Ghost Loops Research Articles

The complex heavy quarkonium potential with the Gribov–Zwanziger action

Gribov–Zwanziger prescription in Yang–Mills theory improves the infrared dynamics. In this work, we study the static potential of a heavy quark-antiquark pair with the HTL resummed perturbation method within the Gribov–Zwanziger approach at finite temperature. The real and imaginary parts of the heavy quarkonium complex potential are obtained from the one-loop effective static gluon propagator. The one-loop effective gluon propagator is obtained by calculating the one-loop gluon self-energies containing the quark, gluon, and ghost loop. The gluon and ghost loops are modified in the presence of the Gribov parameter. We also calculate the decay width from the imaginary part of the potential. We also discuss the medium effect of heavy quarkonium potential with the localized action via auxiliary fields.

The two-, three- and four-gluon sector of QCD in the Landau gauge

Due to the nonperturbative masslessness of the ghost field, ghost loops that contribute to gluon Green's functions in the Landau gauge display infrared divergences, akin to those one would encounter in a conventional perturbative treatment. This is in sharp contrast with gluon loops, in which the perturbative divergences are tamed by the dynamical generation of a gluon mass acting as an effective infrared cutoff. In this paper, after reviewing the full nonperturbative origin of this divergence in the two-gluon sector, we discuss its implications for the three- and four-gluon sector, showing in particular that some of the form factors characterizing the corresponding Green's functions are bound to diverge in the infrared.

Color Confinement and Screening in the θ Vacuum of QCD.

QCD perturbation theory ignores the compact nature of the SU(3) gauge group that gives rise to the periodic θ vacuum of the theory. We propose to modify the gluon propagator to reconcile perturbation theory with the anomalous Ward identities for the topological current in the θ vacuum. As a result, the gluon couples to the Veneziano ghost describing the tunneling transitions between different Chern-Simons sectors of the vacuum; we call the emerging gluon dressed by ghost loops a "glost." We evaluate the glost propagator and find that it has the form G(p)=(p(2)+χ(top)/p(2))(-1) where χ(top) is the Yang-Mills topological susceptibility related to the η' mass by the Witten-Veneziano relation; this propagator describes the confinement of gluons at distances ∼χ(top)(-1/4)≃1 fm. The same functional form of the propagator was originally proposed by Gribov as a solution to the gauge copies problem that plagues perturbation theory. The resulting running coupling coincides with the perturbative one at p(2)≫√[χ(top)], but in the infrared region either freezes (in pure Yang-Mills theory) or vanishes (in full QCD with light quarks), in accord with experimental evidence. Our scenario makes explicit the connection between confinement and topology of the QCD vacuum; we discuss the implications for spin physics, high energy scattering, and the physics of quark-gluon plasma.

Effects of divergent ghost loops on the Green’s functions of QCD

In the present work we discuss certain characteristic features encoded in some of the fundamental QCD Green's functions, whose origin can be traced back to the nonperturbative masslessness of the ghost field, in the Landau gauge. Specifically, the ghost loops that contribute to these Green's functions display infrared divergences, akin to those encountered in the perturbative treatment, in contradistinction to the gluonic loops, whose perturbative divergences are tamed by the dynamical generation of an effective gluon mass. In d=4, the aforementioned divergences are logarithmic, thus causing a relatively mild impact, whereas in d=3 they are linear, giving rise to enhanced effects. In the case of the gluon propagator, these effects do not interfere with its finiteness, but make its first derivative diverge at the origin, and introduce a maximum in the region of infrared momenta. The three-gluon vertex is also affected, and the induced divergent behavior is clearly exposed in certain special kinematic configurations, usually considered in lattice simulations; the sign of the corresponding divergence is unambiguously determined. The main underlying concepts are developed in the context of a simple toy model, which demonstrates clearly the interconnected nature of the various effects. The picture that emerges is subsequently corroborated by a detailed nonperturbative analysis, combining lattice results with the dynamical integral equations governing the relevant ingredients, such as the nonperturbative ghost loop and the momentum-dependent gluon mass.

