We give predictions for the neutrinoless double beta decay rate in a simple
variant of the A_4 family symmetry model. We show that there is a lower bound
for the neutrinoless double beta decay amplitude even in the case of normal
hierarchical neutrino masses, corresponding to an effective mass parameter
|m_{ee}| >= 0.17 \sqrt{\Delta m^2_{ATM}}. This result holds both for the CP
conserving and CP violating cases. In the latter case we show explicitly that
the lower bound on |m_{ee}| is sensitive to the value of the Majorana phase. We
conclude therefore that in our scheme, neutrinoless double beta decay may be
accessible to the next generation of high sensitivity experiments.Comment: 4 pages, 5 figures, 1 tabl

A. Villanova del Moral

Ernest Ma

J. W. F. Valle

M. Doi

M. Hirsch

T. Yanagida

English

arXiv

Crossref

Predicting neutrinoless double beta decay

We give predictions for the neutrinoless double beta decay rate in a simple variant of the A(4) family symmetry model. We show that there is a lower bound for the beta beta(0 nu) amplitude even in the case of normal hierarchical neutrino masses, corresponding to an effective mass parameter parallel to m(ee)parallel to >= 0.17 root Delta m(ATM)(2). This result holds both for the CP conserving and CP violating cases. In the latter case we show explicitly that the lower bound on parallel to m(ee)parallel to is sensitive to the value of the Majorana phase. We conclude therefore that in our scheme, beta beta(0 nu) may be accessible to the next generation of high sensitivity experiments

Furtado Valle, José Wagner

Hirsch, Martin

Ma, Ernest

Villanova del Moral, A.

