Vacuum Energy from Correlation Functions

In QFT the objects of interest are the n point Correlation functions which contain all the information about the theory and can be used to compute any expectation value in principle. However I cant figure out how to compute the vacuum energy from the correlation functions alone and cant find any sources or articles which discuss this computation.

Is there a way to do it with the n point functions only. I know how to compute the Vacuum Energy in QFT with the field operators but want to know how to compute it with the n point functions alone without the field operators.

Computing the vacuum energy directly from the n-point correlation functions in QFT without using the field operators themselves can be challenging, as the vacuum energy is typically defined in terms of the Hamiltonian operator, which involves the field operators. However, there is a formalism in QFT known as the Schwinger-Dyson equations that allows one to derive relations between the n-point correlation functions. These equations are a set of functional differential equations satisfied by the Green's functions (n-point correlation functions) of the theory. While they are typically derived from the path integral formulation of QFT, they provide a way to relate different correlation functions to each other.