Automatic CP Invariance and Flavor Symmetry

The approximate conservation of CP can be naturally understood if it arises as an automatic symmetry of the renormalizable Lagrangian. We present a specific realistic example with this feature. In this example, the global Peccei-Quinn symmetry and gauge symmetries of the model make the renormalizable Lagrangian CP invariant but allow non zero hierarchical masses and mixing among the three generations. The left-right and a horizontal U(1)H symmetry is imposed to achieve this. The nonrenormalizable interactions invariant under these symmetries violate CP whose magnitude can be in the experimentally required range if U(1)H is broken at very high, typically, near the grand unification scale. pacs#: 11.30.Er, 11.30.Hv, 12.15.Ff, 12.60.-i, 12.60.Cn, 12.60.Fr The SU(3) ⊗ SU(2) ⊗ U(1) symmetry associated with the Standard Model (SM) of electroweak interactions is known to be inadequate for explaining fermionic masses and mixing. The gauge symmetry of the SM can accommodate these masses and in particular the CP violation [1] but does not provide any theoretical understanding of mass hierarchy or of approximate CP conservation. Some understanding of these issues can be obtained by imposing additional symmetries acting in the space of fermionic flavors. Such horizontal symmetries are known [2, 3] to lead to desired patterns of fermionic masses and mixing. It can also help in understanding the approximate conservation of CP . The aim of this note is to discuss this aspect of horizontal symmetry through an example. In this example, the exact conservation of a horizontal U(1) symmetry leads automatically to a CP conserving theory while its breakdown at very high scale leads to the observed CP violation. Ideally one would like to have CP as an automatic symmetry of the renormalizable Lagrangian in analogy with the baryon and the lepton number symmetries which are consequences of the gauge structure and the field content in the standard model. This actually happens in a special case with two generations of fermions [1, 4] . In this case, the most general Lagrangian invariant under the SM interactions is automatically CP invariant if there is only one Higgs doublet or if there are two Higgs doublets but natural flavor conservation is imposed as an additional symmetry [4]. This feature however gets spoiled when one introduces the third generation. In principle the presence of the third generation need not spoil the CP invariance if Yukawa couplings are suitably restricted. To be realistic, these restrictions must however be such that all masses and mixing angles are non-zero and hierarchical in accordance with the observed pattern. This can be accomplished if additional gauge interactions are postulated. We will present an explicit example where the same horizontal symmetry gives Fritzsch structure [5] for the quark mass matrices and also leads automatically to a CP invariant Lagrangian. In realistic case, one needs CP violation as well as deviations from the Fritzsch structure [6]. Both these occur through non-renormalizable interactions when the horizontal symmetry is broken at very high scale. The smallness of CP violation in this case is thus intimately linked to the scale of horizontal symmetry breaking. Our example requires extension of SU(3) ⊗ SU(2) ⊗ U(1) to a left-right symmetric theory [7]. In addition to the GLR = SU(3)⊗ SU(2)L ⊗ SU(2)R ⊗U(1)(B−L) group we need to impose a horizontal symmetry U(1)H and the Peccei-Quinn [8] (PQ) symmetry U(1)PQ in order to get a fully CP invariant theory.