Screened massive expansion of Schwinger-Dyson equations

A general formal derivation of the screened massive expansion is provided by Schwinger-Dyson equations. Some known issues of the expansion are clarified and a more general framework is established for a natural extension of the method to two loop or to amplitudes that are not directly defined by a generating functional. For instance, a one-loop screened expansion is given for the effective gauge-parameter-independent gluon propagator that arises from the pinch technique.

Figure 1

The functionals $\mathrm{\Pi }$ and $\mathrm{\Sigma }$ in the SD equations, Eq. (1). The black dots are dressed vertices, while the gray dots label dressed propagators.

Figure 2

Graphs contributing to the one-loop screened expansion of ${\mathrm{\Pi }}^{\prime }$ and $\mathrm{\Sigma }$. The doubly crossed gluon tadpole, graph (1d) in braces, is not present in the minimal expansion.

Figure 3

All graphs generated by the first one-loop graph of Fig. 1 after a double iteration of the SD equations, according to Eq. (25), limiting to pure gluon terms, neglecting any ghost and vertex insertions for brevity. All straight lines are massive gluon propagators while the black dots label exact, dressed, propagators. Only dressed propagators can receive further insertions in the next iteration.