Gauge-invariant Correlation Functions Research Articles

De Sitter at all loops: the story of the Schwinger model

We consider the two-dimensional Schwinger model of a massless charged fermion coupled to an Abelian gauge field on a fixed de Sitter background. The theory admits an exact solution, first examined by Jayewardena, and can be analyzed efficiently using Euclidean methods. We calculate fully non-perturbative, gauge-invariant correlation functions of the electric field as well as the fermion and analyze these correlators in the late-time limit. We compare these results with the perturbative picture, for example by verifying that the one-loop contribution to the fermion two-point function, as predicted from the exact solution, matches the direct computation of the one-loop Feynman diagram. We demonstrate many features endemic of quantum field theory in de Sitter space, including the appearance of late-time logarithms, their resummation to de Sitter invariant expressions, and Boltzmann suppressed non-perturbative phenomena, with surprising late-time features.

Chromoelectric field correlator for quarkonium transport in the strongly coupled mathcal{N} = 4 Yang-Mills plasma from AdS/CFT

Previous studies have shown that a gauge-invariant correlation function of two chromoelectric fields connected by a straight timelike adjoint Wilson line encodes crucial information about quark-gluon plasma (QGP) that determines the dynamics of small-sized quarkonium in the medium. Motivated by the successes of holographic calculations to describe strongly coupled QGP, we calculate the analog gauge-invariant correlation function in strongly coupled mathcal{N} = 4 supersymmetric Yang-Mills theory at finite temperature by using the AdS/CFT correspondence. Our results indicate that the transition processes between bound and unbound quarkonium states are suppressed in strongly coupled plasmas, and moreover, the leading contributions to these transition processes vanish in both the quantum Brownian motion and quantum optical limits of open quantum system approaches to quarkonia.

Gauge Invariance of Non-Abelian Field Strength Correlators: The Axial Gauge Puzzle.

Many transport coefficients of the quark-gluon plasma and nuclear structure functions can be written as gauge invariant correlation functions of non-Abelian field strengths dressed with Wilson lines. We discuss the applicability of axial gauge n·A=0 to calculate them. In particular, we address issues that appear when one attempts to trivialize the Wilson lines in the correlation functions by gauge fixing. We find it is always impossible to completely remove the gauge fields n·A in Wilson lines that extend to infinity in the n direction by means of gauge transformations. We show how the obstruction appears in an explicit example of a perturbative calculation, and we also explain it more generally from the perspective of the path integral that defines the theory. Our results explain why the two correlators that define the heavy quark and quarkonium transport coefficients, which are seemingly equal in axial gauge, are actually different physical quantities of the quark-gluon plasma and have different values. Furthermore, our findings provide insights into the difference between two inequivalent gluon parton distribution functions.

Hamiltonian analysis and positivity of a new massive spin-2 model

Recently a new model has been proposed to describe free massive spin-2 particles in D dimensions in terms of a non symmetric rank-2 tensor e μν and a mixed symmetry tensor B μ[αβ]. The model is invariant under linearized diffeomorphisms without Stueckelberg fields. It resembles a spin-2 version of the topologically massive spin-1 BF model (Cremmer–Scherk model). Here we apply the Dirac–Bergmann procedure in order to identify all Hamiltonian constraints and perform a complete counting of degrees of freedom. In D = 3 + 1 we find 5 degrees of freedom corresponding to helicities ±2, ±1, 0 as expected. The positivity of the reduced Hamiltonian is proved by using spin projection operators. We have also proposed a parent action that establishes the duality between the Fierz–Pauli and the new model. The equivalence between gauge invariant correlation functions of both theories is demonstrated.

One-loop quantum gravitational corrections to the scalar two-point function at fixed geodesic distance

We study a proposal for gauge-invariant correlation functions in perturbative quantum gravity, which are obtained by fixing the geodesic distance between points in the fluctuating geometry. These correlation functions are non-local and strongly divergent, and we show how to renormalise them by performing a ‘wave function renormalisation’ of the geodesic embedding coordinates. The result is finite and gauge-independent, but displays unusual features such as double logarithms at one-loop order.

