Radiative and Seesaw Threshold Corrections to the S 3 Symmetric Neutrino Mass Matrix
RRadiative and Seesaw Threshold Correctionsto the S Symmetric Neutrino Mass Matrix
Shivani Gupta a and C. S. Kim b Department of Physics and IPAP, Yonsei University, Seoul 120-749, Korea
Pankaj Sharma c Korea Institute of Advanced Study, Seoul 130-722, Korea
Abstract
We systematically analyze the radiative corrections to the S symmetric neutrino mass matrixat high energy scale, say the GUT scale, in the charged lepton basis. There are significant correc-tions to the neutrino parameters both in the Standard Model (SM) and Minimal SupersymmetricStandard Model (MSSM) with large tan β , when the renormalization group evolution (RGE) andseesaw threshold effects are taken into consideration. We find that in the SM all three mixing anglesand atmospheric mass squared difference are simultaneously obtained in their current 3 σ rangesat the electroweak scale. However, the solar mass squared difference is found to be larger thanits allowed 3 σ range at the low scale in this case. There are significant contributions to neutrinomasses and mixing angles in the MSSM with large tan β from the RGEs even in the absence ofseesaw threshold corrections. However, we find that the mass squared differences and the mixingangles are obtained in their current 3 σ ranges at low energy when the seesaw threshold effects arealso taken into account in the MSSM with large tan β . PACS numbers: 14.60.Pq, 14.60.St a [email protected] b [email protected] c [email protected] a r X i v : . [ h e p - ph ] A ug . INTRODUCTION The neutrino oscillation experiments have enriched our knowledge of masses and mixingsof neutrinos and thus flavor structure of leptons. These developments aspire theorists toconstruct models for unraveling the symmetries of lepton mass matrices. For three neutrinoflavors the solar and atmospheric mass squared differences are given as, ∆ m ≈ − eV and | ∆ m | ≈ − eV , where ∆ m ij = m j − m i and m i,j are the mass eigenvalues ofneutrinos, respectively. With the evidence of nonzero value of reactor mixing angle θ [1]we now have information of all three mixing angles contrary to earlier studies where onlyan upper bound on θ existed. The lepton flavor mixing matrix comprising of three mixingangles and a CP violating phase is given as U = c c s c s e − iδ CP − s c − c s s e iδ CP c c − s s s e iδ CP s c s s − c c s e iδ CP − c s − s c s e iδ CP c c . (1)Here c ij = cos θ ij , s ij = sin θ ij ; θ ij are the three mixing angles, δ CP is the Dirac CP phase.There are two additional CP phases if neutrinos are Majorana particles. The best fit valuesalong with their 3 σ ranges of neutrino oscillation parameters [2] are shown in Table I. Diracphase δ CP , which contributes to CP violation in the leptonic sector, is expected to be mea-sured in the long baseline neutrino oscillation experiments. The strength of this leptonic CPviolation is parameterized by Jarlskog rephasing invariant [3] J = c s c s c s sin δ CP .The two Majorana phases, however, contribute to the lepton number violating processeslike neutrinoless double beta decay. The possible measurement of effective Majorana massin the current and upcoming experiments like GERDA, EXO, CUORE, MAJORANA, Su-perNEMO [4] will provide some additional constraints on the two Majorana phases andneutrino mass scale. The cosmological constraint on the sum of neutrino masses by thePlanck Collaboration [5] is Σ m ν i < . − . m ν i = 0.32 ± θ has lead to many studies for the deviation from the assumed2 arameter Best fit 3 σ ∆ m / − eV (NH or IH) 7.54 6.99 – 8.18∆ m / − eV (NH) 2.43 2.19 – 2.62 θ ◦ ◦ – 36.8 ◦ θ ◦ ◦ –10.19 ◦ θ ◦ ◦ – 52.95 ◦ TABLE I. The experimental constraints on neutrino parameters taken from [2]. symmetries that predict the vanishing θ value. Among many possible discrete flavor sym-metries to produce the current data, S has been extensively studied in literature [7]. It isthe smallest discrete non-Abelian group which is the permutation of three objects. Pertur-bations to S