CP Symmetry Breaking, or the Lack of it, in the Strong Interactions

I review the strong CP problem and its solutions Presented at VietNam 2004 5th Rencontres du Vietnam New Views in Particle Physics Hanoi, VietNam August 5–August 11, 2004 ∗Work supported by Department of Energy contract DE–AC02–76SF00515. 1 Background: What causes CP Violation? This is a review talk. I have nothing new to say on this subject. However I do think it is time that we began to think about it again, as it is intrinsically a problem for the Standard Model and all extensions of the theory, and should be included when we discuss the challenges for that theory. I begin with some pedagogical background. I want to stress something that has been implicit in a number of previous talks. CP violation arises from complex relative phases of coupling constants in a physical theory. I begin by reminding you of how this works in a simple case, namely what is called direct CP violation. The reason I do this is simply to give a context in which to discuss how CP violation can occur in the QCD theory of strong interactions, and the lengths we seem to have to go to to prevent it. CP violation can occur in a decay process if two distinct amplitude terms contribute to the decay rate and the coupling constant coefficients in these terms have different phases φi [1]. In addition, we need some other phase in the problem before we are sensitive to this CP violating phase difference of couplings. Now this seems a fairly simple statement, so the younger among you may wonder why it was that for so many years, from the time of Dirac’s equation until CP violation was experimentally discovered in K decays [2], we only considered field theories in which there were no such phase differences of couplings and hence no possible CP violation. The clue is that in the Dirac equation and in pure QED CP symmetry is automatic. Gauge invariance requires that there is only one universal gauge coupling and Hermiticity of the Lagrangian automatically makes the gauge coupling real. In addition any fermion mass term can be made real by a chiral phase rotation of that fermion field, which has no impact on gauge coupling terms. Indeed it turns out that, since all fields can be arbitrarily redefined by phase rotations, most simple field theories with only a few species of particles have no possible CP violation. If you write the most general Lagrangian with all couplings arbitrarily complex, you can use the freedom of phase redefinition of fields to make all couplings real. (If you calculate any rate using the complex form of the couplings you find the rates are all independent of these phases, as they must be if you can redefine the phases away.) However, except for gauge couplings, as you add new species of particles to your theory, you get more possible new independent couplings than the number of added fields, and so, with a sufficiently rich theory, CP violating coupling phases can appear. Those who study CP violation in the quark sector will tell you that the two generation standard Model (with massless neutrinos) has no possible CP violation while with three generations there is one remaining phase in the CKM matrix that controls all CP violating effects. This is true, provided only that you have first gotten rid of a peculiar type of CP violation that is intrinsic to QCD, and that is the subject of my talk.