Abstract
We present the detailed derivation of the longitudinal part of the three-gluon vertex from the Slavnov-Taylor identities that it satisfies, by means of a nonperturbative implementation of the Ball-Chiu construction; the latter, in its original form, involves the inverse gluon propagator, the ghost dressing function, and certain form factors of the ghost-gluon kernel. The main conceptual subtlety that renders this endeavor nontrivial is the infrared finiteness of the gluon propagator, and the resulting need to separate the vertex into two pieces, one that is intimately connected with the emergence of a gluonic mass scale, and one that satisfies the original set of Slavnov-Taylor identities, but with the inverse gluon propagator replaced by its "kinetic" term. The longitudinal form factors obtained by this construction are presented for arbitrary Euclidean momenta, as well as special kinematic configurations, parametrized by a single momentum. A particularly preeminent feature of the components comprising the tree-level vertex is their considerable suppression for momenta below 1 GeV, and the appearance of the characteristic "zero-crossing" in the vicinity of 100-200 MeV. Special combinations of the form factors derived with this method are compared with the results of recent large-volume lattice simulations as well as Schwinger-Dyson equations, and good overall agreement is found. A variety of issues related to the distribution of the pole terms responsible for the gluon mass generation are discussed in detail, and their impact on the structure of the transverse parts is elucidated. In addition, a brief account of several theoretical and phenomenological possibilities involving these newly acquired results is presented.
Highlights
The three-gluon vertex of QCD, to be denoted by IΓαμν, is inseparably linked with the non-Abelian nature of the theory [1], and is crucial for its most celebrated perturbative property, namely asymptotic freedom [2,3]
We present the detailed derivation of the longitudinal part of the three-gluon vertex from the SlavnovTaylor identities that it satisfies, by means of a nonperturbative implementation of the Ball-Chiu construction; the latter, in its original form, involves the inverse gluon propagator, the ghost dressing function, and certain form factors of the ghost-gluon kernel
The main conceptual subtlety that renders this endeavor nontrivial is the infrared finiteness of the gluon propagator, and the resulting need to separate the vertex into two pieces, one that is intimately connected with the emergence of a gluonic mass scale, and one that satisfies the original set of Slavnov-Taylor identities, but with the inverse gluon propagator replaced by its “kinetic” term
Summary
The three-gluon vertex of QCD, to be denoted by IΓαμν, is inseparably linked with the non-Abelian nature of the theory [1], and is crucial for its most celebrated perturbative property, namely asymptotic freedom [2,3]. The reader must note, an important caveat, spelled out in all works cited above: the massless poles contained in IΓαμν, comprising a term to be denoted by Vαμν, must be of a very special type They must be “longitudinally coupled,” i.e., appear exclusively in the form qα=q2, rμ=r2, or pν=p2, or products thereof [see Eq (2.8)] [36,37,38,39,40,41]; and this is clearly not the case for the poles induced from the naive use of JBCðqÞ. In Appendices A and B we present the one-loop results for the form factors in the “totally symmetric” and “asymmetric” configurations, and the transformation rules connecting the BC and the naive bases
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