Faddeev-Popov Ghost Research Articles

Soft synchronous gauge in the perturbative gravity

An attempt to directly use the synchronous gauge ([Formula: see text]) in perturbative gravity leads to a singularity at [Formula: see text] in the graviton propagator. This is similar to the singularity in the propagator for Yang–Mills fields [Formula: see text] in the temporal gauge ([Formula: see text]). There the singularity was softened, obtaining this gauge as the limit at [Formula: see text] of the gauge [Formula: see text], [Formula: see text]. Then the singularities at [Formula: see text] are replaced by negative powers of [Formula: see text], and thus we bypass these poles in a certain way. Now consider a similar condition on [Formula: see text] in perturbative gravity, which becomes the synchronous gauge at [Formula: see text]. Unlike the Yang–Mills case, the contribution of the Faddeev–Popov ghosts to the effective action is nonzero, and we calculate it. In this calculation, an intermediate regularization is needed, and we assume the discrete structure of the theory at short distances for that. The effect of this contribution is to change the functional integral measure or, for example, to add nonpole terms to the propagator. This contribution vanishes at [Formula: see text]. Thus, we effectively have the synchronous gauge with the resolved singularities at [Formula: see text], where only the physical components [Formula: see text] are active and there is no need to calculate the ghost contribution.

Quantization of Yang–Mills theory in de Sitter spacetime

In this paper, we analyze the quantization of Yang–Mills theory in the de Sitter spacetime. It is observed that the Faddeev–Popov ghost propagator is divergent in this spacetime. However, this divergence is removed by using an effective propagator, which is suitable for perturbation theory. To show that the quantization of Yang–Mills theory in the de Sitter is consistent, we quantize it using first-class constraints in the temporal gauge. We also demonstrate that this is equivalent to quantizing the theory in the Lorentz gauge.

Field redefinition invariant Lagrange multiplier formalism with gauge symmetries

It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant.

A condition for the reduction of couplings in the [formula omitted] supersymmetric theories

We demonstrate that in the P=13Q supersymmetric theories the renormalization group invariance of the ratio λijk/e (of the Yukawa couplings to the gauge coupling) is equivalent to a simple relation between the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields, which should be valid in each order of the perturbation theory. In the one- and two-loop approximations it is verified explicitly. Presumably, in higher orders this relation can be satisfied for the planar supergraphs under a certain renormalization prescription. Assuming that it is valid we rewrite the exact equation for the (corresponding contribution to the) anomalous dimension of the matter superfields in the theories under consideration in a different (but equivalent) form.

Five loop minimal MOM scheme field and quark mass anomalous dimensions in QCD

We determine the anomalous dimensions of the gluon, Faddeev–Popov ghost and quark in the minimal momentum subtraction scheme to five loops for a general colour group when quantum chromodynamics is fixed in a linear covariant gauge. The quark mass anomalous dimension is also constructed in the same scheme.

Ludwig Dmitrievich Faddeev. 23 March 1934—26 February 2017

Ludwig Faddeev was a preeminent mathematical physicist of the second half of the twentieth and early twenty-first century. He made fundamental contributions to mathematics and theoretical physics. These included the solution of the quantum three-body scattering problem, groundbreaking work on the quantization of gauge theories, the formulation of Hamiltonian methods for classical integral models, and the creation, with colleagues, of the quantum inverse scattering method which led to the discovery of quantum groups. An incomplete list of important concepts and results named after him include the Faddeev equations, the Faddeev–Popov determinant, Faddeev–Popov ghosts, the Faddeev–Green function, the Gardner–Faddeev–Zakharov bracket, Faddeev–Zamolodchikov algebra and the Faddeev quantum dilogarithm. Faddeev was a charismatic teacher and a natural leader at the Steklov Institute of the Russian Academy of Sciences and the International Mathematical Union.

Analysis of unitarity in conformal quantum gravity

We perform a canonical quantization of Weyl’s conformal gravity by means of the covariant operator formalism and investigate the unitarity of the resulting quantum theory. After reducing the originally fourth-order theory to second-order in time derivatives via the introduction of an auxiliary tensor field, we identify the full Fock space of quantum states under a Becchi–Rouet–Stora–Tyutin (BRST) construction that includes Faddeev–Popov ghost fields corresponding to Weyl transformations. This second-order formulation allows the formal tools of operator-based quantum field theory to be applied to quadratic gravity for the first time. Using the Kugo–Ojima quartet mechanism, we identify the physical subspace of quantum states and find that the subspace containing the transverse spin-2 states comes equipped with an indefinite inner product metric and a one-particle Hamiltonian that possesses only a single eigenstate. We construct the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula for the S-matrix in this spin-2 subspace and find that unitarity is violated in scattering events. The explicit way in which this violation occurs represents a new view on the ghost-problem in quadratic theories of quantum gravity.

