A Prediction from the Type III See-saw Mechanism

Simple unified models based on SO(10) and related groups can lead to the so-called “type III see-saw mechanism” for neutrino masses [1]. In the most general case the type III mechanism leads to a light neutrino mass matrix given by the formula Mν = −(MNH +H TMT N )(u/Ω), where MN is the Dirac mass matrix of the neutrinos, H is a dimensionless complex three-by-three matrix and u/Ω is the ratio of a weak-scale vacuum expectation value to a GUT-scale vacuum expectation value (VEV). In a subsequent paper the type III see-saw mechanism was shown to have certain advantages for leptogenesis, in particular allowing resonant enhancement without fine-tuning the form of neutrino mass matrices [2]. In the simplest case, where a minimal set of Higgs fields breaks B − L, one has H = I and the type III see-saw formula takes the simple form

Simple unified models based on SO(10) and related groups can lead to the so-called "type III see-saw mechanism" for neutrino masses [1]. In the most general case the type III mechanism leads to a light neutrino mass matrix given by the formula M ν = −(M N H + H T M T N )(u/Ω), where M N is the Dirac mass matrix of the neutrinos, H is a dimensionless complex three-by-three matrix and u/Ω is the ratio of a weak-scale vacuum expectation value to a GUT-scale vacuum expectation value (VEV). In a subsequent paper the type III see-saw mechanism was shown to have certain advantages for leptogenesis, in particular allowing resonant enhancement without fine-tuning the form of neutrino mass matrices [2]. In the simplest case, where a minimal set of Higgs fields breaks B − L, one has H = I and the type III see-saw formula takes the simple form The main problem in constructing predictive models of neutrino masses and mixings with the usual "type I" see-saw formula [3], M ν = −M N M −1 R M T N , is to relate the Majorana mass matrix of the right-handed neutrinos M R , with its six complex parameters, to measurable quantities.
There are very special models, such as the recently much studied "minimal SO(10) models", where there is such a relationship [4]. (For an exhaustive list of references on "minimal SO(10) models" see [5].) And the study of leptogenesis may tell us something about the structure of M R (although leptogenesis has only a single data point to work with). In general, however, the lack of information about M R is a problem for the predictivity of type I see-saw models. (The so-called "type II see-saw mechanism" [6] assumes the existence of SU (2) L -triplet Higgs fields with small VEVs that couple directly to ν L ν L . About the type II mechanism we have nothing to say in this paper.) What makes the simplest version of the type III formula, given in Eq. (1), so remarkable and appealing is that it does not involve the masses of the superheavy right-handed neutrinos at all.
As a consequence, the simplest type III formula opens the possibility of constructing models of quark and lepton masses that are extremely predictive. In particular, in models based on SO (10) or other groups that unify an entire family within a single multiplet, the Dirac mass matrix of the neutrinos M N is typically closely related by the grand-unification symmetries to the mass matrices (also of Dirac type, of course) of the up quarks, down quarks and charged leptons, which we will denote respectively as M U , M D , and M L . It is therefore possible in many models (for examples, see [7,8,9]) to predict the matrix M N from a knowledge of the masses and mixings of the quarks and the masses of the charged leptons. This would allow, if Eq. (1) holds, the complete prediction of the mass ratios and mixing angles of the neutrinos with no free parameters.
In this paper we will not be so ambitious. We have not found so far a full three-family model that is as predictive as that and where all the predictions (or "postdictions") are consistent with experiment. Rather, as an illustration of the possibilities of the type III framework, we will present here a simple ansatz for the heavier two families that is well motivated by group-theoretical considerations. This ansatz leads to two interesting predictions that are consistent with present experimental data. Before presenting the ansatz, we very briefly review the type III see-saw mechanism and formula.
In models based on SO(10), there are two ways that the right-handed neutrinos N c i (i = 1, 2, 3) can get mass, either through a renormalizable term such as 16 i 16 j 126 H , or through a higher-dimension effective operator such as 16 i 16 j 16 H 16 H /M GU T . The former allows automatic conservation of "matter parity", whereas the latter makes do with smaller multiplets of Higgs fields.
