Table Of Content

UH-511-1063-2004 LBNL-56686 QLC relation and neutrino mass hierarchy Javier Ferrandis ∗ MEC postdoctoral fellow at the Theoretical Physics Group Lawrence Berkeley National Laboratory One Cyclotron Road, Berkeley CA 94720 Sandip Pakvasa † Department of Physics & Astronomy University of Hawaii at Manoa 2505 Correa Road Honolulu, HI, 96822 5 0 0 Latest measurements have revealed that the deviation from a maximal solar mixing angle is 2 approximately the Cabibbo angle (i.e. QLC relation). We argue that it is not plausible that this deviation from maximality, be it a coincidence or not, comes from the charged lepton mixing. n a Consequently we have calculated the required corrections to the exactly bimaximal neutrino mass J matrix ansatz necessary to account for the solar mass difference and the solar mixing angle. We pointoutthattherelativesizeofthesetwocorrectionsdependsstronglyonthehierarchycaseunder 7 consideration. We find that the inverted hierarchy case with opposite CP parities, which is known 2 to guarantee the RGE stability of the solar mixing angle, offers the most plausible scenario for a 2 highenergyorigin ofaQLC-correctedbimaximalneutrinomassmatrix. Thispossibility mayallow v ustoexplain theQLCrelation in connection with theorigin ofthecharged fermion mass matrices. 8 3 0 I. INTRODUCTION sometimes called the quark-lepton complementarity re- 2 lation, hereafter referred to as QLC relation. There is 1 Duringthelastyearourknowledgeoftheleptonicmix- a similar relation satisfied by the leptonic angle θ23 and 4 ing matrix has reached the precision level. The most thecorrespondingangleinthequarksector,althoughthe 0 / recent 90% C.L. experimental results [1, 2, 3] and sev- errors are somewhat larger. Based on the experimental h eral global fits [4, 5, 6, 7] have improved our knowledge datait is convenientto define the followingparametriza- p of the neutrino mass differences and indicate that the tion [12] of the mixing angles, - p atmospheric mixing is almost maximal while the solar 1 s = +ǫ λ2, (6) e mixing deviates from maximality in a particular way. In 23 √2 A ν h the standard notation, Xiv: ssiinnθθ12 == 00..5730±00..0141,, ((12)) ss12 == ǫ√12λ(cid:0)12,−λν +ǫSλ2ν(cid:1), ((87)) r 23 ± 13 CP ν a sinθ13 < 0.15, (3) where sij = sinθij and the coefficients ǫA, ǫS and ǫCP ∆m2 = ∆m2 =(8.2 0.6) 10 5eV2, (4) are at most of order < 4, as indicated by the experi- ∆m2sun = ∆m221 =(2.4±5 0.×55) −10 3eV2, (5) mental uncertainities. ∼We note that we have defined the atm 32 ± × − deviation from a maximal solar mixing angle as λ and (cid:12) (cid:12) (cid:12) (cid:12) ν We (cid:12)note tha(cid:12)t the(cid:12)mixin(cid:12)g angle θ13 is constrained to be not λ = θC to emphasize that λν may not be exactly θ13 <0.15bythenon-observationofneutrinooscillations the Cabibbo angle. Therefore the MNSP matrix can be attheCHOOZexperiment[3]andafittotheglobaldata written to leading order in powers of λ as, ν [7]. Thissubstantialimprovementhasconfirmedthatthe 1 (1+λ ) 1 (1 λ ) 0 leptonic mixing matrix, heretoafter called MNSP matrix √2 ν −√2 − ν [8], is nearly bimaximal [9, 10] and the deviation from VMNSP = 12(1−λν) 12(1+λν) −√12 +O(λ2ν) bimaximality observedhas revealeda surprising relation  12(1−λν) 12(1+λν) √12  between the Cabibbo angle, θC and the solar mixing an- (9) gle [11], ThemainimplicationoftheQLCrelationisfairlysimple: the MNSP matrix is to first order bimaximal [9] and the θ +θ =45.1 2.4 (1σ), C 12 ◦± ◦ deviation from the exact bimaximality is a correction of the order of the Cabibbo angle, i.e. around 20%. This resemblesincertainwaythesituationinthequarksector, where it is known that to first order the CKM matrix is Electronic address: [email protected]; ∗ URL:http://homepage.mac.com/ferrandis the unitymatrixwhile the maincorrectionis exactlythe Electronicaddress: [email protected] Cabibbo angle. † 2 Explaining the QLC relation is a real challenge that This relation has been recently analyzed with precision anyfuture theoryofflavormustaddress. Alongwiththe byoneofthe authorswhonotedthatindeedthe relation extremesmallnessoftheneutrinomasses,this isanother surprisingly works at the level of 16%, as the following ± featurewhichqualitativelydistinguishestheneutrinosec- ratio shows (see Ref. [15] for details), torfromthechargedfermionsector. Thechargedfermion spectra is very hierarchical, i.e. the third generation m 1/2 m 1/2 d e : = 3.06 0.48. (12) masses are much heavier than the first and second gen- (cid:20)m (cid:21) (cid:20)m (cid:21) ± s µ eration fermion masses. Therefore we expect that there is a basis, probably the flavor basis (also known as la- The relation between the Cabibbo angle and the down- grangianorsymmetry basis),where the chargedfermion strange quark mass ratio can be simply explained, as diagonalizationmatricesareapproximatelydiagonal. On knownfrom the ’70’s[16], if the down quark mass is gen- theotherhand,ithasbeenknownforsometimethatthe eratedfromthemixingbetweenthefirstandsecondfam- leptonic mixing matrix is nearly bimaximal. It was ex- ilies. Analogously, the relation between the Cabibbo an- pected that this distinctive feature could be explained if gle and the electron-muon mass ratio can also be simply the mechanism of neutrino mass generation is somehow explainedifthe electronmassisgeneratedfromthemix- disconnected from the mechanism generating the flavor ing between the first and second lepton families. This structure in the charged fermion sector. This may ex- implies that there is a leptonic basis where the charged plain why many people, surprised by the appearance of lepton mass matrix is given to leading order by, the Cabibbo angle in the leptonic mixing matrix, have 1 proposed to explain the QLC relation as a contamina- 0 mµme 2 (λ3) ttiiooInnnsctoohfmitsihnepgaQfprLeorCmwrteehleawtciihollnarafgnoeradlmylzeoepdtesolonsmomefinxgeienungterrmiincaotimrmixap.slisceas-. Mcl = (cid:16)mmµ(m2τλe3)(cid:17)12 (cid:16)(cid:16)mmm(2τλµτ2(cid:17))(cid:17) OO(1λ2) . (13) In Sec. II we argue that it is not plausible that the QLC  O O  relation is explained by effects arising from the charged Here λ = θ . The order of magnitude in the coeffi- C leptonmixingsector. InSec.IIIweanalyzetheformand cients ( ) and ( ) can be obtained by requiring l 13 l 23 M M relative size of the corrections to the bimaximal three these entriesnotto affectthe leadingorderterms forthe neutrino mass matrix necessary to account for the QLC chargedclepton masscratios. From the matrix in Eq. 13 relation. InSec.IVweanalyzetheeffectsoftheneutrino and the empirical relation in Eq. 11 it follows that the mass hierarchy on the stability of the QLC relation and chargedleptonmixingmatrixtoleadingorderisgivenin the implications for the scale of neutrino mass genera- this leptonic basis by, tion. In Sec. V we analyze the possiblity that the solar mass difference being zero at a high energy scale is RGE 1 λ/3 (λ3) generated, triggered by a high energy origin of the QLC l  λ/3 1 O(λ2). (14) relation. In Sec. VI we summarize the main results of VL ≈ (λ3) (λ2) O1 O O  this paper. To sum up, Eq. 11 necessarily implies that there is a leptonic basis and a quark basis where the charged lepton mass matrix adopts the form given by Eq. 13 while II. THE QLC RELATION CANNOT ARISE the down-type quark mass matrix adopts a similar form FROM CHARGED LEPTON MIXING with m /m =3m /m . It is very plausible that this is µ τ s b the flavor basis in some