symmetric leptonic mass matrices have been used to study the mass spectraof the leptons and predict well known democratic [8] and tri-bimaximal neutrino mixingscenario [9]. The effective light neutrino mass matrix, M ν , invariant under S is given as M ν = pI + qD, (2)where I = , D = . (3)Here D is democratic matrix, p and q are in general complex parameters. There are quite afew studies about the breaking of S symmetry in the leptonic sector [10–12] to produce thecurrent observed neutrino oscillation data. It will be interesting to study order of breaking inthe S symmetric neutrino mass matrix that arise due to the RGE and the seesaw thresholdcorrections between the GUT scale Λ g and the electroweak scale Λ ew in the SM and MSSM.In the light of non zero θ our main aim is to study the viability of producing the currentneutrino oscillation data at Λ ew scale in an S symmetric neutrino mass matrix M ν at Λ g when both radiative and seesaw threshold corrections are taken into account.3arlier studies have shown that there are significant RGE corrections to neutrino massesand mixing angles particularly for quasi-degenerate neutrino spectrum in the MSSM withlarge tan β . The SM is extended by three heavy right handed neutrinos at high energy scale togenerate neutrino masses in Type I seesaw mechanism [13]. The seesaw threshold correctionsarise due to subsequent decoupling of these heavy right handed Majorana neutrinos attheir respective masses. The structure of the Dirac mass matrix M D is proportional to theneutrino Yukawa coupling matrix Y ν . We take a general Y ν and scan the parameter spaceto obtain the desired mixing pattern. The right handed Majorana mass matrix M is foundby inverting the Type I seesaw formula at Λ g . The charged lepton mass matrix M l at Λ g istaken to be diagonal. We study the running behavior of neutrino masses and mixings fromΛ g down to Λ ew using the RGE for the Yukawa couplings in the S symmetric M ν both inthe SM and MSSM. Above the heaviest seesaw scale ( M ) there is a full theory and thusRGEs for Yukawa couplings Y e , Y ν and mass matrix M are considered. However, since ourright handed neutrino mass matrix M is hierarchical ( M < M < M ), we also considerthe seesaw threshold effects and thus the respective set of effective theories in between thesescales, arising from the subsequent decoupling of heavy right handed fields at their respectivemasses. In the SM we find that the atmospheric mass squared difference (∆ m ) along withthe neutrino mixing angles are generated in their present 3 σ ranges at the low energy scale.However, the solar mass squared difference (∆ m ) is greater than its allowed value ( ≈ − )in the SM. We find that it is possible to radiatively generate the current neutrino masses andmixing angles from the S invariant neutrino mass matrix M ν in the charged lepton basis,when the seesaw threshold effects are taken into account in the MSSM with large tan β . Inthe MSSM with large tan β the solar mass squared difference ∆ m can be produced in itscurrent range along with the other neutrino oscillation parameters at Λ ew in the presence ofthese threshold corrections.In section II we give the form of lepton mass matrices considered at Λ g . In the subsequentsection, we give the RGE equations governing from Λ g to Λ ew , in presence of the seesawthreshold effects both in the SM and MSSM. In section IV we study the order of correctionsto the neutrino mass matrix in the presence of seesaw threshold effects. Section V gives ournumerical results for both cases under consideration. We conclude in the last section.4 I. FORM OF LEPTON MASS MATRICES AT THE GUT SCALE
We consider the basis where charged lepton mass matrix ( M l ) is diagonal and the effectivelight neutrino mass matrix ( M ν ) is S symmetric as given in Eq. (2). The Yukawa couplingmatrix for charged leptons is given as Y e = 1 v m e m µ
