Revamped Bi-Large neutrino mixing with Gatto-Sartori-Tonin like relation
RRevamped Bi-Large neutrino mixing withGatto-Sartori-Tonin like relation
Subhankar Roy a,1, ∗ , K. Sashikanta Singh b , Jyotirmoi Borah c a Department of Physics, Gauhati University, Guwahati-781014, India b Department of Physics, Manipur University, Imphal, Manipur-795003, India c Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, India
Abstract
The Gatto Sartori Tonin (GST) relation which connects the Cabibbo angle and the quarkmass ratio: θ C = (cid:112) m d /m s , is instituted as θ = (cid:112) m /m to a Bi-large motivated leptonmixing framework that relies on the unification of mixing parameters: θ = θ C and θ = θ .This modification, in addition to ruling out the possibility of vanishing θ , advocates fora nonzero lowest neutrino mass and underlines the normal ordering of the neutrino masses.The framework is further enhanced by the inclusion of a charged lepton diagonalizing matrix U lL with ( θ l ∼ θ C ). The model is framed at the Grand unification theory (GUT) scale.To understand the universality of the GST relation and the Cabibbo angle, we test theobservational mixing parameters at the Z boson mass scale. Keywords:
Neutrino mixing, Quark mixing, Cabibbo angle, Renormalization GroupEquations, Bilarge neutrino mixing.
1. Introduction
The neutrinos are the most elusive fundamental particles available in Nature. The Stan-dard model (SM) of particle physics fails to give a vivid picture of the same. The quest tounderstand the underlying first principle working behind the neutrino masses and mixingmechanism takes us beyond the SM. In this article, we emphasize on the significance ofthe simple unification schemes in terms of the common parameters and phenomenologicalrelation that both the lepton and quark sectors may share.The SM witnesses only the left-handed flavor neutrinos and the corresponding flavoreigenstates ( ν eL , ν µL and ν τL ) are not identical to their mass eigenstates ( ν L , ν L and ν L ).If the charged lepton Yukawa mass matrix Y l is diagonal, the neutrino flavor eigenstates are ∗ corresponding author Email addresses: [email protected], [email protected] (Subhankar Roy), [email protected] (K. Sashikanta Singh), [email protected] (Jyotirmoi Borah)
Preprint submitted to Elsevier October 19, 2020 a r X i v : . [ h e p - ph ] O c t xpressed as a linear superposition of the neutrino mass eigenstates in the following way, ν αL = (cid:88) i =1 ( U ν ) αi ν iL , ( α = e, µ, τ ) , (1)where, the matrix U ν is known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) ma-trix [1] and it preserves the information of the Lepton mixing. The matrix U ν is testable inthe oscillation experiments and to parametrize U ν , we require three angles and six phases.Out of the six phases, three are absorbed by the redefinition of the left handed charged lep-ton fields ( e L , µ L , and τ L ). If the original framework carries a non-diagonal charged leptonYukawa matrix Y l , then the U ν suffers a substantial amount of correction and the PMNSmatrix is redefined as, U = U † lL .U ν , (2)where, the U lL is the left handed unitary matrix that diagonalizes, Y † l .Y l . The U carriessix observable parameters: three neutrino mixing angles: θ , θ and θ which are known assolar, atmospheric and reactor angles respectively, the Dirac-type CP violating phase ( δ ) andtwo Majorana phases ( ψ and ψ ). Following the particle data group (PDG) parametrization,the U appears as shown below [2], U = R ( θ ) .W ( θ ; δ ) .R ( θ ) .P, (3)where, P = diag ( e − i ψ , e − i ψ , ψ and ψ and the above parametrization ensures thisfact. Moreover, the proper ordering and exact information of the neutrino mass eigenvaluesare unavailable as the oscillation experiments are sensitive only to the parameters: ∆ m = m − m and | ∆ m | = | m − m | . In short, the experimental results suggest: θ ≈ , θ ≈ , θ ≈ , ∆ m = 7 . × − eV , | ∆ m | = 2 . × − eV and δ ∼ [3].A specific model predicts a testable U . For example, the very popular mixing scheme,Tri-Bimaximal (TBM) [4] predicts the mixing angles within U as, θ = 35 . and θ = 45 .These predictions fit well within the 3 σ bound [3] of experimental data. Hence, TBM mixingis still relevant as a first approximation. However, the former projects θ as zero and thispossibility is strictly ruled out by the recent experiments [5, 6].The experiments show that θ ∼ O ( θ C ) , (4)where, the parameter, θ C is the Cabibbo angle [7] and θ C ∼ ◦ . Hence, we expect acorrection to the TBM model [8] which is of the order of θ C . Another example of mixingscheme that