Predictions from High Scale Mixing Unification Hypothesis
Gauhar Abbas, Saurabh Gupta, G. Rajasekaran, Rahul Srivastava
aa r X i v : . [ h e p - ph ] M a y Predictions from High Scale Mixing Unification Hypothesis
Gauhar Abbas, ∗ Saurabh Gupta, † G. Rajasekaran,
1, 2, ‡ and Rahul Srivastava § The Institute of Mathematical Sciences, Chennai 600 113, India Chennai Mathematical Institute, Siruseri 603 103, India
We investigate the renormalization group evolution of masses and mixing angles ofMajorana neutrinos under the ‘High Scale Mixing Unification’ hypothesis. Assumingthe unification of quark-lepton mixing angles at a high scale, we show that all theexperimentally observed neutrino oscillation parameters can be obtained, within3- σ range, through the running of corresponding renormalization group equationsprovided neutrinos have same CP parity and are quasi-degenerate. One of the novelresults of our analysis is that θ turns out to be non-maximal and lies in the secondoctant. Furthermore, we derive new constraints on the allowed parameter spacefor the unification scale, SUSY breaking scale and tan β , for which the ‘High ScaleMixing Unification’ hypothesis works. PACS numbers: 14.60.Pq, 11.10.Hi, 11.30.Hv, 12.15.Lk
I. INTRODUCTION
The quest for a unified theory of quarks and leptons is one of the main goals of beyondstandard model physics. To this end, the unification of mixing angles of quarks and leptons,at a high scale, seems to be an exciting possibility. In the past, it has been investigatedunder the hypothesis referred to as ‘High Scale Mixing Unification’ (HSMU) for the caseof Majorana neutrinos [1–5] and, recently, for the case of Dirac neutrinos [6]. A similarpossibility has also been investigated in [7]. Within the HSMU hypothesis, the observedvalues of oscillation parameters at low energies are obtained through the renormalization ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] group (RG) evolution of these parameters from the unification scale (high scale) to the lowscale M Z (mass of the Z boson).In addition, the HSMU hypothesis also provides hints about the type and nature of theunderlying unified theory that might exist at the unification scale. One of the key predictionsof the HSMU hypothesis is the small non-zero value of θ [1–5]. At the time of the earlierwork on HSMU, only an upper bound on θ existed and it was not known whether θ waszero or non-zero.The recent results from different experiments have established the value of θ to be non-zero [8–12]. This precise measurement can be used to test predictions of various models andput stringent constraints on them. The current global scenario of the neutrino oscillationparameters (for normal hierarchy pattern) [13, 14] is summarized in the Table I. Since θ isfairly well determined now, it is important to check whether HSMU is consistent with thismeasurement. Quantity Best Fit ± σ σ Range∆ m (10 − eV ) 7 . +0 . − . m (10 − eV ) 2 . +0 . − . θ / ◦ . +0 . − . θ / ◦ . +2 . − . ⊕ . +1 . − . θ / ◦ . +0 . − . Furthermore, with the operation of the Large Hadron Collider (LHC), two importantdevelopments have occurred. What is presumably the long awaited Higgs boson has beendiscovered with a mass around 125 GeV [15, 16] and so far, no signature of supersymmetry(SUSY) has been observed [17–19]. Both of these, especially the second one, can haveimportant repercussions on the implementation of HSMU.In the earlier works on HSMU hypothesis, the issue of variation of SUSY breaking scaleas a function of tan β was explored in the split SUSY scenario [4]. In the present work, wederive new constraints on the allowed ranges of SUSY breaking scale and tan β in the case ofMinimal Supersymmetric Standard Model (MSSM). It