Erratum: Analytic properties of the vertex functions in gauge theories. II

Abstract

Errata Erratum: Dragging effect on the inertial frame and the contribution of matter to the gravitational "constant" in a closed cosmological model of the Brans-Dicke theory [Phys. Rev. D 19,2861 (1979)l

A covariant representation of the Ball–Chiu vertex

“…Had we taken the validity of the replacement rules off-shell for granted from the beginning, our method of calculating the three-point vertex would have been even much more efficient; in fact, incomparably more efficient than the combined effort of [4,6,7] that was necessary to arrive at an explicit result for the scalar, spinor and gluon loop contributions to the three-gluon vertex with standard field theory methods. Before applying it to higher-point vertices, it will thus be important to show the validity of this off-shell extension in general.…”

“…Note that commutator terms are always generated by boundary terms in the IBP, and that our three-point results allow us to predict certain terms in the higher-point gluon amplitudes using the knowledge that any "abelian" field strength tensor in the nonabelian effective action must appear as part of the full nonabelian field strength tensor including the commutator term. Note also that the tensor structure multiplying the function S in the Ball-Chiu decomposition does not correspond to anything in the expansion (7.1), which is one way of understanding why S turned out to be zero in the calculations of [4,6]. Since the structure of the effective action is independent of the loop order, this observation allows us also to predict that the vanishing of S is not a one-loop accident, and will be found to persist at higher loop orders.…”

“…When calculated in the standard formalism using any covariant gauge, it satisfies rather complicated Slavnov-Taylor identities involving not only the gluon propagator, but also the ghost propagator and the gluon-ghost-ghost vertex (see, e.g., [12,4,7]). The scalar and fermion loop contributions, on the other hand, satisfy the simple QED-like Ward identity…”

“…Those diagrams add to produce a factor of two. By an analysis of the nonabelian gauge Ward identities, Ball and Chiu [4] in 1980 found a form factor decomposition of this vertex which is valid at any order in perturbation theory, and also independent of whether the particle in the loop is a scalar, fermion or gluon, with the only restriction that a covariant gauge be used. It can be written as + cyclic permutations of (p 1 , µ 1 ), (p 2 , µ 2 ), (p 3 , µ 3 ) .…”

“…However, explicit calculations of the three-vertex have so far been essentially restricted to the one-loop level [2,3,4,5,6,7] (at two loops, the three-gluon vertex has been obtained so far only for some very special momentum configurations [8,9,10]). In this paper we will recalculate, in a simple and unifying way, the scalar, spinor and gluon loop contributions to the one-loop three-vertex (with "gluon" we mean any nonabelian gauge boson).…”