Faddeev-Popov Ghost Research Articles

Erratum: one loop gluon form factor and freezing of α s in the Gribov-Zwanziger QCD Lagrangian

We use the Gribov-Zwanziger Lagrangian in QCD to evaluate the one loop correction to the gluon propagator as a function of the Gribov volume and verify that the propagator vanishes in the infrared limit. Using the corresponding correction to the Faddeev-Popov ghost propagator we construct the renormalization group invariant coupling constant, alpha^eff_S(p^2), from the gluon and ghost form factors and verify, using the Gribov mass gap condition, that it freezes out at zero momentum to a non-zero value. This is qualitatively consistent with other approaches. We also show that there is an enhancement of the propagator of one of the Zwanziger ghosts at two loops similar to that which occurs for the Faddeev-Popov ghosts. From the exact evaluation of the form factors we examine power corrections for the gluon propagator and the effective coupling. We find that both have the same qualitative behaviour in that the leading power correction is O(1/p^2) and not O(1/(p^2)^2).

Infrared Behavior of the Ghost Propagator in the Landau Gauge Yang-Mills Theory

We prove that the Faddeev-Popov ghost dressing function in the Yang-Mills theory is non-zero and finite in the limit of vanishing momenta and hence the ghost propagator behaves like free in the deep infrared regime, within the Gribov-Zwanziger framework of the $D$-dimensional SU(N) Yang-Mills theory in the Landau gauge for any $D>2$. This result implies that the Kugo-Ojima color confinement criterion is not satisfied in its original form. We point out that the result crucially depends on the explicit form of the non-local horizon term adopted. The original Gribov prediction in the Landau gauge should be reconsidered in connection with color confinement.

BRST STRUCTURE OF NONLINEAR SUPERALGEBRAS

In this paper we analyze the structure of the BRST charge of nonlinear superalgebras. We consider quadratic nonlinear superalgebras where a commutator (in terms of (super-)Poisson brackets) of the generators is a quadratic polynomial of the generators. We find the explicit form of the BRST charge up to cubic order in Faddeev–Popov ghost fields for arbitrary quadratic nonlinear superalgebras. We point out the existence of constraints on structure constants of the superalgebra when the nilpotent BRST charge is quadratic in Faddeev–Popov ghost fields. The general results are illustrated by simple examples of superalgebras.

Faddeev-Popov ghosts

In the terminology of theoretical physics, the term ghost is used to identify an object that has no real physical meaning. The name Faddeev–Popov is given to the fictitious fields that were originally introduced in the construction of a manifestly Lorentz covariant quantization of the Yang–Mills field. Later, these objects acquired more widespread application, including in string theory. The necessity of ghosts is associated with gauge invariance. In gauge invariant theories, one usually has to deal with local fields, whose number exceeds that of physical degrees of freedom. For example in electrodynamics, in order to maintain manifest Lorentz invariance, one uses a four component vector potential Aμ(x), whereas the photon has only two polarizations. Thus, one needs a suitable mechanism in order to get rid of the unphysical degrees of freedom. Introducing fictitious fields, the ghosts, is one way of achieving this goal.

Gribov copies and anomalous scaling

Nonperturbative and lattice methods indicate that Gribov copies modify the infrared behavior of gauge theories and cause a suppression of gluon propagation. We investigate whether this can be implemented in a modified perturbation theory. The minimal modification proceeds via a nonlocal generalization of the Fadeev-Popov ghost that automatically decouples from physical states. The expected scale invariance of the physics associated with Gribov copies leads to the emergence of a nontrivial infrared fixed point. For a range of a scaling exponent the gauge bosons exhibit unparticlelike behavior in the infrared. The confining regime of interest for QCD requires a larger scaling exponent, but then the severity of ghost dominance upsets naive power counting for the infrared scaling behavior of amplitudes.

Faddeev-Popov-ghost propagators for Yang-Mills theories and perturbative quantum gravity in the covariant gauge in de Sitter spacetime

The propagators of the Faddeev-Popov (FP) ghosts for Yang-Mills theories and perturbative quantum gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. We point out, however, that the modes responsible for these divergences will not contribute to loop diagrams in computations of time-ordered products in either Yang-Mills theories or perturbative quantum gravity. Therefore, we propose that the IR-divergent FP-ghost propagator should be regularized by a small mass term that is sent to zero in the end of any perturbative calculations. This proposal is equivalent to using the effective FP-ghost propagators, which we present in an explicit form, obtained by removing the modes responsible for the IR divergences. We also make some comments on the corresponding propagators in anti-de Sitter spacetime.