was also shown that this hypothesisworks for a wide range of unification scales [1]. We investigate it further and derive newconstraints on the variation of unification scale. In view of the availability of more precisevalues of the neutrino oscillation parameters [13, 14] these investigations are likely to serveas important tests of HSMU hypothesis. A detailed discussion of these constraints is one ofthe main features of this paper.This paper is organized in the following manner. In section II, we provide a generalformalism of the RG running of Majorana neutrino masses and mixing angles. SectionIII, contains our results for the neutrino oscillation parameters at low energy within theframework of HSMU hypothesis. In section IV, we discuss various predictions originatingfrom our analysis. The constraints on the unification scale, SUSY breaking scale and tan β are derived in section V. Finally, in section VI, we summarize our results and give futuredirections. II. RENORMALIZATION GROUP EVOLUTION OF NEUTRINO MASSESAND MIXING ANGLES
We present, in this section, the RG equations used in our analysis. Our basic assumptionis that the neutrinos are Majorana type and mass eigenstates m i , ( i = 1 , ,
3) are ofsame CP parity. We also ignore CP violating phases in the mixing matrix. With theseassumptions, the real PMNS matrix can be parametrized as U = c c s c s − s c − c s s c c − s s s s c s s − c c s − c s − s c s c c , (1)with c ij = cos θ ij and s ij = sin θ ij ( i, j = 1 , , U matrix diagonalizes the neutrinomass matrix M in the flavor basis, i.e. U T M U = diag( m , m , m ).Here, we take a model independent approach and assume that the new physics operatingat the unification scale results in the unification between CKM and PMNS mixing angles.In order to get the low scale values, we work in type-I seesaw scenario. For the RG runningfrom unification scale to seesaw scale, we use the standard MSSM RG equations withinthe framework of type-I seesaw mechanism [20]. Below the seesaw scale all right handedneutrinos are integrated out and the masses of left handed neutrinos are generated by adimension 5 operator added to the standard SM/MSSM Lagrangian. We have numericallychecked our results by varying seesaw scale from 10 − GeV and we find that our analysisdepends weakly on the chosen value of seesaw scale. Thus, for the sake of illustration anddefiniteness, we have taken seesaw scale of O (10 ) GeV throughout this work.At this juncture, we would like to point out that, for our analysis, we do not need anydetails of the theory operating at the unification scale. Although one such high energy theoryhas already been discussed in literature (see, e.g. [1] for details). Moreover, RG equationspresented here are at one loop level and only dominant terms are shown (cf. (2) and (5)below). However, in numerical computations, we have used full two-loop RG equations [21].The RG evolution of neutrino masses m i , below seesaw scale, is determined by the fol-lowing equations [20–23] d m i d t = m i π (cid:2) α + Cf τ F i (cid:3) , (2)where t = ln( µ/µ ), µ is the renormalization scale and F i (with i = 1 , ,
3) are defined as F = 2 s s − s sin 2 θ sin 2 θ + 2 s c c ,F = 2 c s + s sin 2 θ sin 2 θ + 2 s s c ,F = 2 c c . (3)In SM and MSSM, α , f τ and C are α MSSM = − g − g + 6 y t sin β ,α SM = − g + 2 y τ + 6 (cid:0) y t + y b (cid:1) + λ ,f τ, MSSM = y τ cos β , f τ, SM = y τ ,C = 1 in MSSM , C = −
