Faddeev-Popov Ghost Research Articles

Origin of the pure spinor and Green-Schwarz formalisms

The pure spinor formalism for the superstring was recently obtained by gauge-fixing a purely bosonic classical action involving a twistor-like constraint ∂x m (γ m λ) α = 0 where λ α is a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro constraint ∂x m ∂x m = 0, and the Green-Schwarz fermionic spacetime spinor variables θα arise as Faddeev-Popov ghosts for this constraint. In this paper, the purely bosonic classical action is simplified by replacing the classical d=10 pure spinor λ α with a d=10 projective pure spinor. The pure spinor and Green-Schwarz formalisms for the superparticle and superstring are then obtained as different gauge-fixings of this purely bosonic classical action, and the Green-Schwarz kappa symmetry is directly related to the pure spinor BRST symmetry. Since a d=10 projective pure spinor parameterizes $$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$ , this action can be interpreted as a standard ĉ = 5 topological action where one integrates over the $$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$ choice of complex structure. Finally, a purely bosonic action for the d=11 supermembrane is proposed which reduces upon double-dimensional reduction to the purely bosonic action for the d=10 Type IIA superstring.

Yang-Mills theory in the maximal Abelian gauge in presence of scalar matter fields

We address the issue of the all order multiplicative renormalizability of SU(2) Yang-Mills theories quantized in the maximal Abelian gauge in presence of scalar matter fields. The non-linear character of the maximal Abelian gauge requires the introduction of quartic interaction terms in the Faddeev-Popov ghosts, a well known feature of this gauge. We show that, when scalar matter fields are introduced, a second quartic interaction term between scalar fields and Faddeev-Popov ghosts naturally arises. A BRST invariant action accounting for those quartic interaction terms is identified and proven to be multiplicative renormalizable to all orders by means of the algebraic renormalization procedure.

Removing the Faddeev-Popov zero modes from Yang-Mills theory in spacetimes with compact spatial sections

It is well known that in de Sitter space the (free) minimally coupled massless scalar field theory does not admit any de Sitter--invariant Hadamard state. Related to this is the fact that the propagator for the massive scalar field corresponding to the de Sitter--invariant vacuum state diverges in the massless limit, with the infrared-divergent term being a constant. Since the Faddeev-Popov ghosts for the covariantly quantized Yang-Mills theory are minimally coupled massless scalar fields, it might appear that de Sitter symmetry would be broken in the ghost sector of Yang-Mills theory in de Sitter space. It is shown in this paper that the modes responsible for de Sitter symmetry breaking can be removed in a way consistent with BRST invariance and that a de Sitter--invariant theory can be constructed. More generally, it is shown that the spatially constant modes (the zero modes) of the Faddeev-Popov ghosts can be disposed of in a wide class of spacetimes with compact spatial sections. Then, the effective theory obtained by removing the zero modes, which contains a nonlocal interaction term, is shown to be equivalent to the theory corresponding to using a Faddeev-Popov-ghost propagator with the constant infrared-divergent term removed, provided that one can freely integrate by parts in the spacetime integral at the vertex for the ghost interaction term.

BRST-symmetry breaking and Bose-ghost propagator in lattice minimal Landau gauge

The Bose-ghost propagator has been proposed as a carrier of the confining force in Yang-Mills theories in minimal Landau gauge. We present the first numerical evaluation of this propagator, using lattice simulations for the SU(2) gauge group in the scaling region. Our data are well described by a simple fitting function, which is compatible with an infrared-enhanced Bose-ghost propagator. This function can also be related to a massive gluon propagator in combination with an infrared-free (Faddeev-Popov) ghost propagator. Since the Bose-ghost propagator can be written as the vacuum expectation value of a BRST-exact quantity and should therefore vanish in a BRST-invariant theory, our results provide the first numerical manifestation of BRST-symmetry breaking due to restriction of gauge-configuration space to the Gribov region.

