Particles and scaling for lattice fields and Ising models

The conjectured inequality γ⁽⁶⁾≦0 leads to the existence of φ{symbol}_d⁴ fields and the scaling (continuum) limit for d-dimensional Ising models. Assuming γ⁽⁶⁾≦0 and Lorentz covariance of this construction, we show that for d≧6 these φ{symbol}_d⁴ fields are free fields unless the field strength renormalization Z^-1 diverges. Let λ be the bare charge and ε the lattice spacing. Under the same assumptions (γ⁽⁶⁾≦0, Lorentz covariance and d≧6) we show that if λε^4-d is bounded as ε→0, then Z^-1 is bounded and the limit field is free.