Bound States of the Faddeev-Popov Ghost in Euclidean Gauge Theories

Using the Bethe-Salpeter equation, we study the bound states of the Faddeev-Popov ghost in the Euclidean SU(2) gauge theory. A Lagrangian in a nonlinear gauge has an additional ghost interaction. It is shown that this interaction yields the massless bound state ¯ × c at the scale ΛQCD. We also show that it is impossible to obtain the bound states c × c and ¯ × ¯, even if an additional interaction exists.

Supersymmetric Yang-Mills theory from lorentzian three-algebras

We show that by adding a supersymmetric Faddeev-Popov ghost sector to the recently constructed Bagger-Lambert theory based on a Lorentzian three algebra, we obtain an action with a BRST symmetry that can be used to demonstrate the absence of negative norm states in the physical Hilbert space. We show that the combined theory, expanded about its trivial vacuum, is BRST equivalent to a trivial theory, while the theory with a vev for one of the scalars associated with a null direction in the three-algebra is equivalent to a reformulation of maximally supersymmetric 2+1 dimensional Yang-Mills theory in which there is a formal SO(8) superconformal invariance.

Equivariant symplectic geometry of gauge fixing in Yang–Mills theory

The Faddeev–Popov gauge fixing in Yang–Mills theory is interpreted as equivariant localization. It is shown that the Faddeev–Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form, and a certain closed equivariant 4-form, which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localization.

Massive Yang-Mills theory based on the nonlinearly realized gauge group

We propose a subtraction scheme for a massive Yang-Mills theory realized via a nonlinear representation of the gauge group [here $SU(2)$]. It is based on the subtraction of the poles in $D\ensuremath{-}4$ of the amplitudes, in dimensional regularization, after a suitable normalization has been performed. Perturbation theory is in the number of loops, and the procedure is stable under iterative subtraction of the poles. The unphysical Goldstone bosons, the Faddeev-Popov ghosts, and the unphysical mode of the gauge field are expected to cancel out in the unitarity equation. The spontaneous symmetry breaking parameter is not a physical variable. We use the tools already tested in the nonlinear sigma model: hierarchy in the number of Goldstone boson legs and weak-power-counting property (finite number of independent divergent amplitudes at each order). It is intriguing that the model is naturally based on the symmetry $SU(2{)}_{L}$ local $\ensuremath{\bigotimes}$ $SU(2{)}_{R}$ global. By construction the physical amplitudes depend on the mass and on the self-coupling constant of the gauge particle and moreover on the scale parameter of the radiative corrections. The Feynman rules are in the Landau gauge.

Exploring the infrared structure of QCD with the Gribov-Zwanziger Lagrangian

We review recent one and two loop ${\overline{\mbox{MS}}}$ Landau gauge calculations using the Gribov-Zwanziger Lagrangian. The behaviour of the gluon and Faddeev-Popov ghost propagators as well as the renormalization group invariant effective coupling constant is examined in the infrared limit.

BRST Symmetric Gaugeon Formalism for the Higgs Model

We reinvestigate Yokoyama’s gaugeon formalism for the spontaneously broken Abelian gauge theory. Within the framework of the covariant linear gauges, we give a general gaugefixing Lagrangian which includes the gauge field, the Goldstone mode, the multiplier B-field and Yokoyama’s gaugeon fields (as well as Faddeev-Popov ghosts). As special choices of the values of the gauge-fixing parameters, our theory includes the usual covariant gauges and Rξ-like gauges. Although some of the gauge-fixing parameters can be shifted by the q-number gauge transformation, the ξ parameter cannot be shifted in any of the Rξ-like gauges.

Interacting non-Abelian vector spinor and nilpotent spinor charges

We formulate an interacting theory of a vector-spinor field associated with anti-commuting Majorana spinor charges in arbitrary spacetime dimensions. The field content of the system is , where is a vector spinor in the adjoint representation of an arbitrary gauge group, and is its gauge field, while χαIJ is an extra spinor with antisymmetric adjoint indices IJ. Contrary to common wisdom, the consistency of the vector-spinor field equation is maintained, despite its non-trivial interactions and nilpotency of the spinor charges. After establishing the classical system, we perform its quantization with gauge-fixing and Faddeev–Popov ghost terms, confirming the BRST invariance of the total action.

Lorentz violation and Faddeev-Popov ghosts

We consider how Lorentz-violating interactions in the Faddeev-Popov ghost sector will affect scalar QED. The behavior depends sensitively on whether the gauge symmetry is spontaneously broken. If the symmetry is not broken, Lorentz violations in the ghost sector are unphysical, but if there is spontaneous breaking, radiative corrections will induce Lorentz-violating and gauge-dependent terms in other sectors of the theory.