00 0 m τ , (4)where the Higgs vacuum expectation value (VEV) v is taken to be 246 GeV in the SM and246 · cos β GeV in the MSSM. The Yukawa coupling matrix Y ν for the light neutrinos is takenof the form Y ν = y ν U ν D as given in [14] where D is the diagonal matrix Diag( r , r , y ν , r and r are real, positive dimensionless parameters that characterizeeigenvalues of Y ν . The unitary matrix U ν is the product of the three rotation matrices R ( ϑ ) · R ( ϑ e − iδ ) · R ( ϑ ) having one CP violating phase δ . Thus, Y ν has seven unknownparameters viz. three eigenvalues, three mixing angles and one CP phase. We vary the threehierarchy ( y ν , r , r ) parameters and, though they are completely arbitrary, but assumed tobe < O (1). Three angles ϑ , ϑ , ϑ and δ are varied in the range of (0-2 π ).The right handed mass matrix M is found by inverting the Type I seesaw formula as M = − Y ν M − ν Y Tν , (5) i.e. M in our study is found from Y ν and the S symmetric neutrino mass matrix M ν at Λ g by inverting seesaw formula. The three right handed neutrino masses M , M and M areobtained by diagonalizing the right handed Majorana mass matrix M . The light neutrinomass matrix M ν can be diagonalized by the unitary transformation R as R T M ν R . One ofthe possible form of R can be R = U T BM = √ √ − √ √ − √ − √ √ √ . (6)The mass eigenvalues of M ν are p , p + 3 q and p : corresponding to the light neutrino masses m , m and m , respectively. Due to the degeneracy in the mass eigenvalues m and m , thediagonalizing matrix R is not unique. Degeneracy of masses implies that R is arbitrary up to5rthogonal transformation R ( φ ), where φ is in 1-3 plane. Thus, most general diagonalizingmatrix R is U T BM R ( φ ), which implies the same physics as U T BM . In this work we set φ =0without loss of generality [10, 12]. From the neutrino oscillation data we know ∆ m ≈ − and thus, there is small difference in the mass eigenvalues m and m which is a possibleobjection to this scenario of S invariant approximation as here m and m are degenerate.The possible solution to this problem is given in [10, 12] where the complex numbers p and p + 3 q are considered to have same magnitude but different directions. As shown in [10], q can be chosen completely imaginary and p is taken to be | p | e − i α . The magnitudes of p and q can be written in terms of parameter x as | p | = x sec α , | q | = 23 x tan α , (7)where x is a real free parameter and allowed range of α is 0 ≤ α < π . The magnitude of p and p + 3 q can be made equal by adjusting the phase α . The parameter x vanishes when α =180 ◦ and thus this value is disallowed. Substituting the values of p and q given in Eq.(7), the magnitudes of the mass eigenvalues are given as | m | = | m | = | m | = x sec α . (8)This results in equal magnitude of all three mass eigenvalues and thus a degenerate spectrumof neutrinos to begin with at Λ g . As pointed out earlier in [10] that the phase α affects therate of neutrinoless double beta decay but will not affect neutrino oscillation parameters.Thus, this phase is of Majorana type. When we run these masses from Λ g to highest seesawscale M , the degeneracy of the mass eigenvalues is lifted by RGE corrections. We considerthe normal hierarchical spectrum of masses where m is the lowest mass. The other twomasses are given as m = (cid:112) m + ∆ m and m = (cid:112) m + ∆ m . Since the three masseigenvalues at Λ g have equal magnitude the two mass squared differences are vanishing tobegin with. Once the degeneracy of three mass eigenvalues is lifted, their non zero values aregenerated. In subsequent sections we will explore the generation of the solar and atmosphericmass squared differences, together with the three mixing angles in their current 3 σ limit atΛ ew through the radiative corrections from S invariant neutrino mass matrix at Λ g .6 II. RGE EQUATIONS WITH SEESAW THRESHOLD EFFECTS
