VOLUME 66, NUMBER 13 PHYSICAL REVIEW LETTERS 1 APRIL 1991 Method for Constructing Models with Strong CP Invariance Paul H. Frampton Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, 1Vorth Carolina 27599-3255 Thomas W. Kephart Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235 (Received 5 July 1990; revised manuscript received 28 January 1991) A method is given which leads to a wide class of models with strong CP invariance. These models con- tain a spontaneously broken U(1) symmetry. If this symmetry were global, its breaking would lead to a massless scalar. Because the U(1) is anomaly free, it can be gauged, leading by the Higgs mechanism to a massive gauge boson, the aspon, which might be detectable. PACS numbers: 11.30.Er, 11.15.Ex, 11.30.Rd, 12. 10.Dm Quantum chromodynamics (QCD), the theory of strong interactions in particle and nuclear physics, ex- perienced' both a major advance and a significant set- back in 1976. The advance came when it was realized that the long-standing problem of the g meson mass arises from topological field configurations or instantons and hence resolved the so-called U(1) problem. At the same time, the discovery of instantons created the strong CP problem which remains unresolved. The strong CP problem is that instantons introduce a new parameter OQco into QCD and that this parameter must be fine tuned to 1 part in 10' to avoid disagree- ment with experiment. In particular, instanton effects may be summarized by an additional term in the QCD Lagrangian, with coefficient OQgD, which violates P and CP conservation. Experimental strong CP-violation ef- fects depend on 0 =OQQD+ OQFD where OQFD is the phase of the determinant of the quark mass matrix. The present Letter discusses the strong CP problem from a new perspective, provides a new solution, and should facilitate appropriate model building. We first recall two approaches which have attracted considerable attention previously. The first, and the more popular, is to introduce a color-anomalous U(1)pg which allows 0 to relax to zero; the spontaneous breaking of this U(1)po gives rise to a light boson, the axion, which obtains a mass from instanton effects. A second approach is to assume CP symmetry of the Lagrangian so that OQ+D 0; after spontaneous breaking of CP, the value of OQFD is kept small by arranging a real deter- minant of the quark mass matrix at lowest order. Both methods appear to offer acceptable solutions. One reason that more attention has been given to the ax- ion scenario is that it gives rise to so many additional questions which require further research. A second reason is that there is a systematic method to construct Peccei-Quinn models. As searches for a physical axion remain frustrated, it is worth examining what other ob- servable phenomena or particles might be associated with solution of the strong CP problem. In order to set the scene, and to introduce a more gen- eral approach to model building, let us consider one fam- ily of quarks and their (T3, Y) values under the elec- troweak group: ( — —, , —, ) dL, (0, —, ' ) dt, ( —, ', —, ') uL, (0, — —, ') uL. We introduce in our model a U(1)„,„symmetry and as- sign charge Q„, „=O to all of the above quark states and to the leptons, although the latter do not play a sig- nificant role in solving strong CP. The second and third families have the parallel assignments under the same U(1).... In our model there is also a real representation of ex- otic "heavy" quarks corresponding to a complex repre- sentation C and its conjugate C. In C the exotic heavy quarks have quantum numbers exactly like some of the usual quarks; for example, in C there may be one doublet (2, 6 ) UL. These have charge Q„, „=+h. In representation C we shall then have ( —, ', — —. ' ) Dg~, ( — —, ', — —, ') U;. These have Q„, „= — h. The Higgs sector has one complex doublet y (+ —, ', — —, '), Q„, „=o, and two complex singlets @|2 (0, 0), Q„, „=+h. The gauge group is SU(3)sx SU(2)L x U(1) y [ x U(I )„,„ if it is gauged]. In breaking the symmetry, we give a 1666 1991 The American Physical Society