BRST ghost-vertex operator in Witten's cubic open string field theory on multiple Dp-branes

The Becchi–Rouet–Stora–Tyutin (BRST) ghost field is a key element in constructing Witten's cubic open string field theory. However, to date, the ghost sector of the string field theory has not received a great deal of attention. In this study, we address the BRST ghost on multiple Dp-branes, which carries non-Abelian indices and couples to a non-Abelian gauge field. We found that the massless components of the BRST ghost field can play the role of the Faddeev–Popov ghost in the non-Abelian gauge field, such that the string field theory maintains the local non-Abelian gauge invariance.

Quantum Yang-Mills theory in de Sitter ambient space formalism

We present the quantum Yang-Mills theory in the four-dimensional de Sitter ambient space formalism. In accordance with the SU(3) gauge symmetry the interaction Lagrangian is formulated in terms of interacting color charged fields in curved space-time. The gauge-invariant field equations are obtained in an independent coordinate description, and their corresponding color conserved currents are computed. It is shown that the Faddeev-Popov ghost fields appear similarly to their Minkowskian counterparts. We obtain that the free ghost fields are massless minimally coupled scalar fields. The problems of the vacuum state, namely the breaking of de Sitter invariance, and the appearance of infrared divergence in its quantization procedure, are discussed. The existence of an axiomatic quantum Yang-Mills theory within the framework of the Krein space quantization is examined. The infrared divergence regularization of the interaction between the gauge vector fields and the ghost fields is studied in the one-loop approximation. Two different regularization methods are discussed: cut-off regularization and Krein space regularization. A mass term for the gauge vector fields is obtained, which may explain the mass gap and the color confinement problems at the quantum level in de Sitter background. The large curvature limit at the early universe or inflationary epoch is considered.

Using Covariant Polarisation Sums in QCD

Covariant gauges lead to spurious, non-physical polarisation states of gauge bosons. In QED, the use of the Feynman gauge, sum _{lambda } varepsilon _mu ^{(lambda )}varepsilon _nu ^{(lambda )*} = -eta _{mu nu }, is justified by the Ward identity which ensures that the contributions of non-physical polarisation states cancel in physical observables. In contrast, the same replacement can be applied only to a single external gauge boson in squared amplitudes of non-abelian gauge theories like QCD. In general, the use of this replacement requires to include external Faddeev–Popov ghosts. We present a pedagogical derivation of these ghost contributions applying the optical theorem and the Cutkosky cutting rules. We find that the resulting cross terms mathcal {A}(c_1,bar{c}_1;ldots )mathcal {A}(bar{c}_1,c_1;ldots )^* between ghost amplitudes cannot be transformed into (-1)^{n/2}|mathcal {A}(c_1,bar{c}_1;ldots )|^2 in the case of more than two ghosts. Thus the Feynman rule stated in the literature holds only for two external ghosts, while it is in general incorrect.

Quantization of gravity in the black hole background

We perform a covariant (Lagrangian) quantization of perturbative gravity in the background of a Schwarzschild black hole. The key tool is a decomposition of the field into spherical harmonics. We fix Regge-Wheeler gauge for modes with angular momentum quantum number $l \geq 2$, while for low multipole modes with $l$ $=$ $0$ or $1$ -- for which Regge-Wheeler gauge is inapplicable -- we propose a set of gauge fixing conditions which are 2D background covariant and perturbatively well-defined. We find that the corresponding Faddeev-Popov ghosts are non-propagating for the $l\geq2$ modes, but are in general nontrivial for the low multipole modes with $l = 0,1$. However, in Schwarzschild coordinates, all time derivatives acting on the ghosts drop from the action and the low multipole ghosts have instantaneous propagators. Up to possible subtleties related to quantizing gravity in a space with a horizon, Faddeev's theorem suggests the possibility of an underlying canonical (Hamiltonian) quantization with a manifestly ghost-free Hilbert space.

