Novel sum rules for the three-point sector of QCD

Abstract

For special kinematic configurations involving a single momentum scale, certain standard relations, originating from the Slavnov-Taylor identities of the theory, may be interpreted as ordinary differential equations for the “kinetic term” of the gluon propagator. The exact solutions of these equations exhibit poles at the origin, which are incompatible with the physical answer, known to diverge only logarithmically; their elimination hinges on the validity of two integral conditions that we denominate “asymmetric” and “symmetric” sum rules, depending on the kinematics employed in their derivation. The corresponding integrands contain components of the three-gluon vertex and the ghost-gluon kernel, whose dynamics are constrained when the sum rules are imposed. For the numerical treatment we single out the asymmetric sum rule, given that its support stems predominantly from low and intermediate energy regimes of the defining integral, which are physically more interesting. Adopting a combined approach based on Schwinger–Dyson equations and lattice simulations, we demonstrate how the sum rule clearly favors the suppression of an effective form factor entering in the definition of its kernel. The results of the present work offer an additional vantage point into the rich and complex structure of the three-point sector of QCD.

Gluon dynamics from an ordinary differential equation

“…In this section we present a novel derivation of the ODE that governs J (q 2 ), by resorting to a procedure that, even though closely related to that of [83], exploits a distinct aspect of the fundamental STI.…”

“…As in [83], the starting point of our considerations is the typical quantity considered in lattice simulations of the (transversely projected) three-gluon vertex [11,[90][91][92][93][94][95][96][97][98][99], namely…”

“…The key observation that facilitates the ensuing derivation is that, in the Landau gauge, the ghost-gluon kernels appearing in T μα (r, p, q) and T αμ (q, p, r ) may be written as [81,83] H σ α (r, p, q) = Z 1 g σ α + p ρ K σ αρ (r, p, q) , H σ μ (q, p, r ) = Z 1 g σ μ + p ρ K σ μρ (q, p, r ) , (2.5) where the kernels K do not contain poles as p → 0. Z 1 is the finite constant renormalizing H σ α (r, p, q); in particular, in the so-called Taylor scheme [100,101], Z 1 = 1.…”

“…As has been recently pointed out [83], for certain special kinematic configurations involving a single momentum scale, the BC solutions constitute ordinary differential equations (ODEs) for J (q 2 ). In the simplest such case, corresponding to the so-called "asymmetric" configuration, the closed form of the solution for J (q 2 ) involves (a) the value of the ghost dressing function at the origin, F(0); (b) a special projection of αμν (q, r, p), to be denoted by L asym (q 2 ) [see Eqs.…”

“…Similarly, from the lattice simulations of [88,89], L asym (q 2 ) is known over a wide range of momenta. As for (c), in the absence of lattice simulations for the ghostgluon kernel, the function W(q 2 ) is approximately determined within the one-loop dressed approximation of the corresponding SDE, while Z 1 has been estimated through a oneloop calculation [83].…”