Solution of Schwinger-Dyson equations for $\mathrm{PT}$-symmetric quantum field theory

In recent papers it has been observed that non-Hermitian Hamiltonians, such as those describing $\mathrm{ig}{\phi }^3$ and $-g{\phi }^4$ field theories, still possess real positive spectra so long as the weaker condition of $\mathrm{PT}$ symmetry holds. This allows for the possibility of new kinds of quantum field theories that have strange and quite unexpected properties. In this paper a technique based on truncating the Schwinger-Dyson equations is presented for renormalizing and solving such field theories. Using this technique it is argued that a $-g{\phi }^4$ scalar quantum field theory in four-dimensional space-time is renormalizable, is asymptotically free, has a nonzero value of $〈0|\phi |0〉,$ and has a positive definite spectrum. Such a theory might be useful in describing the Higgs boson.