A Single Source for All Flavor Violation

In a model proposed in 2012, all flavor mixing has a single source and is governed by a single"master matrix."This model was shown to give several predictions for quark and lepton masses and mixing angles and for mixing angles within SU(5) multiplets that are observable in proton decay. Here it is shown that the same master matrix controls the flavor-changing processes mediated by a singlet scalar that exists in the model, giving predictions for tau to mu + gamma, tau to e + gamma, and mu to e + gamma.


Introduction
In a 2012 paper [1], we proposed a model in which all flavor changing effects, including CKM mixing [4] and MNS mixing [3], are controlled by a single "master matrix." In that paper, the model was shown to give several predictions for neutrino properties, including the Dirac neutrino CP phase, as well as post-dictions for quantities that are still not precisely known, such as the atmospheric and solar neutrino mixing angles, and m s /m d . In a subsequent paper [2] it was pointed out that the same model predicts all the mixing angles that come into gauge-boson-mediated proton decay, thus giving further tests.
In this paper, we show that the same model gives predictions for flavor-changing effects produced by the exchange of a Standard-Model-singlet scalar field that exists in the model. In principle, therefore, certain parameters of the model could be measurable in three independent ways: by precise determination of neutrino and quark properties, by proton decay branching ratios, and through flavor-changing decays such as τ → µγ, τ → eγ, and µ → eγ.
The model is based on two assumptions: (1) that SU (5) symmetry relates quarks and leptons, and (2) that all flavor violation comes from mixing between three chiral fermion families that we shall denote 10 i + 5 i , i = 1, 2, 3, and N vector-like fermion multiplets that we shall denote 5 ′ A + 5 of flavor changing is in the 5 sector, it shows up more strongly in left-handed leptons (which are in 5 multiplets) than in left-handed quarks (which are in 10 multiplets). This gives a simple and elegant explanation of the fact that the MNS angles are much larger than the CKM angles, as is the basic idea of so-called "lopsided models" [5].
The effect of the mixing of 5 i and the 5 ′ A shows up in the effective low-energy theory of the known quarks and leptons as a 3 × 3 non-diagonal "master matrix," which we call A, that appears in their mass matrices. By field and parameter re-definitions this master matrix can be brought to a simple triangular form (which we call A ∆ ), which contains only one complex and two real parameters, whose values can be completely determined from CKM mixing. This allows predictions for all other flavor-changing effects.
In section 2, we will review the model and show how it leads to predictions for flavor changing in the lepton sector and in proton decay. In section 3, we will analyze the flavor-changing effects that arise from the exchange of a Standard-Model-singlet scalar that exists in the model.

Review of the model
We shall now review the model and its predictions for masses and mixing matrices. More details can be found at [1,2]. The Yukawa terms of the model are where the subscript H denotes Higgs multiplets. Repeated indices (i, A, or B) are summed over. The first two lines contain typical Yukawa terms that give realistic quark and lepton masses. The third line is the effective Weinberg dim-5 operator that gives mass to the neutrinos in either the type-I or type-II see-saw mechanisms. All the terms in the first three lines involve only the multiplets 10 i + 5 i and are therefore flavor-diagonal. The fourth line of Eq. (1) contains the Yukawa terms that give mass to the vector-like fermions and mix 5 i with 5 ′ A . These masses, coming from SU (5)-singlet Higgs fields, can be much larger than the weak scale, and indeed can even be of order the GUT scale. The only assumption that is required to fit the CKM and MNS mixing angles is that the masses generated by these two terms are of the same order, which we shall refer to as the "heavy scale" M * . This scale must be large enough to explain why these new fermions have not been observed. The fermions that do not get mass of order M * , which consist of the 10 i and three linear combinations of 5 i and 5 ′ A , are the known quarks and leptons, which we will call the "light fermions".
There are many abelian family symmetries that could enforce the flavor-diagonal form of the terms in the first three lines of Eq. (1). A simple example (though not the simplest) is that given in [1], namely G F = K 1 × K 2 × K 3 , where (for a given i equal to 1, 2, or 3) K i is a Z 2 symmetry under which 10 i , 5 i , and 1 ′ Hi are odd and all other fields even. Note that the vacuum expectation values of 1 ′ Hi spontaneously break the abelian family symmetries; so that the last term in Eq. (1), which mixes the 5 i and 5 ′ A , does not respect the family symmetries and can give flavor violation. It is important for the predictivity of the model that the last two terms in Eq. (1) involve only SU (5)-singlet Higgs fields, as otherwise the "master matrix" would be different for quarks and leptons. This can be ensured by another abelian symmetry that prevents the SU (5) adjoint Higgs field from coupling in these terms [1].