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

arXiv:hep-ph/0507148v1  12 Jul 2005IFIC/05-32Predicting Neutrinoless Double Beta DecayM. Hirsch,∗ A. Villanova del Moral,† and J. W. F. Valle‡AHEP Group, Instituto de F´ısica Corpuscular – C.S.I.C./Universitat de Vale`nciaEdificio Institutos de Paterna, Apt 22085, E–46071 Valencia, SpainErnest Ma§Physics Department, University of California, Riverside, CA 92521, USA, andInstitute for Particle Physics Phenomenology, University of Durham, Durham, DH1 3LE, UK(Dated: September 24, 2013)We give predictions for the neutrinoless double beta decay rate in a simple variant of the A4family symmetry model. We show that there is a lower bound for the ββ0ν amplitude even in thecase of normal hierarchical neutrino masses, corresponding to an effective mass parameter |mee| ≥0.17√∆m2atm. This result holds both for the CP conserving and CP violating cases. In the lattercase we show explicitly that the lower bound on |mee| is sensitive to the value of the Majoranaphase. We conclude therefore that in our scheme, ββ0ν may be accessible to the next generation ofhigh sensitivity experiments.PACS numbers: 11.30.Hv, 14.60.Pq, 12.60.Fr, 12.60.-i, 23.40.-sAre neutrinos their own antiparticles? This is one ofthe most basic current unknowns in neutrino physics. Ofall possible manifestations of neutrino masses, neutrino-less double beta decay (ββ0ν , for short) offers – to date– the only potentially viable way to answer this question.If ββ0ν exists, then neutrino masses are Majorana in na-ture, irrespective of their ultimate origin [1]. Togetherwith cosmology [2] and direct kinematical searches in tri-tium decay [3], ββ0ν offers one of the three main com-plementary ways to probe the absolute scale of neutrinomasses.Current experimental limits from ββ0ν on the effec-tive Majorana mass of the neutrino mee are of ordermee ≤ 0.3 − 1 eV [4, 5]. We note that a claim for afinite ββ0ν rate has been published in [6], but this hasso far not been confirmed by any other experiment. Arecent proposal [7] aims explicitly at testing the half-life range suggested in [6]. However, future ββ0ν exper-iments may be able to reach down to much lower massscales. Many experiments sensitive to mee ≃ 0.05 eVhave already been discussed; see for example [8, 9]. Inthe longer-term future, even mee ≃ 0.01 eV [10, 11] doesnot seem impossible.The historic confirmation of neutrino oscillations overthe last few years [12], together with some basic the-ory, suggests that ββ0ν is expected, although in generalno lower bound on the magnitude of the expected ef-fect can be given. Theoretical input is therefore needed.Currently the origin of neutrino masses is completely un-known. The basic dimension–five operator which leadsto neutrino masses [13] can arise from a variety of mech-∗Electronic address: Martin.Hirsch@ific.uv.es†Electronic address: Albert.Villanova@ific.uv.es‡Electronic address: valle@ific.uv.es§Electronic address: ma@phyun8.ucr.eduanisms characterized by vastly different scales. Alter-natives include the seesaw mechanism [14, 15, 16] andlow–energy R−parity violating supersymmetry [17]. Inneither case is it possible, in general, to establish a lowerbound on the magnitude of the ββ0ν rate.Here we consider a simple phenomenological modelbased on a new realization of the A4 family symme-try [18, 19, 20] in which no SU(3)⊗ SU(2)⊗U(1) sin-glet neutrinos are introduced. Instead, the small neutrinomasses arise from the small induced vacuum expectationvalues (VEVs) generated for the neutral components oftriplet Higgs bosons [15, 21], transforming nontriviallyunder the A4 family symmetry. The lepton and Higgsparticle content and their transformation properties un-der A4 and SU(2) ⊗ U(1) are specified in Table I. WithFields L lc φ1 φ2 φ3 η1 η2 η3 ξA4 3 3 1 1′1′′1 1′1′′3SU(2)L 2 1 2 3 3Y –1 2 –1 2 2TABLE