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 MN u=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 MT ijij, 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 MR ij Nj interaction gauge group SU 2 L U 1 Y.] 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 YR ij16i16j126H 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 YR ij16i16j16H16H=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. MR ijNi 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 MN ijiNj c malizable effective operator, must itself come from ‘‘in- YN ijiNj 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 2 L-triplet Higgs field, T, with mass mT of order MG that couples both to neutrinos [ YT ijijT] and to the SU 2 L U 1 Y- 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 MN ij 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 16i1m 16Ha, which contains c 0 no a priori reason why u should vanish. If it does not, then PimFim Ni Sm PimFim iSm u, 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 iSm u. Equation (4) then becomes 0 10 1 no longer proportional to each other. It is then not possible 0 M F u to null out the S block of M by a simple flavor- B N ij in C j c @ T A@ c A ÿ1 u= L mass i;Ni ;Sm MN ij 0 Fin Nj : independent rotation by angle tan , as in the spe- T T cial case discussed above. Consequently, the effective FmjuFmj Mmn Sn light-neutrino mass matrix is more complicated. In the (7) ~ ÿ1 ~ T most general case it can be written M ÿMNMR MN c 0 ÿ1 0T This can be simplified by a rotation in the iN plane by F u M F u , where MR is given by Eq. (6) as before, p i 0 ÿ1 T ÿ1 0 u c 2 c0 and M~ N MN F u M F . However, a great sim- angle tan u=p : i i ÿ Ni = 1 u= , Ni plification results if one assumes that the number of Nc u = 1 u= 2, which has the effect of elim- i i species of singlet fermions S is three, i.e., one for each inating the S entries. It also replaces the 0 in the entry m family. (If there were less than three, not all the Nc would by i get superlarge mass, and some of the light neutrinos III T u would have masses of order v.) With three species of M ÿ MN MN ; (8) 0 Sm, the matrices F and F are (generally) invertible and
101601-2 101601-2 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 10 12 MARCH 2004 0 1 then M MI MIII, where MI is given as before by 00 @ A Eq. (5) and MN 00 mU; (10) u 0 ÿ 1 MIII ÿ M H HTMT ;H F0Fÿ1 T: (9) N N 0 0 ÿ5 In this the generalized type III seesaw formula, the di- wherep mU mt, mu=mt 0:6 10 ,and 0 0 mensionless 3 3 matrix H introduces many unknown 3 mc=mt 0:14. (Superscripts here refer to quantities parameters, more, indeed, than does MR in the type I evaluated at MG.) In this model there is a very large seesaw. However, as we shall now show by an example, in (namely, tanÿ11:8) contribution to the atmospheric neu- SO 10 models it may be easier to obtain a realistic trino mixing angle coming from the charged lepton mass pattern of neutrino masses and mixings without fine- matrix ML, which is completely known. However, as MR tuning the parameters in the generalized type III seesaw is not known, it is impossible to predict the neutrino mass than in the type I seesaw. ratios and the other neutrino mixing angles (or even the The SO 10 model of Ref. [5] gives an excellent fit to atmospheric angle precisely) within the framework of the the quark masses and mixings and the charged lepton type I seesaw. Nevertheless, one can ask whether what we masses, fitting 13 real quantities with eight real parame- know about these neutrino masses and mixings can be ters. This fit uniquely determines the neutrino Dirac mass accommodated in the model with a reasonable form for ÿ1 ÿ1 matrix (at the unification scale) to be MR. Parametrizing that matrix by MR ij aijmR ÿ1 ajimR , the type I seesaw formula gives 0 1 2 a11 a13 a13 ÿ a12 @ 2 A 2 M a13 a33 a33 ÿ a23 mU=mR: (11) 2 a13 ÿ a12 a33 ÿ a23 a33 ÿ 2a23 a22 Neglecting the relatively small first row and column, the condition that the ratio m2=m3 of the two heaviest neu- addition of just four new parameters was used in [6] to trino masses be equal to some value r is that obtain a very good fit to the large mixing angle (LMA) r 1 1 2 solution for the SO 10 model of [5]. Such fine-tuning is a a ÿ a2 a ÿ 2a a a : 22 33 23 1 r2 2 33 2 23 22 33 typically required in SO 10 models relying on the type I seesaw mechanism [7]. (12) It can be seen from Eq. (11) that to fit the LMA solar 4 ÿ4 2 2 2 It is evident that r naturally is of order 4 10 .For solution a11 = , a12 =,anda13 =. Thus, 0 r to be of order (as indicated by experiment, which the correlation between the hierarchies of MR and MN gives r 1=6) the elements must be somewhat ‘‘tuned.’’ extends also to the first family. ÿ1 0 For example, setting a23=a33 p O and By contrast, a satisfactory pattern of neutrino masses ÿ2 ÿ1 a22=a33 q O , Eq. (12) gives the condition and mixings can be achieved without any fine-tuning in 1 2p q 0. In other words, not only must the 23 this model if the type III seesaw mechanism dominates. block of MR have a hierarchy that is correlated with the There are two interesting cases. Suppose, first, that all the 0 hierarchy of the 23 block of MN, but it must also satisfy a elements of F and of F are of order f, a dimensionless nontrivial numerical relation among its elements. This parameter of order or smaller than 1. Then all the ele- 0 ÿ1 T kind of mild fine-tuning of the 23 block of MR with the ments of H F F will be of order one. From Eq. (9), neglecting terms of order , 0 1 0 H H ÿ H 31 31 21 m u M @ H 2H H H ÿ H A U : (13) 31 32 32 33 22 H31 ÿ H21 H32 H33 ÿ H22 2 H33 ÿ H23 Here it is clear that without any fine-tuning jrj (jm2=m3j) is somewhat less than one, as desired. More MG, something that is impossible in the type I seesaw. 