The Yukawa terms in the first three lines of Eq. (1) give rise to the following diagonal mass matrices m u , m d , m ℓ , and m ν : These are not the mass matrices of the known fermions, because we have not yet taken into account the mixing of the 5 multiplets, of which N linear combinations are "heavy" and 3 are "light". A block-diagonalization to separate the heavy and light 5 fermion states is needed in order to find the effective mass matrices for the light fermions.
The block diagonalization is carried out by a bi-unitary transformation of the (3 + N ) × (3 + N ) mass matrix: Here the elements of G are small and U L is approximately diagonal, because the elements of m d are very small compared to those of M and ∆. One can give exact expressions for the matrices A, B, C, D, and G, which will be useful in section 3. Defining T ≡ M −1 ∆, one can write Since the elements of ∆ and M are of the same order, the elements of T are of O (1), and the matrices A, B, C, D have off-diagonal elements of O(1). By simply multiplying out Eq. (4) one sees that the effective mass matrix of the three "light" down-type quarks, namely M d , is given by The reason that m d gets multiplied on the right by the matrix A is that the matrix m d originally appears in a term 10 i (m d ) ij 5 j , and the matrix A represents the mixing of the 5 multiplets. The mass matrix m ℓ of the charged leptons appears in a term 5 i (m ℓ ) ij 10 j , as can be seen from Eq. (2), and so gets multiplied on the left by A T . Therefore, the effective mass matrix of the three light charged leptons is M ℓ = A T m ℓ . The up quark mass matrix m u appears in a term 10 i (m u ) ij 10 j , which involves no 5 multiplets, and so does not get multiplied by any factors of A. Therefore, M u = m u . Finally, the mass matrix of the neutrinos, which comes from the dim-5 Weinberg effective operator, appears in a term 5 i (m ν ) ij 5 j , and so gets multiplied on both the right by A and the left by A T . Hence, we have altogether We thus see that all flavor violation is controlled by A. Moreover, the matrix A can be brought to a simple form in the following way. By multiplying A on the right by a unitary matrix, the elements below the main diagonal of A can be made zero. Then by rescaling the rows by multiplying from the left by a complex diagonal matrix, the diagonal elements of A can be set to 1. That is, A can be written where D is a complex diagonal matrix, U is a unitary matrix, and A ∆ is a matrix of the form where a, b, and c are real. It is easily seen that the matrix U can be absorbed into redefined righthanded down quarks and the left-handed lepton doublets. Similarly, the phases in D can be absorbed into redefined fields. The diagonal real matrix |D| can be absorbed into redefinitions of the original diagonal mass matrices as follows: m d ≡ m d |D|, m ℓ ≡ m ℓ |D|, m ν ≡ m ν |D| 2 , and m u ≡ m u . Thus, after these redefinitions, the mass matrices of the three light families take the new form It is easy to see that to a very good approximation the elements of the diagonal matrix m d are just the eigenvalues of M d , i.e. the physical masses of the d, s, and b quarks. Therefore, in the basis of Eq. (10), the mass matrices of the up quarks and down quarks look as follows One sees immediately that Similarly, the elements of the diagonal matrix m ℓ in Eq. (10) are to a very good approximation the masses of the e, µ, and τ . In the basis of Eq. (10), therefore, the charged lepton mass matrix M ℓ has the form This is not diagonal, but the rotations required to diagonalize it are very small for left-handed charged leptons (namely, of . Thus, to a very good approximation, in the basis where the charged lepton mass matrix is diagonal, the effective neutrino mass matrix M ν has the form (from Eqs. (9), (10) and (12)) We have scaled out an overall mass scale µ ν and parametrized the diagonal matrix m ν as diag(qe iβ , pe iα , 1). There are nine neutrino observables: three masses, three MNS mixing angles, the Dirac CP-violating phase, and two Majorana CP-violating phases. These are determined by five model parameters, p, q, α, β, and µ ν . Therefore there are four predictions, which we may take to be (M ν ) ee (which comes into neutrino-less double beta decay), and the three CP-violating phases. In [1], it is found that the model's best-fit values are δ Dirac = 1.15π radians, and (M ν ) ee = 0.002 eV.
