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New Type of Seesaw Mechanism for Neutrino Masses

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New Type of Seesaw Mechanism for Neutrino Masses PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 10 12 MARCH 2004 New Type of Seesaw Mechanism for Neutrino Masses S. M. Barr Bartol Research Institute, University of Delaware, Newark, Delaware 19716, USA (Received 22 September 2003; revised manuscript received 18 November 2003; published 12 March 2004) In a wide class of unified models there is an additional (and possibly dominant) term in the neutrino T mass formula that under the simplest assumption takes the form M MN MNu=MG, where MN is the neutrino Dirac mass matrix, and u O MW. This makes possible highly predictive models. A generalization of this form yields realistic neutrino masses and mixings more readily than the usual seesaw formula in some models. The conditions for resonant enhancement of leptogenesis can occur naturally in such models. DOI: 10.1103/PhysRevLett.92.101601 PACS numbers: 14.60.Pq, 12.10.Dm, 98.80.Cq Grand unified theories (GUTs) provide an elegant ex- effective mass operator MTijij, with MT / 2 2 planation of the magnitude of neutrino masses. In GUTs, YThHui =mT v =MG. the exchange of fields whose mass is of order MG ’ 2 The type I and type II seesaw formulas can be under- 1016 GeV, the unification scale, can lead to light-neutrino stood as arising from block diagonalizing the complete 2 ÿ3 masses of order the ‘‘seesaw’’p scale v =MG ’ 10 eV. mass matrix of the neutrinos and antineutrinos: [The parameter of v 2G ÿ1=2 174 GeV is the F M M vacuum expectation value (VEV)which breaks the weak c T ij N ij j L mass i;Ni T c ; (2) M ij MRij Nj interaction gauge group SU 2L U 1Y.] The seesaw N scale accords well with the order of magnitude of the 2 with MR MG, MN v, MT v =MG,givingM neutrino masses inferredp from atmospheric and solar I II ÿ1 T 2 ÿ2 M M ÿMNMR MN MT, neglecting terms qneutrino oscillations ( matm ’ 5 10 eV [1] and higher order in v=MG. m2 ’ 8:5 10ÿ3 eV [2]). So far, two kinds of seesaw c sol In SO 10 models, the i and Ni both are contained in mechanisms have been widely discussed in the literature: the spinor multiplet 16i. Under the SU 5 subgroup that the conventional or type I seesaw [3] and the triplet or contains the standard model group, the i are in 5 and the type II seesaw [4]. These differ in the kinds of O M c G Ni are in 1. Thus, we write, in an obvious notation, i fields whose exchange leads to light-neutrino masses. In c 5 16i and Ni 1 16i. The mass matrix MR can arise in this Letter we observe that a third kind of seesaw mecha- two simple ways in SO 10, therefore. It can come from a nism can operate in a wide class of unified theories based renormalizable term of the form YRij16i16j126H or on SO 10 or larger groups. We argue that this ‘‘type III’’ from an effective nonrenormalizable term of the form seesaw mechanism, as we call it, leads to a form of the light-neutrino mass matrix, M, that may more readily fit O R YRij16i16j16H16H=MG: (3) the observed pattern of neutrino masses and mixings. The type III mechanism may also have advantages for lepto- Both possibilities are much discussed in the literature. In genesis. We will begin by reviewing type I and type II the former case, the type II mechanism can operate, along seesaw mechanisms. with type I, since the Higgs multiplet 126H contains the In the type I or conventional seesaw, there are ‘‘right- triplet T. In the latter case, however, one expects MT 0 c in Eq. (2), since the 16H does not contain the triplet T.Itis handed’’ neutrinos, Ni , which have O MG Majorana masses with each other, described by the term this latter case that can lead to the type III seesaw c c mechanism, as we shall now show. MRijNi Nj ,andO v Dirac masses with the ordinary c The operator shown in Eq. (3), since it is a nonrenor- neutrinos, described by the term MNijiNj c malizable effective operator, must itself come from ‘‘in- YNijiNj hHui.