In Type I seesaw the SM is extended by introducing three heavy right handed neutrinosand keeping the Lagrangian of electroweak interactions invariant under SU (2) L × U (1) Y gauge transformation. In this case, the leptonic Yukawa terms of the Lagrangian are writtenas − L ν = ¯ l e φY e e R + ¯ l e ˜ φY ν ν R + 12 ¯ ν cR M ν R + h.c.. (9)Here φ is the SM Higgs doublet and ˜ φ = iσ φ ∗ . e R and ν R are right handed charged leptonand neutrino singlets, Y e and Y ν are the Yukawa coupling matrices for charged leptons andDirac neutrinos, respectively. The last term of Eq. (9) is the Majorana mass term for theright handed neutrinos.Quite intensive studies have been done in literatures [15, 16] regarding the general featuresof RGE of neutrino parameters. At the energy scale below seesaw threshold i.e. when all theheavy particles are integrated out, the RGE of neutrino masses and mixing angles is describedby the effective theory which is same for various seesaw models. But above the seesaw scalefull theory has to be considered and thus, there can be significant RGE effects due to theinterplay of heavy and light sector. The RGE equations and subsequent decoupling of heavyfields at their respective scales are elegantly given in Refs. [17–19]. The comprehensivestudy of the RGE and seesaw threshold corrections to various mixing scenarios is recentlydone in [20].The effective neutrino mass matrix M ν above M is given as M ν ( µ ) = − v Y Tν ( µ ) M − ( µ ) Y ν ( µ ) , (10)where v = 246 · sin β GeV in the MSSM and µ is the renormalization scale. Y ν and M are µ dependent. Since we study the evolution of leptonic mixing parameters from Λ g to Λ ew scalein a generic seesaw model we need to take care of the series of effective theories that ariseby subsequent decoupling of the heavy right handed fields M i ( i =1,2,3) at their respectivemass thresholds. The Yukawa couplings Y ν and M are dependent on the energy scale Λ.At the GUT scale we consider the full theory and the one loop RGEs for Y e , Y ν and M aregiven as ˙ Y e = 116 π Y e [ α e + C H e + C H ν ] , Y ν = 116 π Y ν [ α ν + C H e + C H ν ] , ˙ M = 116 π C (cid:2) ( Y ν Y † ν ) M + M ( Y ν Y † ν ) T (cid:3) , (11)where ˙ Y i = dY i dt ( i = e, ν ), t= ln ( µ/µ ) with µ ( µ ) being the running (fixed) scale, and H i = Y † i Y i ( i = e, ν ). The coefficients are C = , C = − , C = − , C = , C = 1 in the SMand C = 3 , C = 1 , C = 1 , C = 3 , C = 2 in the MSSM, respectively. The expressions for α e and α ν in the SM and MSSM are explicitly given as α e ( SM ) = T r (3 H u + 3 H d + H e + H ν ) − ( 94 g + 94 g ) ,α ν ( SM ) = T r (3 H u + 3 H d + H e + H ν ) − ( 920 g + 94 g ) ,α e ( MSSM ) = T r (3 H d + H e ) − ( 95 g + 3 g ) ,α ν ( MSSM ) = T r (3 H u + H ν ) − ( 35 g + 3 g ) , (12)where g , are the U (1) Y and SU (2) L gauge coupling constants.The heavy right handed mass matrix M obtained from Eq. (5) is non diagonal and thusis diagonalized by the unitary transformation U R as U TR M U R = Diag ( M , M , M ) . (13)The Yukawa coupling Y ν is accordingly transformed as Y ν U ∗ R . At the highest seesaw scale M , the effective operator κ (3) is given by the matching condition as κ (3) = 2 Y Tν M − Y ν , (14)in the basis where M is diagonal. The Yukawa coupling Y ν above is a 3 × M . At the scale lower than M ( µ < M ) the effectiveneutrino mass matrix M ν is given as M ν = − v { κ (3) + 2 Y Tν (3) M − Y ν (3) } . (15)As can be seen that M ν is the sum of κ (3) given in Eq. (14) and the seesaw factor which isobtained after decoupling M . Thus, Y ν (3) is 2 × M (3) is 2 × M and M is governed by the running of κ (3) , Y ν (3) and M (3) . The runningof κ (3) is given as˙ κ (3)( SM ) = 116 π (cid:2) ( C H e + C H ν (3) ) T κ (3) + κ (3) ( C H e + C H ν (3) ) + α (3) κ (3) (cid:3) , (16)8 κ (3)( MSSM ) = 116 π (cid:2) ( H e + C H ν (3) ) T κ (3) + κ (3) ( H e + C H ν (3) ) + α (3) κ (3) (cid:3) , where C = and H ν (3) = Y † ν (3) Y ν (3) . α (3) in the SM and MSSM is explicitily given as α (3)( SM ) = 2 T r (3 H d + 3 H u + H e + H ν (3) ) − g + λ,α (3)( MSSM ) = 2 T r (3 H u + H ν (3) ) − g − g . The effective operator κ (2) at M is given by the matching condition as κ (2) = κ (3) + 2 Y Tν (2) M − Y ν (2) . (17)where all the parameters are set to seesaw scale M . M ν at the scale below M has the RGEequation given in Eq. (16). In their RGE expression we have Y ν (2) which is 1 × M (2) which is 1 ×
