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We consider hamiltonians HB = H0 + ßH1(g), where H0 is the hamiltonian of a free Bose field <¿(x) of mass m > 0 in two-dimensional space-time, H¡(g) = j" g(x): P(tj>(x)): dx where g^O is a spatial cutoff and P is an arbitrary polynomial which is bounded below, and the coupling constant ß is in the cut plane, i.e. ß^ negative real. We show that HB generates a semigroup with hypercontractive properties and satisfies higher order estimates of the form \\H0N'Rse\\ <<x>, where N is the number operator, Rs = (Hß + b)~1, r a positive integer, and ß, s, and b are suitably chosen. For any Og0<7r, Rs converges in norm to R0 as |j8| ->0 with |argj8|á®. Finally we discuss applications of these results and establish asymptotic series and Borel summability for various objects in the real ß theory.

The $(\phi\sp{2n})\sb{2}$ field Hamiltonian for complex coupling constant