carries vanishing θ is the democratic mixing pattern which predicts large solarand atmospheric angles: θ = 45 ◦ and θ (cid:39) . ◦ . But interestingly, in order to convergeto the reality conditions, all these three mixing angles require corrections of the order of θ C .In Ref. [9], the natural perturbation is implemented on the democratic mixing matrix, U and the PMNS matrix is defined as, U = U X , where the X is the correction matrix such2hat: X = X ( θ x , θ x , θ x ). From, the model-building point of view, on choosing θ x = 0 andsticking to the ansatz : θ x (cid:39) − θ x (cid:39) θ C , one may see that except θ , the other two anglesremain slightly outside the 3 σ bounds [3]. But whether U arises from the neutrino sectoror charged lepton sector (or both) is model-dependent.On the other hand, the promising mixing schemes termed as Bi-large(BL) neutrino mix-ing [10–16] shelters θ C as an inherent parameter within the neutrino sector. Also, it assumes,large (and equal) values of θ and θ . The angle, θ is visualized as: sin θ ∼ λ , where λ = sin θ C ≈ .
22, is called the Wolfenstein parameter [17]. The geometrical origin of the BLmodel is explored originally in Ref. [11]. We know that θ C is a significant parameter withinthe quark sector and realization of the same within the neutrino sector extends the possibili-ties for new unification schemes. The BL framework is further strengthened by the fact thatin the SO (10) or SU (5) inspired GUTs, a single operator generates the Yukawa matricesfor both: down type quarks and the charged leptons ( Y d and Y l respectively) [18–24]. In thiscontext, the matrix elements of Y l are proportional to those of Y d and this results in, U lL ∼ V CKM (5)where, the V CKM is called the Cabibbo- Kobayashi- Maskawa matrix [17, 25] and thePMNS matrix is redefined as U ∼ V † CKM .U ν [26] in a basis where Y l is diagonal. Interestingly,the role of the Cabibbo angle is not limited to the quark mixing only, but it describes thequark masses also. We see that, ratio of up and charm quark masses: m u /m c ∼ λ andthat between charm and top quarks: m c /m t ∼ λ . Also, the ratio of down and strangequarks and that between strange and bottom quarks are m d /m s ∼ λ and m s /m b ∼ λ respectively [27]. The former case is called the Gatto-Sartori-Tonin (GST) relation and isexpressed as shown below [28]: sin θ C (cid:39) (cid:114) m d m s , (6)The above relation is derived in many occasions starting from the study of discrete flavorsymmetry groups [29–32]. The question appears whether in case of lepton sector, the massesand the mixing angles are somehow related or not. Indeed, the quark and the neutrinosector differ a lot than being similar. The V CKM is too close to an Identity matrix, whereasthe PMNS matrix U , is far from being an Identity matrix. Although the mixing schemesdiffer a lot, but believing on the unification framework like GUT, there lies enough reasonsto explore similar signatures in both quark and lepton sectors. We see that in the leptonsector the ratios of the charged lepton masses: m e /m µ ∼ λ and m µ /m τ ∼ λ [27]. So, wesee that even though there are differences, yet the parameter θ C (or λ ) finds its existencein both quark and lepton sectors. But unlike the charged leptons, the exact masses of theneutrino mass eigenstates are not yet known.Following the footprints of GST relation in eq.(6), the viability of a similar GST likerelation in the neutrino sector: sin θ ij = (cid:114) m i m j , (7)3s explored in our earlier work [33]. In this analysis, the Y l is assumed to be diagonal and theCP violation is ignored. The phenomenology shows that there is a single GST like relation,sin θ = (cid:114) m m , (8)or, sin θ = (cid:114) m m , (9)which is possible in the neutrino sector. Needless to mention that the two relations cannotbe experienced simultaneously. The first relation seems more appealing when we are tryingto explore the unification possibilities. This is because θ and θ C are of the same orderand in this work we assume that θ unifies with θ C at the GUT scale. Also, this relationadvocates for the normal ordering of the neutrino masses and this possibility is indicatedrecently by the experimental results [3]. On the other hand, the second relation concerning θ favors inverted ordering of neutrino masses. The vindication of nonzero θ , its proximitytowards the Cabibbo angle and the hint for normal ordering of neutrino masses make thefoundation of unification schemes stronger. In the next section we shall try to explore howthe GST relation can be invoked in the framework of Bi-large neutrino mixing.