1. The low energy effective theory operator κ (1) is obtained after integratingout all three heavy right handed fields. The one loop RGE for κ (1) from lowest seesaw scale M down to Λ e scale is given as˙ κ (1) = ( C H Te ) κ (1) + κ (1) ( C H e ) + ακ (1) , (18)where α = 2 T r (3 H u + 3 H d + H e ) − g + λ in the SM , (19) α = 2 T r (3 H u ) − g − g in the MSSM.When the Higgs field gets VEV, the light neutrino mass matrix is obtained from κ (1) as M ν = κ (1) v . We diagonalize M ν to obtain neutrino masses, mixing angles and CP phases. IV. NEUTRINO MASSES AND MIXINGS
The RGE above the highest seesaw scale M depends on more parameters than below thelowest seesaw scale M due to presence of the neutrino Yukawa couplings ( Y ν ). The RGEequations consist of H e , M and H ν out of which latter can be large. In the basis wherecharged lepton mass matrix is diagonal, M ν at two different energy scales Λ ew and Λ g arehomogeneously related as [18, 21, 22] M Λ ew ν = I K · I T · M Λ g ν · I. (20)9ere I K is flavor independent factor arising from gauge interactions and fermion antifermionloops. It does not influence the mixing angles. The matrix I has the form I = Diag ( e − ∆ e , e − ∆ µ , e − ∆ τ ) , (cid:119) Diag (1 − ∆ e , − ∆ µ , − ∆ τ ) + O (∆ e,µ,τ ) , (21)where ∆ j = 116 π (cid:90) [3( H j ) − ( H ν j )] dt, (22)where j = e, µ, τ . Numerically, ∆ SMτ can be of the order of 10 − when Y τ ∼ .
01 and Y ν = 0 . µ and µ are 10 and 10 , respectively. In the MSSM, ∆ MSSMτ ∼ − (1 + tan β ) for the same values of Y ν and Y τ . In the absence of seesaw thresholdeffects, ∆ τ is small ≈ − in the SM for the above mentioned values of Y τ .At above seesaw scale appreciable deviations may occur only for large values of Y τ or Y ν .In the absence of seesaw threshold effects i.e. when there is no H ν term in Eq. (22) theradiative corrections are governed by ∆ τ term as ∆ e,µ is too small. On the other hand, dueto the presence of large H ν , the seesaw threshold effects play a crucial role in the runningof mixing angles. Below the seesaw scales, deviations are obtained from Y τ ∼ √ m τ /v ≈ O (10 − ) in the SM, and ∼ √ m τ v cos β in the MSSM. There can be significant deviations in theMSSM with large tan β which enhances Y τ . In the presence of seesaw threshold, appreciabledeviations are possible as it is natural to have Y ν ∼ O (1). The analytic expressions for therunning neutrino mixing angles, masses and CP phases are quite long and have been earlierderived in literatures [19, 21, 23]. As shown earlier in [16, 24], the mass squared differencesin the denominator of the RGEs play a significant role in the evolution of neutrino mixingangles and phases, if the left-handed neutrinos are nearly degenerate. V. NUMERICAL RESULTS AND DISCUSSIONS
At the GUT scale Λ g we have seven free parameters in Y ν and two free parameters α and x in M ν . The three mixing angles in ϑ , ϑ and ϑ and phase δ are allowed to take the valuesin the range (0–2 π ). The hierarchy parameters y ν , r , r and x are randomly varied and areexpected to be < O (1). The physical range of phase α is from (0– π ). The mass spectrum atthe high scale is degenerate and thus we have vanishing solar and atmospheric mass squareddifference to begin with. The parameter space at the Λ g with which the low energy neutrino10 arameters SM MSSMInput r × − × − r δ ◦ ◦ y ν θ ◦ ◦ θ ◦ ◦ θ ◦ ◦ x ( eV ) 7.46 × − . × − α ◦ ◦ Output m ( eV ) 9 . × − . × − θ ◦ ◦ θ ◦ ◦ θ ◦ ◦ ∆ m ( eV ) 4.18 × − . × − ∆ m ( eV ) 2.48 × − . × − M (GeV) 3.1 × . × M (GeV) 4.43 × × M (GeV) 2.83 × . × J -2.99 × − − . × − | m ee | ( eV ) 9.2 × − . × − TABLE II. Numerical values of input and output parameters that are radiatively generated viathe RGE and seesaw threshold effects both in the SM and MSSM. The input parameters are takenat Λ g = 2 × GeV and tan β =55 in the MSSM. ew is illustrated in Table II. The set of input parameters in thatparticular parameter space are also given in the table. We choose the set of parameters inparameter space at the high scale for which maximum value of θ is obtained and the othermixing angles and mass squared differences are simultaneously obtained in current 3 σ rangeat Λ ew . However, the parameter space under consideration is only for illustration and notunique. Search for complete parameter space is an elaborate study and thus independentfuture work. A. Radiative Threshold corrections in the SM