The high-temperature phase of Landau-gauge Yang-Mills theory

The properties of the high-temperature phase of Yang-Mills theory in Landau gauge are investigated by extending an earlier study on the infinite-temperature limit to finite temperatures. To this end the Dyson-Schwinger equations for the propagators of the gluon and the Faddeev-Popov ghost are solved analytically in the infrared and numerically at non-vanishing momenta. Gluons, polarized transversely with respect to the heat bath are found to comply with the Gribov-Zwanziger and the Kugo-Ojima scenario, while longitudinally polarized gluons are screened. Therefore the high-temperature phase is strongly interacting. It is furthermore conjectured that Yang-Mills theory undergoes a first-order phase transition. Indications are found that at high temperatures the thermodynamic properties are nearly those of an ideal gas, although long-range interactions prevail.

Background field method in stochastic quantization of [formula omitted] supersymmetric Yang–Mills theory

In the previous works, we proposed the stochastic quantization method (SQM) approach to N = 1 supersymmetric Yang–Mills theory (SYM). In four dimensions, in particular, we obtained the superfield Langevin equation and the corresponding Fokker–Planck equation which describe the underlying stochastic process manifestly preserving the global supersymmetry as well as the local gauge symmetry. The stochastic gauge fixing procedure was also applied to SYM 4 in the superfield formalism. In this note, we apply the background field method to SYM 4 in terms of the stochastic action principle in the SQM approach. The one-loop β-function for the gauge coupling agrees with that given by the path-integral approach, thereby confirming that the stochastic gauge fixing procedure with the background local gauge invariant Zwanziger's gauge fixing functions simulates the contributions from the Nielsen–Kallosh ghost as well as the Faddeev–Popov ghost at the one-loop level. We also show the equivalence of the effective stochastic action in the background field method to the standard one in SQM.

High-temperature limit of Landau-gauge Yang-Mills theory

The infrared properties of the high-temperature limit of Landau-gauge Yang-Mills theory are investigated. In a first step the high-temperature limit of the Dyson-Schwinger equations is taken. The resulting equations are identical to the Dyson-Schwinger equations of the dimensionally reduced theory, a three-dimensional Yang-Mills theory coupled to an effective adjoint Higgs field. These equations are solved analytically in the infrared and ultraviolet, and numerically for all Euclidean momenta. We find infrared enhancement for the Faddeev-Popov ghosts, infrared suppression for transverse gluons and a mass for the Higgs. These results imply long-range interactions and over-screening in the chromomagnetic sector of high-temperature Yang-Mills theory while in the chromoelectric sector only screening is observed.

THE MOST GENERAL AND RENORMALIZABLE MAXIMAL ABELIAN GAUGE

We construct the most general gauge fixing and the associated Faddeev–Popov ghost term for the SU(2) Yang–Mills theory, which leaves the global U(1) gauge symmetry intact (i.e. the most general Maximal Abelian gauge). We show that the most general form involves eleven independent gauge parameters. Then we require various symmetries which help to reduce the number of independent parameters for obtaining the simpler form. In the simplest case, the off-diagonal part of the gauge fixing term obtained in this way is identical to the modified maximal Abelian gauge term with two gauge parameters which was proposed in the previous paper from the viewpoint of renormalizability. In this case, moreover, we calculate the beta function, anomalous dimensions of all fields and renormalization group functions of all gauge parameters in perturbation theory to one-loop order. We also discuss the implication of these results to obtain information on low-energy physics of QCD.

Physical states at the tachyonic vacuum of open string field theory

We illustrate a method for computing the number of physical states of open string theory at the stable tachyonic vacuum in level truncation approximation. The method is based on the analysis of the gauge-fixed open string field theory quadratic action that includes Fadeev–Popov ghost string fields. Computations up to level 9 in the scalar sector are consistent with Sen's conjecture about the absence of physical open string states at the tachyonic vacuum. We also derive a long exact cohomology sequence that relates relative and absolute cohomologies of the BRS operator at the non-perturbative vacuum. We use this exact result in conjunction with our numerical findings to conclude that the higher ghost number non-perturbative BRS cohomologies are non-empty.

Quantum stochastic differential equations for boson and fermion systems—method of non-equilibrium thermo field dynamics

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the Itô and the Stratonovich types, is presented within the framework of non-equilibrium thermo field dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schrödinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev–Popov ghosts to dissipative open situations is carried out within NETFD.