I: Lepton and scalar boson quantum numbersthese transformation properties, the charged lepton massmatrix is already diagonal in the flavor basis, withme = h1v1 + h2v2 + h3v3mµ = h1v1 + ωh2v2 + ω2h3v3mτ = h1v1 + ω2h2v2 + ωh3v3where hi are charged lepton Yukawa couplings, vi = 〈φ0i 〉and ω is a complex cubic root of unity satisfying 1+ω+ω2 = 0. The neutrino mass matrix is then of the formMν = a+ b+ c f ef a+ ωb+ ω2c de d a+ ω2b + ωc (1)2where the only non-diagonal entries are those of the A4triplet ξ. Here we have defineda = λ1〈η01〉b = λ2〈η02〉c = λ3〈η03〉d = κ〈ξ01〉e = κ〈ξ02〉f = κ〈ξ03〉(2)where λi, κ and 〈η0i 〉, 〈ξ0i 〉 are triplet Yukawa couplings,and VEVs, respectively. Let us further assume thatthe conditions b = c, and d = e = f hold. Whereasthe former is an ad hoc assumption, the latter canbe maintained naturally because of a residual Z3symmetry. Note that Mν has a very remarkable (andpossibly unique) property here in that each entry isrenormalized by the charged-lepton Yukawa couplingsin the same way, i.e. |h1|2 + |h2|2 + |h3|2, instead ofbeing proportional to m2i +m2j as in the Standard Model.It is easy to see that in the above limit we have thepredictionθ23 = pi/4, θ13 = 0which matches well with the neutrino oscillationdata [12]. Furthermore it can be shown that, in the limitwhere the solar mass splitting is neglected, b and d can bemade real, so that the atmospheric neutrino mass split-ting takes on a very simple form∆m232= 6bd ≡ ∆m2atm(3)The solar neutrino mass splitting ∆m2sol≪ ∆m2atmcanbe expressed as∆m221 =√T 21+ T 22+ T 23≡ ∆m2sol(4)whereT1 ≡ 6√2|b||d| sin(φ2) (5)T2 ≡ 2√2|d|(2|a| cos(φ1) + |b| cos(φ2) + |d|)(6)T3 = −3|b|2 + |d|2 − 6|a||b| cos(φ1 + φ2) (7)+ 2|a||d| cos(φ1)− 2|b||d| cos(φ2)with φ1 ≡ φa − φd and φ2 ≡ φd − φb, where a =|a| exp(iφa), b = |b| exp(iφb) and d = |d| exp(iφd). Thecondition in Eq. (4) leads to three inequalities|Ti| ≤ ∆m2solwhich, normalized by ∆m2atm, can be expressed in termsof the small parameter α ≡ ∆m2sol/∆m2atmas√2| sin(φ2)| ≤ α (8)√23|b|∣∣∣2|a| cos(φ1) + |b| cos(φ2) + |d|∣∣∣ ≤ α (9)16|b||d|∣∣∣− 3|b|2 + |d|2 − 6|a||b| cos(φ1 + φ2) (10)+2|a||d| cos(φ1)− 2|b||d| cos(φ2)∣∣∣ ≤ αwhere the current allowed values of α are shown inFig.13 of Ref. [12].As for the solar mixing angle, it is given here byt2S ≡ tan(2θ12) = 2√2d3b− d (11)which reduces totan2 θ12 = 1/2 (12)in two ways, namely,b = 0, b = 2d/3 (13)Current fits of solar, reactor, atmospheric and acceleratorneutrino oscillation data lead to a best fit point to thesolar mixing angle for which tan2 θ12 is slightly less than1/2. Performing a series expansion of tan2 θ12 around thetwo solutions in Eq. (13), we gettan2 θ12 ≃ 12+bd(14)tan2 θ12 ≃ 12− 1d(b− 23d)(15)These two branches are depicted in Fig. 1. The positive-0.2 0 0.2 0.4 0.6 0.8 1b/d0.30.40.50.6tan2(θ 12)Normal HierarchyInverse HierarchyFIG. 1: Solar mixing angle, tan2 θ12, vs. b/d.horizontal axis corresponds to a normal hierarchy (NH)neutrino spectrum, while the negative part correspondsto inverse hierarchy (IH), as depicted in Fig. 1. Thisbehavior is also recognized in the model of [22].We now turn to neutrinoless double beta decay. Theneutrino–exchange amplitude for this process is simplygiven by〈mν〉 = mee = a+ 2b (16)In the case of real parameters, we can solve the following3system of equations∆m2sol≡ ∆m221= |2a+ b+ d|√(d− 3b)2 + 8d2∆m2atm≡ ∆m232 = 6bdt2S ≡ tan(2θ12) = 2√2d3b− dand express the parameters a, b and d in terms of ex-perimentally measurable ones ∆m2sol, ∆m2atmand t2S .Substituting in Eq. (16) we can therefore express 〈mν〉in terms of these measured observables. We then obtain,up to an overall sign,mee√∆m2atm= Sign[∆m2atm]1√2√2t2S + t22S(17)±Sign[2√2t2S + t22S ]α√2√2t2S + t22S4√1 + t22SThe calculated values of |mee/√∆m2atm| as functions oft2S according to Eq. (17) are shown in Fig. 2. It can2 2.5 3 3.5 4 t2S0.10.20.512510mee / √∆m2 ATM Normal HierarchyInverse Hierarchy2 2.5 3 3.5 4 t2S0.10.20.512510mee / √∆m2 ATM Normal HierarchyInverse HierarchyFIG. 