majorana phases, increase (See Fig.(<ref>)). 




 §.§ For varying m_s and SS scale:


The discussion concerned so far, is true only for the SS scale: 10^14 GeV. We try to see how a changing SS scale, along with m_s, can affect the physical parameters at the EW scale as per the step(4) mentioned above. We note down the following. To exemplify, let us choose the input data set B5, which is capable of producing, observable parameters at the EW scale consistent within 3σ, with m_s being fixed at 5 TeV, and the SS scale at 10^14 GeV. With the SS scale fixed, first we vary m_s and we get certain plot, which shows how the numerical value of that observable parameter at EW scale changes against m_s. We redo the same to get another plot, but at a different SS scale, for same input data set. We observe the ascent or descent of the plots against the different SS scale.

 


  (a) Among the three mixing angles, θ_13 at EW scale decreases if the SS scale is increased, whereas θ_12 and θ_23 increase. For wide ranges of m_s and the SS scale, the output values stay within the 3σ bound. However, for different input data sets concerned, the exclusion of certain m_s values or SS scales are also possible, depending upon the 3σ bound of the concerned mixing angles. For example, consider the case of θ_12 at the EW scale, against a fixed input data set B5. If we believe SS scale to be 10^10 GeV, then, from the plots, it is evident that the  susy breaking scale should not be more than 7 TeV (See Figs.(<ref>)).  For the other two mixing angles, (θ_13), and (θ_13) see Figs.<ref> and <ref> respectively.


  (b) With all the conditions being the same as before, the δ increases if the SS scale is increased, whereas the reverse is true for the Majorana phases. (See Figs. (<ref>)-(<ref>)).



  (c) We observe certain interesting results in concern with Δ m_21^2 and Δ m_31^2. The mass squared differences are found highly  sensitive to the initial data set, m_s and the SS scale. The Δ m_31^2 remains more or less stable against m_s, but crosses 3σ bound if SS scale is varied. On the contrary, the Δ m_21^2 fluctuates more with m_s, but less with SS scale. It is interesting to note that against a fixed input data set (say, B5), with respect to 3σ range of Δ m_21^2, one can even find a bound over the m_s scale. This bound shifts to the right, i.e, towards a higher m_s region as we take the input numerals as per the initial data sets from B1 to B13 (see Figs. <ref> and <ref>).    




 §.§ The SC relation and the mass ratios

In addition to the physical observables, we try to see how the certain parameters/ relation  evolve against the varying energy scale. The neutrino oscillation experiments hint not for individual neutrino masses, the study of individual parameters and how they evolve carry physical insight. This study is relevant from the model building point of view.  




  (a)As stated earlier, we have assumed that at the SS scale ,the three mixing angles are connected via a complementarity relation (See Eq. (<ref>)). We see that for a fixed m_s and a chosen SS scale, with all the input parameters fixed to a certain data set (say,B5), the angles evolve (except θ_13 which is almost stable), but the SC relation connecting the mixing angles, remains almost invariant against the radiative evolution. This stability is achievable, even if we vary the SS scale or m_s. We have shown the radiative evolution of the angles along with the SC relation for both varying m_s (with a fixed SS scale) and varying SS scale (with fixed m_s). For details, see Figs. (<ref>)-(<ref>). The SC relation is a phenomenologically motivated relation like the QLC relation<cit.> that connects the quark and lepton sectors. A relation of this kind bears the signature of a certain hidden symmetry. As pointed out in our analysis, that which reflects the invariance of the former against radiative evolution may turn out as a fruitful information for the model builders.  


  (b) Like the mixing angles, we try to see how the mass parameters respond to radiative evolution. Instead of concentrating on individual neutrino masses, we focus on the three mass ratios as such: m_2/m_1, m_3/m_1, and m_3/m_2. This is inspired by the phenomenology of the quark sector. Where, we see that the mass ratio between the down and strange quarks is naturally related to the quark mixing angle (Cabibbo angle) which plays an important role in describing the mixing among the quarks<cit.>. To exemplify, we fix the m_s at 5 TeV and the input data set at B5. Following this, we see how the three neutrino mass ratios vary against the changing SS scale. The details are shown in Fig.(<ref>). One sees that the ratio m_3/m_1 or m_3/m_2, though remains invariant in the SUSY region, yet changes after crossing the m_s scale. But, interestingly, the ratio m_2/m_1 remains almost invariant and tries to maintain a constant numerical value as such: m_2/m_1∼ 2. A summarized version of the different types of effect each neutrino parameters receive due to the variation of m_s and SS are given in Table <ref>.