Gluon mass through ghost synergy

In this work we compute, at the "one-loop-dressed" level, the nonperturbative contribution of the ghost loops to the self-energy of the gluon propagator, in the Landau gauge. This is accomplished within the PT-BFM formalism, which guarantees the gauge-invariance of the emerging answer. In particular, the contribution of the ghost-loops is automatically transverse, by virtue of the QED-like Ward identities satisfied in this framework. Using as nonperturbative input the available lattice data for the ghost dressing function, we show that the ghost contributions have a rather sizable effect on the overall shape of the gluon propagator, both for $d=3,4$. Then, by exploiting a recently introduced dynamical equation for the effective gluon mass, whose solutions depend crucially on the characteristics of the gluon propagator at intermediate energies, we show that if the ghost loops are removed from the gluon propagator then the gluon mass vanishes. These findings strongly suggest that, at least at the level of the Schwinger-Dyson equations, the effects of gluons and ghosts are inextricably connected, and must be combined suitably in order to reproduce the results obtained in the recent lattice simulations.

Witten index in supersymmetric 3d theories revisited

We have performed a direct calculation of Witten index in N = 1,2,3 supersymmetric Yang-Mills Chern-Simons 3d theories. We do it in the framework of Born-Oppenheimer (BO) approach by putting the system into a small spatial box and studying the effective Hamiltonian depending on the zero field harmonics. At the tree level, our results coincide with the results of Witten, but there is a difference in the way the loop effects are implemented. In Witten's approach, one has only take into account the fermion loops, which bring about a negative shift of the (chosen positive at the tree level) Chern-Simons coupling k. As a result, Witten index vanishes and supersymmetry is broken at small k. In the effective BO Hamiltonian framework, fermion, gluon and ghost loops contribute on an equal footing. Fermion loop contribution to the effective Hamiltonian can be evaluated exactly, and their effect amounts to the negative shift k -> k - h/2 for N =1 and k -> k - h for N = 2,3 in the tree-level formulae for the index. In our approach, with rather natural assumptions on the structure of bosonic corrections, the shift k -> k + h brought about by the gluon loops also affects the index. Since the total shift of k is positive or zero, Witten index appears to be nonzero at nonzero k, and supersymmetry is not broken. We discuss possible reasons for such disagreement.

Gauge-invariant truncation scheme for the Schwinger-Dyson equations of QCD

We present a new truncation scheme for the Schwinger-Dyson equations of QCD that respects gauge invariance at any level of the dressed loop expansion. When applied to the gluon self-energy, it allows for its nonperturbative treatment without compromising the transversality of the solution, even when entire sets of diagrams (most notably the ghost loops) are omitted or treated perturbatively.

Quantization of the first-order two-dimensional Einstein–Hilbert action

A canonical analysis of the first-order two-dimensional Einstein–Hilbert action has shown it to have no physical degrees of freedom and to possess an unusual gauge symmetry with a symmetric field ξμν acting as a gauge function. Some consequences of this symmetry are explored. The action is quantized and it is shown that in a ‘Landau gauge’ all loop diagrams beyond one-loop order vanish. Furthermore, explicit calculation of the one-loop two-point function shows that it too vanishes, with the contribution of the ghost loop cancelling that of the ‘graviton’ loop.

A numerical algorithm for efficiently obtaining a Feynman parameter representation of one-gluon loop QCD Feynman diagrams for a large number of external gluons

A numerical program is presented which facilitates a computation pertaining to the full set of one-gluon loop diagrams (including ghost loop contributions), with M attached external gluon lines in all possible ways. The feasibility of such a task rests on a suitably defined master formula, which is expressed in terms of a set of Grassmann and a set of Feynman parameters. The program carries out the Grassmann integration and performs the Lorentz trace on the involved functions, expressing the result as a compact sum of parametric integrals. The computation is based on tracing the structure of the final result, thus avoiding all intermediate unnecessary calculations and directly writing the output. Similar terms entering the final result are grouped together. The running time of the program demonstrates its effectiveness, especially for large M. Program summary Program title:DILOG2 Program identifier:ADXN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADXN_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Programming language:FORTRAN 90 Computer(s) for which the program has been designed:Personal Computer Operating system(s) for which the program has been designed: Windows 98, XP, LINUX Number of processors used:one No. of lines in distributed program, including test data, etc.:2000 No. of bytes in distributed program, including test data, etc.:16 249 Distribution format:tar.gz External routines/libraries used:none CPC Program Library subprograms used:none Nature of problem:The computation of one gluon/ghost loop diagrams in QCD with many external gluon lines is a time consuming task, practically beyond reasonable reach of analytic procedures. We apply recently proposed master formulas towards the computation of such diagrams with an arbitrary number ( M) of external gluon lines, achieving a final result which reduces the problem to one involving integrals over the standard set, for given M, of Feynman parameters. Solution method:The structure of the master expressions is analyzed from a numerical computation point of view. Using the properties of Grassmann variables we identify all the different forms of terms that appear in the final result. Each form is called “structure”. We calculate theoretically the number of terms belonging to every “structure”. We carry out the calculation organizing the whole procedure into separate calculations of the terms belonging to every “structure”. Terms which do not contribute to the final result are thereby avoided. The final result, extending to large values of M, is also presented with terms belonging to the same “structure” grouped together. Restrictions:M is coded as a 2-digit integer. Overflow in the dimension of used array is expected to appear for M ⩾ 20 in a processor that uses 4-bytes integers or for M ⩾ 34 in a processor with 8-bytes integers. Running time:Depends on M, see enclosed figures.