2. Modified bilarge ansatz
Several BL schemes are proposed in the Refs. [10–16], and out of which we adopt theone [10, 14] which in addition to unifying the reactor angle and Cabibbo angle, θ ν = θ C stresses further on the unification of the atmospheric and solar mixing angles such thatwithin the neutrino sector, sin θ ν = sin θ ν = ψλ . Here, ψ is a free parameter and as perthe earlier analysis [10, 13, 14], ψ ≈
3. In our previous works [13, 14], it is shown that sucha BL mixing framework can be made more promising by incorporating a CKM-like chargedlepton diagonalizing matrix. But none of the works related to BL mixing mentioned abovetakes the neutrino mass parameters into consideration. Emphasizing on the possibility thatthere may lie a correlation between the reactor angle and the mass ratio, m /m , and thisrelation exists naturally in a unification framework defined at the GUT scale ( ∼ GeV),the Bi-large ansatz in the lepton sector is modified as presented below. θ ν = θ C = (cid:114) m m = (cid:114) m d m s , (10) θ ν = θ ν = sin − ( ψλ ) , (11) θ l (cid:39) θ C , (12) θ l = Aλ , (13)where, the θ νij ’s and θ lij ’s are the mixing angles that parametrize the U ν and U lL respec-tively. Here, A is one of the Wolfenstein parameters that appears in the CKM matrix [34].It is worth mentioning that this modified BL framework favors the normal ordering of the4eutrino masses. At the same time, we see that the provision of strict normal hierarchywhich insists on m = 0 is ruled out as θ ν is nonzero. The last two ansatze involving themixing angles from charged lepton sector indicates for a CKM-like U lL .Based on the above discussion, at the GUT scale we design the neutrino mixing matrixas shown below, U ν = c − cλ s − sλ e − iδ λ − cs (cid:0) e iδ λ + 1 (cid:1) c − e iδ s λ s − sλ s − c e iδ λ − cs (cid:0) e iδ λ + 1 (cid:1) c − cλ .P, (14)where, s = ψλ and s = cos(sin − ( ψλ )). The δ is a free phase parameter within theneutrino sector at the GUT scale. We establish the effective light neutrino mass matrix, m ν as shown below, m ν ( m , m , ψ, ψ , ψ , δ , λ ) = U ∗ ν .diag { λ , m (cid:48) , } .U † ν m , (15)where, m (cid:48) = m /m . The m ν contains four free parameters: m , m , ψ , δ and two Majoranaphases ψ and ψ . The m ν depends on the renormalization energy scale µ . As m ν appearingin the equation above is defined at M GUT , we have to run it down upto testable low energyscale, at M Z , the Z boson mass scale to extract the information of the oscillation parametersfrom the former. The running of m ν involves several complicated steps. We believe that m ν results from the see saw mechanism [35, 36] which again involves three heavy right-handedMajorana neutrino mass eigen states, N i =1 , , R with the mass eigenvalues M , , R respectively.Here, the N R is the heaviest Right-handed eigenstate and M R < M GUT . The effective lightneutrino mass matrix, m ν ( µ ) is related to heavy right handed neutrino mass matrix, M R ( µ )and light Dirac neutrino Yukawa matrix, Y ν ( µ ) in the following way, m ν ( µ ) = − v Y ν ( µ ) T M − R ( µ ) Y ν ( µ ) , (16)where, v is the Higgs vev. In our analysis, we assume that this parameter does notrun. We shall be working in the light of minimal super symmetric extension of the SM(MSSM) [37–39] and thus we take, v = 246 GeV sin β . While running down the m ν , theheavy right-handed states are to be integrated out at different thresholds.Between M GUT and M R , the following Renormalization Group Equation (RGE) holdsgood [40–50]. 