In particle physics, the family symmetries or horizontal symmetries are various discrete, global, or local symmetries between quark-lepton families or generations. In contrast to the intrafamily or vertical symmetries (collected in the conventional Standard Model and Grand Unified Theories) which operate inside each family, these symmetries presumably underlie physics of the family flavors. They may be treated as a new set of quantum charges assigned to different families of quarks and leptons.
Spontaneous symmetry breaking of these symmetries is believed to lead to an adequate description of the flavor mixing of quarks and leptons of different families. This is certainly one of the major problems that presently confront particle physics. Despite its great success in explaining the basic interactions of nature, the Standard Model still suffers from an absence of such a unique ability to explain the flavor mixing angles or weak mixing angles (as they are conventionally referred to) whose observed values are collected in the corresponding Cabibbo–Kobayashi–Maskawa matrices.
While being conceptually useful and leading in some cases to the physically valuable patterns of the flavor mixing, the family symmetries are not yet observationally confirmed.
Introduction
The Standard Model is based on the internal symmetries of the unitary product group the members of which have a quite different nature. The color symmetry has the vectorlike structure due to which the lefthanded and righthanded quarks are transformed identically as its fundamental triplets. At the same time, the electroweak symmetry consisting of the weak isospin and hypercharge is chiral. So, the lefthanded components of all quarks and leptons are the doublets,
whereas their righthanded components are its singlets
Here, the quark-lepton families are numbered by the index both for the quark and lepton ones. The up and down righthanded quarks and leptons are written separately and for completeness the righthanded neutrinos are also included.
Many attempts have been made to interpret the existence of the quark-lepton families and the pattern of their mixing in terms of various family symmetries – discrete or continuous, global or local. Among them, the abelian and non-abelian and family symmetries seem to be most interesting. They provide some guidance to the mass matrices for families of quarks and leptons, leading to relationships between their masses and mixing parameters. In the framework of the supersymmetric Standard Model, such a family symmetry should at the same time provide an almost uniform mass spectrum for superpartners, with a high degree of the family flavor conservation, that makes its existence even more necessary in the SUSY case.
The U(1) symmetry case
This class of the family symmetry models was first studied by Froggatt and Nielsen in 1979 and extended later on in. In this mechanism, one introduces a new complex scalar field called flavon whose vacuum expectation value (VEV) presumably breaks a global family symmetry imposed. Under this symmetry different quark-lepton families carry different charges . Aсcordingly, the connection between families is provided by inclusion into play (via the relevant see-saw mechanism) some intermediate heavy fermion(s) being properly charged under the family symmetry . So, the effective Yukawa coupling constants for quark-lepton families are arranged in a way that they may only appear through the primary couplings of these families with the messenger fermion(s) and the flavon field . The hierarchy of these couplings is determined by some small parameter , which is given by ratio of the flavon VEV to the mass of the intermediate heavy fermion, (or , if the messenger fermions have been integrated out at some high-energy cut-off scale). Since different quark-lepton families carry different charges the various coupling constants are suppressed by different powers of being primarily controlled by the postulated fermion charge assignment.