32 in SM . (4)Here y f , ( f = τ, t, b ) represents the Yukawa coupling for τ -lepton, top and bottom quarks,respectively. The gauge couplings are denoted by g i and λ stands for the Higgs self-couplingin SM.The RG equations which govern evolution of mixing angles are given as [20–23]d θ d t = − Cf τ π sin 2 θ s ( m + m ) ∆ m + O ( θ ) , d θ d t = − Cf τ π sin 2 θ sin 2 θ m ∆ m (1 + ξ ) [( m − m ) + ξ ( m + m )] + O ( θ ) , d θ d t = − Cf τ π sin 2 θ m (cid:20) c ( m + m ) + s ( m + m ) ξ (cid:21) + O ( θ ) , (5)with ξ = ∆ m ∆ m , ∆ m = m − m , ∆ m = m − m . (6)In this work, Dirac as well as Majorana phases of the PMNS mixing matrix are taken tobe zero. The results with non-zero phases will be presented in a future publication [24]. In(2) and (5), for sake of brevity, we have given only the dominant terms of the RG equationsat one loop level. The full two loop RG equations, used in this work, can be found in [21].The numerical computations, at two loop, are done using a MATHEMATICA based packageREAP [20]. III. MAGNIFICATION OF MIXING ANGLES VIA RG EVOLUTION
The HSMU hypothesis is implemented in two steps. We first follow a bottom-up approachand take the known values of gauge couplings, Yukawa couplings and CKM matrix elementsat the low scale ( M Z ) [25] and evolve them up to the SUSY breaking scale ( M SUSY ) usingthe standard SM RG equations [23]. From the SUSY breaking scale to the unification scale,the evolution of these parameters is governed by MSSM RG equations [21, 23].At the unification scale, following the HSMU hypothesis, we assume that the PMNS mix-ing angles ( θ , θ , θ ) are identical to the CKM mixing angles ( θ ,q , θ ,q , θ ,q ). In addition tothis, we choose initial neutrino masses to be quasi-degenerate with normal hierarchy patternand PMNS phases to be zero. The requirements of normal hierarchy and quasi-degeneracyof neutrinos are essential ingredients to achieve large mixing angle magnification (within the3- σ range at the low scale) [1].We then follow a top-down approach and run down the neutrino masses and mixing anglesfrom unification scale to the seesaw scale using MSSM RG equations within the frameworkof type-I seesaw mechanism [20]. From seesaw scale to SUSY breaking scale, the runningis done using MSSM RG equations with dimension-5 operator [21, 23]. Below the SUSYbreaking scale to the low scale, RG running is governed by the SM RG equations.In the earlier works on HSMU hypothesis [1–5], the SUSY breaking scale was taken as1 TeV. At present, this is not favored by direct SUSY searches at the LHC [17, 18]. Inview of this, we have taken the SUSY breaking scale as 2 TeV. The working of HSMUhypothesis requires large values of tan β which is also consistent with constraints imposedby SUSY searches [18, 19, 26–28]. Therefore, in this section, we have taken tan β to be 55.Moreover, we have taken unification scale to be 2 × GeV which is a generic scale forGrand Unified Theories (GUTs). The dependence of our analysis on these parameters isdiscussed in section V.
TABLE II: Radiative magnification to bilarge mixings at low energies for input values of θ = θ ,q = 13 . , θ = θ ,q = 2 . and θ = θ ,q = 0 . . We have taken the unification scale= 2 × GeV, M SUSY = 2 TeV and tan β = 55. The various entries in the table also highlightthe correlations between low scale neutrino oscillation parameters.I II III IV V m (eV) 0.4152 0.3972 0.4344 0.4102 0.4240 m (eV) 0.4186 0.4005 0.4380 0.4137 0.4275 m (eV) 0.4825 0.4617 0.5049 0.4769 0.4928 m (eV) 0.3577 0.3422 0.3742 0.3534 0.3653 m (eV) 0.3583 0.3428 0.3749 0.3541 0.3659 m (eV) 0.3620 0.3463 0.3788 0.3578 0.3697∆ m (eV ) RG . × − . × − . × − . × − . × − ∆ m (eV ) RG . × − . × − . × − . × − . × − M ˜ e /M ˜ µ, ˜ τ m (eV ) th − . × − − . × − − . × − − . × − − . × − ∆ m (eV ) th − . × − − . × − − . × − − . × − − . × − ∆ m (eV ) 7 . × − . × − . × − . × − . × − ∆ m (eV ) 2 . × − . × − . × − . × − . × − θ / ◦ .
00 54 .
00 54 .
00 53 .
84 54 . θ / ◦ .
67 8 .
67 8 .
67 8 .
66 8 . θ / ◦ .
38 33 .
38 33 .
38 31 .