But the model actually is considerably more predictive than counting parameters suggests, due to the fact that the expressions for observables in terms of model parameters are very non-linear. It is found that for good fits, certain quantities that have already been measured (such as, θ atm , θ sol , and m s /m d ) must be in a restricted part of their present experimental range. For example, a value of the atmospheric angle smaller than π/4 is preferred by the fits, and a value of m s /m d less than the median value of 20 is somewhat preferred. (See [1] for details.) As shown in [2] the model also makes non-trivial predictions for branching ratios in proton decay, which we will not review here.

Flavor Changing from Singlet Scalar Exchange
In this section we consider the effects of the scalar field 1 H that couples to the vector-like fermions to produces the N × N mass matrix M AB = Y ′ AB 1 H . We will henceforth call this singlet Higgs field Ω = Ω +Ω. The exchange of theΩ will mediate flavor-changing processes. For these effects to be observable in practice, we must assume that the scale M * , which characterizes the mass and vacuum expectation value of Ω, is not too much larger than the weak scale. We will assume that it is of order 1 to several TeV.
Let us look first at the Yukawa couplings ofΩ to the down-type quarks. In the same notation of Eq. (3), the Yukawa couplings ofΩ to the down-type quarks is given by When one block-diagonalizes to separate the light and heavy fermion stats, this Yukawa matrix is transformed by the unitary matrices U L and U R as in Eq. (4): So the effective Yukawa coupling ofΩ to the three light down-type quarks d, s, and b, is given by Remarkably, this Yukawa matrix, which we will call Y d can be written simply in terms of the master matrix A. Using Eq. (5), one gets In going from line 3 to line 4, we have used the fact that (I + T T † T ) −1 T = T (I + T † T ) −1 , as can easily be seen by expanding out the expressions is parentheses as power series. In the last line, we have used the fact that A is hermitian. Let us rewrite this expression in terms of the triangular matrix A ∆ , since that is the matrix whose elements are known. Using Eq, (8) we have The factor U on the right will be absorbed by the re-definition of the right-handed down-quark fields that was discussed after Eq. (9). Doing this re-definition, and using the fact that m d DA ∆ = m d A ∆ ≡ M d , the Yukawa coupling matrix takes the form The mass matrix M d is diagonalized by a bi-unitary transformation to give m s , m b ). From Eq. (11). One sees that the matrix V L is the CKM matrix, while the matrix Obviously, only the second term in the brackets leads to flavor changing. Let us parametrize the unknown matrix D as diag(δ, ǫ, ζ). The flavor-changing Yukawa coupling matrix of theΩ to the physical down-type quarks is of the form where ∆ ds = ∆ sd = |δ| 2 b, ∆ db = ∆ * bd = |δ| 2 ce iθ , ∆ sb = ∆ * bs = |ǫ| 2 a + |δ| 2 bce iθ .
Note that the flavor-changing (i.e. off-diagonal) elements of Y F C d depend only on two unknown combinations of parameters: |δ| 2 / Ω and |ǫ| 2 / Ω . Note also that ∆ ds and ∆ sd are real in the physical basis of the quarks, so that the ǫ parameter of the K 0 − K 0 system does not put constraints on flavor changing coming from the singlet scalar exchange.
The charged-lepton sector is identical except for a left-right transposition. So writing the flavorchanging Yukawa coupling matrix of theΩ to the physical charged leptons as where The flavor-changing Yukawa couplings come into the processes ℓ 1 → ℓ 2 γ through-two loop diagrams, as shown in [6]. The specific diagrams that dominate in this model have the vector-like fermions running around the loop that gives an effectiveΩ-photon-photon coupling. The resulting branching ratios for the flavor-changing lepton decays can be expressed in terms of the quantities given in Eq. (24) as follows [7]: One prediction is that If one assumes that the expression for ∆ µτ in Eq. (24) is dominated by the |δ| 2 term, then one would also have the prediction BR(τ → µγ) ∼ = |c| 2 BR(µ → eγ) ∼ = 16 · BR(µ → eγ).
The flavor-changing processes involving quarks do not get large enough contributions from the exchange of the singlet scalarΩ to stand out from Standard Model contributions. For instance, the coefficient of (sd)(sd) operators is found from from Eqs. Let us now consider the parameters δ, ǫ, ζ. While the matrix D = diag(δ, ǫ, ζ) is not known a priori, it is nevertheless possible to derive strict upper bounds on the parameters |δ|, |ǫ|, and |ζ| from the properties of the master matrix A. From the fact that A ≡ (I + T † T ) −1/2 and that A = DA ∆ U , one has that Computing the matrix on the left side of the above equation, one obtains