We note that the emerging features of lepton mixing can be reproduced if,
with inverted neutrino mass ordering, both the smallest neutrino mass and the
$\tau\tau$ element of the neutrino mass matrix vanish. Then, the atmospheric
neutrino mixing angle is less than maximal and the Dirac phase $\delta$ is
close to $\pi$. We derive the correlations among the mixing parameters and show
that there is a large cancellation in the effective mass responsible for
neutrinoless $\beta\beta$ decay. Three simple seesaw models leading to our
scenario are provided.Comment: 11 pages, 4 figure

Lavoura, Luís

Rodejohann, Werner

Watanabe, Atsushi

Physics Letters B

English

arXiv

We note that the emerging features of lepton mixing can be reproduced if, with inverted neutrino mass ordering, both the smallest neutrino mass and the $\tau\tau$ element of the neutrino mass matrix vanish. Then, the atmospheric neutrino mixing angle is less than maximal and the Dirac phase $\delta$ is close to $\pi$. We derive the correlations among the mixing parameters and show that there is a large cancellation in the effective mass responsible for neutrinoless $\beta\beta$ decay. Three simple seesaw models leading to our scenario are provided

Lavoura, L.

Rodejohann, W.

Watanabe, A.

MPG.PuRe

arXiv:1307.6421v1  [hep-ph]  24 Jul 2013CFTP/13-016Reproducing lepton mixing in a texture zero modelLu´ıs Lavoura,a∗ Werner Rodejohann,b† and Atsushi Watanabeb‡aUniversity of Lisbon, Instituto Superior Te´cnico, CFTP,1049-001 Lisboa, PortugalbMax–Planck–Institut fu¨r Kernphysik,Saupfercheckweg 1, D–69117 Heidelberg, GermanyAbstractWe note that the emerging features of lepton mixing can be reproduced if, withinverted neutrino mass ordering, both the smallest neutrino mass and the ττ elementof the neutrino mass matrix vanish. Then, the atmospheric neutrino mixing angle isless than maximal and the Dirac phase δ is close to pi. We derive the correlationsamong the mixing parameters and show that there is a large cancellation in theeffective mass responsible for neutrinoless ββ decay. Three simple seesaw modelsleading to our scenario are provided.∗email: balio@cftp.ist.utl.pt†email: werner.rodejohann@mpi-hd.mpg.de‡email: atsushi.watanabe@mpi-hd.mpg.deThe increasingly precise determination of the lepton mixing parameters makes a the-oretical interpretation of the data a necessity. In this Letter we propose a form of theneutrino mass matrix that accommodates, and in fact correlates, two emerging struc-tures [1] of lepton mixing: (i) the value of the atmospheric mixing angle θ23 is significantlyless than maximal; (ii) the CP -violating Dirac phase δ is close to pi. Table 1 shows themixing parameters obtained from a global fit to the neutrino oscillation data.1Parameter best-fit 1σ range 3σ range∆m2sol [10−5 eV2] 7.54 7.32 – 7.80 6.99 – 8.18∆m2atm [10−3 eV2] 2.42 2.31 – 2.49 2.17 – 2.61sin2 θ12 0.307 0.291 – 0.325 0.259 – 0.359sin2 θ13 0.0244 0.0219 – 0.0267 0.0171 – 0.0315sin2 θ23 0.392 0.370 – 0.431 0.335 – 0.663δ 1.09pi (0.83 – 1.47)pi 0 – 2piTable 1: Values of the lepton mixing parameters in the case of an inverted neutrino massordering. This table was taken in abridged form from Ref. [1].While features (i) and (ii) are not fully established yet, they seem interesting