Generalized Loop Space and Evolution of the Light-Like Wilson Loops

Equations of motion for the light-like QCD Wilson loops are studied in the generalized loop space (GLS) setting. To this end, the classically conformal-invariant non-local variations of the cusped Wilson exponentials lying (partially) on the light-cone are formulated in terms of the Fréchet derivative. The rapidity and renormalization-group behaviour of the gauge-invariant quantum correlation functions (in particular, the three-dimensional parton densities) are demonstrated to be connected to certain geometrical properties of the Wilson loops defined in the GLS.

Gauge turbulence, topological defect dynamics, and condensation in Higgs models

The real-time dynamics of topological defects and turbulent configurations of gauge fields for electric and magnetic confinement are studied numerically within a 2+1D Abelian Higgs model. It is shown that confinement is appearing in such systems equilibrating after a strong initial quench such as the overpopulation of the infrared modes. While the final equilibrium state does not support confinement, metastable vortex defect configurations appear in the gauge field which are found to be closely related to the appearance of physically observable confined electric and magnetic charges. These phenomena are seen to be intimately related to the approach of a non-thermal fixed point of the far-from-equilibrium dynamical evolution, signalled by universal scaling in the gauge-invariant correlation function of the Higgs field. Even when the parameters of the Higgs action do not support condensate formation in the vacuum, during this approach, transient Higgs condensation is observed. We discuss implications of these results for the far-from-equilibrium dynamics of Yang-Mills fields and potential mechanisms how confinement and condensation in non-abelian gauge fields can be understood in terms of the dynamics of Higgs models. These suggest that there is an interesting new class of dynamics of strong coherent turbulent gauge fields with condensates.

On the definition and interpretation of a static quark-antiquark potential in the color-adjoint channel

We study possibilities to define a static quark-antiquark pair in a color-adjoint orientation based on Wilson loops with generator insertions, using both lattice QCD and leading-order perturbation theory in various gauges. Nonperturbatively, the only way to obtain nonzero results while maintaining positivity of the Hamiltonian is by using either temporal gauge or Coulomb gauge with some additional constraint on the temporal links removing the residual gauge symmetry. In this case the correlator is equivalent to a gauge invariant correlation function of a static quark-antiquark pair and a static adjoint quark; the resulting three-point potential is attractive. Saturating open color indices with color magnetic fields instead also leads to a gauge invariant correlator. However, this object is found to couple to the singlet sector only and results in a hybrid potential. None of the considered lattice observables reproduces the repulsive adjoint potential predicted by perturbation theory in Lorenz or Coulomb gauges.

Heavy-light diquark masses from QCD sum rules and constituent diquark models of tetraquarks

Diquarks with ${J}^{P}={0}^{\ifmmode\pm\else\textpm\fi{}}$, ${1}^{\ifmmode\pm\else\textpm\fi{}}$ containing a heavy (charm or bottom) quark and a light quark are investigated using QCD Laplace sum rules. Masses are determined using appropriately constructed gauge invariant correlation functions, including for the first time next-to-leading order perturbative contributions. The ${J}^{P}={0}^{+}$ and ${1}^{+}$ charm-light diquark masses are, respectively, found to be $1.86\ifmmode\pm\else\textpm\fi{}0.05$ and $1.87\ifmmode\pm\else\textpm\fi{}0.10\text{ }\text{ }\mathrm{GeV}$, while those of the ${0}^{+}$ and ${1}^{+}$ bottom-light diquarks are both determined to be $5.08\ifmmode\pm\else\textpm\fi{}0.04\text{ }\text{ }\mathrm{GeV}$. The sum rules derived for heavy-light diquarks with negative parity are poorly behaved and do not permit unambiguous mass predictions, in agreement with previous results for negative parity light diquarks. The scalar and axial vector heavy-light diquark masses are degenerate within uncertainty, as expected by heavy quark symmetry considerations. Furthermore, these mass predictions are in good agreement with masses extracted in constituent diquark models of the tetraquark candidates $X(3872)$ and ${Y}_{b}(10890)$. Thus these results provide QCD support for the interpretation of the $X(3872)$ and ${Y}_{b}(10890)$ as ${J}^{\mathrm{PC}}={1}^{++}$ tetraquark states composed of diquark clusters. Further implications for tetraquarks among the heavy quarkoniumlike XYZ states are discussed.

