Abstract
We extend earlier studies of transverse Ward-Fradkin-Green-Takahashi identities in QED, their usefulness to constrain the transverse fermion-boson vertex and their importance for multiplicative renormalizability, to the equivalent gauge identities in QCD. To this end, we consider transverse Slavnov-Taylor identities that constrain the transverse quark-gluon vertex and derive its eight associated scalar form factors. The complete vertex can be expressed in terms of the quark’s mass and wave-renormalization functions, the ghost-dressing function, the quark-ghost scattering amplitude and a set of eight form factors. The latter parametrize the hitherto unknown nonlocal tensor structure in the transverse Slavnov-Taylor identity which arises from the Fourier transform of a four-point function involving a Wilson line in coordinate space. We determine the functional form of these eight form factors with the constraints provided by the Bashir-Bermudez vertex and study the effects of this novel vertex on the quark in the Dyson-Schwinger equation using lattice QCD input for the gluon and ghost propagators. We observe significant dynamical chiral symmetry breaking and a mass gap that leads to a constituent mass of the order of 500 MeV for the light quarks. The flavor dependence of the mass and wave-renormalization functions as well as their analytic behavior on the complex momentum plane is studied and as an application we calculate the quark condensate and the pion’s weak decay constant in the chiral limit. Both are in very good agreement with their reference values.
Highlights
Scale, solving the Dyson-Schwinger equation (DSE) for the quark propagator in QCD [3, 4] and making use of the three-body Faddeev equation yields the proton’s mass and that of the Roper, the nucleon’s parity partners and the ∆ baryons in a consistent symmetrypreserving truncation [5,6,7,8,9,10,11,12,13,14]
We extend earlier studies of transverse Ward-Fradkin-Green-Takahashi identities in QED, their usefulness to constrain the transverse fermion-boson vertex and their importance for multiplicative renormalizability, to the equivalent gauge identities in QCD
We determine the functional form of these eight form factors with the constraints provided by the Bashir-Bermudez vertex and study the effects of this novel vertex on the quark in the Dyson-Schwinger equation using lattice QCD input for the gluon and ghost propagators
Summary
The most prominent occurrence of the quark-gluon vertex is in the DSE, which is nothing else than the relativistic equation of motion of the quark in QCD formulated in a nonperturbative manner. We do not intend to solve the DSE of the vertex in a given truncation scheme but rather make use of three STIs presented in sections 2.2 and 2.3 This allows us to derive the scalar form factors associated with the tensor structure of the vertex. It is known that the ghost-gluon vertex is not ultraviolet divergent in Landau gauge and that the quark-ghost kernel is finite at one-loop order [121]. This is the case in the dressed-perturbative approach [90] we employ and we choose ZH = 1. With all the 12 tensor structures of the fully dressed quark-gluon vertex we derive
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