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.
Highlights
As is well-known, the fundamental Slavnov–Taylor identities (STIs) [81,82] impose crucial constraints between the two- and three-point sectors of the theory [83,84,85,86,87]
We offer a novel point of view inspired by these profound relations, which, for the special kinematic conditions that we consider, give rise to two relatively simple sum rules
We have presented a couple of new sum rules, which originate from the STIs that connect the two- and three-point sectors of quenched QCD, in the Landau gauge
Summary
As is well-known, the fundamental Slavnov–Taylor identities (STIs) [81,82] impose crucial constraints between the two- and three-point sectors of the theory [83,84,85,86,87]. One may reverse this point of view entirely, and consider the relations described above as a means of obtaining J (q2), once the lattice results for Lasym(q2) or Lsym(s2) have been used as inputs If this alternative perspective is adopted, it becomes immediately clear that these relations may be interpreted as a first-order linear differential equations for J (q2), whose solution may be written in exact closed form. It is well known that the physical J (q2) does not possess any type of pole at q2 = 0; instead, as it has been established in a series of works, the massless ghost loop entering into the SDE of the J (q2) forces it to diverge logarithmicaly as q2 → 0 [58,59,97] These unphysical poles may be eliminated from the solution for J (q2) by means of an appropriate expansion around the origin, provided that certain integral conditions hold exactly. We conclude with two Appendices: in the first, we implement the transition from the Taylor scheme to the MOM-type scheme used in the lattice simulations of Lsym(q2); in the second, we present the steps necessary for the one-loop dressed determination of the function W(q2)
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