Holographic geometry of the renormalization group and higher spin symmetries

We consider the Wilson-Polchinski exact renormalization group applied to the generating functional of single-trace operators at a free-fixed point in $d=2+1$ dimensions. By exploiting the rich symmetry structure of free field theory, we study the geometric nature of the RG equations and the associated Ward identities. The geometry, as expected, is holographic, with $AdS$ spacetime emerging correspondent with RG fixed points. The field theory construction gives us a particular vector bundle over the $d+1$-dimensional RG mapping space, called a jet bundle, whose structure group arises from the linear orthogonal bi-local transformations of the bare fields in the path integral. The sources for quadratic operators constitute a connection on this bundle and a section of its endomorphism bundle. Recasting the geometry in terms of the corresponding principal bundle, we arrive at a structure remarkably similar to the Vasiliev theory, where the horizontal part of the connection on the principal bundle is Vasiliev's higher spin connection, while the vertical part (the Faddeev-Popov ghost) corresponds to the $S$-field. The Vasiliev equations are then, respectively, the RG equations and the BRST equations, with the RG beta functions encoding bulk interactions. Finally, we remark that a large class of interacting field theories can be studied through integral transforms of our results, and it is natural to organize this in terms of a large $N$ expansion.

Ghosts in Keldysh-Schwinger formalism

We discuss how to introduce Faddeev-Popov ghosts to the Keldysh-Schwinger formalism describing equilibrium and nonequilibrium statistical systems of quantum fields such as the quark-gluon plasma, which is considered. The plasma is assumed to be homogeneous in a coordinate space but the momentum distribution of plasma constituents is arbitrary. Using the technique of the generating functional, we derive the Slavnov-Taylor identities and one of them expresses the ghost Green's function, which we look for, through the gluon one. As an application, the Green's function of ghosts is used to compute the gluon polarization tensor in the hard loop approximation that appears to be automatically transverse, as required by the gauge invariance.

Quantization ofncoupled scalar field theory

We study a model of $n$ coupled scalar fields in Minkowski spacetime in which all masses degenerate, which is considered as a toy model of polycritical gravity on anti-de Sitter spacetime. We quantize this model within the Becchi--Rouet--Stora--Tyutin scheme by introducing $n$ Faddeev--Popov (FP) ghost fields. Extending a Becchi--Rouet--Stora--Tyutin quartet generated by two scalars and two FP ghosts to $n$ scalars and $n$ FP ghosts, there remains a physical subspace with positive norm for odd $n$, but there exists only the vacuum for even $n$. This clearly shows a nontriviality of odd higher-order derivative scalar field theories. This is helpful to understand the truncation mechanism, which is used to obtain a unitary conformal field theory dual to linearized polycritical gravity. It turns out that the truncation mechanism is nothing but a general quartet mechanism that appears when introducing the FP ghost action.

Faddeev-Popov ghosts in quantum gravity beyond perturbation theory

We study the Faddeev-Popov ghost sector of asymptotically safe quantum gravity, which becomes non-perturbative in the ultraviolet. We point out that nonzero matter-ghost couplings and higher-order ghost self-interactions exist at a non-Gaussian fixed point for the gravitational couplings, i.e., in the ultraviolet. Thus the ghost sector in this non-perturbative ultraviolet completion does not keep the structure of a simple Faddeev-Popov determinant. We discuss implications of the new ghost couplings for the Renormalization Group flow in gravity, the form of the ultraviolet completion, and the relevant couplings, i.e., free parameters, of the theory.

Localization in abelian Chern-Simons theory

Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected, and abelian. The abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. The partition function is then formally computed using the technique of non-abelian localization. This study leads to a natural identification of the abelian Reidemeister-Ray-Singer torsion as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections for the class of Sasakian three-manifolds. The torsion part of the abelian Chern-Simons partition function is computed explicitly in terms of Seifert data for a given Sasakian three-manifold.

Pure spinors, twistors, and emergent supersymmetry

Starting with a classical action whose matter variables are a d=10 spacetime vector $x^m$ and a pure spinor $\lambda^\alpha$, the pure spinor formalism for the superstring is obtained by gauge-fixing the twistor-like constraint $\partial x^m (\gamma_m \lambda)_\alpha =0$. The fermionic variables $\theta^\alpha$ are Faddeev-Popov ghosts coming from this gauge-fixing and replace the usual (b,c) ghosts coming from gauge-fixing the Virasoro constraint. After twisting the ghost-number such that $\theta^\alpha$ has ghost-number zero and $\lambda^\alpha$ has ghost-number one, the BRST cohomology describes the usual spacetime supersymmetric states of the superstring.