14 35 . In Table II, we present five sets of neutrino oscillation parameters at low and high energyscales obtained within HSMU hypothesis. Each column in the table depicts some specific setof values for neutrino oscillation parameters chosen in a way to show correlations betweenthem. In order to highlight the correlation between any two low scale parameters we choosethe unification scale neutrino masses such that all other parameters, at the low scale, remainclose to their best fit values . In column I, all the low scale parameters are obtained closeto their best fit values, except θ which is 54 ◦ . In column II, keeping θ and θ close totheir best fit values at the low scale, the values of ∆ m and ∆ m are obtained at their 3- σ upper and lower edge respectively. For this pattern, θ again turns out to be 54 ◦ , i.e. non-maximal. Whereas, in column III, ∆ m and ∆ m are respectively kept at their 3- σ lowerand upper edge. The rest of the results are similar to the previous ones. In columns IV andV, θ is taken to its lower and upper 3- σ limit, respectively, keeping all other parameters(except θ ) close to their best fit values at the low scale. We see that θ always remainsabove 45 ◦ and lies in the second octant. Moreover, as is clear from Table II, for a fixed valueof θ , the correlation between θ and θ is weak.The RG evolution of the three PMNS and CKM mixing angles from the unification scale(2 × GeV) to the low scale ( M Z ) is shown in Figure 1. As clear from the figure, owingto the quasi-degeneracy of neutrino masses, large angle magnification occurs in the PMNSsector. The magnification of CKM mixing angles ( θ qij , i, j = 1 , ,
3) is almost negligiblebecause of the hierarchical nature of quark masses. We also observe that the major part ofmagnification occurs near SUSY breaking scale which, in this case, is chosen to be M SUSY =2 × GeV. The SM RG equations lead to negligible angle magnification as clear from theflatness of curves below M SUSY .The RG evolution of neutrino masses from unification scale to M Z is shown in Figure 2.It is clear that all the masses decrease as we move from unification scale to low scale (cf.Figure 2). Initially, at unification scale, the splitting among the masses is relatively largebut after RG evolution the splitting gets narrowed down and they acquire nearly degeneratemass at M Z . Low energy threshold corrections to neutrino masses
It is evident from table II that only one (i.e. ∆ m ) out of two mass squared differences, atthe low scale, lies within experimental 3- σ range. This discrepancy can easily be accountedfor by threshold corrections [3, 5]. In the case of quasi-degenerate neutrinos, the low energy The RG evolution of θ and θ is correlated in the HSMU hypothesis. Therefore, at the low scale, bothcannot be obtained near their best fit values simultaneously. µ (GeV) ° ° ° ° ° ° ° θ ij θ θ θ θ q12 θ q13 θ q23 FIG. 1: The RG evolution of CKM and PMNS mixing angles with respect to RG scale ( µ ). Thisfigure corresponds to the neutrino oscillation parameters quoted in the first column of Table II. MSSM threshold corrections can result in a significant contribution, as shown in [29–32].These threshold corrections are given by following equations [3, 5]:(∆ m ) th = 2m cos 2 θ [ − e + T µ + T τ ] , (∆ m ) th = 2m sin θ [ − e + T µ + T τ ] , (∆ m ) th = 2m cos θ [ − e + T µ + T τ ] . (7)Here, m = ( m + m + m ) is the mean mass of the quasi-degenerate neutrinos and T α ( α = e, µ, τ ) is the one-loop factor. Its form has been previously calculated in [29, 32]and given by T α = g π (cid:20) x µ − x α y µ y α + ( y α − y α ln ( x α ) − ( y µ − y µ ln ( x µ ) (cid:21) , (8)where g is the SU (2) coupling constant and y α = 1 − x α with x α = M α /M ˜ w ; M ˜ w standsfor wino mass, M α represents the mass of charged sleptons. Moreover, without any loss ofgenerality, the loop-factor has been defined to give T µ = 0 (cf. [3, 5] for details). µ (GeV)0.360.380.40.420.440.460.48 m i ( e V ) m m m FIG. 2: The RG evolution of neutrino masses ( m i ) with respect to RG scale ( µ ). This figurecorresponds to the values in the first column of Table II. At the LHC, for simplified scenarios, chargino masses are excluded up to 750 GeV in thepresence of light sleptons and up to 300 GeV in the case of heavy sleptons [17, 18]. In theview of above constraints, here we have taken the wino mass to be 800 GeV.After the inclusion of threshold corrections, along with the RG-evolution effects, the finalexpression for mass squared differences is given as∆ m ij = (∆ m ij ) RG + (∆ m ij ) th . (9)It is clear from table II, that the RG effects, along with threshold corrections, result ingood agreement between the predictions of HSMU hypothesis and the present experimentallyallowed range of neutrino oscillation parameters (cf. Table I). At this juncture, we wouldlike to point out that, although the threshold corrections for mass square differences aresignificant yet they are negligibly small compare to the mean mass of neutrinos. The sameis also true for the threshold corrections to mixing angles [3, 5].0 IV. PREDICTIONS FROM HSMU HYPOTHESIS
Within the framework of the HSMU hypothesis, the low energy oscillation data can beused to put stringent constraints on the allowed parameter range for the neutrino massesand mixing angle. The aim of this section is to discuss the predictions from our analysiswhich are obtained after imposing these constraints. These predictions can be tested inpresent and future experiments as discussed in this section.
