Recent measurements of the neutrino mixing angles cast doubt on the validity
of the so-far popular tri-bimaximal mixing ansatz. We propose a parametrization
for the neutrino mixing matrix where the reactor angle seeds the large solar
and atmospheric mixing angles, equal to each other in first approximation. We
suggest such bi-large mixing pattern as a model building standard, realized
when the leading order value of the reactor angle equals the Cabibbo angle.Comment: 4 pages, 2 figs. v2: matches version appearing in Phys.Rev.D, rapid
  communication

A. Gouvea

J. W. F. Valle

M. Tórtola

R. Nichol

S. M. Boucenna

S. Morisi

Y. Itow

English

arXiv

Recent measurements of the neutrino mixing angles cast doubt on the validity of the so-far popular tri-bimaximal mixing ansatz. We propose a parametrization for the neutrino mixing matrix where the reactor angle seeds the large solar and atmospheric mixing angles, equal to each other in first approximation. We suggest such bi-large mixing pattern as a model building standard, realized when the leading order value of the reactor angle equals the Cabibbo angle

Boucenna, Mohamed Sofiane

Morisi, S.

Tórtola Baixauli, María Amparo

Furtado Valle, José Wagner

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

IFIC/12-42Bi-large neutrino mixing and the Cabibbo angleS. M. Boucenna,1, ∗ S. Morisi,1, † M. Tórtola,1, ‡ and J. W. F. Valle1, §1 AHEP Group, Instituto de Física Corpuscular – C.S.I.C./Universitat de ValènciaEdificio de Institutos de Paterna, Apartado 22085, E–46071 València, Spain(Dated: August 27, 2012)Recent measurements of the neutrino mixing angles cast doubt on the validity of the so-far populartri-bimaximal mixing ansatz. We propose a parametrization for the neutrino mixing matrix wherethe reactor angle seeds the large solar and atmospheric mixing angles, equal to each other in firstapproximation. We suggest such bi-large mixing pattern as a model building standard, realizedwhen the leading order value of θ13 equals the Cabibbo angle λC .PACS numbers: 11.30.Hv 14.60.-z 14.60.Pq 14.80.Cp 23.40.BwINTRODUCTIONUnderstanding the structure of neutrino mixing fromfirst principles is part of the flavor problem of the Stan-dard Model, one of the deepest in particle physics. Apartial useful strategy is to formulate attractive mixingpatterns that may help to seek for possible underlyingflavor symmetries. The tri-bimaximal mixing (TBM)ansatz [1] for describing the neutrino mixing matrix [2]assumes a maximal atmospheric angle and zero reactorangle, as was suggested by the experimental data. Suchstriking features point towards an underlying symmetryand indeed, many models based on (mostly discrete) non-abelian flavour symmetries were successful in reproduc-ing this ansatz [3, 4]. Small deviations from the TBMare expected on theoretical grounds.However the recent results published by the Double-Chooz [5], Daya Bay [6], RENO [7], T2K [8] and MI-NOS [9] collaborations, indicate that the the reactor an-gle is relatively large so that corrections to the TBMpattern should be, in fact, quite large, casting doubt onits validity as a good first approximation reproducing theneutrino mixing pattern. To be more precise, on theo-retical grounds a small deviation of order of the squareof the Cabibbo angle was expected for the reactor angle,while recent observations indicate a much larger valueof about the order of the Cabibbo angle. To evade thisproblem, different ansatz have been considered like thebimaximal mixing [10] or the Golden ratio, see [11] for areview. However, most of these models assume a µ − τ -invariant structure in order to predict a maximal atmo-spheric mixing angle. On the other hand, at the Neutrino2012 conference the MINOS collaboration also gave hintsfor a non-maximal atmospheric mixing angle.Here we propose a different approach where we takethe reactor mixing angle as the fundamental parameter.As will be shown below, the resulting