The “tetrad only” theory space: nonperturbative renormalization flow and asymptotic safety

We set up a nonperturbative gravitational coarse graining flow and the corresponding functional renormalization group equation on the as to yet unexplored "tetrad only" theory space. It comprises action functionals which depend on the tetrad field (along with the related background and ghost fields) and are invariant under the semi-direct product of spacetime diffeomorphisms and local Lorentz transformations. This theory space differs from that of Quantum Einstein Gravity (QEG) in that the tetrad rather than the metric constitutes the fundamental variable and because of the additional symmetry requirement of local Lorentz invariance. It also differs from "Quantum Einstein Cartan Gravity" (QECG) investigated recently since the spin connection is not an independent field variable now. We explicitly compute the renormalization group flow on this theory space within the tetrad version of the Einstein-Hilbert truncation. A detailed comparison with analog results in QEG and QECG is performed in order to assess the impact the choice of a fundamental field variable has on the renormalization behavior of the gravitational average action, and the possibility of an asymptotically safe infinite cutoff limit is investigated. Implications for nonperturbative studies of fermionic matter coupled to quantum gravity are also discussed. It turns out that, in the context of functional flow equations, the "hybrid calculations" proposed in the literature (using the tetrad for fermionic diagrams only, and the metric in all others) are unlikely to be quantitatively reliable. Moreover we find that, unlike in perturbation theory, the non-propagating Faddeev-Popov ghosts related to the local Lorentz transformations may not be discarded but rather contribute quite significantly to the beta functions of Newton's constant and the cosmological constant.

Monodromies and functional determinants in the conformal field theory driven quantum cosmology

We apply the monodromy method for the calculation of the functional determinant of a special second order differential operator $F=-d^2/d\tau^2+{\ddot g}/g$, $\ddot g= d^2g/d\tau^2$, subject to periodic boundary conditions with a periodic zero mode $g=g(\tau)$. This operator arises in applications of the early Universe theory and, in particular, determines the one-loop statistical sum for the microcanonical ensemble in cosmology generated by a conformal field theory (CFT). This ensemble realizes the concept of cosmological initial conditions by generalizing the notion of the no-boundary wavefunction of the Universe to the level of a special quasi-thermal state which is dominated by instantons with an oscillating scale factor of their Euclidean Friedmann-Robertson-Walker metric. These oscillations result in the multi-node nature of the zero mode $g(\tau)$ of $F$, which is gauged out from its reduced functional determinant by the method of the Faddeev-Popov gauge fixing procedure. The calculation is done for a general case of multiple nodes (roots) within the period range of the Euclidean time $\tau$, thus generalizing the previously known result for the single-node case of one oscillation of the cosmological scale factor. The functional determinant of $F$ expresses in terms of the monodromy of its basis function, which is obtained in quadratures as a sum of contributions of time segments connecting neighboring pairs of the zero mode roots within the period range.

Two loop QCD vertices at the symmetric point

We compute the triple gluon, quark-gluon and ghost-gluon vertices of QCD at the symmetric subtraction point at two loops in the MSbar scheme. In addition we renormalize each of the three vertices in their respective momentum subtraction schemes, MOMggg, MOMq and MOMh. The conversion functions of all the wave functions, coupling constant and gauge parameter renormalization constants of each of the schemes relative to MSbar are determined analytically. These are then used to derive the three loop anomalous dimensions of the gluon, quark, Faddeev-Popov ghost and gauge parameter as well as the beta-function in an arbitrary linear covariant gauge for each MOM scheme. There is good agreement of the latter with earlier Landau gauge numerical estimates of Chetyrkin and Seidensticker.

On the functional determinant of a special operatorwith a zero mode in cosmology

The functional determinant of a special second orderquantum-mechanical operator is calculated with its zero mode gaugedout by the method of the Faddeev-Popov gauge fixing procedure. Thisoperator subject to periodic boundary conditions arises inapplications of the early Universe theory and, in particular,determines the one-loop statistical sum in quantum cosmologygenerated by a conformal field theory (CFT). The calculation is donefor a special case of a periodic zero mode of this operator havingtwo roots (nodes) within the period range, which corresponds to theclass of cosmological instantons in the CFT driven cosmology withone oscillation of the cosmological scale factor of its EuclideanFriedmann-Robertson-Walker metric.