2: |mee/√∆m2atm| vs. |t2S | according to Eq. (17). Thebound depends slightly on the value of α: the left panel cor-responds to α = 0.022 and the right one to α = 0.065. Thedark (red) lines correspond to normal hierarchy, while the grey(green) line is for the inverse hierarchy case. The vertical linecorresponds to the horizontal line in Fig. 1 at tan2 θ12 = 1/2.be seen that given the currently allowed experimental 3σrangetan2 θ12 ∈ [0.30, 0.61] (18)we can set lower bounds for |mee/√∆m2atm|:0.17 < |mee/√∆m2atm| for NH (19)0.70 < |mee/√∆m2atm| for IH (20)Note that the lower bound on ββ0ν for the NH case isespecially relevant, as it is absent in the generic case. Itresults from this specific realization of the A4 symmetryand is also in contrast with previous A4–based modelsthat led to quasi–degenerate neutrinos [19, 23].Moreover, should future precision experiments narrowdown the experimental range for tan2 θ12 then we mightbe able to distinguish between both neutrino mass hi-erarchies. For example, if tan2 θ12 ≤ 1/2 could ever beestablished, then we would have0.23 < |mee/√∆m2atm| < 0.41 for NH (21)0.70 < |mee/√∆m2atm| for IH (22)It can also be seen that, up to order α corrections, meecan only be zero if tan2 θ12 = 1, now strongly rejectedexperimentally. Note that the two solutions in Fig. 2correspond to the two branches depicted in Fig. 1. Onecan see that only in the branch corresponding to the so-lution b = 0 and for values of tan2 θ12 that are less than1/2, there is a relative minus sign between b and d, whichis the condition for inverse hierarchy, as can be seen fromEq. (3).In the general case of complex parameters, lowerbounds on |mee/√∆m2atm| can also be established foreach hierarchy. This task is simplified greatly by takinginto account the reliable approximations sinφ2 ≃ 0, and2|a|cosφ1+ |b|cosφ2+ |d| ≃ 0. From Fig. 3, we see that inthe complex case, lower bounds on |mee/√∆m2atm| areindeed also established for each hierarchy. By compar-ing Fig. 3 with the right panel in Fig. 2 we find thatthese lower bounds are in fact exactly the same as ob-tained in the real case. The robustness of these boundsis easily understood. It follows from the fact that themaximum degree of destructive interference between thethree neutrino–exchange contributions occurs in the realcase with appropriate CP parities [24].2 2.5 3 3.5 4 t2S0.10.20.512510mee / √∆m2 ATMFIG. 3: |mee/√∆m2atm| vs. |t2S | for α = 0.065 when complexparameters are allowed.The left-rising lines correspond to inverse hierarchy,while all others refer to the case of normal hierarchy. Thelower–lying pairs of solid lines correspond to the lowerbounds already discussed in Fig. 2. Again, the verticalline corresponds to tan2 θ12 = 1/2. Finally, the remain-ing lines correspond to non-zero values of the relevantCP-violating phase varying cos(φ1) over the range [0,1]in equally–spaced steps. It is conceptually interesting tonote that this phase is “Majorana type” [15, 25], as weare still considering the case θ13 = 0.4Last, but not least, we find it very interesting alsothat the lower bound on |mee/√∆m2atm| which we haveobtained depends on the value of the Majorana violat-ing phase | cos(φ1)|. We can see from Fig. 4 that thelower bounds on |mee/√∆m2atm| for each hierarchy be-come weaker for cos(φ1) = 1. Finally, we mention that,0 0.2 0.4 0.6 0.8 1cos(φ1)0.10.20.512510mee / √∆m2 ATMNormal HierarchyInverse HierarchyFIG. 