parametrizationdoes not reproduce the TBM pattern as a limiting case,though a maximal atmospheric angle can be obtained.The main idea is that, since the reactor angle is the onlysmall mixing parameter for the leptons, we can use it toseed both the solar and atmospheric mixing angles, asfollows,sin θ13 = λ ;sin θ12 = s λ ;sin θ23 = a λ ,(1)where the small parameter λ is the reactor angle, whiles ' a are free parameters of order few. Solar and at-mospheric mixings are expressed in terms of a linear de-pendence on the reactor angle. In the limit where λ→ 0neutrinos are unmixed.Using the general symmetric parametrization of theneutrino mixing matrix [2] one can trivially obtain a sim-ple approximate description by expanding only in thesmall parameter λ. For example, the Jarlskog-like in-variant describing CP violation in neutrino oscillations isthen given by:JCP ≈ a s λ3√1− a2λ2√1− s2λ2 sin(φ13 − φ12 − φ23)(2)given explicitly in terms of the rephase-invariant Diraccombination. Likewise, the effective mass parameter de-scribing the amplitude for neutrinoless double-beta decayis given in terms of the two Majorana CP phases.In what follows, for simplicity, we take all parametersto be real. In order to fix the values of the large param-eters s and a we consider the latest experimental resultson neutrino oscillation parameters. At the Neutrino 2012conference in Kyoto the MINOS Collaboration has re-ported a non-maximal value for the atmospheric mixingangle [9]:sin2 2θ23 = 0.94+0.04−0.051 (3)with maximal mixing disfavoured at 88% C.L. This re-sult comes from the analysis of νµ disappearance in the1 In a three-neutrino analysis this quantity will be equal to4|Uµ3|2(1− |Uµ3|2) = 4 cos2 θ13 sin2 θ23(1− cos2 θ13 sin2 θ23).arXiv:1206.2555v2  [hep-ph]  24 Aug 20122MINOS accelerator beam and corresponds to two degen-erate points for sin2 θ23, namelysin2 θ23 = 0.38 and sin2 θ23 = 0.62. (4)This comes from the fact that the disappearance chan-nel is octant-symmetric and therefore MINOS data bythemselves can not show a preference for a given octantof θ23. However, one may expect that in combinationwith the searches for electron-neutrino appearance at thelong-baseline experiments MINOS and T2K, and withthe recent measurements of θ13 at reactor experimentsthe degeneracy in Eq. (3) will be broken and one oc-tant will be preferred over the other. Global analysis sofar have not been able to give a completely clear pictureabout the true octant of θ23. The analysis given in [12]indicates small deviations of maximality, with the octantpreference correlated with the mass hierarchy. This cor-relation is also seen in the recent analysis of atmosphericneutrino data in Super-Kamiokande [13]. However, theanalyses given in [14, 15] have shown a preference forθ23 < pi/4 with different levels of significance. All pre-vious neutrino oscillation global fits agree in getting θ23in the first octant for normal mass hierarchy, so for thepurpose of this article we will assume that this is thecase.BI-LARGE MIXINGHere we discuss the recent neutrino oscillation resultsin terms of our proposed ansatz. As already pointedout above, the recent experimental data provide a robustmeasurement of a relatively large θ13.Using the best fit values of the mixing angles inRef. [12] or Ref. [14], we can fix the three parametersin Eq. (1): λ ∼ 0.16 (0.15), a ∼ 4.13 (4.21) and s ∼3.53 (3.65). Then, from the data we can directly readthatsin θ12 = O(sin θ23) , (5)Now we go a step further and assume the followingsin θ12 = sin θ23 , (6)which in our parametrization meanss = a. (7)Since both solar and atmospheric angles are large we callthis case bi-large mixing ansatz.Suppose now that we are given some model predictingbi-large mixing a = s at leading order. Next-to-leadingorder operators in the Lagrangian in general induce de-viations from the reference values in Eq. (1) which maybe reliably determined within a given model. Here wepresent a simple model-independent