One-loop effects of extra dimensions on theWWγandWWZvertices

The one-loop contribution of the excited Kaluza-Klein (KK) modes of the $SU_L(2)$ gauge group on the off-shell $WW\gamma$ and $WWZ$ vertices is calculated in the context of a pure Yang-Mills theory in five dimensions and its phenomenological implications discussed. The use of a gauge-fixing procedure for the excited KK modes that is covariant under the standard gauge transformations of the $SU_L(2)$ group is stressed. A gauge-fixing term and the Faddeev-Popov ghost sector for the KK gauge modes that are separately invariant under the standard gauge transformations of $SU_L(2)$ are presented. It is shown that the one-loop contributions of the KK modes to the off-shell $WW\gamma$ and $WWZ$ vertices are free of ultraviolet divergences and well-behaved at high energies. It is found that for a size of the fifth dimension of $R^{-1}\sim 1TeV$, the one-loop contribution of the KK modes to these vertices is about one order of magnitude lower than the corresponding standard model radiative correction. This contribution is similar to the one estimated for new gauge bosons contributions in other contexts. Tree-level effects on these vertices induced by operators of higher canonical dimension are also investigated. It is found that these effects are lower than those generated at the one-loop order by the KK gauge modes.

Alternative refined Gribov-Zwanziger Lagrangian

We consider the implications of the condensation of a general local BRST invariant dimension two operator built out of the localizing ghost fields of the Gribov-Zwanziger Lagrangian which is a localized Lagrangian incorporating the Gribov problem in the Landau gauge. For different colour tensor projections of the general operator, the properties of a frozen gluon propagator and unenhanced Faddeev-Popov ghost propagator, which are observed in lattice computations, can be reproduced. The alternative possibilities are distinguished by the infrared structure of the propagators of the spin-1 fields, other than those of the gluon and Faddeev-Popov ghost, for which there is no numerical simulation data to compare with yet.

Loop calculations in the three dimensional Gribov–Zwanziger Lagrangian

The three dimensional Gribov-Zwanziger Lagrangian is analysed at one and two loops. Specifically, the two loop gap equation is evaluated and the Gribov mass is expressed in terms of the coupling constant. The one loop corrections to the propagators of all the fields are determined. It is shown that when the gap equation is satisfied the Faddeev-Popov ghost and both Bose and Grassmann localizing ghosts all enhance in the infrared limit at one loop. This verifies that the Kugo-Ojima confinement criterion holds to this order and we also show that both Grassmann ghosts are enhanced at two loops. For the Bose ghost we determine the full form of the propagator in the zero momentum limit for both the transverse and longitudinal pieces and confirm Zwanziger's recent general analysis for the low energy behaviour. We provide an alternative but equivalent version of the horizon condition expressing it as the vacuum expectation value of an operator involving only the localizing Bose ghost field. The one loop static potential is also determined.

BRST INVARIANCE AND DE RHAM TYPE COHOMOLOGY OF 't HOOFT–POLYAKOV MONOPOLE

We exploit the 't Hooft–Polyakov monopole to construct closed algebra of the quantum field operators and the BRST charge Q BRST . In the first-class configuration of the Dirac quantization, by including the Q BRST -exact gauge-fixing term and the Faddeev–Popov ghost term, we find the BRST invariant Hamiltonian to investigate the de Rham type cohomology group structure for the monopole system. The Bogomol'nyi bound is also discussed in terms of the first-class topological charge defined on the extended internal two-sphere.

Spectral sum for the color-Coulomb potential inSU(3)Coulomb gauge lattice Yang-Mills theory

We discuss the essential role of the low-lying eigenmodes of the Faddeev-Popov (FP) ghost operator on the confining color-Coulomb potential using $SU(3)$ quenched lattice simulations in the Coulomb gauge. The color-Coulomb potential is expressed as a spectral sum of the FP ghost operator and has been explored by partially summing the FP eigenmodes. We take into account the Gribov copy effects that have a great impact on the FP eigenvalues and the color-Coulomb potential. We observe that the lowest eigenvalue vanishes in the thermodynamic limit much faster than that in the Landau gauge. The color-Coulomb potential at large distances is governed by the near-zero FP eigenmodes; in particular, the lowest one accounts for a substantial portion of the color-Coulomb string tension comparable to the Wilson string tension.

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 ghosts" 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.