In the latter case, the effective d = 5 operator arises most simply from integrating out three or more SO(10)-singlets, which we will denote by 1 a or S a , that have the couplings F ia 16 i 1 a 16 H and (M S ) ab 1 a 1 b . If only the Standard-Model-singlet component of the 16 H has a non-zero VEV, and we denote it by Ω ∼ M GU T , then one has the familiar "double see-saw" mass matrix: By integrating out the superheavy fields N c i and S a , one obtains T . This is just the type I see-saw formula, with an effective M R . Now, if we assume that the SU (2) L -doublet Higgs field contained in 16 H also gets a non-zero VEV (and there is no fundamental reason why it should not), and we denote it by u, then the double see-saw mass matrix takes the form: In this case, it is easy to show that the effective mass matrix of the light neutrinos takes the form: where, as before, The first term is the usual type I see-saw contribution, and the second term is the type III see-saw contribution. (The origin of the type III term can be simply understood as follows. One can eliminate the νS and Sν entries in Eq. (3), i.e. the entries F u and F T u, by doing a rotation of the (ν i , N c i ) basis by an angle θ ∼ = tan θ = u/Ω. That reduces the matrix in Eq. (3) to the same form as Eq. (2), but with the zeros replaced by terms of the type III form.) Both the type I and the type III terms in Eq. (3) are formally of order M 2 W /M GU T . However, since the elements of M N are actually small compared to M W because of small Yukawa couplings (except perhaps for the third family), and since M N comes in quadratically in the type I term but only linearly in the type III term, one might expect the type III term to dominate for generic values of the parameters. Moreover, in the limit that the elements of M S are small compared to the GUT scale, the type I contribution becomes small. As was pointed out in [2], that is a good limit for the purposes of enhancing leptogenesis. It is therefore plausible that one can neglect the type I term, and we shall do so. Now let us turn to the ansatz for the various Dirac mass matrices. Suppose that these have the form (neglecting the small masses of the first family) where the "texture zero" in the 22 elements can be enforced by an abelian family symmetry, either discrete or continuous. We will say more on this later. And further suppose that the entries satisfy the conditions The relations given in Eq. (7) are not arbitrary, but follow from group-theory if the elements of the mass matrices come from no operators except of the following simple types: (1) 16 i 16 j 10 H , (7) is satisfied no matter how many operators there are of any of these types. Any operator of type (1) gives a = g, b = h, c = e, and d = f , thus satisfying Eq. (7). Any operator of type (2) gives contributions that are flavor-antisymmetric (since the 120 is in the antisymmetric product of two spinors). Consequently, it gives a + b = 0, c + d = 0, e + f = 0, and g + h = 0, thus also satisfying Eq. (7) in a trivial way. and so c + d = e + f , satisfying Eq. (7). In the same way it is easily seen that a + b = g + h.
Finally, consider an operator of type (4). One of the spinor Higgs fields (say the unprimed one) gets a superlarge VEV that breaks SO(10) down to SU (5). The effective operator that results is One might ask why we do not include the effects of operators of even higher dimension, such as 16 i 16 j 10 H 45 n H /M n GU T , which are not obviously smaller than the dimension-five operators that we included in our analysis, and which would not satisfy Eq. (7)  Given the simplest type III form (Eq. (1)), and the ansatz of Eqs. (5), (6), and (7), one has  [12,13].) Hence, we drop their running and allow these two inputs to vary within the experimentally allowed range and plot our predictions for m s /m b and m τ /m b as a function of them in Fig. 1.
We take the experimental values of the quarks from Ref. [14], except for m s for which we use the results of lattice calculations as given in Ref. [15] and double the error as suggested in Ref. [16].
The values of the CKM angles and the charged lepton masses are taken from PDG 2004 [17].
In presenting our results for m s /m b and m τ /m b in Fig. 1, we give the percentage by which the predicted GUT values differ from the RGE-evolved experimental central values.
In doing the renormalization group running we assume that all the sparticles have mass of 1 TeV. From M Z to 1 TeV, the running is done at one loop, assuming the Standard Model with two Higgs doublets. From 1 TeV to the GUT scale (taken to be 2 × 10 16 GeV) we do a two-loop running assuming the MSSM. The gauge coupling constants are taken from PDG 2004 [17]. We present one example of the RGE evolution in Table I.
It should be noted that, even with the assumption that we are making that the parameters a, b, c, Note that the value of a is very small. It is this that accounts for the smallness of m c /m t . One way that a might be small naturally (i.e. without fine-tuning) using only the set of operators that satisfy Eq.   The values of m s /m b that we predict are satisfyingly close to the experimental (lattice) results.
A couple of things should be noted in this regard. First, it was long thought that the Georgi-  A second point is that inclusion of the first family is likely to push up the predicted value of m s /m b by about 5%. The reason is that empirically the relation for the Cabbibo angle θ C ≃ m d /m s is known to work very well [19]. As is well-known, this formula arises naturally if the 11 element of the down quark mass matrix vanishes and the 12 and 21 elements are approximately equal [20]. But then diagonalizing the 12 block of the down quark mass matrix will push up the value of the 22 element by a factor of (1 + |m d /m s |).
In any event, we see that further improvement in the measurement of the θ atm , δm 2 atm , δm 2 sol ,