4: Lower bound on |mee/√∆m2atm| vs. | cos(φ1)|, givencurrent t2S uncertainty. The lines in dark (red) and grey(green) correspond to normal and inverse hierarchy respec-tively.even though for simplicity we have focused on the caseb = c, all results can be generalized to the case b 6= c, inwhich case θ13 is allowed to be nonzero.In summary, we have given predictions for the neu-trinoless double beta decay rate in a simple hierarchicalvariant of the A4 family symmetry model. We showedthat there is a lower bound for the ββ0ν amplitude evenin the case of normal hierarchical neutrino masses. Wehave seen that the bound is robust as it holds irrespectiveof whether CP is conserved or not. In the latter case weshow explicitly how the lower bound on |mee| is sensitiveto the value of the Majorana phase. Our scheme suggeststhat neutrinoless double beta decay may be within reachof the next generation of high sensitivity experiments.This work was supported by the Spanish grantBFM2002-00345 and by the EC Human Potential Pro-gramme RTN network MRTN-CT-2004-503369. M. H.is supported by the Ramon y Cajal programme. A. V.M. is supported by Generalitat Valenciana. The work ofE.M. was supported in part by the U.S. Department ofEnergy under Grant No. DE-FG03-94ER40837.[1] J. Schechter and J. W. F. Valle, Phys. Rev. D25, 2951(1982); for a recent short review see B. Kayser, eConfC040802 (2004) L004 [hep-ph/0506165].[2] S. Hannestad, Nucl. Phys. Proc. Suppl. 145, 313 (2005),[hep-ph/0412181]; S. Pastor, hep-ph/0505148.[3] G. Drexlin, Nucl. Phys. Proc. Suppl. 145, 263 (2005).[4] L. Baudis et al., Phys. Rev. Lett. 83, 41 (1999)[hep-ex/9902014].[5] C. E. Aalseth et al. [IGEX Collaboration], Phys. Rev.D65, 092007 (2002) [hep-ex/0202026].[6] H. V. Klapdor-Kleingrothaus, A. Dietz, I. V. Krivosheinaand O. Chkvorets, Nucl. Instrum. Meth. A522, 371(2004) [hep-ph/0403018].[7] I. Abt et al., hep-ex/0404039.[8] Expression of Interest in the Super-NEMO double betadecay experiment http://nemo.in2p3.fr/supernemo/[9] R. Ardito et al., hep-ex/0501010.[10] R. Gaitskell et al. [Majorana Collaboration],nucl-ex/0311013.[11] D. Akimov et al., Nucl. Phys. Proc. Suppl. 138, 224(2005).[12] For an updated review of the status of global fits to neu-trino oscillations and references to the recent, thoughalready classic, experimental papers, see M. Maltoni,T. Schwetz, M. A. Tortola and J. W. F. Valle, NewJ. Phys. 6, 122 (2004), [hep-ph/0405172].[13] S. Weinberg, Phys. Rev. D22, 1694 (1980); E. Ma, Phys.Rev. Lett. 81, 1171 (1998), [hep-ph/9805219].[14] M. Gell-Mann, P. Ramond and R. Slansky, (1979),Print-80-0576 (CERN); T. Yanagida, (KEK lectures,1979), ed. Sawada and Sugamoto (KEK, 1979); see alsoP. Minkowski, Phys. Lett. B67, 421 (1977).[15] J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227(1980); Phys. Rev. D25, 774 (1982).[16] R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44,912 (1980); Phys. Rev. D23, 165 (1981); G. Lazarides,Q. Shafi and C. Wetterich, Nucl. Phys.B181, 287 (1981).[17] M. Hirsch and J. W. F. Valle, New J. Phys. 6, 76 (2004),[hep-ph/0405015].[18] E. Ma and G. Rajasekaran, Phys. Rev. D64, 113012(2001), [hep-ph/0106291].[19] K. S. Babu, E. Ma and J. W. F. Valle, Phys. Lett. B552,207 (2003), [hep-ph/0206292].[20] E. Ma, Phys. Rev. D70, 031901 (2004),[hep-ph/0404199]; G. Altarelli and F. Feruglio,hep-ph/0504165; E. Ma, hep-ph/0505209.[21] E. Ma and U. Sarkar, Phys. Rev. Lett. 80, 5716 (1998),[hep-ph/9802445].[22] S.-L. Chen, M. Frigerio, and E. Ma, hep-ph/0504181.[23] M. Hirsch et al, Phys. Rev. D69, 093006 (2004),[hep-ph/0312265].[24] L. Wolfenstein, Phys. Lett. B107, 77 (1981). J. Schechterand J. W. F. Valle, Phys. Rev. D24, 1883 (1981), Err.D25, 283 (1982).[25] M. Doi, T. Kotani, H. Nishiura, K. Okuda and E. Taka-sugi, Phys. Lett. B102, 323 (1981); S. M. Bilenky,J. Hosek and S. T. Petcov, Phys. Lett. B94, 49 (1980);J. Schechter and J. W. F. Valle, Phys. Rev. D23, 1666(1981); for a recent review see R. N. Mohapatra et al.,hep-ph/0412099.

Predicting Neutrinoless Double Beta Decay

Predicting Neutrinoless Double Beta Decay

Abstract

Similar works

Full text

Available Versions

Crossref

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