ε′ and the decay η→ππ in a theory with both explicit and spontaneousCPviolation

In an extended version of the standard model, containing three Higgs doublets but with a CP-invariant strong interaction sector, there are two sources of CP violation. For the parameter ε′, of the K→ππ system, such a theory allows any value less than or equal to its present experimental mean value. If the present experimental magnitude of ε′ is used one obtains that the rate for the CP-violating decay η→ππ is ${10}^{12}$ times larger than the corresponding rate in a theory without spontaneous CP violation. Our estimates also yield that the upper limit on the rate of η→ππ in such a theory is about a factor of 16 larger than the corresponding present upper limit obtained in a theory where CP is violated in strong interactions due to the θ term.