A. Predictions for Masses, h M β i and h M ββ i at M Z As clear from Table II, the neutrino masses at M Z lie in the range of 0 . .
38 eV.This range can be probed by various presently running as well as near future experimentsand hence it provides an important test for HSMU hypothesis. For example, the recentresult from GERDA gives an upper bound of 0 . . h M ββ i component of massmatrix [33]. Similarly, EXO-200 provides an upper bound of 0 . .
38 eV on the same [34].Although the present bounds on h M β i from tritium beta decay are comparatively weak ( < . .
08 eV depending on the choice of the priors [39]. The lower limit of Planckis in tension with our hypothesis but it should be noted that the cosmological constraints arehighly model dependent and should be taken in conjunction with other experiments. In viewof the above considerations, the absolute value of neutrino masses provides an importanttest of our hypothesis. We would like to point out that the above mentioned mass range(0 . .
38 eV) is obtained for a specific choice of unification scale, SUSY breaking scale andtan β (cf. Table II for details). The dependence of neutrino masses (at M Z ) with respect tothese parameters is discussed, in detail, in section V. B. Predictions for mixing angles at M Z It is clear from the RG equations (5) that, within HSMU hypothesis, the mixing angles θ and θ are correlated. In Figure 3, we show the explicit dependence of θ on θ keepingother low scale neutrino oscillation parameters fixed near to their best fit values. We observe1that θ turns out to be above 45 ◦ (i.e. lies in the second octant), for the whole 3- σ rangeof θ . This prediction is easily testable in the current and in future experimets, like INO,T2K, NO ν A, LBNE, Hyper-K and PINGU [40–45].Even for the lower edge value of the present 3- σ range of θ , the value of θ is non-maximal and is around 47 ◦ , as evident from Figure 3. The values of θ increase with θ .When θ is around 9 ◦ , θ reaches its upper edge of 3- σ limit and it goes into the disfavoredregion for higher values of θ (which is still within its 3- σ range). This, in turn, putsconstraints on the values of θ , which should lie in the range 7 . ◦ -8 . ◦ . ° ° ° ° θ ° ° ° ° θ θ = 33.36 ° FIG. 3: The variation of θ with respect to θ . For plotting this figure we have kept all otheroscillation parameters to be at their best-fit values. The vertically shaded regions lie outside the3- σ range of θ whereas the horizontally shaded one lies outside 3- σ range of θ [14]. At this point we would like to mention that, the RG evolution of θ also depends on ∆ m .Therefore, it can be varied independently of the other two angles by making an appropriate As shown in table II, θ also depends very weakly on θ . The above quoted range is for θ at its bestfit value. m at unification scale. Hence, within HSMU hypothesis, no effective constraintson its range can be obtained. V. ALLOWED PARAMETER RANGE FOR UNIFICATION SCALE, SUSYBREAKING SCALE AND tan β In this section, we study the variation of unification scale, SUSY breaking scale and tan β and its impact on HSMU hypothesis. We derive constraints on the range of these parametersfor which HSMU hypothesis works. For this purpose, in this section, we have fixed the valuesof experimentally measured quantities θ and θ to their best fit values (i.e. 33 . ◦ and8 . ◦ respectively) at M Z . We also fix ∆ m = 2 . × − eV , which is slightly higher thanits best fit value, so that after adding appropriate threshold corrections, it remains within3- σ range.Since in our hypothesis the quantities θ and ∆ m are fixed in terms of other quantities,we have not put any restrictions on them, apart from the fact that, after adding appropriatethreshold corrections they should remain within 3- σ limit. A. Variation of Unification Scale
