Atmospheric neutrino mixing and b-tau Unification

Extrapolating the tau and b masses in the MSSM tends to give for their ratio at the GUT scale a number around 1.2, for most viable values \tan \beta, rather than the minimal SU(5) prediction of 1. We suggest that this may be due to large off-diagonal elements in the charged lepton mass matrix M_L that can also explain the large atmospheric neutrino mixing angle. Several simple models with definite predictions for m_{tau}/m_b(M_{GUT}) are presented.

In minimal SU(5) and many other simple schemes of grand unification it is predicted that m b = m τ at the unification scale [1,2]. This relation is known to work rather well in the context of the MSSM [3,4]. That is, when the actual masses of the b and τ measured at low energy are extrapolated in the MSSM to M GU T , they are found to be nearly equal.
Nevertheless, for a wide range of tan β, m τ /m b (M GU T ) tends to be about 1.2 rather than 1.0, as is shown in Table I. In this paper we suggest that m τ may be slightly larger than m b at the GUT scale because of the large mixing in the lepton sector between the third family and the lighter families that is seen in atmospheric neutrinos.
First, let us review briefly the situation with regard to b-τ unification in the MSSM. While it is true that for most of the range of tan β the ratio m τ /m b (M GU T ) deviates substantially from unity, it is well known that this deviation is small if tan β is either close to m t /m b or close to 1 (the so-called high tan β and low tan β allowed regions) [5,6,7]. The reason these regions give better agreement with the minimal SU(5) prediction is that they correspond to certain Yukawa couplings becoming large enough to reduce the value of m b slightly through their effect on its renormalization group running. (For tan β ∼ = m t /m b , the bottom Yukawa coupling is nearly equal to that of top, and therefore of order one; and for tan β ∼ = 1, the top Yukawa coupling becomes greater than one due to the fact that v u = v(1 + (tan β) −2 ) −1/2 drops significantly below v.) The high tan β case is attractive from the viewpoint of minimal SO(10) unification, where the two MSSM Higgs doublets come from a single SO(10) multiplet, implying that the top and bottom Yukawa couplings are equal at the GUT scale [8]. However, in more realistic SO(10) models that attempt to fit the quark and lepton mass spectrum it is often not the case that the two MSSM doublets come purely from a single SO(10) multiplet, and consequently tan β becomes a free parameter [9,10]. Moreover, from the point of view of the MSSM, a value of tan β as large as m t /m b involves somewhat of a fine-tuning [11]. As far as the low tan β case is concerned, values of tan β very close to 1 are now excluded in the constrained MSSM [12,13].
Another way to save the minimal SU(5) prediction, besides assuming extreme values of tan β, is to invoke finite radiative corrections to m b coming from gluino and chargino loops (the gluino loop being typically the larger) [14,15]. Since there are contributions to these loops that are proportional to tan β, they can be substantial. For tan β ≈ 30, for instance, m b can easily receive a net correction of 15% to 20%, which, assuming it is negative, would restore agreement with minimal SU (5). However, this possibility is not without difficulties. The sign of the gluino and chargino loops are given, respectively, by sgn(µM 3 ) and −sgn(µA t ). Generally, to lower the value of m b and improve agreement with the minimal SU(5) prediction, one needs that sgn(µM 3 ) be negative, which would typically imply in minimal supergravity that sgn(µA t ) also is negative. However, if sgn(µA t ) is negative, then the chargino-loop contributions to the b → sγ amplitude adds constructively to the charged-Higgs-loop and standard model contributions, giving too large an effect unless sparticle masses are assumed to be large. (If sgn(µA t ) is positive, on the other hand, then the charged-Higgs-loop and the chargino-loop contributions to b → sγ tend to cancel each other, which is good since the experimental result for b → sγ is consistent with the SM prediction.) For a recent discussion of the one-loop SUSY corrections to m b and the constraints on them coming from b → sγ, see [16].
Thus, although there are ways to save the minimal SU(5) prediction for m τ /m b , perhaps the most conservative assumption is that this ratio is indeed slightly larger than unity at the GUT scale. In this paper we explore one reason why this might be so. We shall propose several versions of this idea that make definite predictions for m τ /m b (M GU T ). Eventually, if supersymmetry is discovered, it may be possible to determine the values of tan β and the parameters that are needed to compute the gluino-and chargino-loop corrections to m b , and thus pin down the value of m τ /m b (M GU T ) and test these models.
The basic idea of this paper is that m τ is made slightly larger than m b due to the same mixing effects in the lepton sector that produce the large neutrino mixing angle θ atm .
(Indeed, in some published models that explain the large atmospheric neutrino mixing, it does happen that m τ is slightly larger than m b at the GUT scale. For instance, by a factor of 1.04 in [9] and a factor 1.08 in [10]. These papers to some extent inspired the present work.) Suppose, for example, that the charged-lepton and down-quark mass matrices have these approximate forms at the GUT scale: where ρ ∼ 1. The zeros represent small elements that give the masses of the lighter two families. These matrices imply that m τ /m b ∼ = 1 + ρ 2 . The large off-diagonal element ρ in M L also produces a large mixing angle between τ and µ, namely tan θ ℓ 23 = ρ. If we assume that the neutrino mass matrix is nearly diagonal or hierarchical in this basis, so that the leptonic mixing angles come almost entirely from the charged lepton sector, then tan θ atm ∼ = tan θ ℓ 23 , implying that This kind of "lopsided" form for M L has been suggested by many groups as a simple way to explain the largeness of the atmospheric neutrino mixing angle [17,18,19,20]. A "bimaximal" (or perhaps it is better to say "bi-large") pattern of neutrino mixings can be elegantly explained by a simple extension of this idea [21], namely assuming that a whole column of M L is large: where ρ ′ ∼ ρ ∼ 1. Under the same assumption about the neutrino mass matrix, this gives and Jarlskog long ago [22]. The matrices of Eq. (1) would then become Making the same assumption that the atmospheric mixing is almost entirely coming from M L (i.e. in this basis M ν is nearly diagonal or hierarchical), we then have The same prediction results if this is generalized to bi-maximal mixing.
