I.T. Drummond

The stochastic method for numerical simulations:: Higher order corrections

Abstract We derive discrete versions of stochastic differential equations governing the evolution of some random variable x ( t ) to arbitrary order in Δt , giving explicit formulae to second order. These are tested in the static case by examples where x takes values in the groups U(1) and SU(2).

The two-Reggeon/particle coupling

Abstract A study is made of the two-Reggeon/particle coupling function defined through a production amplitude, with particular reference to its dependence on the Toller angle ω. When one of the Reggeon spins takes a physical value, the coupling function reduces to a polynomial in cos ω. This is understood in several ways: (a) as resulting from the vanishing of non-sense amplitudes in a partial wave decomposition of the production amplitude; (b) in terms of a Fourier transform of the production amplitude; (c) in the perturbation theory; (d) as an extrapolation of the properties of the three-particle coupling that may be deduced from a set of interaction Lagrangians. When the spin of one of the Reggeons in physical the non-vanishing sense amplitudes become equal to the Reggeon/two-particle coupling amplitudes. This is verified in the perturbation theory model for the case of zero spin. The results of the investigation are used to suggest a method of parametrising the two-Reggeon/particle coupling function which allows it to be approximated from a knowledge of ordinary Regge residues. A generalisation of the analysis to the case of parity change at the central vertex is discussed. The implications for non-sense dips in differential cross sections is pointed out. The method of treating external particles with spin is indicated.

Foam diagram summation at finite temperature

We show that large-N φ4 theory is not trivial if one accepts the presence of a tachyon with a truly huge mass, and that it allows exact calculation. We use it to illustrate how to calculate the exact resummed pressure at finite temperature and verify that it is infrared and ultraviolet finite even in the zero-mass case. In three dimensions a residual effect of the resummed infrared divergences is that at low temperature or strong coupling the leading term in the interaction pressure becomes independent of the coupling and is 45 of the free-field pressure. In four dimensions the pressure is well defined provided that the temperature is below the tachyon mass. We examine how rapidly this expansion converges and use our analysis to suggest how one might reorganise perturbation theory to improve the calculation of the pressure for the QCD plasma.

Distortion of line and surface elements in model turbulent flows

Material lines and surfaces transported in a random velocity field undergo bending and stretching. We investigate the time evolution of curvature in line and surface elements both analytically and by numerical simulation for a simple model turbulence. Our analysis is close to that of Pope for the evolution of curvature in surface elements

Turbulent stretching of line and surface elements

Material line and surface elements transported in a turbulent velocity field increase in length or area at an exponential rate. In this paper we investigate how the stretching rates are related to the statistical properties of the velocity field both analytically and numerically in simple models of turbulence. In a Gaussian model the statistics exhibit time-reversal invariance. We demonstrate that, as pointed out by Kraichnan (1974), this leads to equality of line and area stretching rates. We also construct a model which violates the time-reversal property and splits the values of the rates for lines and surfaces. The sign of the splitting depends on the sign of the time-reversal breakdown.

Measuring the aspect ratio renormalization of anisotropic-lattice gluons

Using tadpole-improved actions we investigate the consistency between different methods of measuring the aspect ratio renormalization of anisotropic-lattice gluons for bare aspect ratios {chi}{sub 0}=4,6,10 and inverse lattice spacing in the range a{sub s}{sup -1}=660--840 MeV. The tadpole corrections to the action, which are established self-consistently, are defined for two cases, mean link tadpoles in the Landau gauge and gauge invariant mean plaquette tadpoles. Parameters in the latter case exhibited no dependence on the spatial lattice size L, while in the former, parameters showed only a weak dependence on L easily extrapolated to L={infinity}. The renormalized anisotropy {chi}{sub R} was measured using both the torelon dispersion relation and the sideways potential method. There is general agreement between these approaches, but there are discrepancies which are evidence for the presence of lattice artifact contributions. For the torelon these are estimated to be O({alpha}{sub S}a{sub s}{sup 2}/R{sup 2}), where R is the flux-tube radius. We also present some new data that suggest that rotational invariance is established more accurately for the mean-link action than the plaquette action.

Strong coupling model for string breaking on the lattice

Abstract A model for SU ( n ) string breaking on the lattice is formulated using strong coupling ideas. Although necessarily rather crude, the model gives an explicit picture of string breaking in the presence of dynamical quarks as a mixing phenomenon that involves the string state and a two-meson state. A careful analysis, within the model, of the Wilson loop shows that the evolution of the mixing angle as a function of separation may obscure the expected crossover effect. If a sufficiently extensive mixing region exists then an appropriate combination of transition amplitudes can help in revealing the effect. The sensitivity of the mixing region to the values of the meson energy and the dynamical quark mass is explored and an assessment made of the detectibility of string breaking in a practical lattice simulation.

REGGE-PALATINI CALCULUS

We formulate a Palatini version of the Regge calculus by constructing a discrete torsion field on the simplicial manifold. The action has two components, the original Regge action and an additional action for the torsion field. In the absence of matter the variational equations reduce the torsion field to zero. Matter fields can act as sources of torsion.

Stochastic quantization simulation of φ4 theory

Abstract We present the results of a simulation of φ 4 theory on an 8 4 lattice, using the stochastic quantisation method of Parisi and Wu extended to include slave equations for the estimators of connected Green functions. Our results demonstrate the weakly interacting nature of the renormalised theory for a wide range of bare couplings, and are consistent with the expected triviality of the continuum theory.

QCD PRESSURE AND THE TRACE ANOMALY

Abstract Exact relations between the QCD thermal pressure and the trace anomaly are derived. These are used, first, to prove the equivalence of the thermodynamic and the hydrodynamic pressure in equilibrium in the presence of the trace anomaly, closing a gap in previous arguments. Second, in the temporal axial gauge a formula is derived which expresses the thermal pressure in terms of a Dyson-resummed two-point function. This overcomes the infrared problems encountered in the conventional perturbation-theory approach.