and worthinvestigating. The proposal that we make is that, in the basis of a diagonal charged-leptonmass matrix,2 the following features hold:Mττ = m3 = 0 , (1)where M is the neutrino mass matrix and m3 is the mass of the lightest neutrino when theneutrino mass hierarchy is inverted.3 We shall show that this leads to the correlationtan θ23 ≃ −2 sin θ13 cos δ tan 2θ12 , (2)and therefore implies the desired features that cos δ is negative and large while θ23 is small,hence significantly less than maximal.4 Moreover, our scheme makes predictions: (i) thereis an inverted neutrino mass hierarchy with vanishing smallest mass; (ii) the sole Majoranaphase has a value that leads to a large cancellation in the effective mass on which thelifetime of neutrinoless ββ decay depends, hence that lifetime must lie at the upper end ofits allowed range.The implications of single texture zeros, with and without vanishing smallest neutrinomass, have often been studied [6, 7, 8, 9, 10, 11, 12]. We stress in this Letter how well1Other fits [2, 3] give somewhat different results for θ23 and for δ.2Actually, it is enough that the diagonalization of the charged-lepton mass matrix consists only of arotation in the e–µ plane.3It could already be gathered from Ref. [4] that Eq. (1) is in agreement with the data.4The so-called TM1 scenario, in which the first column of the PMNS matrix satisfies(|Ue1| , |Uµ1| , |Uτ1|) =(√23,√16,√16), displays [5] a similar correlation between θ23 and δ.2the consequences of Eq. (1) fit the current data. We also point out that Eq. (1) can bearranged in models; we illustrate this through three seesaw models furnished with discretesymmetries.We start with the phenomenology of Eq. (1). Defining—as in Ref. [1], from which wehave taken the values of the mixing parameters—∆m2atm = |m23 − (m21 +m22)/ 2|, one has,from m3 = 0, thatm21 = ∆m2atm(1−r2), (3a)m22 = ∆m2atm(1 +r2), (3b)where r ≡ ∆m2sol/∆m2atm ≈ 0.031 and ∆m2sol = m22 −m21. Since the neutrino mass matrix,in the charged-lepton mass basis, is M = U∗ diag (m1, m2, m3)U†, where U is the PMNSmatrix, one hasMττ = 0 ⇔ U2τ1√1−r2= −U2τ2√1 +r2. (4)In the standard parametrization of U ,Uτ1 = s12s23 − c12c23s13eiδ , (5a)Uτ2 =(−c12s23 − s12c23s13eiδ)eiρ/2 , (5b)where sj = sin θj and cj = cos θj for j = 12, 13, 23. To get rid of the Majorana phase ρ onetakes the moduli of both sides of Eq. (4):(s212s223 + c212c223s213 − 2s12c12s23c23s13 cos δ)√1−r2=(c212s223 + s212c223s213 + 2s12c12s23c23s13 cos δ)√1 +r2.(6)Definingε ≡√1− r/2−√1 + r/2√1− r/2 +√1 + r/2= −r4−r364− · · · , (7)equation (6) may be rewritten as(s212 − c212) (s223 − c223s213)− 4s12c12s23c23s13 cos δ + ε(s223 + c223s213)= 0 . (8)Since c212 − s212 ≈ 0.4 while ε ≈ 0.008, the third term in the left-hand side of Eq. (8) maybe neglected relative to the first one and one ends up with5cos δ ≈ −tan θ232 sin θ13 tan 2θ12, (10)5The exact version of Eq. (10) iscos δ = −(c212 − s212)s223 − s213(c212 − s212)c223 + ε(s223 + c223s213)4s12c12s23c23s13, (9)The first term of the numerator is of order 0.15 while the second and third terms are of order 0.006 and0.004, respectively.3and we have demonstrated Eq. (2). Equation (10) indicates that cos δ is negative. More-over, since sin θ13 ≈ 0.16 is small, |cos δ| should be large. In order