Simulating fermion production in1+1dimensional QED

We investigate fermion-antifermion production in $1+1$ dimensional QED using real-time lattice techniques. In this nonperturbative approach the full quantum dynamics of fermions is included, while the gauge field dynamics can be accurately represented by classical-statistical simulations for relevant field strengths. We compute the nonequilibrium time evolution of gauge-invariant correlation functions, implementing ``low-cost'' Wilson fermions. Introducing a lattice generalization of the Dirac-Heisenberg-Wigner function, we recover the Schwinger formula in $1+1$ dimensions in the limit of a static background field. We discuss the decay of the field due to the backreaction of the created fermion-antifermion pairs and apply the approach to strongly inhomogeneous gauge fields. The latter allows us to discuss the striking phenomenon of a linear rising potential building up between produced fermion bunches after the initial electric pulse ceased.

3-loop 3PI effective action for 3D SU(3) QCD

The 3PI method is a technique to resum an infinite class of diagrams, which may be useful in studying nonperturbative thermodynamics and dynamics in quantum field theory. But it has never been successfully applied to gauge theories, where there are serious questions about gauge invariance breaking. We show how to perform the 3PI resummation of QCD in 3 Euclidean spacetime dimensions, a warmup problem to the 4 or 3+1 dimensional case. We present the complete details of the technical problems and how they are overcome. We postpone a comparison of gauge invariant correlation functions with their lattice-determined counterparts to a future publication.

Duality of parity doublets of helicity±2inD=2+1dimensions

In $D=2+1$ dimensions there are two dual descriptions of parity singlets of helicity $\ifmmode\pm\else\textpm\fi{}1$, namely, the self-dual model of first order (in derivatives) and the Maxwell-Chern-Simons theory of second order. Correspondingly, for helicity $\ifmmode\pm\else\textpm\fi{}2$ there are four models ${S}_{\mathrm{SD}\ifmmode\pm\else\textpm\fi{}}^{(r)}$ describing parity singlets of helicities $\ifmmode\pm\else\textpm\fi{}2$. They are of first, second, third, and fourth order ($r=1$, 2, 3, 4), respectively. Here we show that the generalized soldering of the opposite helicity models ${S}_{\mathrm{SD}+}^{(4)}$ and ${S}_{\mathrm{SD}\ensuremath{-}}^{(4)}$ leads to the linearized form of the new massive gravity suggested by Bergshoeff, Hohm, and Townsend (BHT) similarly to the soldering of ${S}_{\mathrm{SD}+}^{(3)}$ and ${S}_{\mathrm{SD}\ensuremath{-}}^{(3)}$. We argue why in both cases we have the same result. We also find out a triple master action which interpolates between the three dual models: linearized BHT theory, ${S}_{\mathrm{SD}+}^{(3)}+{S}_{\mathrm{SD}\ensuremath{-}}^{(3)}$, and ${S}_{\mathrm{SD}+}^{(4)}+{S}_{\mathrm{SD}\ensuremath{-}}^{(4)}$. By comparing gauge invariant correlation functions we deduce dual maps between those models. In particular, we learn how to decompose the field of the linearized BHT theory in helicity eigenstates of the dual models up to gauge transformations.

Calculating the gauge-invariant two-point correlation function of gluonic field strengths

The two-point gauge-invariant correlation function of gluonic field strengths, which is the main input in the stochastic vacuum model, is derived by using its relation to the Green functions of one- and two-gluon gluelumps. These Green functions are found by evaluating the path integrals of one and two gluons confined in the field of an infinitely heavy adjoint source. This allows one to obtain also the terms in the exponential asymptotics of the two-point correlation function, which are produced by fluctuations of the gluons around their mean separation from the source. The pre-exponents are derived as well, demonstrating some novel dependences on the distance.