Study of radiative corrections to S symmetric neutrino mass matrix can be divided intothree regions that are governed by different RGE equations. The first region is above thehighest seesaw scale M to Λ g , where there can be considerable contribution of Y ν . Thesecond region is the region between the three seesaw scales and the third is below the lowestseesaw scale M , where all heavy fields are decoupled. The solar mixing angle θ can havelarge RGE corrections as the running is enhanced by the factor proportional to m ∆ m at theleading order, which can be large for degenerate spectrum. The RGE is comparatively smallfor other two mixing angles θ and θ , where the RGE is proportional to m ∆ m . However,for the degenerate neutrino mass spectrum there can be considerable corrections for thesemixing angles, too. Below the seesaw scale the RGE corrections to the mixing angles inthe SM are negligible as they get contributions only from Y τ . In Fig 1, we show the RGEcorrections to the mixing angles and masses in the SM for the set of input parameters givenin second column of Table II. Fig. 1 shows that below M scale there is no significantcorrections to the mixing angles. Below the lowest seesaw scale the running of the masseigenvalues is significant even in the SM for degenerate as well as hierarchical neutrinos[16] due to the factor α given in Eq. (19), which is much larger than Y τ . The runningof masses is given by a common scaling of the mass eigenvalues [25]. Clearly, the RGE ofeach mass eigenvalue is proportional to the mass eigenvalue itself. The running of massesin Fig. 1 can be seen to start from the degenerate values of masses at Λ g and there aresignificant corrections to the masses below M . Earlier analyses [24] studied the successfulgeneration of mass squared differences and mixing angles for degenerate neutrinos in the SM.In their analysis it is shown that the generation of mass squared differences is very sensitive12o the value of sin θ , which should be greater than 0.99 to fit the mass squared differencessimultaneously with the consistent angles. This limit is, however, completely ruled out bythe current oscillation data. The degeneracy of three mass eigenvalues is lifted by the RGErunning from Λ g to M . Potentially significant breaking of neutrino mass degeneracy isprovided by the RGE effects. The seesaw threshold effects in addition increase the masssplitting between the masses m and m required to fit the masses with the current data in θ ij ( deg ) Log ( µ /GeV) θ θ θ m i ( e V ) Log ( µ /GeV) m m m -7 -6 -5 -4 -3 -2
0 2 4 6 8 10 12 14 16 Δ m ij ( e V ) Log ( µ /GeV) Δ m Δ m FIG. 1. The RGE of the mixing angles, masses and mass squared differences between the GUTscale Λ g and the electroweak scale Λ ew in the SM. The initial values of the parameters are givenin the second column of Table II. The boundaries of three grey shaded areas, i.e. dark, mediumand light denote the points when heavy right handed singlets M , M and M are integrated outrespectively. Y ν . The contribution of Y ν in the RGE can result all the three mixing anglesin their current allowed ranges at the EW scale in the SM. Thus, there are significantcorrections to mixing angles even in the SM in the presence of the seesaw threshold effectswhen there is exactly the degenerate mass spectrum to begin with. As can be seen from Fig.1, the mixing angles at the GUT scale are θ = 45 ◦ , θ = 35 . ◦ and θ = 0 ◦ . For the set ofparameters given in the second column of Table II we get the mixing angles in the allowedrange at the electroweak scale in the SM. θ is found to have values below maximality.The presence of seesaw threshold