CP Violation Makes Left-Right Symmetric Extensions With Non-Hermitian Mass Matrices Appear Unnatural

Following a similar recent analysis for CP violation in the electroweak sector of the standard model, we estimate the naturalness of a magnitude of CP violation (measured by the Jarlskog invariant J) close to the observed value in extensions of the standard model with left-right symmetry, such as the Pati-Salam model, where quark mass matrices are not Hermitian in general. We construct a simple and natural measure on the space of complex matrices which is both geometrically motivated and uses the observed quark mass hierarchy. We find that, unlike in the case of the standard model where the observed value for J seemed rather typical, one would now expect to observe |J|< 10^{-7}, clearly in conflict with the observed value J $\approx$ 3 x 10^{-5}. The crucial difference in the calculation lies in the non-Hermiticity of mass matrices that modifies the measure. We conclude that one would need additional assumptions modifying the measure to reproduce the observed value, and that in this sense the standard model is preferred to certain classes of left-right symmetric extensions: It does not need additional assumptions to explain the magnitude of CP violation.