  In a recent letter by Carl M. Bender and S. P. Klevansky [arXiv:1002.3253] an
elementary field theory is proposed where the authors claimed that it can
account for the existence of family of particles without any employment of
group structures. In this work, we relied on the fact that the Dyson-Schwinger
equations used by the authors stem from a variational principle and tried to
reproduce their results by a mere variational algorithm. We were able to
reproduce exactly their results by the employment of two different variational
calculations of the vacuum energy in the proposed theory. We showed that the
two vacuum solutions obtained by the authors of the commented letter correspond
to two non-degenerate vacuum energies except for the $d=0$ space-time
dimensions where the vacuum states are degenerate. Since any variational
calculation of the vacuum energy should be higher than the true (exact) vacuum
energy, the most accurate result will correspond to the vacuum that lowers the
vacuum energy. We showed that the vacuum with unbroken $Z_{2}$ symmetry has
lower energy than the vacuum with broken $Z_{2}$ symmetry. Accordingly, the
theory will prefer to live with the vacuum of unbroken $Z_{2}$ symmetry in
which the proposed theory is Hermitian. Thus the proposed theory neither has a
preferred non-Hermitian representation nor it describes a family of particles
as claimed by the authors of the commented work.
