Yukawa Textures and the mu-term

We show how with an anomaly-free U(1), simple assumptions concerning the origin of Yukawa textures and the Higgs mu-term lead to the prediction of a new physics scale of 10^8GeV and automatic conservation of baryon number.


Introduction
One of the most appealing approaches to the fermion mass hierarchy problem is provided by the Froggatt-Nielsen (FN) mechanism [1]. According to FN, the hierarchy is produced from Yukawa textures produced by higher dimension terms involving MSSM singlet "flavon" fields θ via terms such as H 2 Q i u c j ( θ M θ ) a ij , where M θ represents the scale of new physics, and a ij = 0, 1, 2... We consider here the case when the MSSM gauge group is extended by a single U ′ 1 group which is broken by θ = 0. An exhaustive analysis of this general approach has been performed recently [2] by Dreiner and Thormeier (DT); this paper also contains a comprehensive list of references. Our assumptions here differ from DT in two critical respects: * We impose cancellation of all mixed U ′ 1 anomalies 2 without invoking the Green-Schwarz mechanism [4].
* We relax the assumption that there is only a single flavon field.
It might appear that our second assumption would rob us of most if not all predictive power; we will show however, that there is a very simple naturalness criterion which results in a constrained framework resulting in definite predictions. This arises as follows. Each Yukawa matrix Y u,d,e gains its texture from a particular flavon, θ u,d,e with U ′ 1 charges −Q u , −Q d , and −Q e , and we will choose Q u = 1 3 . Our naturalness criterion is simply that this state of affairs arises by virtue of the charges of the fields, and is not imposed.
Since we assume that the vevs of the various flavons are approximately the same, with then if we want the (11) entries of Y u and Y d to be of order λ 8 and λ 4 respectively, we could not have Q d = 2Q u since evidently were that so the Y u entry could be made O(λ 4 ) by using θ d instead of θ u . In imposing this criterion we will allow for possible flavon contributions to the Kahler potential. The kinetic term for the quark doublets Q will be, for example, For a recent account of how an anomaly free family-dependent U ′ 1 might be embedded in a replicated gauge group, see Ref. [3]. 3 We might want to assume that each flavon is accompanied by an oppositely charged θpartner; the simplest way to obtain a U ′ 1 D-flat direction, i.e. preventing the quadratic D-terms for the U ′ 1 from generating large masses for all the MSSM fields [5], is to assume the θs exist and have vevs approximately equal to the corresponding θs. We will indicate when this issue affects our discussion subsequently. in superspace, where and k 1 = k 2 + k 3 (or a cyclic permutation). Then we define and D Q is the diagonal matrix whose entries are the square roots of the eigenvalues of K Q . Evidently and the Yukawa matrix Y u , for example, will be replaced by It is important to realise that while the Yukawa terms are holomorphic, so that powers of θ * u,d,e cannot contribute to them, the Kahler terms are not. Note also, as remarked by DT, that the textures of Y u and Y ′ u may well differ, with, for example, texture zeroes being "filled in". We, however, will restrict ourselves to cases when Y u,d,e already have our desired texture, and this texture is preserved by the canonicalisation.
Thus far our analysis of the Kahler term mirrors that of DT. We differ from them in the following respect, however. We claim that quite generally the canonicalisation matrix C can always be chosen (without fine-tuning) to have the same texture as K. DT present an apparent counterexample, based on the matrix but it is easy to construct a further unitary transformation that reduces their canonicalisation matrix to our claimed form; and a unitary transformation obviously preserves the canonical kinetic form. Consider a simple 2 ⊗ 2 example, This matrix is diagonalised by the transformation which is not close to the unit matrix; however the matrix is a perfectly valid canonicalisation matrix, and takes the form , (1.8) in accordance with our assertion.
Let us turn now to a realistic example. Consider the "Wolfenstein" textures (see for example Ref. [6]): The Y u,d textures lead to the Wolfenstein texture for the CKM matrix, and appropriate hierarchies for the quark masses. There is considerable freedom in the choice of Y e texture; the above decision relates to the incorporation of neutrino masses, as will become clear anon. To avoid fine tuning of the leading order contributions we would expect α d ∼ α e and tan β ∼ λ α d −3 ; so we will restrict our attention to all cancel and the above textures are obtained if the following relations are satisfied: Let us now discuss the issue of naturalness we described above (ignoring at first the Kahler potential). Our system will be unnatural if there are solutions for α, β ∈ {0, 1, 2, 3 · · ·} to any of the following system of equations: If we were to assume the existence of θ flavon partners (with similar vevs) then Eq. (1.11a), for example, would be replaced by (1.14) In that case an additional set of Q e values would be unnatural: Eq. (1.12) would now also include the set 9, 10, 11, 12, 13, 14, 9 2 , 11 2 , 13 2 , 10 3 , 11 3 , 9 4 , 11 4 , 9 5 (1.15) and Eq. (1.13) the set If we restrict to α d ≤ α e , then we have three possible solutions. In what follows we will concentrate on α d = 2, α e = 3.
Turning to the Kahler terms, one sees easily that since q 2 = q 1 − 1, L 1 = L 2 + Q e etc., we have providing the Kahler textures are generated by θ u for Q, u c , d c and by θ e for L, e c , and that in each case only one flavon can contribute. For this to be natural we must exclude solutions to (once again for α, β, α, β ∈ {0, 1, 2, 3 · · ·}) Thus for example Q e = −2 is now also seen to be unnatural (this would, of course be unnatural in any case if we were assuming the existence of θ-partners). Note, however, that our anomaly cancellation conditions preclude Q e < 0.
