Flavor Democracy

Prerequisites: Flavour puzzle. Full path (33 pages)

Best explanations

Concrete Mass and flavor mixing schemes of quarks and leptons

Intuitive
Concrete
Abstract
Why is it interesting?
History

Intuitive

The quarks and charged leptons come in three generations with very different masses: in each family of particles the third generation is far heavier than the other two. The Standard Model can accommodate any pattern of masses but does not explain this one (see the flavour puzzle).

Flavor democracy is a proposal for where the pattern comes from. Assume that, at some deeper level, the three generations are treated exactly alike: the Higgs field couples every generation to every other generation with the same strength. No generation is singled out, which is where the name comes from. With all couplings equal, the three generations combine into one particular mixture that gets all of the mass, and two other mixtures that stay massless. The heavy mixture is identified with the third generation.

In this picture the masses of the first two generations, and the mixing between generations seen in weak decays, appear only when the democratic symmetry is broken by small extra terms. Since the observed masses of the lighter generations are indeed small compared to the third, the breaking has to be small, and the model tries to relate the small masses and the small mixing angles to the same small breaking.

Concrete

In the Standard Model the quark masses come from mass matrices $M^u$ and $M^d$ (complex $3\times 3$ matrices in generation space, see flavor physics). The flavor democracy ansatz takes, in some basis, each mass matrix proportional to the matrix with all entries equal: $M_0 = \frac{c}{3}\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} .$ This matrix is invariant under $M \to P_L M P_R^T$ for arbitrary permutation matrices $P_L, P_R$ , so the ansatz follows from an $S(3)_L \times S(3)_R$ permutation symmetry acting on the left- and right-handed fields.

Spectrum. The all-ones matrix has eigenvalues $3, 0, 0$ . Hence $M_0$ has one eigenvalue $c$ and two zero eigenvalues. It is diagonalized by the orthogonal matrix $U = \begin{pmatrix} 1/\sqrt2 & 1/\sqrt6 & 1/\sqrt3 \\ -1/\sqrt2 & 1/\sqrt6 & 1/\sqrt3 \\ 0 & -2/\sqrt6 & 1/\sqrt3 \end{pmatrix}, \qquad U^T M_0\, U = \text{diag}(0, 0, c) .$ The massive state is the symmetric combination $(1,1,1)/\sqrt3$ . With $c \approx m_t$ in the up sector and $c \approx m_b$ in the down sector, the symmetric limit gives a heavy third generation and massless first and second generations.

No mixing in the symmetric limit. If both $M^u$ and $M^d$ are democratic, both are diagonalized by the same $U$ , and the CKM matrix $V = U^\dagger U = 1$ . (The two massless states are degenerate, so any rotation among them is allowed, but it can be absorbed by a field redefinition.) Mixing angles therefore come entirely from the symmetry breaking terms, which explains why they are small if the breaking is small.

Breaking the symmetry. Add a small perturbation $\Delta M$ . To first order in $\Delta M$ , the two light masses are the eigenvalues of $\Delta M$ projected onto the degenerate subspace spanned by the first two columns of $U$ . A simple example is $\Delta M = \text{diag}(0, 0, \delta)$ with $|\delta| \ll c$ . In the basis of $U$ its matrix elements in the light subspace are zero except for the second diagonal entry, $\frac{4}{6}\delta = \frac{2}{3}\delta$ . This gives masses approximately $c + \frac{\delta}{3}, \qquad \frac{2\delta}{3}, \qquad 0 ,$ so this breaking produces the second generation mass and leaves the first generation massless. A further, smaller breaking term is needed for the first generation. Different breaking patterns in the up and down sectors rotate the light states differently and generate the CKM angles, which in such models become functions of the quark mass ratios. The predictions for the mixing angles depend on the chosen breaking pattern and have to be compared with the measured CKM matrix.

Original Paper: Quark masses and Cabibbo angles by Haim Harari et. al.
Essential Review: Mass and flavor mixing schemes of quarks and leptons by Harald Fritzsch, Zhi-zhong Xing

Abstract

Important Papers

Flavour democracy and the lepton-quark hierarchy by H. Fritzsch and J.Plankl
H. Harari, H. Haut and J. Weyers, Phys. Letl. B 78 (1978) 459;
Y. Chikashige, G. Gelmini, R.D. Pcccei and M. Roncadelli, Phys. Lett. B 94 (1980) 499.
C. Jarlskog, in: Proc. Intern. Conf. on Production and decay of heavy hadrons (Heidelberg, 1986).
Y. Koide, Phys. Rev. D 28 (1983) 252; Phys. Left. B 120 (1983) 16l; preprints US-89-01 (1989), US-89-07 (1989).
H. Fritzsch, preprint MPI-PAE 22/88 ( 1988); Proc. XI. Warsaw Symp. on High energy physics (Kazimierz, Poland, 1988); preprint MP1-PAE/PTh 26/89.
L. Laoura, Phys. Lett. B 228 (1989) 245.
C.H. Albright and M. Lindner, preprint, FERMILAB-PUB-89/17 T

Possible Origins of the Democratic Yukawa Matrices

Specific breaking of chiral symmetries
BCS like "pairing forces", see also this paper by Nambu and especially the papers by Meshkov and Kaus.
Radiative corrections involving heavy fermions

Why is it interesting?

Flavor democracy is one of the most attractive approaches to explain the observed mass and mixing angles pattern of the standard model.

It was already noted in 1978 that a Yukawa matrix where all entries are the same, i.e. democratic, yields one important aspect of the flavor puzzle automatically: the heaviness of the third generation. In addition, adding a small perturbation to this democratic ansatz yields successfull relations between the masses and mixing angles.

In addition, there is a close relationship to the (in)famous Koide formula and other popular solutions of the flavor problem, called Fritzsch textures.

History

The notion "democratic family mixing" was introduced by C. Jarlskog in University of Stockholm Report No. 10, 1986