Renormalized couplings and scaling correction amplitudes in theN-vector spin models on the sc and the bcc lattices

For the classical N-vector model, with arbitrary $N,$ we have computed through order β¹⁷ the high-temperature expansions of the second field derivative of the susceptibility χ₄(N,β) on the simple cubic and on the body centered cubic lattices. [The N-vector model is also known as the $O(N)$ symmetric classical spin Heisenberg model or, in quantum field theory, as the lattice $O(N)$ nonlinear σ model.] By analyzing the expansion of χ₄(N,β) on the two lattices, and by carefully allowing for the corrections to scaling, we obtain updated estimates of the critical parameters and more accurate tests of the hyperscaling relation dν(N)+γ(N)-2Δ₄(N)=0 for a range of values of the spin dimensionality $N,$ including $N=0$ (the self-avoiding walk model), $N=1$ (the Ising spin 1/2 model), $N=2$ (the XY model), $N=3$ (the classical Heisenberg model). Using the recently extended series for the susceptibility and for the second correlation moment, we also compute the dimensionless renormalized four point coupling constants and some universal ratios of scaling correction amplitudes in fair agreement with recent renormalization group estimates.