corrections and thus Y ν in the RGE equations make thispossible even in the SM, as can be seen in Fig. 1. The grey shaded area in Fig. 1 and Fig.2 illustrates the ranges of effective theories that emerge when we integrate out heavy righthanded singlets. At each seesaw scales, i.e. M , M and M , one heavy singlet is integratedout and thus (n-1) × Y ν remains. Therefore, the running behavior betweenthese scales can be different from running behavior below or above these scales. Betweenthese scales the neutrino mass matrix comprises of two terms κ ( n ) and 2 Y Tν ( n ) M − R ( n ) Y ν ( n ) , asgiven in Eq. (15). It is shown in [16] that in the SM these two terms between the thresholdsare quite different which can give dominant contribution to the running of mixing angles inthis region. Both θ and θ in Fig. 1 get large corrections between three seesaw scales. θ gets the deviation of ≈ ◦ in the upper direction.Three neutrino mixing angles can be generated in their current allowed ranges startingfrom S symmetric neutrino mass matrix at the Λ g in the SM as there can be large correc-tions when we consider the seesaw threshold effects. In the SM, atmospheric mass squareddifference ∆ m is generated within the current oscillation data limit ( ≈ . × − ) startingfrom the vanishing value at the Λ g since all masses are degenerate, as seen from Fig. 1. Thesolar mass squared difference ∆ m of the order of ≈ − eV is simultaneously generatedat the Λ ew , as shown in Fig.1 which is larger than its present value. The byproduct of thisanalysis is the masses of right-handed neutrinos that are determined from Eqs. (5) and(13) and are not free parameters. The values of effective Majorana mass | M ee | and Jarlskogrephasing invariant ( J ) at the Λ ew are also calculated for particular set of parameters givenin Table II. We conclude that in the SM the RGE in addition to seesaw threshold effectscannot simultaneously produce the solar mass squared difference in its allowed range at low14cale along with the other neutrino oscillation parameters. B. Radiative Threshold corrections in the MSSM
As stated earlier we divide the radiative corrections to S symmetric neutrino mass matrixinto three regions governed by different RGE equations in the MSSM. In the region below θ ij ( deg ) Log ( µ /GeV) θ θ θ m i ( e V ) Log ( µ /GeV) m m m -7 -6 -5 -4 -3 -2
0 2 4 6 8 10 12 14 16 Δ m ij ( e V ) Log ( µ /GeV) Δ m Δ m FIG. 2. The RGE of the mixing angles, masses and mass squared differences between the GUTscale Λ g and the electroweak scale Λ ew in the MSSM with tan β =55. The initial values of theparameters are given in the third column of Table II. The boundaries of three grey shaded areas, i.e. dark, medium and light denote the points when heavy right handed singlets M , M and M are integrated out respectively. β , Yukawa coupling Y τ ∼ β thesecorrections can be larger due to the presence of factor Y τ (1 + tan β ) in the MSSM. We showthe RGE of mixing angles and masses in the MSSM with tan β =55 in Fig. 2 for the set ofinput parameters given in third column of Table II. In the region above the energy scale M ,we get contributions from another Yukawa coupling Y ν which brings in more free parametersin the analysis. In the region for the MSSM with large tan β the presence of seesaw thresholdeffects can enhance the RGE of the mixing angles significantly. As can be seen from Fig.2, we can have all three mixing angles simultaneously in the current limit at the EW scalestarting from S symmetric neutrino mass matrix at the Λ g . Fig. 2 shows that there arelarge corrections to θ ( ∼ ◦ ) between the Λ g and M scale due to the presence of Y ν .As mentioned earlier, the running of neutrino mass matrix between the seesaw thresholdsgets contributions from two terms κ ( n ) and 2 Y Tν ( n ) M − R ( n ) Y ν ( n ) given in Eq. (15). As shownin the SM, these two terms are different and thus results in the enhanced running betweenthe different energy regimes. On the other hand, in the MSSM, as can be seen from Fig. 2,there are not much