Solution of the strong CP problem by color exchange.

In this Letter we propose a new way of solving the strong CP problem. ' The two key ingredients of our approach are that (I) CP invariance is spontaneously broken, and (2) light colored scalar fields mediate the CP-nonconserving interaction. One of us (A.Z.) proposed recently the introduction into the standard SU(3) S SU(2) II U(1) theory of an SU(2)-singlet scalar field X, thus allowing Yukawa couplings such as dz Cd& X, dz Csz X, and sz Csz X. The field X has to transform as 6' under color in order for the couplings to be symmetric in family. This proposal was part of a program to look for new physics in the scalar sector of the standard theory. It was also suggested that X exchange could generate a AS = 2, CP-nonconserving interaction. A similar suggestion was made independently by Nieves8 who constructed a specific model at the SU(3) S SU(2) S U(l) level. We note, however, that the spontaneous breaking of CP invariance at the weak scale leads to a nasty domain-wall problem. We thus propose models at the grand-unified level instead, counting on inflation to resolve the domain-wall problem. The central point of this paper is that the colorexchange approach "naturally" solves the strong CP problem. In our approach, the entire quark-mass matrix (and not only its determinant) is real at tree level. This idea was first proposed some time ago by one of us (S.B.) and Seckel, ' who were unable to implement it in a technically natural way, however. Indeed, the vital element missing in Ref. 10 is the existence of light colored scalars, a point which will be discussed in detail below. We impose CP invariance on the Lagrangean so that the Yukawa couplings are real at tree level. Thus the tree-level quark-mass matrix M0"' is completely real. [The phase of the vacuum expectation value (VEV) of the Weinberg-Salam doublet P is meaningless and may be chosen to be real by a global hypercharge rotation. ] Note that this is a consequence of there being only one Were there two or more, their VEV's could and would have CP-nonconserving relative phases that would show up in the tree-level quark-mass matrix. The tree-level value of OQCD is also zero by CP invariance; thus 0'"' = 0. "" + arg detM'"ee = 0 QCD 0 Most remarkably, as we will see, the quantum numbers of X are such that the one-loop correction to the phase of the quark-mass matrix vanishes identically. A small two-loop correction is generated, and for a certain range of parameters, this correction may be detectable by electric-dipole-moment measurements. CP nonconservation in the X-exchange process d+d X s+s can come either from the X propagator or from the X-quark vertex. In the first case, there must be more than one X, or else the propagator is real by Hermiticity. We now construct grand-unified models to illustrate these two possibilities. The moment that we mention grand unification (GU), we are faced with the hierarchy problem, of course. In line with standard practice, we will have to simply decree that certain components of a grand multiplet are light (i.e., with mass &( MG„) while other components are heavy (i.e. , with mass —Mo„). Both of our models are based on SU(5) with fermions of each family in 10L + 5L. In model 3, the Higgs bosons are in 50, 24tt, 15H, and RHs (A,B = 1, 2, . . .). We decree that the SU(2) doublet in 5H and the color-6 X in each of 15H are light, while the rest are all heavy. Here R can be any representation containing an SU(3) SU(2) U(l) singlet. CP invariance is broken spontaneously by the VEV of the singlet components of R, which can have nontrivial relative phases. This CP nonconservation will be communicated to the entire Higgs sector (through couplings such as A„ttcD15"'15~Re'RD). In particular, the propagator of the light color-6's, X, will be CP nonconserving. [A particularly economical choice, given essentially by Nieves, involves taking RH to be just an SU(5) singlet which transforms as RH —RH under CP. The coupling (io.15 "15 R +H.c.), with cr real and B~3, generates CP nonconservation when R~ acquires a VEV. ] SU(5) allows the Yukawa coupling