By treating the vacuum as a medium, H. Euler and W. Heisenberg estimated the nonlinear interactions between photons well before the advent of quantum electrodynamics. In a modern language, their result is often presented as the archetype of an effective field theory (EFT). In this work, we develop a similar EFT for the gauge bosons of some generic gauge symmetry, valid for example for $SU(2)$, $SU(3)$, various grand unified groups, or mixed $U(1)\otimes SU(N)$ and $SU(M)\otimes SU(N)$ gauge groups. Using the diagrammatic approach, we perform a detailed matching procedure which remains manifestly gauge invariant at all steps, but does not rely on the equations of motion hence is valid off shell. We provide explicit analytic expressions for the Wilson coefficients of the dimension four, six, and eight operators as induced by massive scalar, fermion, and vector fields in generic representations of the gauge group. These expressions rely on a careful analysis of the quartic Casimir invariants, for which we provide a review using conventions adapted to Feynman diagram calculations. Finally, our computations show that at one loop, some operators are redundant whatever the representation or spin of the particle being integrated out, reducing the apparent complexity of the operator basis that can be constructed solely based on symmetry arguments.

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• Higher-dimensional field theories