Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the Connes–Kreimer Hopf algebra applies to any combinatorially non-local field theory which is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank $r>2$.