The NSVZ relations for mathcal{N} = 1 supersymmetric theories with multiple gauge couplings

We investigate the NSVZ relations for mathcal{N} = 1 supersymmetric gauge theories with multiple gauge couplings. As examples, we consider MSSM and the flipped SU(5) model, for which they easily reproduce the results for the two-loop β-functions. For mathcal{N} = 1 SQCD interacting with the Abelian gauge superfield we demonstrate that the NSVZ-like equation for the Adler D-function follows from the NSVZ relations. Also we derive all-loop equations describing how the NSVZ equations for theories with multiple gauge couplings change under finite renormalizations. They allow describing a continuous set of NSVZ schemes in which the exact NSVZ β-functions are valid for all gauge coupling constants. Very likely, this class includes the HD+MSL scheme, which is obtained if a theory is regularized by Higher covariant Derivatives and divergences are removed by Minimal Subtractions of Logarithms. That is why we also discuss how one can construct the higher derivative regularization for theories with multiple gauge couplings. Presumably, this regularization allows to derive the NSVZ equations for such theories in all loops. In this paper we make the first step of this derivation, namely, the NSVZ equations for theories with multiple gauge couplings are rewritten in a new form which relates the β-functions to the anomalous dimensions of the quantum gauge superfields, of the Faddeev-Popov ghosts, and of the matter superfields. The equivalence of this new form to the original NSVZ relations follows from the extension of the non-renormalization theorem for the triple gauge-ghost vertices, which is also derived in this paper.

Bringing Yang-Mills theory closer to quasiclassics

A deformation of pure Yang-Mills theory by a phantom field similar to the Faddeev-Popov ghost is considered. In this theory an ersatz supersymmetry is identified which results in cancellation of quantum corrections up to two-loop order. A quadruplet built from two complex fields in the adjoint---the Faddeev-Popov ghost ${c}^{a}$ and the phantom ${\mathrm{\ensuremath{\Phi}}}^{a}$, all with the wrong statistics---balances four gauge fields ${a}_{\ensuremath{\mu}}^{a}$. At this level, the instanton measure and the $\ensuremath{\beta}$ function are fully determined by quasiclassics. In a simple ${\ensuremath{\phi}}^{4}$ theory with a phantom added I identify a strictly conserved ersatz supercurrent. In the latter theory unitarity of amplitudes persists despite the presence of the phantom. In deformed Yang-Mills it is likely (although not proven) to persist too in all amplitudes with only gluon external legs. It remains to be seen whether this construction is just a device facilitating some loop calculations or if broader applications can be found.

Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e., off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a nonrelativistic (NR) free particle whose spacexand timetvariables are a function of an evolution parameterτ. The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r.t. this evolution parameterτ. We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized byτ) is generalized onto a1,2-dimensional supermanifold which is characterized by the superspace coordinatesZM=τ,θ,θ¯where a pair of the Grassmannian variables satisfy the fermionic relationships:θ2=θ¯2=0,θ θ¯+θ¯ θ=0, andτis the bosonic evolution parameter. In the context of ACSA, we take into account only the1,1-dimensional (anti)chiral super submanifolds of the general1,2-dimensional supermanifold. The derivation of the universal Curci-Ferrari- (CF-) type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly the same as that of the reparameterization invariant SUSY (i.e., spinning) and non-SUSY (i.e., scalar) relativistic particles. This is a novel observation, too.

Non-Perturbative Propagators in Quantum Gravity

We employ non-perturbative renormalisation group methods to compute the full momentum dependence of propagators in quantum gravity in general dimensions. We disentangle all different graviton and Faddeev–Popov ghost modes and find qualitative differences in the momentum dependence of their propagators. This allows us to reconstruct the form factors that are quadratic in curvature from first principles, which enter physical observables like scattering cross sections. The results are qualitatively stable under variations of the gauge fixing choice.