underlying theory of flavor. For The MNSP mixing matrix is given by instance, this could be the basis where quarks and lep- tonsunifyincommonrepresentationsofaGrandUnified VMNSP =(VLl)†Vν (10) group. It is known that some GUT models can explain the relation in Eq. 11 [14]. This could be achieved if the where is the neutrino diagonalization matrix and l Vν VL Higgsfieldgivingmasstothechargedleptonsanddown- isthe lefthandedchargedleptondiagonalizationmatrix, type quarks transforms under particular representations diag = ( l) l . When trying to explain the QLC Ml VL †MlVR oftheGUTgroup: 45intheSU(5)caseor126inSO(10) relationthefirstideathatcomestoourmindisthepossi- models. bility that the QLC relation may arise from the charged It has been recently proposed[17, 18] that, to explain lepton mixing matrix. We will argue that this is not the deviation from a maximal solar mixing angle, one plausible if one wants to understand the well known em- couldassumethat the neutrino mixing matrix inthe fla- pirical relations which connect the electron/muon mass vor basis is exactly or approximately bimaximal, i.e, ratio with the quark sector. There is an empirical rela- tionwhichhasbeenknownforquite alongtime [13,14], 1 1 0 √2 −√2 1 1 = 1 1 1 . (15) md 2 me 2 Vν 2 2 −√2 Vus 3 , (11)  1 1 1  | |≈(cid:20)ms(cid:21) ≈ (cid:20)mµ(cid:21)  2 2 √2  3 Normalized mass matrix zero term solar mass correction QLC correction Eigenvalues M Matm Msol MQLC ν ν ν ν normalchierarchy  00 c021 −021  γ2 −1√12 c−12√12 −12√12  γ2 −40cλν λ0ν λ0ν  (γ0,≈γ,λ1)  0 −21 12   −√12 12 12   0 λν λν  wiitnhvesartmede ChiPerapracrhityies 10 012 021  γ2 −1√12 −12√12 −12√12  γ2 −20λν λ0ν λ0ν  (1,γ(1≈+λ2γ/)2,0) 0 12 21   −√12 12 12   0 λν λν  withinovpeprtoesditehiCerParpchayrities  √012 √012 √012  √γ2−12√12 −212√12 −212√12   2λ0ν −0λν −0λν  (1,γ−(≈1λ+2γ/2),0)  √12 0 0   21 −2√12 −2√12   0 −λν −λν  TABLE I: Bimaximal zero order normalized neutrino mass matrices for the normal and inverted hierarchy cases and their minimal first and second ordercorrections, which are necessary toaccount for thesolar mass differenceand theQLC relation. and that the QLC relation is generated from charged denote the neutrino mass eigenstates by, lepton mixing. We have pointed out above that most diag =(m ,m ,m ) (16) probably the flavor basis of the underlying theory of fla- Mν 1 2 3 vor is the basis where quarks and leptons unify in com- Neglecting the charged lepton mixing, which can only monrepresentations. Inthisbasisweexpectthecharged give a second order contribution to the QLC relation as lepton diagonalization matrix to be given by Eq. 14. we saw in the previous section, the reconstructed neu- Nevertheless, if this was the case we would obtain that trino mass matrix is, θth12is=isπ4qu+iteθ6Cinicnosntesaisdteonft.the observed QLC relation, and Mν =VMNSPMdνiagVM†NSP. (17) If one insists to fully generate the observed deviation This can be written as, frombimaximalityintheMNSPmatrixfromthecharged = BiMax+ QLC, (18) lepton mixing, assuming that the neutrino mixing ma- Mν Mν Mν trix is approximately bimaximal in the flavor basis, the where BiMax isthewellknownbimaximalmassmatrix Mν required mixing in the charged lepton sector would be whose general expression is given by [9], verylargeandasa consequencethe chargedleptonmass 1m 1 ∆ 1 ∆ matrix would adopt a very unnatural form in the flavor 2 12 √2 12 √2 12 basisinordertoreproducethecorrectelectronmass[19]. BiMax = 1 ∆ 1(m +m ) 1(m m ). Mν √2 12 2 12 3 2 12− 3 Thiskindofscenariosdonotprovideaconvincingexpla-  1 ∆ 1(m m ) 1(m +m ) nation of the precise relation that connects the charged  √2 12 2 12− 3 2 12 3  (19) lepton spectra and the quark spectra, see Eq. 11. Here we have defined, Therefore,mostprobablythebulkofthedifferencebe- tween θ12 and π4 is already present in the neutrino mass m12 = 1(m1+m2), ∆12 = 1(m1 m2). (20) matrix in the flavorbasis,or in other wordsthe QLC re- 2 2 − lationmustarisefromthe mechanismthatgeneratesthe The QLC correction, λ = π/4 θ , to the bimaximal ν 12 − neutrino mass matrix and not from the charged lepton ansatz is generically given by, mixing. 