We initiate a new approach to perturbative corrections in QFTs with multiple scalars and show that one-loop RG flows can be described in terms of a commutative but non-associative algebra. Through the use of the algebra and simple scaling arguments we can identify useful large N scalings of the couplings and large N limits. The algebraic concepts of subalgebras and ideals are used to characterise the corrections. We demonstrate this method for example models in 4D with O(N) symmetry, as well as for a multi-scalar theory with M SU(N) adjoint scalars. Using our method we classify all large N limits of these algebras: the standard ‘t Hooft limit, a ‘multi-matrix’ limit, and an intermediate case with extra symmetry and no free parameter. The algebra identifies these limits without the need of diagrammatic or combinatorial analysis.