Exploring new corners of asymptotically safe unimodular quantum gravity

The renormalization group flow of unimodular quantum gravity is investigated within two different classes of truncations of the flowing effective action. In particular, we search for nontrivial fixed-point solutions for polynomial expansions of the $f(R)$-type as well as of the $F({R}_{\ensuremath{\mu}\ensuremath{\nu}}{R}^{\ensuremath{\mu}\ensuremath{\nu}})+RZ({R}_{\ensuremath{\mu}\ensuremath{\nu}}{R}^{\ensuremath{\mu}\ensuremath{\nu}})$ family on a maximally symmetric background. We close the system of beta functions of the gravitational couplings with anomalous dimensions of the graviton and Faddeev-Popov ghosts treated according to two independent prescriptions: one based on the so-called background approximation and the other based on a hybrid approach which combines the background approximation with simultaneous vertex and derivative expansions. For consistency, in the background approximation, we employ a background-dependent correction to the flow equation which arises from the proper treatment of the functional measure of the unimodular path integral. We also investigate how different canonical choices of the endomorphism parameter in the regulator function affect the fixed-point structure. Although we find evidence for the existence of a nontrivial fixed point for the two classes of polynomial projections, the $f(R)$ truncation exhibits better (apparent) convergence properties. Furthermore, we consider the inclusion of matter fields without self-interactions minimally coupled to the unimodular gravitational action and we find evidence for compatibility of asymptotically safe unimodular quantum gravity with the field content of the Standard Model and some of its common extensions.

Infrared problem in the Faddeev-Popov ghost propagator in perturbative quantum gravity in de Sitter spacetime

The propagators for the Faddeev-Popov (FP) ghosts in Yang-Mills theory and perturbative gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. An IR cutoff in the momentum space to regularize these divergences breaks the de Sitter invariance. These IR divergences are due to the spatially constant modes in the Yang-Mills case and the modes proportional to the Killing vectors in the case of perturbative gravity. It has been proposed that these IR divergences can be removed, with the de Sitter invariance preserved, by first regularizing them with an additional mass term for the FP ghosts and then taking the massless limit. In the Yang-Mills case, this procedure has been shown to correspond to requiring that the physical states, and the vacuum state in particular, be annihilated by some conserved charges in the Landau gauge. In this paper we show that there are similar conserved charges in perturbative gravity in the covariant Landau gauge in de Sitter spacetime and that the IR-regularization procedure described above also correspond to requiring that the vacuum state be annihilated by these charges with a natural definition of the interacting vacuum state.

Unimodular quantum gravity: steps beyond perturbation theory

The renormalization group flow of unimodular quantum gravity is computed by taking into account the graviton and Faddeev-Popov ghosts anomalous dimensions. In this setting, a ultraviolet attractive fixed point is found. Symmetry-breaking terms induced by the coarse-graining procedure are introduced and their impact on the flow is analyzed. A discussion on the equivalence of unimodular quantum gravity and standard full diffeomorphism invariant theories is provided beyond perturbation theory.

Construction of a translation-invariant U(N) damped noncommutative gauge model

We have built a noncommutative unitary gauge group model preserving translation invariance. It describes the interaction of the Dirac field with the gauge field. The interaction term is expanded as a power series resulting from the introduction of the inverse covariant derivative. The consistency of the model is sustained by the fact that the Ward identity holds at tree level. The pure Yang–Mills action, including the fixing term and the Faddeev–Popov ghost term were constructed. It is striking that the commutator of our covariant derivative contained the torsion tensor, in addition to the field strength from which the Yang–Mills action was built.

From Slavnov—Taylor Identities to the Renormalization of Gauge Theories

An important, and highly non-trivial, problem is proving the renormalizability and unitarity of quantized non-Abelian gauge theories. Lee and Zinn-Justin have given the first proof of the renormalizability of non-Abelian gauge theories in the spontaneously broken phase. An essential ingredient in the proof has been the observation, by Slavnov and Taylor, of a non-linear, non-local symmetry of the quantized theory, a direct consequence of Faddeev and Popov’s quantization procedure. After the introduction of non-physical fermions to represent the Faddeev-Popov determinant, this symmetry has led to the Becchi-Rouet-Stora-Tyutin fermionic symmetry of the quantized action and, finally, to the resulting Zinn-Justin equation, which makes it possible to solve the renormalization and unitarity problems in their full generality. For an elementary introduction to the discussion of quantum non-Abelian gauge field theories in the spirit of the article, see, for example, L. D. Faddeev, “Faddeev-Popov ghosts,” Scholarpedia 4 (4), 7389 (2009); A. A. Slavnov, “Slavnov-Taylor identities,” Scholarpedia 3 (10), 7119 (2008); C. M. Becchi and C. Imbimbo, “Becchi-Rouet-Stora-Tyutin symmetry,” Scholarpedia 3 (10), 7135 (2008); J. Zinn-Justin, “Zinn-Justin equation,” Scholarpedia 4 (1), 7120 (2009).