2 0 0 MQνLC =0 −1 −1λν∆12. (21) 0 1 1  − −  III. QLC CORRECTED BIMAXIMAL MASS We note that we used λ and not λ = θ to emphasize MATRICES ν C thatλ maynotbeexactlytheCabibboangle. Addition- ν allythebimaximalmassmatrixcanbeseparatedintotwo The charged lepton mixing cannot account for the pieces, observed deviations from the bimaximal ansatz in the BiMax = atm+ sol (22) MNSP matrix. Therefore, it is interesting to study the Mν Mν Mν generic corrections to the bimaximal neutrino mass ma- The expressions for atm, sol and QLC depend on Mν Mν Mν trix that can account for the QLC relation. The form the hierarchy case under consideration. The particular and relative size of these corrections can give us some forms can be found in table I. Next we will comment on insight in the origin of the neutrino mass matrix. Let us the main features of the different hierarchy cases. 4 A. Normal hierarchy case B. Inverted hierarchy case with same CP parities In the normal hierarchy case we obtain the leading In the inverted hierarchy case with same CP-parities order term in the neutrino mass matrix assuming that we obtain the leading order term in the neutrino mass m1 =0 and m2 =0, matrix assuming that m1 =m2 and m3 =0, 0 0 0 m 1 0 0 Maνtm = 23 0 1 −1. (23) Maνtm =m10 21 12 . (30) 0 −1 1  0 21 12  This matrix generates mass for one neutrino, ν , which 3 This matrix generates a degenerate mass for two neutri- using the atmospheric mass difference, corresponds to, nos which corresponds roughly to the atmospheric mass scale, m3 =q∆m2atm =(4.9±0.6)×10−2 eV. (24) To generate the solar mass difference we need to give m1 =m2 = ∆m2atm (31) q masstotheneutrinoν . Tothisendweneedtointroduce 2 asmallperturbationofthepreviousmatrixcontrolledby Inthiscase,togeneratethesolarmassdifferenceweneed the parameter γ = m2/m3 ≪ 1. To be consistent with tobreakthedegeneracybetweenthemassesofν1 andν2. bimaximalmixingweneedtheperturbationmatrixtobe Tothisendweintroduceasmallperturbationoftheform of the form, m =m (1+γ). Tobeconsistentwithbimaximalmixing 2 1 we need the perturbation matrix to be given by, 1 1 1 sol =γm3  1 −1√2 −1√2 . (25) 1 1 1 Mν 2 −√12 21 12  sol = γm  1 −1√2 −1√2 . (32) −√2 2 2  Mν 2 1 −√12 12 12  In the normal hierarchycase γ is related to the neutrino  −√2 2 2  mass differences by, The solar mass difference is given by, (m2 m2) γ2 (m223−−m122) = (1−γ2) ≈γ2. (26) ∆m2sol =(m22−m21)=m21γ(2+γ)≈2m21γ. (33) Using experimental data γ is determined to be, Inthiscase,γ canbedeterminedfromexperimentaldata to be given by, 1 ∆m2 2 γ ≈(cid:18)∆m2astoml (cid:19) =0.18±0.03. (27) γ 1 ∆m2sol 1λ2 0.024 (34) ≈ 2∆m2 ≈ 2 ≈ atm We note that γ is curiously approximately the Cabibbo angle,γ λ,thiswasnoticedearlierinRef.[20]. Finally Finally to generate a deviation from maximality in the ≈ togenerateadeviationfrommaximalityinthesolarmix- solar mixing angle able to account for the QLC relation ing angle able to account for the QLC relation we need we need to introduce a second perturbation given by, to introduce a second perturbation given by, 1 0 0 4 0 0 − MQνLC =γm23λν−0 1 1 (28) MQνLC =γm1λν 00 121 112  (35)  0 1 1  2 2    Therefore, in the inverted hierarchy case with same CP- Therefore, in the normal hierarchy case, the correction otordMeraνγtm≈coλm, iin.eg.fraopmprtohxe2m0%at,riixnMthesνolenistryat(1m1o)stanodf