A way that a different Clebsch might arise is through an effective operator involving an adjoint Higgs (24) field. Specifically, suppose that there is a superheavy vectorlike pair of quark and lepton multiplets, denoted 5 ′ + 5 ′ , and that the superpotential contains the following couplings: Integrating out the heavy vectorlike fields, as shown in Fig We have been discussing b-τ unification in the context of SUSY SU(5) grand unification.
However, as is well known, four-dimensional SUSY SU(5) models typically have difficulties with natural doublet-triplet splitting and proton decay from dimension-5 operators [23].
(The former can be resolved by means of the missing partner mechanism, by introducing Higgs in 50 and 75 dimensional representations [24,25].) As recent papers have shown, these problems can be simply evaded in SUSY GUT models with one or two extra space dimensions compactified on orbifolds, the orbifold fixed points being three-branes [26,27] [28]. In light of these facts, it seems that a more satisfactory context in which to discuss b-τ unification may be five-dimensional models.
We will now discuss a simple five-dimensional model that implements the idea contained in Eq. (1). This model is very similar to the four-dimensional model of Eq. (6). As in that model, there is an extra vectorlike 5 + 5 pair of quark/lepton multiplets that, when integrated out, leads to the off-diagonal element ρ. However, in the five-dimensional model this vectorlike pair does not couple to an adjoint (24) of Higgs, as in Eq. (6), but feels the breaking of SU(5) simply due to the fact that it lives in the five-dimensional bulk.
Imagine an N = 1 supersymmetric SU(5) model in five dimensions, where the fifth dimension is compactified on S 1 /Z 2 × Z ′ 2 . The circle S 1 has coordinate y, with y ≡ y + 2πR. Under the first Z 2 , which maps y → 2πR − y, half of the supersymmetry charges are odd, so that N = 1 supersymmetry is left in the four dimensional effective theory. The second We assume that on the brane at O there are three families of quarks and leptons (i.e. three copies of 10 + 5), which we will give a family index i. In the bulk, in addition to the gauge fields of SU(5), which are in a vector multiplet of N = 1 5d supersymmetry, there are assumed to be two 5 + 5 pairs of hypermultiplets. One of these pairs of hypermultiplets, which we will denote 5 H + 5 H , contains the two Higgs doublets of the MSSM. The other pair of hypermultiplets, which we will denote 5 ′ + 5 ′ , contains extra vectorlike quark and lepton fields that mix with the three families living on the brane at O. In each of the hypermultiplets We see that this model is essentially a minimal SU(5) model in five dimensions, except for the presence of the fields 5 ′ + 5 ′ . The bulk fields are assumed to have the following couplings to the brane fields in the superpotential on the brane at O: Note the similarity to Eq.
There are also terms, not shown, that give mass to the lighter families of quarks and leptons.
In addition, there should be mass terms in the superpotential on O ′ for those components of the fields 5 ′ + 5 ′ that do not vanish on that brane, i.e. for the doublets in the Φ and the triplets in the Φ c : where, again, we define four-dimensional couplings by M 2 = π 2 RM 2 and M 3 = π 2 RM 3 . One may now proceed to integrate out the bulk quarks and leptons in 5 ′ + 5 ′ as shown in Fig. 2. If we take into account only the Kaluza-Klein zero modes in these fields, then integrating them out gives a contribution to M L but not M D , as only the weak doublets in where ρ = (λ 3 /λ 33 ) m The size ofρ compared to ρ depends on how large the masses m, M, M 2 and M 3 that appear in the superpotentials on O and O ′ are compared to the compactification scale M c = 1/R. And this, in turn, depends on where these masses are assumed to come from.
We assume that the five-dimensional model we are discussing is an effective theory below some cutoff scale M s . In that effective theory, one would expect that the dimensionless parameter M = π 2 RM in Eq. (8)  The structure shown in Eq. (10), and the extension of it with large (M L ) 13 , can be embedded in a model of the quark and lepton masses of all three families, as we will now show by a simple example. Suppose that there is a U(1) flavor symmetry, broken spontaneously by the vacuum expectation value of a flavon field, φ f , whose flavor charge is −1. Entries in the quark and lepton mass matrices that violate the U(1) charge by n units will thus "cost" n powers of the flavon field, and thus be proportional to a small symmetry-breaking parameter ǫ n . This is just the usual Froggatt-Nielson kind of scenario. Assign the U(1) charges as follows: 10 1 (+4), 10 2 (+2), 10 3 (0), 5 1 (+4), 5 2 (+4), If we define σ ∼ ρ 2 + ρ ′2 ∼ ǫ 3 , so that the atmospheric neutrino mixing angle is of order one, then we see that m µ ∼ ǫ 5 and m s ∼ ǫ 6 . So this kind of model, being much more lopsided in the charged lepton sector than in the down quark sector, naturally accounts for the fact that m µ is much larger than m s at the GUT scale. The Georgi-Jarlskog factor [22] (m s /m µ ) comes out naturally of order ǫ ∼ 0.2, though in this model it is not precisely predicted.
Note that in this particular model there are contributions to θ atm that are of order ǫ coming from the diagonalization of M ν . This means that no precise and testable relation of the kind we want exists between θ atm and m τ /m b , unless the model is augmented in some way as to make the O(1) coefficients c 33 , c 23 , etc. in Eq. (13) predictable or small.