for it not to exceed 1,θ23 must be as small as possible, hence in the first octant, while both θ13 and θ12 shouldbe largish. To be precise,sin θ13 tan 2θ12 >tan θ23|min(3σ)2= 0.355 (11)must hold.The arguments of the left- and right-hand sides of Eq. (4) should also coincide. Ne-glecting the small terms containing s13 in Eqs. (5), this results in the conditionρ ≈ pi . (12)A consequence of Eq. (12) occurs in the effective mass |Mee| ≡ 〈m〉 that governs the rateof neutrinoless ββ decay [13]. For vanishing m3, and for ρ ≈ pi, this is given by〈m〉 ≈ c213√∆m2atm√1− sin2 2θ12 sin2 ρ/2ρ≈pi≈ c213√∆m2atm cos 2θ12 . (13)This gives 〈m〉 ≈ 0.018 eV for the best-fit values in Table 1. The effective mass thereforelies at the lower end of the range 0.013 eV < 〈m〉 < 0.050 eV generally allowed for theinverted hierarchy. Current limits on 〈m〉 are around 0.3 eV, while future experiments areaiming at entering the regime of the inverted hierarchy by improving current lifetime limitson neutrinoless ββ decay by an order of magnitude.Regarding other neutrino mass observables, KATRIN [14, 15] will not be able to see asignal if our scheme holds, whereas in cosmology [16] there could be detection in sophisti-cated future surveys, since the sum of the light-neutrino masses ism1 +m2 ≈ 2√∆m2atm ≈ 0.1 eV . (14)We display the phenomenology of our scenario in Fig. 1, which confirms our analyticalexpressions. Again, we stress the interesting correlation between the CP -violating Diracphase δ and the atmospheric neutrino mixing angle θ23.Next we turn to model realizations of the scenario under study. We shall present threesuch models. We use the type-I seesaw mechanism with only two right-handed neutrinosνR1,2; this immediately ensures the existence of one massless neutrino. In the first modelwe need two Higgs doublets φ1,2 and a complex scalar gauge singlet S. Let there bea Z4 symmetry under which the fields—including the left-handed lepton doublets DLα(α = e, µ, τ)—transform asDLe → −iDLe , DLµ → −iDLµ , eR → −eR , µR → −µR , (15a)νR1 → −iνR1 , φ1 → iφ1 , S → −iS . (15b)4The Yukawa couplings D¯Leφ1eR and D¯Lµφ1µR give mass to the electron and the muonwhile D¯Lτφ2τR gives mass to the τ .6 The Yukawa couplings to the right-handed neutrinosareLYukawa = −y6 ν¯R1Cν¯TR2 S −(y1D¯Le + y2D¯Lµ)νR1 φ˜2−(y4D¯Le + y5D¯Lµ)νR2 φ˜1 − y3D¯Lτ νR2 φ˜2 +H.c., (16)where y1–6 are coefficients, C is the charge-conjugation matrix in Dirac space, and φ˜k =iσ2φ∗k for k = 1, 2. There is also a bare Majorana mass termLMajorana = −m2ν¯R2 Cν¯TR2 +H.c. (17)Therefore, the neutrino mass matrices, in the notationLνmass = −ν¯RMD νL −12ν¯RMRCν¯TR +H.c., (18)areMD =(y∗1v2 y∗2v2 0y∗4v1 y∗5v1 y∗3v2), MR =(0 y6sy6s m), (19)where s = 〈0 |S| 0〉 and vk = 〈0 |φ0k| 0〉. The effective light-neutrino mass matrix isM = −MTD M−1R MD=−1y26s2 y∗1v2 y∗4v1y∗2v2 y∗5v10 y∗3v2( −m y6sy6s 0)(y∗1v2 y∗2v2 0y∗4v1 y∗5v1 y∗3v2), (20)which evidently has Mττ = 0.A simpler model, which dispenses with the singlet S, is the following. Let there be asymmetry under whichDLe → σDLe DLµ → σDLµ , DLτ → σ−1DLτ , (21a)eR → σ3eR , µR → σ3µR , τR → σ−1τR , (21b)νR1 → σνR1 , νR2 → σ−1νR2 , φ1 → σ−2φ1 . (21c)where |σ| = 1 and σ4 6= 1. Then the Yukawa couplings to the right-handed neutrinos areLνYukawa = −(y1D¯Le + y2D¯Lµ)νR1 φ˜2−(y4D¯Le + y5D¯Lµ)νR2 φ˜1 − y3D¯Lτ νR2 φ˜2 +H.c., (22)6Notice that, without loss of generality, we may assume the charged-lepton mass matrix already tobe diagonal, since its diagonalization only amounts to a redefinition of (DLe, DLµ) and of (eR, µR) inEq. (15a).5where y1–5 are coefficients. There is a bare Majorana mass termLMajorana = −m ν¯R1 Cν¯TR2 +H.c. (23)Defininga ≡ y∗1v2 , b ≡ y∗2v2 , c ≡ y∗3v2 , d ≡ y∗4v1 , f ≡ y∗5v1 , (24)gives for the mass matrixM = −1m a db f0 c( 0 11 0)(a b 0d f c)= −1m 2ad af + bd acaf + bd 2bf bcac bc 0 . (25)This M contains five physical parameters,∣∣∣∣adm∣∣∣∣ ,∣∣∣∣ ba∣∣∣∣ ,∣∣∣∣fd∣∣∣∣ ,∣∣∣ cd∣∣∣ , arg (afb∗d∗) . (26)These parameters can be computed from the matrix elements of M , and thus from theneutrino masses and mixings. Notice for instance that∣∣∣∣y1y2∣∣∣∣ =∣∣∣∣MeτMµτ∣∣∣∣ ,∣∣∣∣y4y5∣∣∣∣ =∣∣∣∣MeeMµτMµµMeτ∣∣∣∣ . (27)Another example of a model leading to Mττ = 0 uses S3 and Z3 discrete groups asflavour symmetries. The multiplication rules of S3 can be found for instance in Ref. [11].The particle content and group assignments are given in Table 2.(DLe, DLµ) DLτ (νR1, νR2) eR µR τR (ξ1, ξ2) χS3 2 1′ 2 1 1 1′ 2 1Z3 1 1 ω ω2 ω 1 ω ωTable 2: Field content of our third model. The standard-model Higgs doublet φ, whichis invariant under both S3 and Z3, has been omitted. The fields ξ1,2 and χ are ‘familons’,i.e. auxiliary scalar gauge singlets. We use ω ≡ exp (i2pi/3).At leading order in the inverse power of the cutoff scale Λ, the neutrino Yukawa cou-plings and the Majorana masses are given byLν = −g12(ν¯R1 Cν¯TR1 ξ∗1 + ν¯R2 Cν¯TR2 ξ∗2)− g2 ν¯R1 Cν¯TR2 χ∗ (28a)−h1(D¯Le νR1 + D¯Lµ νR2)χ∗φ˜Λ(28b)−h2(D¯Le νR2 ξ∗1 + D¯Lµ νR1 ξ∗2) φ˜Λ(28c)−h3 D¯Lτ (νR1 ξ∗1 − νR2 ξ∗2)φ˜Λ+ H.c., (28d)6where g1,2 and h1,2,3 are dimensionless constants. After ξ1,2 and χ obtain vacuum expecta-tion values according to the alignment [17](ξ1ξ2)→(〈ξ1〉0), χ→ 〈χ〉, (29)and after electroweak symmetry breaking with v = 〈φ˜〉, the mass matrices at low energytake the formsMD =a b0 ac 0 , MR =(MA MBMB 0), (30)where a = h1〈χ〉∗v/Λ, b = h2〈ξ1〉∗v/Λ, c = h3〈ξ1〉∗v/Λ, MA = g1〈ξ1〉∗ and MB = g2〈χ〉∗.Performing the seesaw diagonalization, one finds the effective Majorana mass matrixM =1MB2ab a2 bca2 0 acbc ac 0− MAM2Bb2 ab 0ab a2 00 0 0 . (31)The charged-lepton mass sector is written through a combination of higher-dimensionaloperators and a renormalizable operator. At leading order in 1/Λ, it is given byLl = −hτ D¯Lτ τR φ (32a)−he(D¯Le ξ1 + D¯Lµ ξ2)eRφΛ(32b)−hµ(D¯Le ξ∗2 + D¯Lµ ξ∗1)µRφΛ+ H.c., (32c)where he,µ,τ are dimensionless constants. In the vacuum configuration of Eq. (29), thecharged-lepton mass matrix is diagonal:Ml = diag(heΛ〈ξ1〉,hµΛ〈ξ1〉∗, hτ)v∗. (33)The hierarchy between me,µ and mτ is naturally explained by the suppression by the cutoffscale.A dark matter candidate is easily accommodated in this model. For example, we canreplace the S3 charges of DLτ and τR with 1 and introduce a third right-handed neutrino,νR3, which is 1′ of S3 and ω of Z3. Then, νR3 has no Yukawa couplings, so that it is stable.In the early Universe, νR3 can communicate with the thermal plasma via the s-channelexchange of the scalar fields, since it has a νR3νR3χ vertex and the real part of χ mixeswith the usual Higgs field. In this setup, it is known [18] that the