In the previous sections, we have chosen our unification scale as 2 × GeV which isthe typical scale for GUTs. Since our hypothesis does not depend on the details of the highscale theory, it is not necessary to take the unification scale to be same as that of GUT.Thus, in this subsection, we analyze the effect of variation of unification scale.It is clear from Figure 1, that a major part of angle magnification happens only closeto M SUSY . Therefore, it is expected that the desired angle magnification can be achievedeven when the unification scale is not same as the GUT scale. In Figure 4 and 5, we have,respectively, shown the variation of unification scale with respect to high and low scaleneutrino masses. The magnitude of low scale masses (and derived quantities such as h M β i and h M ββ i ) put constraints on the unification scale as evident from the Figure 5 and further In our case, since the neutrinos are quasi-degenerate and phases are absent, the mean mass ( m ) and h M ββ i are almost the same. Hence, in drawing the constraints in Figures 5, 7 and 9 we have neglected the smalldifference in the exact values of m and h M ββ i . Unification Scale (GeV)0.350.40.450.50.550.60.65 m i ( e V ) m m m FIG. 4: Unification scale vs neutrino masses ( m i ) at unification scale. In plotting this figure wehave taken M SUSY = 2 × GeV and tan β = 55. elaborated in Section V D.Our analysis works for a wide range of unification scale from the Planck scale to muchlower scales (cf. Figures 4, 5). The reason for this is that the major part of magnification ofangles happens in a relatively small range near M SUSY . Hence, one can take the unificationscale to be several orders of magnitude lower than the GUT scale and still achieve desiredmagnification at M Z . The noteworthy point is that as we lower the unification scale theinput neutrino masses have to be taken more degenerate because the range of MSSM RGrunning becomes shorter (cf. Figure 4). Thus, to achieve desired magnifications at M Z , onehas to make the input neutrino masses more degenerate to account for the lesser range ofMSSM RG running.This increasing degeneracy of masses, in turn, results in ∆ m approaching its 3- σ rangemuch before M Z . Therefore, to counter this and to keep ∆ m within its 3- σ range at M Z ,one is also forced to increase the mean input mass at unification scale. Furthermore, oncethe input mean mass is increased, it results in a relative increase in the mean mass at M Z ,4 Unification Scale (GeV)0.30.350.40.450.5 m i ( e V ) m m m GERDAEXO-200
FIG. 5: Unification scale vs neutrino masses ( m i ) at M Z . Here we have taken M SUSY = 2 × GeV and tan β = 55. The shaded regions are excluded by 0 νββ decay experiments [33, 34]. partly because now it is higher to begin with and partly because of the small range of MSSMRG running.Thus, the mean mass of neutrinos at unification scale as well as at M Z increases aswe decrease the unification scale. Hence, one can constrain the lowest possible unificationscale using data from various experiments. We will further elaborate on such experimentalconstraints in Section V D. B. Variation of SUSY Breaking Scale
We have, so far, fixed the SUSY breaking scale at 2 × GeV. In this section, we analyzethe effects of variation of SUSY breaking scale. It is clear from Figure 1 that the majorpart of magnification occurs only in and around the SUSY breaking scale. So one shouldexpect to shift the scale of SUSY breaking from the so far chosen value and still be able toachieve the desired magnification. Our analysis works for a wide range of SUSY breaking5 M SUSY (GeV)0.40.450.50.550.60.650.7 m i ( e V ) m m m FIG. 6: M SUSY vs neutrino masses ( m i ) at unification scale. In plotting this figure, we have takenunification scale = 2 × GeV and tan β = 55. scale starting from the TeV scale to much higher scales (as is clear from Figures 6 and 7).While plotting these figures, we have taken the unification scale = 2 × GeV, tan β = 55and the value of observables at M Z to be same as before.As we increase the SUSY breaking