Mapnadrsνiotlieissλ,3ta/ht2emtchooersrtreecosttfi.oonTrdhtoeerMenγt/aνrt2imes≈cionmλt3ihneignQfrtLohmCe ctehonretrrmeycat(it1or1nix), approx. λ/2 in the rest of entries of the matrix. The eonrdtreires4γinλ theλQinLCthecoenrrtercyt(io1n1,)aMndQνaLpCp,raorxe. λat2 tmheosrtesotf. MentQνryL≈C(,11a)reanadt moλs4ttahecroersrte.ctTiohneroeffoorerdfeorr tλh3e/2invinertthede ν ≈ ≈ hierarchy case with same CP-parities to reproduce the Thereforeforthe normalhierarchycasetoreproducethe neutrino data we need the following hierarchy between neutrino data we need the following hierarchy between the different corrections, the different corrections, QLC sol < atm. (29) QLC < sol atm. (36) Mν ≃Mν Mν Mν ∼Mν ≪Mν 5 C. Inverted hierarchy case with opposite CP sume again that the mixing in the charged lepton sec- parities tor in the flavor basis is very small, as a consequence the MNSP matrix is very approximately the left-handed In the inverted hierarchy case with opposite CP- neutrino diagonalization matrix. We obtain, paritieswe obtainthe leadingorderterminthe neutrino mass matrix assuming that m2 =−m1 and m3 =0, MνM†ν =VMNSP(Mdνiag)2VM†NSP. (42) 0 1 1 We can generalize the results of Secs. IIIA and IIIB for m atm = 1 1 0 0. (37) thenormalandinvertedhierarchycases. Inthefirstcase Mν √2 1 0 0 wewillintroducethe sameperturbationrequiredto gen-   erate the solar mass difference, i.e. m = γm . The 2 1 As inthe same parities casewe need to breakthe degen- ( )sol and ( )QLC perturbations can be ob- eracy between the masses of ν1 and ν2 to generate the tMainνeMdfr†νomEqs.2M5aνnMd2†ν8byimplementingthesubstitu- solar mass difference. To this end we introduce a small tionm m2andγ γ2. Intheinvertedhierarchycase perturbation of the form m2 = m1(1+γ). To be con- we will1n→ow i1ntroduc→e the solar mass difference pertur- − sistent with bimaximal mixing we need the perturbation bationinthe form,m2 =m2(1+γ2). Indoingsowe can 2 1 matrix to be given by, obtain the perturbations ( )sol and ( )QLC by implementing the sameMsuνbMsti†νtution, mMνMm†ν2 and sol =γm1 −1√12 121 121 . (38) bγa→tionγ2p,airnamEeqtse.r3γ2wanildl b3e5.deNteervmerinthedeleisns,1tht→hisecpae1sretubry- Mν √2 122 −−22√√122 −−22√√122  γth2e≈no∆rmma2soll/a∆ndm2aintmver≈teλd2.hiTerhaerrcehfyorceawseeswcoilrlroecbttiaoinnsftoor The solar mass difference is again given by, the bimaximal ansatz similar to those in Eqs. 29 and 36 respectively. ∆m2 =(m2 m2)=m2γ(2+γ). 2m2γ. (39) sol 2− 1 1 ≈ 1 Thereforeγ λ2/2. Finallytogenerateadeviationfrom maximality i≈n the solar mixing angle able to account for IV. RADIATIVE STABILITY OF THE QLC the QLC relation we need to introduce a second pertur- RELATION bation given by, It has been known for some time that the RGE ef- 2√2 0 0 m fects can considerably affect the neutrino mixing angles MQνLC = √21λν 0 −√2 −√2 . (40) [21, 22]. These effects can be especially importantin the  0 √2 √2  contextofSUSYSO(10)models,whichareofespecialin-  − −  terestforneutrinophysics,sinceinthiscaseallthethree Therefore, in the inverted hierarchy case with same CP- third generation Yukawa couplings can be large [23, 24]. parities,the correctionto atm comingfromthe matrix Mν The RGE effects also depend crucially on the type of Msνol is at most of order γ/2√2 ≈ λ3/√2. Interestingly neutrino mass hierarchy under consideration [25, 26]. the size of the entries to the QLC correction depends In the normal hierarchy case the RGE effects are upon sign(m2) and in the opposite CP-parities case un- knowntobeverysmallandasaconsequencetheycannot derconsiderationweobtainthatMQνLC isbetween√2λν account for a RGE generation of the QLC and or ∆m2sol and 2√2λν 2/3, i.e. approximately between 30% and that, as we have seen in the previous section, must be ≈ 60% of the leading term. Therefore for the inverted hi- of the same order of magnitude. Interestingly, in the in- erarchy case with opposite CP-parities to reproduce the vertedhierarchycasetheRGEevolutionofthesolarmix- neutrino data we need the following characteristic hier- ingdependscruciallyontheneutrinoCP-parities[25,27]. archy between the different corrections, TheRGEequationforthesolarmixingangleinthiscase adopts a simple form, which is valid for small θ , as sol QLC < atm. (41) 13 Mν ≪Mν ∼Mν experiments indicate, given by [28], Thisisverydifferentfromthehierarchiesrequiredforthe corrections generated in the normal hierarchy case and dθ12 = Ch2τs c s2 ∆m2atm (1+cos(φ φ ))+ (θ ). inverted hierarchy case with same CP-parities. In those dt 8π2 12 12 23 ∆m2 1− 2 O 13 sol two cases the QLC correction was of the same order or (43) smaller than the solar correction respectively. Here t = ln(µ/µ ), µ is the renormalization scale and 0 φ are the neutrino CP-phases. We will assume that 1,2 an exactly bimaximal neutrino mass matrix is generated D. Generalization to the Dirac case at high energies, s = c = s = 1/√2, and that the 12 12 23 solarandatmosphericneutrinomassdifferences arephe- It is straightforward to extend the previous results to nomenologically acceptable, i.e. that ∆m2 /∆m2 the case that neutrinos are Dirac fermions. We will as- λ2. We obtain for the RGE generated shifstolin theatsmola≈r 6 mixing angle, CP-phase, are zero. The RGE for ∆m2 is given in this sol case by a simple expression [28], Ch2 1 Λ ∆θ12 ≈ 32πτ2λ2 (1+cos(φ1−φ2))ln(cid:18)mZ(cid:19). (44) 8π2d∆dmt2sol =α∆m2sol−Ch2τ2s223(m22c212−m21s212)+O(θ13) (47) Here ∆θ =θ (Λ) θ (m ). In the SM C =3/2 and 12 12 12 Z h2 = m2/m2 10 4−and assuming that Λ = 1016 GeV Assuming that at high energies θ12 =π/4 λ and θ23 = τ τ t ≈ − π/4 we obtain for the radiatively generat−ed solar mass we obtain for the radiatively generated ∆θ , 12 difference, ∆θ12|mSMax ≈3×10−4(1+cos(φ1−φ2)) (45) 8π2d∆m2sol =α∆m2 2Cλh2∆m2 . (48) dt sol− τ atm We note that to fit the experimental results we should obtain ∆θ λ. It has already been pointed out [18] This equation has a simple analytical solution. In the 12 ≈ thatintheSMthiscorrectionisverysmallanditcannot SM where C =3/2 we obtain, be the source of the QLC relationnor perturba possible hnieguhtreinneorCgyPo-priagriintioesf.theQLCrelationirrespectiveofthe ∆m2sol(µ) SM ≈ 3αmm2τ2λ∆m2atm(1−e8πα2ln(Λµ)). (49) (cid:12) t In the MSSM the situation is more complicated. In (cid:12) Assuming that Λ = 1016 GeV and µ = m we obtain this case C = 1 and h2 tan2βm2/m2, where tanβ Z is the wellknow−nratioofτM≈SSM Higgτsvactuum expecta- ∆m2sol(mZ) SM ≈ 2.8×10−5∆m2atm, which is too small toaccountfo(cid:12)rtheobservedsolarmassdifference. Onthe tion values. This is relevant in the case of SUSY SO(10) (cid:12) other hand in the MSSM C = 1 and we obtain, modelswhichrequirealargetanβ. Assumingtanβ =50 − we obtain, 2t2m2λ max 1 ∆m2sol(µ)(cid:12)MSSM ≈(−) αβm2tτ ∆m2atm(1−e8πα2ln(Λµ)). ∆θ12|MSSM ≈−2(1+cos(φ1−φ2)) (46) (cid:12) (50) AssumingthatΛ=1016GeV,µ=m andtanβ islarge, Z This shows that for the same CP-parities case the solar tanβ =50,we obtain ∆m2sol(mZ) MSSM ≈−2λ2∆m2atm. mixing angle would be unstable under RGE corrections Therefore the radiatively generate(cid:12)d ∆m2 (m ) is of the (cid:12) sol Z as is well known. We cannot generate radiatively the right magnitude but unfortunately of the wrong sign. QLC relation because the MSSM correction has a sign The experimental data requires that ∆m2 (m ) sol Z ≈ contrary to the required to fit the experimental data, λ2∆m2 . Therefore a RGE generation of ∆m2 (cid:12) atm sol M(cid:12)SSM ∆θ12 λ. Ontheotherhand,Eq.46showsthatthesolar triggered by a very