observed relic density iseasily obtained without contradicting the Higgs properties observed at LHC and the directdetection bounds.7In summary, a simple texture-zero scenario can accommodate the values of the CPphase δ around pi and of the atmospheric mixing angles θ23 sizably less than pi/4. Theframework makes predictions in the form of an inverted hierarchy with a massless neutrinoand a strong cancellation in neutrinoless ββ decay. Simple models are possible to realizethe scenario.AcknowledgmentsThe work of LL is supported through the Marie Curie Initial Training Network “UNILHC”PITN-GA-2009-237920 and also through the projects PEst-OE-FIS-UI0777-2011, PTDC-FIS-098188-2008, and CERN-FP-123580-2011 of the Portuguese FCT. The work of WRwas supported by the Max Planck Society through the Strategic Innovation Fund in theproject MANITOP.References[1] G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo, and A. M. Rotunno, Phys.Rev. D 86 (2012) 013012 [arXiv:1205.5254 [hep-ph]].[2] D. V. Forero, M. To´rtola, and J. W. F. Valle, Phys. Rev. D 86 (2012) 073012[arXiv:1205.4018 [hep-ph]].[3] M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado, and T. Schwetz, J. High Energy Phys.1212 (2012) 123 [arXiv:1209.3023 [hep-ph]].[4] J. Liao, D. Marfatia, and K. Whisnant, arXiv:1306.4659 [hep-ph].[5] C. H. Albright and W. Rodejohann, Eur. Phys. J. C 62 (2009) 599 [arXiv:0812.0436[hep-ph]].[6] P. H. Frampton, S. L. Glashow, and D. Marfatia, Phys. Lett. B 536 (2002) 79[hep-ph/0201008].[7] A. Merle and W. Rodejohann, Phys. Rev. D 73 (2006) 073012 [hep-ph/0603111].[8] W.-L. Guo, Z.-Z. Xing, and S. Zhou, Int. J. Mod. Phys. E 16 (2007) 1[hep-ph/0612033].[9] E. I. Lashin and N. Chamoun, Phys. Rev. D 85 (2012) 113011 [arXiv:1108.4010 [hep-ph]].[10] P. O. Ludl, S. Morisi, and E. Peinado, Nucl. Phys. B 857 (2012) 411 [arXiv:1109.3393[hep-ph]].[11] W. Rodejohann, M. Tanimoto. and A. Watanabe, Phys. Lett. B 710 (2012) 636[arXiv:1201.4936 [hep-ph]].8[12] W. Grimus and P. O. Ludl, JHEP 1212 (2012) 117 [arXiv:1209.2601 [hep-ph]].[13] W. Rodejohann, J. Phys. G: Nucl. Part. Phys. 39 (2012) 124008 [arXiv:1206.2560[hep-ph]].[14] G. Drexlin, V. Hannen, S. Mertens, and C. Weinheimer, Adv. High Energy Phys.2013 (2013) 293986 [arXiv:1307.0101 [physics.ins-det]].[15] D. S. Parno, arXiv:1307.5289 [physics.ins-det].[16] K. N. Abazajian et al., Astropart. Phys. 35 (2011) 177 [arXiv:1103.5083 [astro-ph.CO]].[17] N. Haba and K. Yoshioka, Nucl. Phys. B 739 (2006) 254 [hep-ph/0511108].[18] L. Lopez-Honorez, T. Schwetz, and J. Zupan, Phys. Lett. B 716 (2012) 179[arXiv:1203.2064 [hep-ph]].9Figure 1: Scatter plots displaying the correlations among some neutrino mass and mix-ing observables following from Eq. (1). The horizontal and vertical full lines display the1σ bounds of Ref. [1] on the corresponding observables; the dashed lines display the 3σbounds. The blue (dark) points were found by allowing all the observables to lie withintheir corresponding 1σ ranges of Table 1; the yellow (light) points pertain to the 3σ ranges.〈m〉min and 〈m〉max are the minimal and maximal values of the effective mass in the invertedhierarchy with vanishing neutrino mass.10

Reproducing lepton mixing in a texture zero model

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