Renormalization problem in nonrenormalizable massless Φ4 theory

Nonrenormalizable massless Φ⁴ theory is made finite by regularization via higher derivatives in the kinetic part of the Lagrangean. The theory is shown to remain finite in the infinite cutoff limit if certain integrals over functions of one variable, with computable Taylor expansion at the origin, are finite. The values of these integrals are the only unknowns in the double series in powers ofg andg ^2/ε obtained for the Green's functions in massless (Φ⁴)_4+ε with generic ε. For ε=1 and ε=2, these series reduce to double series in powers ofg and lng. The problems of extension to (Φ⁴)_4+ε with mass, of causality and unitarity, of the relation to the BPHZ formalism, and of the indeterminacy of the result are discussed.