 


§ SUMMARY AND DISCUSSION


In this paper we have studied the radiative evolution of neutrino observational  parameters for varying m_s scale following a top-down approach. We presume the hierarchy of the three neutrino masses to be of normal type. All the nine observational parameters related to neutrino oscillations are allowed to run down from the seesaw scale up to the electroweak scale using their respective RGEs (both MSSM and SM). We also use the RGEs of the three gauge couplings, third generation Yukawa couplings, and quartic Higgs coupling. All the neutrino parameters along with  the other couplings undergo RG evolution, and subsequently, get different RG corrections. The m_s, which appears to be a leading parameter is kept varying between 1 TeV to 13 TeV and the effect of such a variation on the observational parameters at the EW scale is noted. Instead of adhering to a fixed SS scale, we allow the latter to change between 10^10 GeV to 10^15 GeV, and have checked how the observational parameters vary. Besides, the work reveals that the self-complementarity relation among the mixing angles remains stable against the radiative evolution. Also, we have studied how certain parameters like neutrino mass ratios behave during this evolution.  

The relevance of the SUSY is unavoidable in the context of particle physics, as it can answer to certain important theoretical issues like the hierarchy problem, the unification of gauge couplings, the existence of dark matter etc. But, unlike the Standard Model, the SUSY is still lacking the experimental evidences. Although the LHC experiment is running at 13 TeV, it has not yet witnessed any signature of SUSY. This may imply that the SUSY breaks at certain higher energy scale which is not yet achieved by the LHC experiment, or even if it breaks at a low energy, the beam luminosity available in the LHC experiment is not sufficient to detect the same. Hence, there is still a hope that SUSY exists. The SUSY breaking scale, m_s, is an important parameter and influences the neutrino observational parameters. The origin of a neutrino mass owes to the seesaw mechanism, and the scale at which the latter occurred is also unknown. But theoretically one may predict that scale to be lying within the range of 10^10 to 10^15 GeV. In our analysis, these two parameters, the m_s and the SS scale partake a lot. Besides, the input data set (like, Aj or Bj) which are although model independent, plays an important role. Initially, the input parameters in the data sets are chosen such that against a fixed m_s and a fixed SS scale (10^14 GeV), the neutrino observational parameters at the EW scale lie within the 3σ bound. This is to be mentioned that the initial entries in terms of the three mixing angles, follow a self-complementarity relation. 


At EW scale, the three mixing angles, CP violating phases, and Δ m_31^2  try to maintain more or less stability with respect to the 3σ bound if the m_s scale is varied at a fixed SS scale. But the parameter Δ m_21^2 is less stable at lower m_s, whereas the stability increases towards higher m_s. Similar stability is achievable for the three mixing angles if the SS scale is varied. But for Δ m_31^2, the stability is lost. One sees that if the stability of Δ m_21^2 is obtained towards a higher m_s, ruling out of a certain SS scale is possible in the light of 3σ bound of Δ m_21^2. It is worth mentioning that a strong conclusion in view of the optimization of the SUSY breaking and SS scales can not be drawn by observing the plots (See Figs.(<ref>)-(<ref>)), because the Δ m_21^2 at the EW scale is very much sensitive to the initial arbitrary model independent entries available in the data sets (Aj and Bj). Justifying these initial entries under a certain model or framework goes beyond the scope of this article. But through our analysis, one can at least visualize the interplay between the m_s and the SS scale and how these affects the final physical observables. Though in the present analysis, we limit ourselves not to invoke the model dependent ground of these data sets, yet we emphasize on the certain traits that these numerals may carry. We see that the data set are characterized by the SC relation, θ_13+θ_12≈θ_23 and a mass ratio: m_2/m_1∼ 2, which remains almost invariant against radiative evolution. Besides, we have observed the other mass ratios like m_3/m_1 or m_3/m_2, also triy to maintain a stability up to SUSY breaking scale, but after that they change. This study is motivated in the context of the quark sector, where the quark mass ratio m_down/m_strange plays an important role in describing the quark mixing. Relations among certain parameters and their stability during radiative evolution may bear the traits of a certain hidden symmetry present in the lepton sector and may serve as a key to some new models. 



