Averaging out the Einstein equations

A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.     

Soft gluons in logarithmic summations

We demonstrate that all the known single- and double-logarithm summations for a parton distribution function can be unified in the Collins–Soper resummation technique by applying soft approximations appropriate in different kinematic regions to real gluon emissions. Neglecting the gluon longitudinal momentum, we obtain the (double-logarithm) resummation for two-scale QCD processes, and the Balitsky–Fadin–Kuraev–Lipatov (single-logarithm) equation for one-scale processes. Neglecting the transverse momentum, we obtain the threshold (double-logarithm) resummation for two-scale processes, and the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (single-logarithm) equation for one-scale processes. If we keep the longitudinal and transverse momenta simultaneously, we derive a unified resummation for a large Bjorken variable x, and a unified evolution equation appropriate for both intermediate and small x. Received: 9 March 1999 / Revised version: 12 April 1999 / Published online: 3 August 1999     

Collective excitations of gluons

Three gauge invariant antisymmetric tensor fields are introduced in the nonabelian gauge theories. They are certain non-linear combinations of the conjugate field tensor and they obey O(3) algebra. An effective chiral lagrangian for these fields is derived. It describes 3 vector and 3 axial mesons with vacuum quantum numbers. The masses are generated by spontaneous restoration of Lorentz invariance.     

Confinement models for gluons

Confinement model for gluons using a ‘colour super current’ is formulated. An attempt has been made to derive a suitable dielectric function corresponding to the current confinement. A simple inhomogeneous dielectric confinement model for gluons is studied for comparison. The model Hamiltonians are second quantized and the glueball states are constructed. The spurious motion of the centre of confinement is accounted for. The results of the current confinement scheme is found in good agreement with experimental candidates.     

Quarks and gluons in color fields

Production and collective motion ofq \(\bar q\) and gluon pairs in strong color fields is studied using boost-invariant version of collisionless Boltzmann-Vlasov transport equations. It is shown that, for realistic fields, the effective formation time for pair production according to Schwinger mechanism is a small fraction of 1 fermi. Collective oscillations of quarkgluon gas, although damped by pair production and expansion of the system, remain an important feature of the process. The considered mechanism produces quark and gluons in a state which is rather far from equilibrium and thus strong collision effects (collision time ?1 fm) are needed to achieve equilibrated distributions at formation time. Finally, it is observed that, if such equilibrated distribution is produced at later times, the expansion of the system modifies dramatically its properties, as is explicitly shown for small oscillations of quark-gluon plasma in a weak external field.     

Dynamics of confined gluons

Propagation of gluons in the confining vacuum is studied in the framework of background perturbation theory, where nonperturbative background contains confining correlators. Two settings of the problem are considered. In the first, the confined gluon evolves in time together with the static quark and antiquark forming the one-gluon static hybrid. The hybrid spectrum is calculated in terms of string tension and is in agreement with earlier analytic and lattice calculations. In the second setting, the confined gluon is exchanged between quarks and the gluon Green’s function is calculated, giving rise to the Coulomb potential modified at large distances. The resulting screening radius of 0.5 fm presents a problem when confronted with lattice and experimental data. A possible solution of this discrepancy is discussed.     

Properties of gluons in color superconductors

I give an introduction to color superconductivity in cold, dense quark matter. I focus in particular on how the solution to the gap equation is qualitatively different in ordinary BCS theory with local four-fermion interactions, in theories with non-local, but screened boson exchange, and in QCD, where magnetic gluon exchange is not screened at zero temperature. I argue that a reliable computation of the gap parameter requires knowledge of the gluon self-energy in a color superconductor. As a first step to determine the gluon self-energy, I report calculations of the Debye and Meissner masses in two- and three-flavor color superconductors.     

Topological “vector gluons”

It is shown that the axion can be made invisible in the family-unifyingSU(9) model and that all the light fermions become massive due to the radiative correction.     

Elastic diffraction and non-perturbative gluons

Zeitschrift für Physik C Particles and Fields - We consider a simple model for the propagator of a non-perturbative gluon and derive the rules determining its coupling to perturbative gluons...     

Soft gluons in semi-inclusive deep inelastic experiments

The longitudinal and transverse properties of the QCD contribution to semi-inclusive deep inelastic scattering is worked out within the soft gluon resummation procedure. The Λ parameter needed to reproduce the data is in agreement with the value extracted from scaling violations. The intrinsic transverse momentum squared of a quark inside a proton is around 0.45 GeV². Predictions for the transverse correlations $\text{〈}\text{p}{}_{\mathrm{1\perp }}\text{·}\text{p}{}_{\mathrm{2\perp }}\text{〉}$ of two outgoing pions are guven.     

Medium-induced emissions of hard gluons

We present a derivation of the medium-induced gluon radiation spectrum beyond the current limitation of soft gluon emission. Making use of the path integral approach to describe the propagation of high-energy particles inside a medium, we study the limiting case of a hard gluon emission. Analytical and numerical results are presented and discussed within the multiple soft scattering approximation. An ansatz interpolating between soft and hard gluon emissions is provided. The Landau–Pomeranchuk–Migdal effect is observed in the expected kinematic region.     

In-medium QCD and Cherenkov gluons

The equations of in-medium gluodynamics are proposed. Their classical lowest order solution is explicitly shown for a color charge moving with constant speed. For nuclear permittivity larger than 1 it describes emission of Cherenkov gluons resembling the results of classical electrodynamics. The choice of nuclear permittivity and Lorentz invariance of the problem are discussed. Effects induced by the transversely and longitudinally moving (relative to the collision axis) partons at LHC energies are described.     

Composite gluons and magnetic monopoles

We present a static solution to the classical field equations of a purely spinorial model with SU(2) internal symmetry. The model contains composite vector and pseudovector gluon fields. The solution for the vector field is precisely that of the Wu-Yang monopole.     

Elastic forward scattering of gluons

The colour octet and singlet parts of the elasticgggg-scattering amplitude are evaluated in the Regge kinematical regions–t in the LLA, withi-terms taken into account, by constructing and solving a set of the infrared evolution equations.This work was supported, in part, by a Soros Foundation Grant awarded by the American Physical Society     

On granular surface flow equations

Conservation equations are written for surface flows (either fluid or granular). The particularity of granular surface flows is then pointed out, namely that the depth of the flowing layer is not a priori fixed, leading to open equations. It is shown how some hypothesis on the flowing layer allows to close the system of equations. A possible hypothesis, similar to that made for a fluid layer, but inspired from granular flow experiments, is presented. The force acting on the flowing layer is discussed. Averaging over the flowing depth, as in shallow water theory, then allows to transform these conservation laws into equations for the evolution of the profile of a granular pile. Apart from their interest for building models, these conservation laws can be used to measure experimentally the effective forces acting on a flowing layer. Received 25 July 1998 and Received in final form 14 January 1999