Renormalization of Quantum Field Theories on Noncommutative R^d, I. Scalars

A noncommutative Feynman graph is a ribbon graph and can be drawn on a genus $g$ 2-surface with a boundary. We formulate a general convergence theorem for the noncommutative Feynman graphs in topological terms and prove it for some classes of diagrams in the scalar field theories. We propose a noncommutative analog of Bogoliubov-Parasiuk's recursive subtraction formula and show that the subtracted graphs from a class $\Omega_d$ satisfy the conditions of the convergence theorem. For a generic scalar noncommutative quantum field theory on $\re^d$, the class $\Omega_d$ is smaller than the class of all diagrams in the theory. This leaves open the question of perturbative renormalizability of noncommutative field theories. We comment on how the supersymmetry can improve the situation and suggest that a noncommutative analog of Wess-Zumino model is renormalizable.