Covariance of Scattering Amplitudes from Counting Carefully

TL;DR

This paper presents a general, Lagrangian-free combinatorial framework showing that on-shell amputated connected functions $\mathcal{A}_{a_1\dots a_n}$ transform covariantly under field redefinitions. By encoding the problem in the tree-level effective action and exploiting Faà di Bruno's formula alongside Bell polynomials, the authors prove covariance at tree level and establish covariant Feynman rules based on covariant building blocks $\mathcal{R}_n$, yielding a closed covariant expression for $\mathcal{A}_n$ in terms of refined diagram counts. A refined generating-function approach counts all tree topologies, including isomorphisms and vertex-type refinements, enabling an explicit, manifestly covariant closed formula for any $n$. The work sets the stage for loop generalizations and demonstrates a robust, geometry-independent path to manifest covariance in amplitude calculations, with potential implications for SMEFT/HEFT analyses. The key contributions are the explicit covariance proof via Faà di Bruno/Bell polynomials, the development of covariant Feynman rules, and the closed covariant expression for tree-level $\mathcal{A}_n$ together with a comprehensive counting framework using generating functions.

Abstract

Invariance of on-shell scattering amplitudes under field redefinitions is a well known property in field theory that corresponds to covariance of on-shell amputated connected functions. In recent years there have been great efforts to define a formalism in which the covariance is manifest at all stages of calculation, mainly resorting to geometrical interpretations. In this work covariance is analysed using combinatorial methods relying only on the properties of the tree level effective action, without referring to specific formulations of the Lagrangian. We provide an explicit proof of covariance of on-shell connected functions and of the existence of covariant Feynman rules and we derive an explicitly covariant closed formula for tree level on-shell connected functions with any number of external legs.

Figures (7)

• Figure 1: Four point connected amputated function as sum of four point 1PI and gluing of two 3 point 1PI diagrams
• Figure 2: Diagrams contributing to four point connected function
• Figure 3: Diagrams contributing to 6 point connected function with 1 gluing
• Figure 4: Labeling of Feynman diagrams (left) compared to tree graphs (right)
• Figure 5: Tree graph and its sub-trees