The matter density distribution for mesons and baryons

The matter density distribution inside a hadron is evaluated using gauge-invariant correlation functions within the quenched and the unquenched theory. Comparison with the charge density distribution suggests that hadron deformation is a consequence of the relativistic motion of the quarks.

On the coupling of the self-dual field to dynamicalU(1) matter and its dual theory

We consider arbitrary U(1) charged matter non-minimally coupled to the self-dual field in d = 2 + 1. The coupling includes a linear and a rather general quadratic term in the self-dual field. By using both Lagragian gauge embedding and master action approaches we derive the dual Maxwell Chern–Simons-type model and show the classical equivalence between the two theories. At the quantum level the master action approach in general requires the addition of an awkward extra term to the Maxwell Chern–Simons-type theory. Only in the case of a linear coupling in the self-dual field can the extra term be dropped and we are able to establish the quantum equivalence of gauge invariant correlation functions in both theories.

What is `Holography' in the Plane-Wave Limit of the AdS5/SYM4Correspondence?

The issue of holographic principle in the PP-wave limit of the AdS/CFT correspondence is discussed. We show that, in the plane-wave limit, the relation between the partition function in the bulk and the gauge-invariant correlation functions on the boundary should be interpreted on the basis of a tunneling picture in the semi-classical approximation which is appropriate for the plane-wave limit. This leads to a natural relation between Euclidean S-matrix in the bulk and the operator-product expansion of the so-called BMN operators on the boundary.

Gauge-invariant field strength correlators in full QCD

We study, by numerical simulations on a lattice, the behaviour of the gauge-invariant correlation functions of the gauge field strengths in the QCD vacuum with four flavours of dynamical staggered fermions.

Magnetic screening in the high temperature phase of the standard model

Non-perturbative effects in the high-temperature phase of the electroweak theory are characterized by a magnetic screening length. Its size influences the range of validity of perturbation theory, and it also determines the critical Higgs boson mass where the first-order phase transition changes to a crossover. We propose a gauge-invariant definition of the magnetic screening length and discuss its role in several gauge-dependent and gauge-invariant correlation functions.

Gauge-invariant effective potential: Equilibrium and nonequilibrium aspects.

We propose a gauge invariant formulation of the effective potential in terms of a gauge invariant order parameter, for the Abelian Higgs model. The one-loop contribution at zero and finite temperature is computed explicitly, and the leading terms in the high temperature expansion are obtained. The result is contrasted to the effective potential obtained in several covariant gauge-fixing schemes, and the gauge invariant quantities that can be reliably extracted from these are identified. It is pointed out that the gauge invariant effective potential in the one-loop approximation is complex for {\em all values} of the order parameter between the maximum and the minimum of the tree level potential, both at zero and non-zero temperature. The imaginary part is related to long-wavelength instabilities towards phase separation. We study the real-time dynamics of initial states in the spinodal region, and relate the imaginary part of the effective potential to the growth rate of equal-time gauge invariant correlation functions in these states. We conjecture that the spinodal instabilities may play a role in non-equilibrium processes {\em inside} the nucleating bubbles if the transition is first order.

Topological defects on fluctuating surfaces: General properties and the Kosterlitz-Thouless transition.

We investigate the Kosterlitz-Thouless transition for hexatic order on a free fluctuating membrane and derive both a Coulomb gas and a sine-Gordon Hamiltonian to describe it. The Coulomb-gas Hamiltonian includes charge densities arising from disclinations and from Gaussian curvature. There is an interaction coupling the difference between these two densities, whose strength is determined by the hexatic rigidity, and an interaction coupling Gaussian curvature densities arising from the Liouville Hamiltonian resulting from the imposition of a covariant cutoff. In the sine-Gordon Hamiltonian, there is a linear coupling between a scalar field and the Gaussian curvature. We discuss gauge-invariant correlation function for hexatic order and the dielectric constant of the Coulomb gas. We also derive renormalization group recursion relations that predict a transition with decreasing bending rigidity $\kappa$.