16 π dm ν dt = (cid:18) − g − g + 2 T r ( Y † ν Y ν + 3 Y † u .Y u ) (cid:19) m ν +( Y † l Y l + Y † ν Y ν ) T m ν + m ν ( c l Y † l Y l + c ν Y † ν Y ν ) , (17)where, t = ln ( µ/µ ). Here, Y u is the up quark yukawa matrix. Apart from the knowledgeof m ν , Y ν , and Y u at the GUT scale, we require the information of Yukawa matrix of down5uark ( Y d ) as the texture of the Y l is dependent on how we parametrize Y d . To define thetextures of the respective Yukawa matrices, we shall draw the motivation from SU(5) GUTphenomenology discussed in Refs. [51–56].First, we parametrize Y d as shown below, Y d = d λ d λ d λ − d λ d λ d λ d λ , (18)where, the d (cid:48) ij s are O (1) coefficients and these are tabulated in Table. (1).The SU(5) GUT suggests that a general element of the charged lepton Yukawa matrix,( Y l ) ij is proportional to ( Y Td ) ij , ( Y l ) ij = α ( Y d ) ji . Here, the proportionality constant α is not arbi-trary, rather it’s choice is strictly guided by SU(5) GUT phenomenology and is constrainedto limited numbers of integers and fractions described in the Refs. [51–54]. We choose α interms of allowed entries: {− , , , } and portray the charged lepton Yukawa matrix asshown below, Y l = d λ − d λ d λ d λ − d λ d λ − d λ T . (19)In developing the above two yukawa matrices, we take care of the ratio ( y µ y d ) / ( y s y e ) whichcomes out to be 11.40 and this complies with the bound prescribed by the Ref. [54]. Wechoose the non-diagonal up-quark yukawa matrix, Y u in the following manner, Y u = u λ u λ − u λ u λ u , (20)where, u ij s are O (1) coefficients (see Table. (1)). We diagonalize the above matrices followingthe RL convention such that, U † ( x ) R .Y ( x ) .U ( x ) L = Y diag ( x ) , where, x = d, u and l . The matrices U ( x ) L are recognized by diagonalizing Y † ( x ) Y ( x ) such that U † ( x ) L .Y † ( x ) Y ( x ) .U ( x ) L = Y diag ( x ) . Weidentify, U uL ≈ − A λ A λ , (21) U dL ≈ − λ λ − λ − λ
00 0 1 , (22)and, U lL ≈ − a λ aλ − aλ − a λ − A λ A λ , (23)6or Y u , Y d and Y l respectively. Here a = 1 .
03 and the V CKM matrix is identified as, V CKM = U † uL .U dL ≈ − λ λ − λ − λ A λ − A λ (24)We see that the parameter a appearing in U lL shifts a little from unity and thus the U lL is not exactly equal to the V CKM matrix, but U lL ≈ V CKM . The parameter a = 1, is true ifthe correlation between Y l and Y d were, Y l = Y Td . The choice of the Dirac neutrino Yukawamatrix, Y ν is arbitrary and we fix it as per Ref.[57] as shown below, Y ν = 12 ν λ ν λ ν λ
00 0 1 , (25)where, the coefficients ν ij ’s are illustrated in Table. (1).The m ν suffers further Quantum corrections in the intervals, M R < µ < M R and M R <µ < M R [58–64]. In order to deal with the RGE evolution of the effective neutrino massmatrix, the Yukawa matrices, gauge couplings, to integrate out the heavy neutrino singletsand to derive the information of the neutrino oscillation parameters at different energyscales, we use the mathematica package REAP (Renormalisation group Evoluion of Anglesand Phases) [57]. The analysis involves a parameter known as supersymmetry breaking scale( m susy ) which is still unknown. Theoretically, m s ranges from a few Tev to hundred Tev.