TWO-LOOP GHOST–ANTIGHOST CONDENSATION FOR SU(2) YANG–MILLS THEORIES IN THE MAXIMAL ABELIAN GAUGE

In the framework of the formalism of Cornwall et al. for composite operators, we study the ghost–antighost condensation in SU (2) Yang–Mills theories quantized in the maximal Abelian gauge and derive analytically a condensating effective potential at two ghost loops. We find that in this approximation the one-loop pairing ghost–antighost is not destroyed but no mass is generated if the ansatz for the propagator suggested by the tree level Hubbard–Stratonovich transformations is used.

Inertia Orbifolds, Configuration Spaces and the Ghost Loop Space

In this paper we define and study the ghost loop orbifold LsX of an orbifold X consisting of those loops that remain constant in the coarse moduli space of X .W econstruct a configuration space model for LsX using an idea of G. Segal. From this we exhibit the relation between the Hochschild and cyclic homologies of the inertia orbifold of X (that generate the so-called twisted sectors in string theory) and the ordinary and equivariant homologies of LsX .W ealso show how this clarifies the relation between Chen–Ruan orbifold cohomology, Hochschild homology, and periodic cyclic homology.

Inertia Orbifolds, Configuration Spaces and the Ghost Loop Space

In this paper we define and study the loop of an orbifold $X$ consisting of those loops that remain constant in the coarse moduli space of $X$. We construct a configuration space model for the ghost loop orbifold using an idea of G. Segal. From this we exhibit the relation between the Hochschild and cyclic homologies of the inertia orbifold of $X$ (that generate the so-called twisted sectors in string theory) and the ordinary and equivariant homologies of the ghost loop orbifold. We also show how this clarifies the relation between orbifold K-theory, Chen-Ruan orbifold cohomology, Hochschild homology, and periodic cyclic homology.

Multiplicative renormalizability of gluon and ghost propagators in QCD

We reformulate the coupled set of continuum equations for the renormalized gluon and ghost propagators in QCD, such that the multiplicative renormalizability of the solutions is manifest, independently of the specific form of full vertices and renormalization constants. In the Landau gauge, the equations are free of renormalization constants, and the renormalization point dependence enters only through the renormalized coupling and the renormalized propagator functions. The structure of the equations enables us to devise novel truncations with solutions that are multiplicatively renormalizable and agree with the leading order perturbative results. We show that, for infrared power law behaved propagators, the leading infrared behavior of the gluon equation is not solely determined by the ghost loop, as concluded in previous studies, but that the gluon loop, the three-gluon loop, the four-gluon loop, and even massless quarks also contribute to the infrared analysis. In our new Landau gauge truncation, the combination of gluon and ghost loop contributions seems to reject infrared power law solutions, but massless quark loops illustrate how additional contributions to the gluon vacuum polarization could reinstate these solutions. Moreover, a schematic study of the three-gluon and four-gluon loops shows that they too need to be considered in more detail before a definite conclusion about the existence of infrared power behaved gluon and ghost propagators can be reached.