estimate of such cor-rections, obtained as follows. Typically it is expectedthat the corrections to the three mixing angles from nextto leading order terms are of the same order, that issin θij → sin θij ±  where we have introduced a newparameter  to characterize the magnitude of the correc-tion. In this case our bi-large mixing gets corrections ofthe same order for the three mixing angles (given by )and which may either increase or decrease the startingbi-large values of the mixing angles. For definiteness letus consider an example where bi-large mixing is correctedassin θ13 = λ−  ;sin θ12 = sλ−  ;sin θ23 = aλ+  .(8)where we take s = a as in Eq. (7). Since we have threefree parameters, we can fix them using the best fit valuesreported by global analysis of neutrino oscillation data inRefs. [12, 14]. The results are given in Table I.Ref. λ s Forero et al. [12] 0.23± 0.04 2.8+0.5−0.4 0.067+0.035−0.025Fogli et al. [14] 0.19+0.03−0.02 3.0+0.5−0.3 0.038+0.019−0.018TABLE I: Best fit values and 1σ ranges for the parametersλ, s and  in Eq. (8) according to the global fits to neutrinooscillations.In order to quantitatively clarify the role of the rela-tion in Eq. (7) with respect to the reactor mixing angle,we consider here the most generical case given by Eq. (8)where the three angles are given in terms of four param-eters instead of three. Three of these parameters can befixed from the three measured mixing angles, leaving onefree parameter that we choose to be λ. In order to quan-tify the deviation from our exact bi-large mixing ansatzdefined in Eq. (7) we plot the combination (a−s)/(a+s)as a function of the expansion parameter λ in Fig. 1. Thecolored/shaded bands are calculated from the two andthree sigma allowed ranges for the neutrino oscillationparameters obtained in the current global fits. The solidand dashed lines indicate the best fits of Refs. [12] and[14], respectively. It is remarkable that the strict bi-largeansatz in Eq. (7) holds when λ ' λC where λC ≈ 0.22.This means that λC is the leading order value of sin θ13.In Fig. 2 we show the average value of a and s, that is(a+ s)/2, as a function of λ. As displayed in the figure,the correlation is such that (a+ s)/2 ∼ 3 when λ ∼ λC .It follows that one possible form of our bi-large ansatz,3FIG. 1: Deviation from the bi-large ansatz versus the ex-pansion parameter λ at two and three sigma in the neutrinooscillation parameters. The solid and dashed lines indicatethe best fits of Refs. [12] and [14], respectively. The strictbi-large ansatz holds when λ ' λC (vertical line).FIG. 2: Average of solar and atmospheric angles versus theexpansion parameter λ at two and three sigma in the neutrinooscillation parameters. The solid and dashed lines indicatethe best fits of Refs. [12] and [14], respectively.which can be useful for model building, issin θ13 = λ;sin θ12 = 3λ;sin θ23 = 3λ .(9)It is remarkable to see how such a simple form is nearlyconsistent with current global neutrino oscillation data.Now a few words on model building. Given a particu-lar mixing matrix U , the structure of the neutrino massmatrix is fixed asmν = U ·D · UTwhereD is a diagonal matrix. For the sake of illustration,we consider in our mixing ansatz in Eq. (9) a normal andstrongly hierarchical spectrum (mν1 = 0) for neutrinomasses and fix the square mass differences at their bestfit values, as given in Ref. [12]. We find that the resultingweak-basis neutrino mass matrix mν has the formmν ∼ 0.20 0.32 0.150.75 0.701 ∼ λC λC λC1 11 (10)where in the last step the Cabibbo angle is used as theexpansion parameter and we do not specify numericalcoefficients of order one.From the form obtained in Eq. (10) one sees that theparameter λC appears only in the first row. On theother hand in the “atmospheric sector” there seems to be“democracy” in the choice of the neutrino mass entries.Altogether the above indicates two general features re-garding the neutrino mass generation mechanism.