On the SL(2, Bbb R) symmetry in Yang-Mills Theories in the Landau, Curci-Ferrari and Maximal Abelian Gauge

Abstract

The existence of a SL(2, R) symmetry is discussed in SU (N ) Yang-Mills in the maximal Abelian Gauge. This symmetry, also present in the Landau and Curci-Ferrari gauge, ensures the absence of tachyons in the maximal Abelian gauge. In all these gauges, SL(2, R) turns out to be dynamically broken by ghost condensates.It is widely believed that the dual superconductivity mechanism [1, 2] can be at the origin of color confinement. The key ingredients of this mechanism are the Abelian dominance and the monopoles condensation. According to the dual superconductivity picture, the low energy behavior of QCD should be described by an effective Abelian theory in the presence of monopoles. The condensation of the monopoles gives rise to the formation of Abrikosov-Nielsen-Olesen flux tubes which confine all chromoelectric charges. This mechanism has received many confirmations from lattice simulations in Abelian gauges, which are very useful in order to characterize the effective relevant degrees of freedom at low energies.Among the Abelian gauges, the so called maximal Abelian gauge (MAG) plays an important role. This gauge, introduced in [2, 3], has given evidences for the Abelian dominance and for the monopoles condensation, while providing a renormalizable gauge in the continuum. Here, the Abelian degrees of freedom are identified with the components of the gauge field belonging to the Cartan subgroup of the gauge group SU(N). The other components correspond to the (N 2 − N) off-diagonal generators of SU(N) and, being no longer protected by gauge invariance, are expected to acquire a mass, thus decoupling at low energies. The understanding of the mechanism for the dynamical mass generation of the off-diagonal components is fundamental for the Abelian dominance.A feature to be underlined is that the MAG is a nonlinear gauge. As a consequence, a quartic self-interaction term in the Faddeev-Popov ghosts is necessarily required for renormalizability [4,5]. Furthermore, as discussed in [6,7] and later on in [8], the four ghost interaction gives rise to an effective potential whose vacuum configuration favors the formation of off-diagonal ghost condensates cc , cc and cc 1 . However, these ghost condensates were proven [9] to originate an unwanted effective tachyon mass for the off-diagonal gluons, due to the presence in the MAG of an off-diagonal interaction term 1 The notation for the ghost condensates cc , cc and cc stands for f iab c a c b , f iab c a c b and f iab c a c b , where f iab are the structure constants of the gauge group. The index i runs over the Cartan generators, while the indices a, b correspond to the off-diagonal generators.

Ghost condensates and dynamical breaking ofSL(2,R) in Yang–Mills in the maximal Abelian gauge

“…As already observed [18], the breaking of SL(2, R) can actually occur in different channels, according to which generators are broken. More specifically, the three generators of SL(2, R), namely δ, δ and δ F P are known [19] to obey the algebra δ, δ = δ F P , where δ F P denotes the ghost number.…”

“…It is useful to remind that the ghost number generator δ F P is part of SL(2, R), which is present in the MAG for any gauge group SU(N) [18].…”

“…All these gauges, including the Maximal Abelian Gauge, possess a global SL(2, R) symmetry [18], a feature which seems to be deeply related to the ghost condensation.…”

Boundary effects in super-Yang–Mills theory

“…This new effective Lagrangian obtained from a sum of the original classical Lagrangian with the gauge fixing and the ghost terms is invariant under the BRST transformations [1,2]. The BRST symmetry has been studied for various different gauges [3][4][5][6][7], and it has been applied for analyzing various aspects of different supersymmetric theories [8][9][10][11][12]. a e-mail: mirfaizalmir@gmail.com…”

Some novel features in 2D non-Abelian theory: BRST approach

“…We have focused on the coupled (but equivalent) Lagrangian densities [9,10] in the Curci-Ferrari gauge [11,12] which respect the proper (anti-)BRST symmetries on the constrained hypersurface (in the 2D spacetime manifold) where the CF-condition [13] is satisfied (cf. Eqn.…”