deviations in the mixing angles between the energy scales. It is becausein the MSSM the two contributions are almost identical and thus cancel each other resultingin minimum deviation in those regions. The only significant corrections occurs in the regionabove the highest seesaw scale M due to relatively large Y ν .The mixing angle θ does not have much corrections and θ receives the correction of2.5 ◦ in the upper direction and is thus above maximal. The running of masses in the MSSM(Fig. 2) is much larger than the SM due to the presence of tan β which in our case is large.The dominant effect, however, is the corrections in the range M ≤ µ ≤ Λ g where the flavordependent terms ( Y l and Y ν ) can be large. The interesting dependence of α ν (MSSM) andtan β on the running contributions of flavor dependent terms is given in [16]. For large tan β the contribution of Y e and Y ν become important. We also show the radiative corrections tothe solar and atmospheric mass squared difference from the Λ g to the Λ ew for degeneratemasses at the GUT scale in the MSSM with tan β =55. To begin with both the mass squareddifferences are zero at the GUT scale. From the mass squared differences shown in Fig 2,we see that the RGE in combination with seesaw threshold corrections can result both masssquared differences in their current 3 σ ranges at the low scale. The hierarchy parameters y ν , r and r though arbitrary but are found to be < O (1). For given set of input parameters16n Table II, the value of effective Majorana mass | M ee | is ≈ − eV and that of J is ≈ × − . Thus, we find that it is possible to simultaneously obtain the neutrino oscillationparameters at the electroweak scale for S mass matrix at the Λ g in the MSSM with largetan β . VI. CONCLUSIONS
We studied the RGE corrections to the S symmetric neutrino mass matrix in the presenceof seesaw threshold corrections both in the SM and MSSM. In the absence of seesaw thresholdeffects there are negligible corrections to the mixing angles in the SM and MSSM withlow tan β . However, large corrections are possible in neutrino parameters once the seesawthreshold effects are taken into consideration both in the SM and MSSM. Above all seesawscales there is the contribution of the Yukawa coupling, Y ν in addition to Y l and thus theRGE depends on more parameters than below the seesaw scales. Thus, we can say thecorrections depend on the form of Y ν which has free parameters. In the SM we found thatthe mixing angles can be obtained in their current 3 σ range at the electroweak scale whenwe begin with S neutrino mass matrix at the GUT scale Λ g . The significant running occursbetween and above the seesaw threshold scales. Below the lowest seesaw scale there are nosignificant corrections as the only contribution comes from Y τ which is small. However, inthis case of exactly equal magnitude of mass eigenvalues, the two mass squared differenceare not simultaneously generated in the current range at the electroweak scale. The solarmass squared difference is found to be large ( O (10 − )) in comparison to its allowed value atthe electroweak scale.There can be large radiative corrections in the MSSM with tan β =55 when thresholdeffects are taken into consideration. The large corrections to the mixing angles occur at thescale above the seesaw threshold where the Yukawa coupling, Y ν , is present and has largefree parameters which can enhance running for large tan β . All the mixing angles and themass squared differences are obtained in their current 3 σ range at the low scale in this case.The free input parameters y ν , r and r are expected to be small < O (1). Thus, for the SMit is not possible to radiatively generate the solar mass squared difference at the electroweakscale in its current range. In the MSSM with large tan β we can generate all the masses andmixing angles in the current allowed range at the electroweak scale, starting from exactly17egenerate mass spectrum at the GUT scale. VII. ACKNOWLEDGEMENTS
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