high energy generation of the(cid:12)QLC mixin≈g angle in the case of an inverted neutrino spectra perturbedbimaximalscenario,assumingandinvert(cid:12)edhi- with a maximal CP-parity phase difference between the erarchy with opposite CP-parities, does not seem to be heaviesteigenvalueswillbeespeciallystablesinceinthat in agreement with the data. case cos(φ φ )= 1 and as a consequence dθ /dµ= 1 2 12 − − 0. We note that the term (θ ) which has not been 13 O included in the RGE for θ12 also cancels for opposite VI. CONCLUSIONS CP-parities[28]. This opensthe possibilitythatthe QLC relation is generated at a high energy scale, remaining We have studied several model independent implica- stable all the way down to the electroweak scale. tions of the measured deviation from maximality in the solar mixing angle. We have pointed out that it is not plausible that this deviation is generated in the charged V. A QLC TRIGGERED ∆m2sol? lepton mixing matrix. We have studied the generic low energy corrections to the exactly bimaximal ansatz nec- Let us assume that a QLC corrected bimaximal neu- essarytoaccountforboththesolarmassdifferenceanda trinomassmatrixisgeneratedatsomehighenergyscale. non-maximalsolarmixingangle. Wepointedoutthatthe Wehaveseenintheprevioussectionthatifthereisanin- relative size of these correctionsdepends strongly on the verted neutrino hierarchy with opposite CP-parities, i.e. neutrino hierarchy under consideration. For the normal m = m , the QLC relation will remain stable under and inverted hierarchy with same CP parities it seems 1 2 − RGE evolution. It is interesting to study if an initial very difficult to understand the origin of the QLC re- high-energy deviation from maximality in the solar mix- lation independently from the origin of ∆m2 since the sol ing, like the one given by the QLC relation, can trigger respective corrections are of the same order of magni- the generationof the correct solar mass difference radia- tude. In that case the QLC relation is most probably a tively through RGE running. In some cases the solar coincidenceunlessthe neutrinomassmatrixisgenerated massdifference,aspointedoutsometimeago[30], could at low energy scales. be fully generated by RGE corrections. We will assume On the other hand, for an inverted hierarchy with op- a limit case where at high energy θ and δ, the Dirac positeCPparitiesthecorrectiontothebimaximalansatz 13 7 necessarytoexplaintheQLCrelationisofthesameorder be the most interesting possibility from a model build- but smaller than the leading term of the bimaximal ma- ing point of view when searching for a non-coincidental, trix and both are much larger than the correctionneces- high-energy explanation of the QLC relation. saryto generate∆m2 . Additionally the leading bimax- sol imal term as well as the QLC perturbation could both have a high energy origin since the solar mixing angle is verystableunderRGEeffects. Thisraisesthepossibility Acknowledgments to link the origin of the QLC relation with the origin of the chargedfermion mass matrices. Although this setup does not allow us to radiatively generate ∆m2 entirely This work is supported by: the Ministry of Science sol byRGEcorrectionsthereareotherpossibleexplanations of Spain under grant EX2004-0238, the US Department available in the literature for the origin of the measured of Energy under Contracts: DE-AC03-76SF00098, DE- ∆m2 . We believe, as our analysis indicates, that the FG03-91ER-40676andDE-FG03-94ER40833andby the sol inverted hierarchy case with opposite CP-parities may NationalScienceFoundationundergrantPHY-0098840. [1] Y. Ashie et al. [Super-Kamiokande Collaboration], D 66, 113006 (2002); P. H. Frampton, S. T. Petcov [arXiv:hep-ex/0404034] and