# Dihedral Invariant Polynomials in the effective Lagrangian of QED

## TL;DR

This paper introduces a novel group-theoretical method using dihedral invariant polynomials to compute weak field expansions in QED, providing new insights into the structure of the three-loop effective Lagrangian.

## Contribution

It develops a new technique based on dihedral group invariants for calculating Feynman diagram expansions in QED, revealing potential closed-form expressions for coefficients.

## Key findings

- Computed first coefficients of three-loop effective Lagrangian in 1+1 QED
- Identified dihedral symmetry in the problem
- Suggested possible closed-form involving zeta functions

## Abstract

We present a new group-theoretical technique to calculate weak field expansions for some Feynman diagrams using invariant polynomials of the dihedral group. In particular we show results obtained for the first coefficients of the three loop effective lagrangian of 1+1 QED in an external constant field, where the dihedral symmetry appears. Our results suggest that a closed form involving rational numbers and the Riemann zeta function might exist for these coefficients.