scale, the input neutrino masses have to be taken tobe more degenerate. The reason for this is that by increasing the SUSY breaking scale therange of MSSM RG running becomes shorter. Thus, to achieve desired magnifications at M Z one has to make the input neutrino masses more degenerate in order to counter thelesser range of MSSM RG running. At the same time, we have to increase the mean massof neutrinos at unification scale in order to keep the ∆ m within its 3- σ range at M Z .Since the mean mass of the neutrinos increases with increasing SUSY breaking scale, onecan constrain the highest possible SUSY breaking scale using data from various experiments.Moreover, the lower ranges of SUSY breaking scale are constrained from SUSY searches atthe LHC [17, 18]. We further discuss these constraints in the Section V D.6 M SUSY (GeV)0.340.360.380.40.420.440.460.48 m i ( e V ) m m m GERDAEXO-200
FIG. 7: M SUSY vs neutrino masses ( m i ) at M Z . In plotting this figure, we have taken unificationscale = 2 × GeV and tan β = 55. The vertically shaded region is disfavored by the LHC SUSYsearches [18] whereas the horizontal ones are excluded by 0 νββ decay experiments [33, 34]. C. Variation of tan β In MSSM, the RG running of angles gets enhanced by a factor of (1 + tan β ) [cf. (5) fordetails]. Therefore, the larger values of tan β enhance the magnification at M Z . This is thereason for choosing tan β = 55 in the previous sections of this work. We have, so far, fixedtan β = 55 but in this section we will vary tan β to obtain the lower limits on it for desiredmagnification.It is clear from Figures 8 and 9 that the mixing angle magnification happens for a widerange of tan β . Although we have not shown this in the figure, the desired angle magnifica-tions can be obtained for values of tan β as low as 4 or 5. But, for low tan β the masses ofneutrinos become very high at low scale. Furthermore, if we take low values of tan β , theinput neutrino masses, at unification scale, have to be taken more degenerate and the mean7
25 30 35 40 45 50 55 60tan β m i ( e V ) m m m FIG. 8: Variation of tan β vs neutrino masses ( m i ) at unification scale. In plotting this figure, wehave taken unification scale = 2 × GeV and M SUSY = 2 × GeV. mass should also be higher. The reason is that with decreasing tan β the factor (1 + tan β )becomes small. Thus, to achieve desired magnifications at M Z one has to make the in-put neutrino masses more degenerate to account for the smaller contribution coming from(1 + tan β ) term. At the same time to keep ∆ m within its 3- σ range at M Z , one is forcedto increase the mean input mass at unification scale. Since the mean mass of the neutrinosincreases with decreasing tan β , one can constrain the range of allowed tan β from variousexperiments, as discussed in Section V D. D. Experimental Constraints
As is clear from previous discussion, the mean mass of neutrinos varies with the variationof the unification scale, SUSY breaking scale and tan β . Therefore, one can constrain therange of these parameters by using data from various experiments, as discussed below.(i) Constraints from Tritium Beta Decay:
The present constraints on h m β i coming fromtritium beta decay are h m β i <
25 30 35 40 45 50 55 60tan β m i ( e V ) m m m GERDAEXO-200
FIG. 9: Variation of tan β vs neutrino masses ( m i ) at M Z . In plotting this figure, we have takenunification scale = 2 × GeV and M SUSY = 2 × GeV. The shaded regions are excluded by0 νββ decay experiments [33, 34]. of the neutrinos and thus the whole mass range of the Figures 5, 7 and 9 easily comes underthis limit. Hence, the tritium beta decay constraints are relatively weak. They allow muchlower values of the unification scale, tan β and much higher values of SUSY breaking scalethan those plotted in the above figures. However, in future, the KATRIN experiment isexpected to probe h m β i as low as 0.2 eV [38] and hence will be able to put much tighterconstraints on the allowed range of these parameters.