Renormalising Feynman diagrams with multi-indices

In this work, we provide a method to obtain the renormalised measure in quantum field theory directly from the renormalisation of the expansion of the original measure. Our approach is based on BPHZ renormalisation via multi-indices, a combinatorial structure extremely successful for describing scalar-valued singular SPDEs. We propose the multi-indices counterpart to the Hopf algebraic program initiated by Connes and Kreimer for the renormalisation of Feynman diagrams. This new Hopf algebra also bridges the gap between the analysis of "pre-Feynman diagrams" and traditional diagrammatic methods. The construction relies on a well-chosen extraction-contraction coproduct of multi-indices equipped with a correct symmetry factor. We illustrate our method by the $\Phi^4$ measure example.