The present study is devoted to a simple visualization, concerning the interplay between the m_s and SS scale and its effect on the physical observables and certain phenomenological relations. The two  Majorana phases are not yet been measured experimentally, and to simplify the analysis, we have considered both of them as equal. Again, we have restricted ourselves only to the normal hierarchy of neutrino masses. The consideration of a  degenerate spectrum for all sparticles that we have adopted in our work, is an idealized situation and is true if m_s≫ m_t,m_Z <cit.>. In principle, a general study can be made by minimizing the number of assumptions in order to get a more generalized result.     

 





§ APPENDIX



 §.§ RGEs for gauge couplings

The two loop renormalization group equations for gauge couplings are <cit.> as follows:


    dg_i/dt=b_i/16π^2g_i^3+(1/16π^2)^2[∑_j=1^3b_ij g_i^3  g_j^2 - ∑_j=t, b,  τa_ij g_i^3  h_j^2],


where, t=ln μ and  b_i,  b_ij,  a_ij are β function coefficients in MSSM,

    b_i 
    	
    	=
    	[ 6.6, 1.0, -3.0 ]
    	,  b_ij =
    	[  7.96  5.40 17.60;  1.80 25.00 24.00;  2.20  9.00 14.00 ]
    	,  a_ij =
    	[ 5.2 2.8 3.6; 6.0 6.0 2.0; 4.0 4.0 0.0 ]
 
	and, for non-supersymmetric case, we have

    b_i 
    	
    	=
    	[ 4.100, -3.167, -7.000 ]
    	,  g_ij =
    	[  3.98  2.70   8.8;  0.90  5.83  12.0;  1.10  4.50 -26.0 ]
    	,  a_ij =
    	[ 0.85  0.5  0.5; 1.50  1.5  0.5; 2.00  2.0  0.0 ].
	



 §.§ RGEs for Yukawa couplings

At two-loop level for MSSM, <cit.>

    dh_t/dt   =   h_t/16π^2(6h_t^2+h_b^2-∑_i=1^3c_i  g_i^2)+ 
          h_t/(16π^2)^2[∑_i=1(c_ib_i+c_i^2/2)g_i^4 +g_1^2g_2^2+136/45g_1^2g_3^2+ 8g_2^2g_3^2.+ 
          . (6/5g_1^2+6g_2^2+16g_3^2)h_t^2 +2/5g_1^2h_b^2-22h_t^4-5h_b^4-5h_t^2h_b^2-h_b^2h_τ^2], 
    
    dh_b/dt   =   h_b/16π^2(6h_b^2+h_τ^2+h_t^2-∑_i=1^3 c_i^' g_i^2 )+ 
          h_b/(16π^2)^2[∑_i=1(c'_ib_i+c_i^'2/2)g_i^4+g_1^2g_2^2+8/9g_1^2g_3^2+8g_2^2g_3^2+( 2/5g_1^2+6g_2^2+16g_3^2). h_b^2 
          . +4/5g_1^2h_t^2+6/5g_1^2h_τ^2-22h_b^4-3h_τ^4-5h_t^4-5h_b^2h_t^2-3h_b^2h_τ^2 ], 
    
    dh_τ/dt   =   h_τ/16π^2(4h_τ^2+3h_b^2-∑_i=1^3c_i^” g_i^2) 
           +h_τ/(16π^2)^2[∑_i=1(c”_ib_i+c_i^”2/2)g_i^4+9/5g_1^2g_2^2+(6/5g_1^2+6g_2^2)h_τ^2 . 
           +.(-2/5g_1^2+16g_3^2)h_b^2 + 9h_b^4-10h_τ^4-3h_b^2h_t^2-9h_b^2h_τ^2 ],

where


    c_i =
    	[ 13/15, 3, 16/13 ]
    	,  c_i^' =
    	[ 7/15, 3, 16/3 ]
    	,  c_i^” =
    	[ 9/5, 3, 	0 ].