World-line approach to the Bern-Kosower formalism in two-loop Yang-Mills theory

Based on the world-line formalism with a sewing method, we derive the Yang-Mills effective action in a form useful to generate the Bern-Kosower-type master formulae for gluon scattering amplitudes at the two-loop level. It is shown that four-gluon ($\Phi^4$ type sewing) contributions can be encapsulated in the action with three-gluon ($\Phi^3$ type) vertices only, the total action thus becoming a simple expression. We then derive a general formula for a two-loop Euler-Heisenberg type action in a pseudo-abelian $su(2)$ background. The ghost loop and fermion loop cases are also studied.

Canonical formalism and the Leibbrandt-Mandelstam prescription for noncovariant gauges.

A very simple and elegant approach to the Leibbrandt-Mandelstam regularization is given within the canonical formalism. For any value of {ital n}{sup 2} with {ital n}{sub 0} {ital and} {bold n} different from zero, it consists of introducing the hyperbolic operator {ital n}{sup *}{center dot}{partial derivative}n{center dot}{partial derivative}=(n{sub 0}{partial derivative}{sub 0}){sup 2} {minus}({ital Rn}{center dot}{partial derivative}){sup 2} inside the field equations. The Cauchy problem is solved in the free field theory leading to a propagator regularized by means of the Leibbrandt-Mandelstam prescription. In the non-Abelian theory, these gauges are now on the same footing as relativistic gauges but the contribution of ghost loops vanishes.

Ward identities for the stochastic quantization of gauge fields

Ward identities of BRS or nil-potent type are derived which express the content of local gauge invariance for stochastically quantized gauge theories. First the Langevin equation for stochastically quantized non-abelian gauge theories with gauge fixing is shown to be invariant under a restricted class of local, classical gauge transformations. Next the solution of the Langevin equation is expressed as a functional integral with a local action which is shown to be invariant under the restricted classical gauge transformations. Grassmann variables or ghosts are introduced which make the action invariant under a nil-potent transformation. They play no dynamical role, but only serve as a book-keeping device to calculate the BRS transform of non-ghost amplitudes. (All closed ghost loops vanish.) Finally the Ward identities are solved to obtain the renormalized action and restrictions on renormalization constants.

Gauge invariance of the quantum Wilson loop.

We define the quantum operator for the Wilson loop in QCD and prove that it is gauge invariant. We also derive diagrammatically the Feynman rules in various gauges for the perturbative calculation of the Wilson loop. Our result solves the puzzle whether the Feynman rules in the axial gauge are consistent with those in other gauges, to all order of the coupling constant and for all contours in the space-time region. In particular, we show that one must include the contribution of ghost loops if the principal-value prescription is used for the axial-gauge propagator.

Spacelike axial-gauge Feynman rules: A consistent nonambiguous derivation.

In addition to ambiguities in the propagator definition, the axial gauge nxA = 0 is known to suffer from a fundamental inconsistency. We give here a nonambiguous approach to the Feynman rules to be used in the axial gauge. This gauge appears as the limit for a critical value of consistent theories to which the Becchi-Rouet-Stora quantization scheme and all its consequences can be applied. The poles at nxk = 0 are automatically regularized but not by the usual controversial principal-value prescription. Ghost loops get a priori nonvanishing contributions, which, up to now, have been overlooked.

Inconsistency of Feynman rules derived via path integration.

We show that the Feynman rules in the covariant $\ensuremath{\alpha}$ gauge, the Coulomb gauge, and the axial gauge, in the generally adopted form derived via the Faddeev-Popov formalism, are inconsistent with one another. We suggest that they be replaced by the canonically derived ones, which are consistent. In particular, the Feynman rules in the Coulomb gauge contain the contribution of the anomalous Coulomb interaction, and the Feynman rules in the axial gauge with the principal-value prescription contain the contribution of ghost loops.

Power-counting theorem for infrared logarithms in non-Abelian gauge theories

We have developed a general diagrammatic method for precisely determining the power of infrared logarithms for diagrams of arbitrary order in non-Abelian gauge theories. It is applied specifically to the study of the infrared structure of the gauge-group singlet quark form factor for fixed momentum transfer and nonvanishing quark mass. All leading infrared logarithms of the form factor are found to arise from a class of planar quantum-electrodynamic-type diagrams in which, however, the gluon propagators are fully dressed by gluon and ghost loops. Thus the infrared behavior of the massive-quark sector reflects strongly that of the massless-gluon sector, which in turn is controlled by the ultraviolet structure of the gluon self-interaction.