(i; j are family indices.) Figure 1 shows how the exchange of the Nc leads to effective masses for tegrating out’’ some heavy fields, as shown in the Fig. 2 i diagram. We assume for simplicity that the fields inte- the i. These masses are given by the famous type I seesaw formula grated out, denoted Sm,areSO 10 singlets, 1m, though I ÿ1 T M ÿMNMR MN: (1) In the type II seesaw, there is an SU 2L-triplet Higgs field, T, with mass mT of order MG that couples both to neutrinos [ YTijijT] and to the SU 2L U 1Y- FIG. 1. Diagram that gives the light neutrinos type I seesaw 2 breaking Higgs Hu. Integrating out the T leads to an masses of order v =MG. 101601-1 0031-9007=04=92(10)=101601(4)$22.50 2004 The American Physical Society 101601-1 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 10 12 MARCH 2004 neglecting, as always, terms higher order in v=MG.This is the type III seesaw contribution. Otherwise, the result- ing matrix has the same form as Eq. (4). Therefore, the full result for M is given by the sum of Eqs. (5) and (8). The relative size of the two contributions to M is model dependent. Since M is related to the up quark FIG. 2. Diagram that produces the effective operator N mass matrix M by SO 10, one would expect the entries 16 16 16 16 =M , which generates M . U i j H H G R for the first and second families to be very small com- pared to v. Consequently, because of the fact that MN I III nothing in the our later discussion depends on this. The comes in squared in M but only linearly in M , the latter III couplings needed to produce the diagram in Fig. 2 are should dominate, except perhaps for the third family. M evidently Mmn1m1n and Fim16i1m16H. Altogether, this would also dominate if the elements of Mmn were small gives the well-known ‘‘double seesaw’’ mass matrix compared to ’ MG, as Eqs. (5) and (6) show. 0 10 1 That MIII dominates is an interesting possibility, as 0 M 0 B N ij C j remarkably predictive SO 10 models of quark and lepton c @ T A@ c A L mass i;Ni ;Sm MNij 0 Fin Nj ; masses would then be constructable. Usually the most one 0 FT M S mj mn n can achieve in models where M is given by the type I (4) seesaw formula is predictions for the mass matrices of the up quarks, down quarks, and charged leptons MU;MD; h1 16 i i; j ; ; m; n where H , and where 1 2 3 and ML, and for the Dirac mass matrix of the neutrinos ; ;N N S 1 ... is the number of species of singlets m.Itis (MN), since these four matrices are intimately related to easy to show that the effective mass matrix M of the each other by symmetry. [For example, in the ‘‘minimal light neutrinos is given, up to negligible corrections SO 10 model’’ they all come from one term Yij16i16j10H higher order in v=MG,by and have exactly the same form.] However, sharp predic- ÿ1 T ÿ1 T tions for neutrino masses and mixings are hard to achieve M ÿMN F M F MN: (5) because of the difficulty in constraining the form of MR, In other words, one has the usual type I seesaw formula which comes from different terms. On the other hand, if with the type III seesaw contributions are dominant, then the matrix M is irrelevant; a knowledge of M and M is ÿ1 T R N L MR F M F : (6) sufficient to determine the neutrino mass ratios and mix- ing angles. The name double seesaw refers to the fact that MR arises c In this Letter we are not be so ambitious. Rather, we from a seesaw involving N and Sm, and then produces i look at a version of the type III seesaw that is less masses for the light neutrinos via a seesaw involving i c predictive but still has certain attractive features. In the and Ni . So far, we have only taken into account the VEVof the foregoing, we assumed that there was only a single 16 of Higgs fields that contributed to neutrino masses. If there SU 5-singlet component of the 16H, which we called . However, there is an SU 2 -doublet Higgs in the 5 16 is more than one,P then their coupling to neutrinos comes L H a that can have an O v VEV, which we shall call u. There is fromP the term aimP Fim 16i1m16Ha, which contains c 0 no a priori reason why u should vanish. If it does not, then PimFim Ni Sm PimFim iSmu, whereP Fim a 0 a 2 1=2 the term F 16 1 16 not only produces the O M mass aFimP a= , Fim aFimua=u, a a ,and im i m H G 2 1=2 c u ua . Then Eq. (7) is modified by having term Fim N Sm , but also an O v mass term a i Yukawa matrices in the S and NcS blocks that are Fim iSmu.
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