	
	
Yukawa RGEs for non-supersymmetric case,

    dh_t/dt   =   h_t/16π^2(3/2 h_t^2-3/2h_b^2+Y_2(S)-∑_i=1^3c_i  g_i^2)+ 
          h_t/(16π^2)^2[1187/600g_1^4-23/4g_2^4-108g_3^4-9/20g_1^2g_2^2+19/15g_1^2g_3^2+9g_2^2g_3^2 . 
           +. (223/80g_1^2+135/16g_2^2+16g_3^2)h_t^2-(43/80g_1^2-9/16g_2^2+16g_3^2 )h_b^2. 
           +. 5/2Y_4(S)-2λ(3h_t^2+h_b^2 )+3/2h_t^4-5/4h_t^2h_b^2+11/4h_b^4. 
           +. Y_2(S)(5/4h_b^2-9/4h_t^2)- χ_4(S)+3/2λ^2],



    dh_b/dt   =   h_b/16π^2(3/2h_b^2-3/2h_t^2+Y_2(S)-∑_i=1^3 c_i^' g_i^2 )+ 
          h_b/(16π^2)^2[-127/600g_1^4-23/4g_2^4-108g_3^4-27/20g_1^2g_2^2+31/15g_1^2g_3^2+9g_2^2g_3^2. 
          . -(79/80g_1^2-9/16g_2^2+16g_3^2)h_t^2+(187/80g_1^2+135/16g_2^2+16g_3^2 )h_b^2 . 
          . +5/2Y_4(S)-2λ(h_t^2+3h_b^2 )+3/2h_b^4-5/4h_t^2h_b^2+11/4h_t^4. 
          . +Y_2(S)(5/4h_t^2-9/4h_b^2)-χ_4(S)+3/2λ^2],



    dh_τ/dt   =   h_τ/16π^2(3/2h_τ^2+Y_2(S)-∑_i=1^3c_i^” g_i^2)+ 
          h_τ/(16π^2)^2[1371/200g_1^4-23/4g_2^4-27/20g_1^2g_2^2+ (387/80g_1^2+135/16g_2^2)h_τ^2 +5/2Y_4(S) . 
          . -6λ h_t^2+3/2h_τ^4 -9/4Y_2(S)h_τ^2-χ_4(S)+3/2λ^2 ],



    dλ/dt   =   1/16π^2[9/4(3/25g_1^4+2/5g_1^2g_2^2+g_2^4)-(9/5g_1^2+9g_2^2)λ+4Y_2(S)λ-4H(S)+12λ^2]+ 
          1/(16π^2)^2[-78λ^3+18(3/5g_1^2+3g_2^2)λ^2+(-73/8g_2^4+117/20g_1^2g_2^2+1887/200g_1^4)λ.
          .+305/8g_2^6-867/120g_1^2g_2^4-1677/200g_1^4g_2^2-3411/1000g_1^6-64g_3^2(h_t^4+h_b^4). 
          . -8/5g_1^2(2h_t^4-h_b^4+3h_τ^4)-3/2g_2^4Y_2(S)+10λ Y_4(S)+
    3/5g_1^2(-57/10g_1^2+21g_2^2)h_t^2 . 
          . +(3/2g_1^2+9g_2^2)h_b^2+(-15/2g_1^2+11g_2^2)h_τ^2
    -24λ^2Y_2(S)-λ H(S)+6λ h_t^2h_b^2 . 
          . +20(3h_t^6+3h_b^6+h_τ^6)-12(h_t^4h_b^2+h_t^2h_b^4)],


where

    Y_2(S)   =   3h_t^2+3h_b^2+h_τ^2,  
     
    Y_4(S)   =   1/3[3∑ c_ig_i^2h_t^2+3∑ c^'_ig_i^2h_b^2+3∑ c^”_ig_i^2h_τ^2], 
    χ_4(S)   =   9/4[3h_t^4+3h_b^4+h_τ^4-2/3h_t^2h_b^2], 
    
    H(S)   =   3h_t^4+3h_t^4+h_τ^4, 
    λ   =   m_h^2/V^2, is  the  Higgs  self  coupling  (m_h=  Higgs  mass ).



with the values of beta function coefficients for non-SUSY case and 

	
    c_i =
    	[ 0.85, 2.25, 	8.00 ]
    	,  c_i^' =
    	[ 0.25, 2.25, 	8.00 ]
    	,  c_i^” =
    	[ 2.25, 	2.25, 0.00 ].






§ TABLES
  

























 





§ GRAPHS

























§ ACKNOWLEDGEMENTS

One of the author (K.S. Singh) would like to thank UGC for their financial support (grant no. F.7-65/2007(BSR)).