3. Numerical Analysis
To exemplify, we choose M GUT = 4 . × GeV . At this scale we set, λ = 0 . A = 0 .
705 and the free parameters as shown below, ψ = 2 . , m = 0 . eV, m = 0 . eV,δ = 318 ◦ , ψ = ψ = 360 ◦ g = 0 . , g = 0 . , g = 0 . , and fix tan β at 60. We choose m susy = 3 TeV as one out of many possibilities. The RGEsare run from M GUT upto M Z = 91 . GeV and we extract the necessary information ofthe observable neutrino mixing parameters. We obtain, θ = 34 . ◦ , θ = 8 . ◦ , θ = 48 . ◦ ,δ = 274 . ◦ , ∆ m sol = 7 . × − eV , ∆ m atm = 2 . × − eV , (cid:80) m ν i = 0 . eV,ψ = 359 . ◦ , ψ = 4 . ◦
7e see that the two angles θ and θ comply well within the 1 σ bound, θ is consistentwithin the 2 σ and the solar and the atmospheric mass squared differences agree to stay withinthe 1 σ bound [3, 65]. According to the recent analysis in Refs. [66, 67], the observationalparameter, (cid:80) m ν i has got an upper bound of 0 . eV to 0 . eV and the most stringentupper bound is 0 . eV as per Ref. [68]. The lower bound is predicted as 0 . eV in Refs.[67, 68] or 0 . eV according to the Ref. [2]. We see that prediction of (cid:80) m ν i in our analysislies slightly above the prescribed lower bound.We see that the θ at M Z , unlike the other mixing angles, varies appreciably if the freeparameter δ varies at the GUT scale. To illustrate, keeping all the input parameters fixed,if δ is changed a little from 318 ◦ to 323 ◦ , we see that the θ at the M Z scale changes from8 . ◦ to 7 . ◦ (which lies outside the 3 σ range).Similarly, if m susy is varied a little, the mass parameters at M Z are also affected. Toillustrate, we study how the different observational parameters at the M Z scale evolve againstthe variation of δ , for different values of m s ranging from 1 T ev to 14
T eV . The analysisrequires the knowledge of numerical values of the three gauge coupling constants and threeYukawa couplings at the GUT scale. For this, we use the Refs [49, 50], where the requiredinput parameters are obtained by running the RGE s following a bottom-up approach atdifferent values of m susy , (See Table. (2)).As the observable θ at M z varies a lot with respect to the unphysical parameter δ , werestrict the latter (See Fig.(1a)) with respect to the 3 σ bound of the former,[3]. We see either,33 . ◦ (cid:54) δ (cid:54) . ◦ or 318 . ◦ (cid:54) δ (cid:54) . ◦ . But the first bound predicts a numericalrange of the Dirac CP violating phase, δ at M Z which lies outside the 3 σ region and henceit is rejected (See Fig. (1b)). The other bound of δ predicts, 276 ◦ (cid:54) δ ( M Z ) (cid:54) ◦ andthis is true with respect to the 2 σ range [3]. Now, in view of this allowed range of δ , onefinds, the mixing angles θ and θ at M Z scale lie within the 1 σ bound (See Figs. (2a) and(2b)). It is found that the mixing angles are less sensitive towards m susy . On the contrary,the mass parameters and hence the related observational parameters drift appreciably if m s is varied (See Figs. (3a), (3b), (3c), (2c) and (2d)). We see that although ∆ m sol changes as m susy varies from 1 T eV to 14