2 1 2 precisely, ÿr= 1 r 4 H32=H33 O . Moreover, This is an attractive possibility, but would create problems the LMA solution naturally emerges. For Ue3 to be con- for thermal leptogenesis [8]. 0 sistent with present limits, H21 must approximately A second interesting case is that F and F both have the cancel H31 in the 13 and 31 elements of M. However, form all the other elements of H can be of order one. 0 1 2 Note that a satisfactory pattern of light-neutrino = = = 0 @ A masses and mixings emerges with no hierarchy among F; F = 11; (14) the superheavy neutrinos, which all have masses of order = 11
101601-3 101601-3 PHYSICAL REVIEW LETTERS week ending VOLUME 92, NUMBER 10 12 MARCH 2004 as might arise naturally if the first family of both 16i and Majorana neutrinos with masses approximately given by 1 1 1i had a different Abelian family charge than the other M1 2 M11 and ÿM1 2 M11. No special constraint on 0 families. Then H has the form the forms of Fij, Fij,orHij is required to have this near 0 1 degeneracy condition satisfied except that M be small 1 = = ij compared to F in the original basis (a condition that H @ = 11A: (15) ij also makes the type III contribution to M dominant). = 11 Hence, no clash between the requirements of leptogenesis M Therefore, by Eq. (9), M has the form and of realistic occurs. Indeed, in Ref. [10] it is shown 0 1 that in the realistic model of Ref. [5], which we have been m u using as an example, sufficient leptogenesis can be ob- M @ 1 A U ; (16) tained without any fine-tuning in the type III seesaw. 11 that is, the same form as the previous case, except that Ue3 is automatically of order . The superheavy neutrinos do not consist here of three Majorana fermions, as in the standard type I seesaw [1] Super-Kamiokande Collaboration, M. Shiozawa, in mechanism, but of six Majorana fermions that combine Neutrino 2002: Proceedings of the International (if M is small compared to F , as we are assuming) to Conference on Neutrino Physics and Astrophysics, ij ij Munich, Germany (North-Holland, Amsterdam, 2002). form three pseudo-Dirac pairs. In the basis where Fij c [2] Super-Kamiokande Collaboration, nucl-ex/0309004. is diagonal, there areP the mass terms M1N1S1 c c [3] M. Gell-Mann, P. Ramond, and R. Slansky, in M2N2S2 M3N3S3 ijMijSiSj, where M1 F11 Supergravity, Proceedings of the Workshop, Stony Brook, 2 8 = MG 10 GeV,andM2 F22 and M3 F33 New York, 1979, edited by P. van Nieuwenhuizen and 16 are of order MG 10 GeV. The lightest of these states D. Z. Freedman (North-Holland, Amsterdam, 1979), are of sufficiently small mass to allow thermal lepto- p. 315; T. Yanagida, in Proceedings of the Workshop on genesis with a reheating temperature that is low enough Unified Theory and the Baryon Number of the Universe, to avoid the cosmological gravitino problem. Tsukuba, Japan, 1979, edited by O. Sawada and The type III seesaw has a feature that is advantageous A. Sugramoto (KEK, Ibaraki, Japan, 1979), p. 95; for leptogenesis in certain types of models. It is often R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. necessary in order to get sufficient leptogenesis in real- 44, 912 (1980); S. L. Glashow, in Quarks and Leptons, Carges, 1979,editedM.Levyet al. (Plenum, New York, istic SO 10 models for there to be a resonant enhance- 1980), p. 707. ment [9] caused by the two lightest ‘‘right-handed’’ [4] G. Lazarides, Q. Shafi, and C. Wetterich, Nucl. Phys. neutrinos forming a pseudo-Dirac pair (i.e., equivalently, B181, 287 (1981); R. N. Mohapatra and G. Senjanovic, a pair of Majorana fermions with nearly equal and oppo- Phys. Rev. D 23, 165 (1981). site mass). Having such a pseudo-Dirac pair in the stan- [5] C. H. Albright and S. M. Barr, Phys. Lett. B 452, 287 dard type I seesaw imposes a nontrivial constraint on the (1999); C. H. Albright, K. S. Babu, and S. M. Barr, Phys. form of the matrix MR. This constraint can clash with Rev. Lett. 81, 1167 (1998). what is required in order to get a realistic M through the [6] C. H. Albright and S. M. Barr, Phys. Rev. D 64, 073010 type I seesaw formula. Indeed, this is the case in the (2001). realistic fermion mass model of Ref. [5] that we have [7] S. M. Barr and I. Dorsner, Nucl. Phys. B585, 79 (2000). been using as an illustration: a severe fine-tuning of M is [8] W. Buchmu¨ller, P. Di Bari, and M. Plu¨macher, Nucl. R Phys. B665, 445 (2003); E. Akhmedov, M. Frigerio, required to have both satisfactory leptogenesis and real- and A.Yu. Smirnov, J. High Energy Phys. 0309 (2003) istic M, as shown in Ref. [10]. In the type III seesaw, on 021. the other hand, there are pseudo-Dirac pairs of neutrinos [9] M. Flanz, E. A. Paschos, U. Sarkar, and J. Weiss, Phys. automatically present. And, under the assumption stated Lett. B 389, 693 (1996); L. Covi and E. Roulet, Phys. above about the smallness of Mij, the pseudo-Dirac pair Lett. B 399, 113 (1997). c N1;S1 is slightly split, so that it is equivalent to two [10] C. H. Albright and S. M. Barr (to be published).
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