W. Rodejohann, Nucl. Phys. B 687, 31 (2004); [2] T. Araki et al. [KamLAND Collaboration], S. Antusch and S. F. King, [arXiv:hep-ph/0402121] [arXiv:hep-ex/0406035] G. Altarelli, F. Feruglio and I. Masina, Nucl. Phys. [3] M. Apollonio et al.,Eur. Phys. J. C 27, 331 (2003) B 689, 157 (2004); S. Antusch and S. F. King, [4] J.N.Bahcall,M.C.Gonzalez-GarciaandC.Pena-Garay, Phys. Lett. B 591, 104 (2004); M. Raidal, Phys. [arXiv:hep-ph/0406294] Rev. Lett. 93, 161801 (2004); C. A. de S. Pires, [5] M. C. Gonzalez-Garcia and M. Maltoni, [arXiv:hep-ph/0404146]; S. Antusch and S. F. King, [arXiv:hep-ph/0406056] [arXiv:hep-ph/0405272] P. H. Frampton and R. N. Mo- [6] A. Bandyopadhyay,S. Choubey, S. Goswami, S. T. Pet- hapatra, [arXiv:hep-ph/0407139] cov and D.P. Roy,[arXiv:hep-ph/0406328] [18] H.MinakataandA.Y.Smirnov,[arXiv:hep-ph/0405088] [7] M. Maltoni, T. Schwetz, M. A. Tortola and [19] P.H.Frampton,S.T.PetcovandW.Rodejohann,Nucl. J. W. F. Valle, [arXiv:hep-ph/0405172] Phys. B 687, 31 (2004) [8] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. [20] P. M. Fishbaneand P.Kaus, J. Phys.G 26, 295 (2000) Phys. 28 (1962) 870; V. N. Gribov and B. Pontecorvo, [21] F. Vissani and A. Y. Smirnov, Phys. Lett. B 341, 173 Phys. Lett. B 28, 493 (1969); B. W. Lee, S. Pakvasa, (1994) R.E.ShrockandH.Sugawara, Phys.Rev.Lett.38,937 [22] P.H.ChankowskiandZ.Pluciennik,Phys.Lett.B 316, (1977) [Erratum-ibid. 38, 1230 (1977)] 312 (1993); K. S. Babu, C. N.Leung and J. Pantaleone, [9] V.D.Barger,S.Pakvasa,T.J.WeilerandK.Whisnant, Phys. Lett. B 319, 191 (1993) Phys. Lett. B 437, 107 (1998) A. J. Baltz, A. S. Gold- [23] M.Tanimoto,Phys.Lett.B360,41(1995);M.K.Parida haber and M. Goldhaber, Phys. Rev. Lett. 81, 5730 and N.N. Singh,Phys. Rev.D 59, 032002 (1999) (1998) [24] J. R.Ellis and S.Lola, Phys. Lett. B 458, 310 (1999) [10] M. Jezabek and Y. Sumino, Phys. Lett. B 440, [25] N. Haba and N. Okamura, Eur. Phys. J. C 14, 347 327 (1998); F. Vissani, JHEP 9811, 025 (1998); (2000);N.Haba,Y.Matsui,N.OkamuraandM.Sugiura, D.V. Ahluwalia, Mod. Phys. Lett.A 13, 2249 (1998) Prog. Theor. Phys. 103, 145 (2000); N. Haba, Y. Mat- [11] W. Rodejohann, Phys. Rev. D 69, 033005 (2004); sui and N. Okamura, Eur. Phys. J. C 17, 513 (2000); A. Y. Smirnov, “Neutrinos: ’Annus mirabilis’,” [arXiv:hep-ph/0005075]. [arXiv:hep-ph/0402264]; S. Pakvasa (unpublished) [26] R. N. Mohapatra, M. K. Parida and G. Rajasekaran, [12] W. Rodejohann, Phys.Rev.D 69, 033005 (2004); Phys. Rev.D 69, 053007 (2004) [13] R. Gatto, G. Sartori and M. Tonin, Phys. Lett. B 28 [27] P.H.Chankowski,W.KrolikowskiandS.Pokorski,Phys. (1968)128;N.CabibboandL.Maiani,Phys.Lett.B28, Lett. B 473, 109 (2000); J. A. Casas, J. R. Espinosa, 131 (1968); R.J. Oakes,Phys. Lett. B 29, 683 (1969). A.IbarraandI.Navarro,Nucl.Phys.B573,652(2000); [14] H.GeorgiandC.Jarlskog,Phys.Lett.B86,297(1979). K. R. S. Balaji, R. N. Mohapatra, M. K. Parida and [15] J. Ferrandis, Eur.J.Phys.C 38, (2004); E. A.Paschos, Phys. Rev.D 63, 113002 (2001) [16] S.Weinberg,Trans.NewYorkAcad.Sci.38,185(1977); [28] S. Antusch, J. Kersten, M. Lindner and M. Ratz, Nucl. J.Bjorken(Unpublished);A.deRujula,H.GeorgiandS. Phys. B 674, 401 (2003) L. Glashow, Annals Phys. 109, 258 (1977); H. Fritzsch, [29] J. Ferrandis and S. Pakvasa, Phys. Lett. B 603, 184 Phys.Lett. B 70, 258 (1977). (2004) [17] M. Tanimoto, Phys. Rev. D 59, 017304 (1999); M. Jez- [30] J. A. Casas, J. R. Espinosa, A. Ibarra and I. Navarro, abek and Y. Sumino, Phys. Lett. B 457, 139 (1999); Nucl.Phys.B556,3(1999);J.A.Casas,J.R.Espinosa, T. Ohlsson and G. Seidl, Nucl. Phys. B 643, 247 A. Ibarra and I. Navarro, Nucl.Phys. B 569, 82 (2000) (2002); C. Giunti and M. Tanimoto, Phys. Rev. D 66, 053013 (2002); C. Giunti and M. Tanimoto, Phys. Rev.