T eV , yet it agrees well within the 3 σ range. In contrast, thesame for ∆ m atm goes outside the 3 σ range if m s (cid:62) T eV . The (cid:80) m ν i (at M Z ), varies leastwith respect to m s and stays within the experimental bound (see Fig. (3d)).We wish to add a few notes on quark masses and mixing parameters obtained at the M z scale. With the same input parameters at the GUT scale as described towards the begin-ning of this section along with m susy fixed at 3 T eV , it is found that: m d ( M z ) = 2 . M eV , m s ( M z ) = 45 . M eV , m b ( M z ) = 3 . GeV , m u ( M z ) = 1 . M eV , m c ( M z ) ≈ . GeV and m t ( M z ) = 172 . GeV . These results agree well with the bounds predicted in Ref. ([69]).In addition, we evaluate the CKM mixing parameters at the M Z scale: | V ud | = 0 . | V us | = 0 . | V cs | = 0 . | V cb | = 0 . | V ts | = 0 . | V tb | = 0 . | V cb | , lies within the 1 σ range, the | V ts | and | V tb | lie within the 2 σ range, whereasthe | V ud | , | V us | and | V cs | lie within the 3 σ bound [34]. The | V cd | is found to lie slightly be-low the 3 σ lower limit(0 . | V ub | ∼ O (10 − ) and | V td | = 0 . | V ub | ∼ . | V td | = 0 . Y d and Y u at8UT scale (see Eqs. (18) and (20)), the 1-3 rotation within the V CKM is not taken into con-sideration. We believe that imparting a little perturbation to the textures of Y d and Y u maylead to the necessary changes and further precision to the CKM mixing parameters. Thisis beyond the scope of the present work. On the other hand we obtain the charged leptonmasses as, m e ( M Z ) ≈ . eV , m µ ( M Z ) ≈ . M eV and m τ ( M Z ) ≈ . M eV which are found in agreement with the Ref. ([69]).
4. Summary
The present work tries to address the problem of neutrino masses and mixing by lookinginto the simple unification possibilities and testing the same against the experimental results.We have shown that by relating the smallness of the reactor angle with the Cabibbo angle,unifying both of them at the GUT scale and an extension of the ansatze with a Cabibbomotivated GST relation: θ ν = θ C = (cid:112) m /m for neutrinos as a signature of unification,have got far reaching consequences. Also, this framework considers the unification of θ ν and θ ν . Based on the SU(5) GUT phenomenology, we suggest the textures of Y d and Y l such that θ l = 1 . θ C . We run the neutrino mass matrix from M GUT scale to M Z scalefollowing the RGE and explore the oscillation parameters at the the M Z scale. The GSTrelation within the lepton sector ensures the normal ordering of the neutrino masses. Atthe M Z scale, the Dirac CP phase is predicted to lie within 276 ◦ (cid:54) δ ( M Z ) (cid:54) ◦ andthe θ lies in the second octant. We see that the mixing angles are sensitive to the freeparameter δ whereas, the mass parameters response substantially towards the variation ofsuper symmetry breaking scale. Acknowledgment
SR and KSS thank N. Nimai Singh, Manipur University for the useful discussions. JBthanks Gauhati University for providing him a chance to be a part of the work. SR wishesto thank FIST(DST) grant SR/FST/PSI-213/2016(C) for the necessary support.
