The impact of transverse Slavnov-Taylor identities on dynamical chiral symmetry breaking

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.

От тождеств Славнова-Тейлора к перенормировке калибровочных теорий

Доказательство перенормируемости и унитарности квантованных неабелевых калибровочных теорий является важной и крайне нетривиальной задачей. Ли и Зинн-Жюстен дали первое доказательство перенормируемости неабелевых калибровочных теорий в спонтанно нарушенной фазе. Их доказательство существенно опиралось на обнаруженную Славновым и Тейлором нелинейную нелокальную симметрию квантованной теории, которая является прямым следствием процедуры квантования Фаддеева-Попова. После введения нефизических фермионов для представления детерминанта Фаддеева-Попова эта симметрия привела к фермионной симметрии Бекки-Руэ-Стора-Тютина квантованного действия и в конечном итоге к результирующему уравнению Зинн-Жюстенa, которое позволяет решать задачи перенормировки и унитарности в наиболее общем виде. Элементарное введение в обсуждение квантовых неабелевых калибровочных теорий поля в духе настоящей работы можно найти, например, в следующих статьях: Faddeev L.D. Faddeev-Popov ghosts // Scholarpedia. 2009. V. 4, N 4. Art. 7389; Slavnov A.A. Slavnov-Taylor identities // Scholarpedia. 2008. V. 3, N 10. Art. 7119; Becchi C.M., Imbimbo C. Becchi-Rouet-Stora-Tyutin symmetry // Scholarpedia. 2008. V. 3, N 10. Art. 7135; Zinn-Justin J. Zinn-Justin equation // Scholarpedia. 2009. V. 4, N 1. Art. 7120.

BRST in the exact renormalization group

We show, explicitly within perturbation theory, that the quantum master equation and the Wilsonian renormalization group flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naïvely break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov–Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov–Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to the choice of the renormalization scheme.