  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.
