Gauge-fixing dependence of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Φ</mml:mi></mml:math>-derivable approximations

Abstract

We examine the problem of gauge dependence of the two-particle irreducible (2PI) effective action and its $\ensuremath{\Phi}$-derivable approximations in gauge theories. The dependence on the gauge-fixing condition is obtained. The result shows that $\ensuremath{\Phi}$-derivable approximations, defined as truncations of the 2PI effective action at a certain order, have a controlled gauge dependence, i.e. the gauge dependent terms appear at higher order than the truncation order. Furthermore, using the stationary point obtained for the approximation to evaluate the complete 2PI effective action boosts the order at which the gauge dependent terms appear to twice the order of truncation. We also comment on the significance of this controlled gauge dependence.