• a Frogatt-Nielsen-like flavour symmetry [16] thatcould perhaps generate the required pattern givenin Eq. (10).• some gravity-like [17] “flavor-blind” mechanism op-erating within the “atmospheric sector”. The prob-lem in this case is that, for reasonable values ofthe coefficients of the dimension five operator, theinduced neutrino masses are too small. Howeverthere may be other well-motivated “anarchy”-typeschemes [18, 19].It is beyond the scope of this paper to provide a def-inite model realization, but rather to stress the simplic-ity of the ansatz in Eq. (10) which may provide a freshmodel-building guideline that may perhaps replace thetri-bimaximal ansatz.SUMMARY AND OUTLOOKWe have argued that recent experimental results onneutrino oscillations are very well-described by the ansatzin Eq. (1) with small deviations, as in Eqs. (8). It is trulyremarkable that with the expansion parameter λ taken asλ ' λC , the Cabibbo angle characterizing the quark mix-ing matrix, we obtain the simplest bi-large limit, s = a,given in Eq. (7). This appears to be an important nu-merological “coincidence” which may drastically changeour theoretical approach for constructing neutrino massmodels, by moving from a geometrical interpretation ofthe neutrino mixing angles to one in which these are nolonger associated to Clebsh-Gordan coefficients of anysymmetry, in sharp contrast to the previous paradigm 2.The simple correlations between ansatz deviations and2 We should stress, however, that the TBM pattern may still betenable if the underlying theory is capable of providing suffi-ciently large corrections to θ13 without affecting too much thesolar angle, which is in principle possible.4λ are illustrated in Figs. 1 and 2. The bi-large ansatzholds for λ ' λC (vertical line). It remains to be seenwhether nature is perhaps telling us something profoundregarding the ultimate theory of flavour.AcknowledgmentsWe thank Martin Hirsch for discussions. Thiswork was supported by the Spanish MINECO undergrants FPA2011-22975 and MULTIDARK CSD2009-00064 (Consolider-Ingenio 2010 Programme), by Prome-teo/2009/091 (Generalitat Valenciana), by the EU ITNUNILHC PITN-GA-2009-237920. M.T. acknowledgesfinancial support from CSIC under the JAE-Doc pro-gramme, co-funded by the European Social Fund. S.M.acknowledges Juan de la Cierva contract.∗ Electronic address: boucenna@ific.uv.es† Electronic address: morisi@ific.uv.es‡ Electronic address: mariam@ific.uv.es§ Electronic address: valle@ific.uv.es[1] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys.Lett. B530, 167 (2002).[2] J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227(1980); W. Rodejohann and J. W. F. Valle, Phys.Rev.D84, 073011 (2011), [1108.3484].[3] K. S. Babu, E. Ma and J. W. F. Valle, Phys. Lett. B552,207 (2003), [hep-ph/0206292].[4] G. Altarelli and F. Feruglio, Nucl. Phys. B720, 64(2005), [hep-ph/0504165].[5] DOUBLE-CHOOZ Collaboration, Y. Abe et al.,Phys.Rev.Lett. 108, 131801 (2012), [1112.6353].[6] DAYA-BAY Collaboration, F. An et al., Phys.Rev.Lett.108, 171803 (2012), [1203.1669].[7] RENO collaboration, J. Ahn et al., 1204.0626.[8] T2K Collaboration, K. Abe et al., Phys.Rev.Lett. 107,041801 (2011), [1106.2822].[9] R. Nichol, Plenary talk at the Neutrino 2012 conference,http://neu2012.kek.jp/.[10] G. Altarelli, F. Feruglio and L. Merlo, JHEP 0905, 020(2009), [0903.1940].[11] C. H. Albright, A. Dueck and W. Rodejohann,Eur.Phys.J. C70, 1099 (2010), [1004.2798].[12] D. Forero, M. Tortola, J. W. F. Valle, arXiv:1205.4018,this updates previous results in New J. Phys. 13 (2011)109401 and New J.Phys. 13 (2011) 063004[13] Y. Itow, Plenary talk at the Neutrino 2012 conference,http://neu2012.kek.jp/.[14] G. Fogli et al., arXiv:1205.5254.[15] M. Gonzalez-Garcia, M. Maltoni and J. Salvado, JHEP1004, 056 (2010), [1001.4524].[16] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B147,277 (1979).[17] A. de Gouvea and J. W. F. 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