Systematic investigation of effects of heavy particles via factorized local operators and the renormalization group. I. General formulation in quantum electrodynamics

In any renormalizable theory without spontaneously broken symmetry, where there are both heavy ($M$) and light ($m$) particles, we can show that when $M$ is larger than any scale, then all the $O\left(\frac{1}{M^2}\right)$ heavy-particle effects in the general $n$-point proper amputated light Green's functions can be written as ${\Gamma }_{\mathrm{full}\mathrm{theory}}^n={\Gamma }_{\mathrm{light}\mathrm{theory}}^n+\left(\frac{1}{M^2}\right){\Sigma }_N{C}_N{\Gamma }^n{\left(O_N\right)}_{\mathrm{light}\mathrm{theory}}$, where ${\Gamma }_{\mathrm{light}\mathrm{theory}}^n$ and ${\Gamma }^n{\left(O_N\right)}_{\mathrm{light}\mathrm{theory}}$ are Green's functions and the corresponding operator inserted ones, calculated within the same theory without the heavy particles. $O_N$ are a finite set of operators with densities of dimensions ≤ 6. The universal coefficient functions $C_N$ have all the dependence on heavy particles, such as coupling constants and $\ln \left(\frac{M}{m}\right)$, and are calculable via a set of Callan-Symanzik-type equations. In this article we specialize to a general formulation in QED and in an ensuing article we shall solve the scaling equations with the relevant anomalous dimensions explicitly evaluated to one-loop order.