Three loop bar Mbar S renormalization of QCD in the maximal abelian gauge

Abstract

Abstract. We determine the three loop anomalous dimensions of the quark, centre and offdiagonal gluons, centre and off-diagonal ghosts and the gauge fixing parameters in the maximal abelian gauge for an arbitrary colour group in the MS renormalization scheme at three loops. We show that the three loop MS β-function emerges from the renormalization of the centre gluon and also deduce the anomalous dimension of the BRST invariant dimension two mass operator. Moreover, we demonstrate that in the limit that the dimension of the centre of the group tends to zero, the anomalous dimensions of the quarks, off-diagonal gluons and off-diagonal ghosts tend to those of the quarks, gluons and ghosts of the Curci-Ferrari gauge respectively.

Maximal Abelian and Curci-Ferrari gauges in momentum subtraction at three loops

“…In other words if one takes the formal limit N d A → 0 then the Curci-Ferrari gauge emerges from (2.18). This property was noted in [47] for the three loop MS renormalization group functions but as indicated above, this is also evident from the nature of the gauge fixing. We will exploit this observation later in our computations as a non-trivial check.…”

“…(We use the notation of [47] throughout.) Here N A is the dimension of the adjoint representation of the colour group, N d A is the dimension of the diagonal subgroup with N o A being the dimension of the off-diagonal sector.…”

“…As noted in [47] the gauge fixed part of the MAG Lagrangian is generated automatically via a computer algebra routine from the BRST variation of the defining functional. This is to ensure that definitions and conventions are correctly implemented without error as well as to be confident that the resulting Feynman rules are derived correctly using symbolic manipulation.…”

“…In [47] we used the A a µc i c b vertex for this renormalization. As the vertices are ordinarily used to extract the coupling constant renormalization this means that to determine the lth loop anomalous dimension for c i one has to renormalize the A a µc i c b vertex at the (l + 1)th order, [15,47]. This is on the assumption that the coupling constant renormalization constant has already been set in a particular scheme.…”

Including four-gluon interactions into Landau / maximally-Abelian gauge Dyson-Schwinger studies

“…It is the minimal truncation of the corresponding DSEs which contains the leading UV and IR behaviour. The corresponding oneloop terms give the correct UV behavior [9], where as the sunset diagrams give the IR behavior [11]. Also there are only two vertices which have to be modeled to close the system, the three-point (Acc)-and the four point (AAcc)-vertex.…”