It is easy to verify that, as we asserted earlier, the canonicalisation matrices corresponding to all the K-matrices in Eq. (1.17) have precisely the same texture as the corresponding K-matrix, and that the engendered transformations preserve the form of the textures in Eq. (1.9).
We turn now to the Higgs µ-term. If we suppose that it is generated in the same way as the Yukawa textures, that is via a term of the form M µ H 1 H 2 λ a µ , can we place any constraint on a µ ? Clearly we have a µ = α µ + β µ where Now in our example, we see from Eq. (1.10c) that to obtain Q e > 1 we would need α d > 3α e /2 + 3, which would again be difficult to reconcile with the fact that m b > m τ .
So we may assume Q e < 1, and hence manifestly the smallest attainable value of a µ is obtained for β µ = 0 and is a µ = ∆ = α d + 6. So in our favoured case (α d , α e ) = (2, 3) we have a µ = 8 corresponding to M µ ≈ 10 8 GeV if we set µ = 500GeV. This conclusion is not altered if we assume the existence of θ flavon partners.
For α d,e = (2, 3) we list the various U ′ 1 charges in Table 1. An immediate consequence is that the dimension 3,4 R-parity violating operators of the form LH 2 , LLe c , QLd c , u c d c d c , L * H 1 are all forbidden, not only in the sense that they are not U ′ 1 invariant, but also in that they cannot be flavon generated: for example L 1 +h 2 = 1223/300, which manifestly can not be produced by a linear combination of Q u = 1 and Q e = 4/15.   In all cases, given Table 1, these operators are not U ′ 1 invariant; moreover (like the Rparity violating dimension 3,4 operators) they cannot cannot be flavon generated. In fact because our scale of new physics is so (comparatively) low, B-violating operators with dimension up to at least 8 are potentially dangerous. The number of such operators is large so we do not list them. We have, however, verified that there are no operators through dimension 9 violating either B or L that can be generated by any combination of our two flavon charges. Remarkably enough, this conclusion was reached by simply examining all B, L violating operators with U 1 hypercharge zero, without worrying whether they are SU 3 ⊗SU 2 invariant: this set manifestly contains the genuine SU 3 ⊗SU 2 ⊗U 1 operators. We conclude therefore that with θ-charges Q u,d,e = 1, 1, 4 15 and a physics scale M θ ∼ 10 8 GeV, we can explain the matter mass hierarchy, the CKM matrix texture, and the magnitude of the Higgs µ-term. So B and L-violation associated with M θ are highly suppressed; but since this includes the dimension 5 operator χ ij = H 2 L i H 2 L j associated (generally via the see-saw mechanism) with neutrino masses, at this stage we have no explanation for the origin of the neutrino masses. However, the matrix of charges corresponding to χ ij is easily constructed: (1.20) Then it is easy to see that if we introduce one more flavon with charge −Q ν such that Q ν = 13 150 , we obtain a neutrino mass matrix with texture where the λ 10 arises because 7 + 2.(4/15) + 13/150 = 381/50. This texture, as shown in Ref. [8], is compatible with current knowledge of the neutrino spectrum and mixing angles, without excessive fine-tuning (for a recent review of neutrino mass patterns see Ref. [9]).
Moreover, even with the introduction of this new flavon, it remains the case that Bviolation remains suppressed to at least the dimension 9 level. Because of the λ 10 factor in Eq. (1.21), we are thus able to generate neutrino masses with the same scale, M θ , as both the Yukawa couplings and the Higgs µ-term. It is easy to check from Eq. (1.21) that the largest neutrino mass is compatible with the "normal hierarchy" neutrino spectrum.
We turn now to an alternative texture form which we previously employed in the context of Anomaly Mediation [10] [11]: The Kahler textures are now given by: where β d,e = α d,e + 2. Here we have assumed ∆ = 0; for ∆ = 0 we require instead of It is easy to show that (using the fact that α d,e ∈ {0, 1, 2, 3 · · ·}) α d = α e = 0 is the only possible solution to Eq. (1.25). This case was analysed in Ref. [11] in the AMSB context.
Unfortunately, however, since for α d = α e = ∆ = 0, we have that Q u = Q d = 1, Q e = −2, it is easy to show that this case is clearly unnatural when we take into account the Kahler textures (or introduce θ-flavons), so from the point of view of the present paper is unsatisfactory. It is interesting, therefore, that in the AMSB context we again find ourselves driven to ∆ = 0, and hence a texture-generated µ-term.
Reverting to ∆ = 0, it is straightforward to enumerate the unnatural flavon charge assignments in the same way as we did for the Wolfenstein texture. Thus Yukawa unnaturalness will follow given a solution to any of:  [11] it would require considerable fine-tuning to suppress it sufficiently to prevent an unacceptable proton decay rate from this operator.
In conclusion: the generalisation to several flavon fields relaxes some of the constraints on the anomaly free FN scenario, but it remains predictive if we assume a common mass scale origin for the Yukawa textures and the µ-term. The solution we have described is based on Eq. (1.9), and predicts that M θ ∼ 10 8 GeV. Other lepton textures (for example Y e ∼ Y d ) are also possible; however the choice made in Eq. (1.9) enables us to also accommodate neutrino masses, albeit by means of a somewhat bizarre choice for the neutrino flavon charge. These textures are not satisfactory for the AMSB scenario described in Ref. [11] because of FCNC effects associated with the Fayet-Iliopoulos D-terms; the alternative textures which avoid this problem (Eq. (1.22)) turn out to be unsatisfactory from the point of view of naturalness that we have taken here.