(ii) Constraints from Neutrinoless Double Beta Decay : At present, the EXO-200 andGERDA experiments provide the most stringent constraints on h m ββ i . The latest resultsfrom phase I of the GERDA experiment have given the upper limit on h m ββ i to be 0.20-0.40eV [33], whereas EXO-200 has given an upper limit of 0.14-0.38 eV [34]. This, in turn, putsstringent constraints on the allowed range of various parameters, as given below.(a) The lower limit of unification scale is constrained to be around 10 GeV by GERDAand around 10 GeV by EX0-200 (cf. Figure 5).(b) The results from GERDA constrains the highest possible SUSY breaking scale to be9around 10 GeV, whereas EXO-200 puts a limit of around 10 GeV (cf. Figure 7).(c) The lowest possible value of tan β is constrained to be around 50 (cf. Figure 9).In future, these limits are expected to improve, thus resulting in more tighter constraintse.g. the GERDA phase II is aiming for an increased sensitivity by a factor of about 10 (cf.[33] for details). It should be noted that the above constraints are for the case when all thePMNS phases are taken to be zero. These constraints are likely to change in the presenceof phases. We will analyze them, in detail, in our next work [24].(iii) Cosmological Constraints:
The recent result of the Planck collaboration hasgiven constraints on the sum of neutrino masses to be in the range of 0.23 eV [95%;Planck+WP+highL+BAO] to 1.08 eV [95%; Planck+WP+highL ( A L )] depending on valueschosen for the priors [39]. The limit of 1.08 eV implies that the mean neutrino mass hasto be around 0.36 eV, thus putting similar constraints to those obtained from h m ββ i . Thelowest value (i.e. 0.23 eV [95%; Planck+WP+highL+BAO]) is in tension with our hypoth-esis. However, as noted by the Planck collaboration itself, the cosmological limits are highlydependent on chosen values of priors, so these limits should be taken as indicative and notconclusive.To conclude, in view of the above experimental constraints, for fixed values of otherparameters, (1) The unification scale should be taken around 10 GeV or above, (2) TheSUSY breaking scale should be taken below 10 GeV and (3) The tan β should be takenabove 50. VI. CONCLUSIONS
We have investigated the implications of High Scale Mixing Unification hypothesis in thewake of new data and experimental constraints. This hypothesis leads to the experimentallyobserved mixing angles and mass square differences at low energy scales ( M Z ). The smallbut non-zero value of θ is a natural outcome of this hypothesis which has been recentlyconfirmed by various experiments [8–12]. We found that, in absence of phases, for the present3- σ range of θ HSMU hypothesis uniquely predicts the value of θ to be non-maximal andabove 45 ◦ . The normal hierarchy and quasi-degeneracy of neutrino masses are essentialassumptions to realize HSMU hypothesis. We have also analyzed the allowed parameterrange for other parameters of our hypothesis vis-a-vis various experimental constraints. We0found that (i) the unification scale should be above 10 GeV, (ii) the SUSY breaking scaleshould lie below 10 GeV, and (iii) the value of tan β should be taken above 50.However, it should be noted that all the above conclusions have been drawn by takingDirac as well as Majorana phases of PMNS matrix to be zero. These conclusions may changein the presence of phases. The detailed implications of these phases are under investigationand will be reported in our future publications [24].Moreover, in a recent analysis of the HSMU hypothesis with Dirac type neutrinos, wehave found similar predictions for mixing angles [6]. At the end, we would like to point outthat the above mentioned two scenarios can be distinguished from each other by the scaleof their mean mass (or h m β i ) and h m ββ i measurements. Acknowledgments
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