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Zhou, Impacts of the Higgs mass on vacuum stability, running fermionmasses and two-body Higgs decays, Phys. Rev. D 86 (2012) 013013. arXiv:1112.3112 , doi:10.1103/PhysRevD.86.013013 . able 1: The coefficients of Y d and Y u as shown in eqs. (18) and (20) respectively are described in thistable. O (1) coefficients appearing in Y d,u,ν d = 2 . , d = 2 . , d = 2 . ,d = 1 . , d = 1 . , d = 1 . , d = 2 . u = 0 . , u = 0 . , u = 0 . , u = 1 . , u = 0 . ν = 0 . , ν = 0 . i, ν = 0 . Table 2:
The list of the gauge coupling constants g , g , g and the M GUT for different values of the SUSYbreaking scales ranging from 1
T eV to 14
T eV is given. m s ( T eV ) M GUT (10 GeV ) g g g susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV ( σ region) δ [ ◦ ]( GU T ) θ [ ◦ ] ( M Z ) (a) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV (3 σ region)(1 σ region) δ [ ◦ ]( GU T ) δ [ ◦ ] ( M Z ) (b)Figure 1: ( and (1b) show how the θ ( M Z ) changes with respect to variation of δ at the GUT scalerespectively for different values of the SUSY breaking scale, m susy ranging from 1 T eV to 14
T eV (All thegraphs are merged almost together). In both of the plots, black horizontal line, purple and orange bandssignify the best fit value, 1 σ and 3 σ ranges of the concerned parameter. In Fig. ( ), with respect to the3 σ range [3] of θ , two possible ranges of input parameter δ : 33 . ◦ (cid:54) δ (cid:54) . ◦ and 318 . ◦ (cid:54) δ (cid:54) . ◦ (shown by two vertical grey bands) are obtained. In Fig. ( ) we see that only the second range isallowed in the light of the 3 σ bound of δ . This range 318 . ◦ (cid:54) δ (cid:54) . ◦ predicts the Dirac CP phase ( δ )within the 2 σ bound [3]. susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV ( σ region)( σ region) δ [ ◦ ]( GU T ) θ [ ◦ ] ( M Z ) (a) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV ( σ region)( σ region) δ [ ◦ ]( GU T ) θ [ ◦ ] ( M Z ) (b) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV ( σ region)( σ region) . . . . . . . . δ [ ◦ ]( GU T ) ∆ m s o l [ − e V ] ( M Z ) (c) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV ( σ region)( σ region) . . . . . . . δ [ ◦ ]( GU T ) ∆ m a t m [ − e V ] ( M Z ) (d)Figure 2: (2a) , (2b) , (2c) and (2d) show the variation of θ , θ , ∆ m sol and ∆ m atm at the M Z scalerespectively with respect to the variation of δ at the GUT scale for different values of the SUSY breakingscale, m susy ranging from 1 T eV to 14
T eV . The plots in Figs ( ) and ( ) merge almost together. TheBlack line, purple and the orange bands represent the best-fit, 1 σ and 3 σ bounds [3] respectively for theconcerned observational parameters. The vertical grey band represents the allowed bound of δ at GUTscale which is 318 . ◦ (cid:54) δ (cid:54) . ◦ . The θ is predicted around 34 ◦ (1 σ ) and that for θ is around49 ◦ [3].In Figs. (2c) and (2d), the ∆ m sol and ∆ m atm varies appreciably with respect to both δ and m susy .We note that unlike ∆ m sol , the ∆ atm goes outside the 3 σ range if m susy > T ev .
60 120 180 240 300 3602 . . . . . δ [ ◦ ]( GU T ) m [ − e V ] ( M Z ) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV (a) . . . . . . δ [ ◦ ]( GU T ) m [ − e V ] ( M Z ) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV (b) . . . . . . . . δ [ ◦ ]( GU T ) m [ − e V ] ( M Z ) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV (c) m susy = 1 TeVm susy = 3
TeVm susy = 5
TeVm susy = 7
TeVm susy = 9
TeVm susy = 11
TeVm susy = 14
TeV (Most stringentupper bound on P m ν i )(0.078 eV )(0.058 eV )(Lower bound on P m ν i ) δ [ ◦ ]( GU T ) PPP m ν i [ − e V ] ( M Z ) (d)Figure 3: (3a) , (3b) , (3c) and (3d) show the variation of m , m , m and (cid:80) m ν i at the M Z scalerespectively with respect to the variation of δ at the GUT scale for different values of the SUSY breakingscale, m susy ranging from 1 T eV to 14
T eV . The vertical grey band represents the allowed bound of δ atGUT scale which is 318 . ◦ (cid:54) δ (cid:54) . ◦ . In Fig. ( ), the bound on (cid:80) m ν i is prescribed with respectto the ref. [68].is prescribed with respectto the ref. [68].