The numerical approach to quantum field theory in a non-commutative space

Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is presented, and their implications are reviewed. In addition, we also discuss how related numerical techniques have been recently applied in computer simulations of dimensionally reduced supersymmetric theories.Comment: 15 pages, 6 figures, invited talk presented at the Humboldt Kolleg "Open Problems in Theoretical Physics: the Issue of Quantum Space-Time", to appear in the proceedings of the Corfu Summer Institute 2015 "School and Workshops on Elementary Particle Physics and Gravity" (Corfu, Greece, 1-27 September 2015

Thermodynamics of the strongly interacting gluon plasma in the large-N limit

We report on our recent study of equilibrium thermodynamic observables in SU(N) gauge theories with N=3, 4, 5, 6 and 8 colors at temperatures T in the range from 0.8 T_c to 3.4 T_c (where T_c denotes the critical deconfinement temperature). The results, which show a very weak dependence on the number of colors, are compared with gauge/gravity models of the QCD plasma, including the improved holographic QCD model proposed by Kiritsis and collaborators, and with the supergravity prediction for the entropy density deficit. Furthermore, we investigate the possibility that the trace anomaly may receive contributions proportional to T^2 at temperatures close to T_c. Finally, we present the extrapolated results for the pressure, trace anomaly, energy and entropy densities in the limit for N going to infinity.Comment: 7 pages, 8 eps figures, presented at the XXVII International Symposium on Lattice Field Theory, July 26-31 2009, Peking University, Beijing, China; v2: added reference

Quantum Field Theory in a Non-Commutative Space: Theoretical Predictions and Numerical Results on the Fuzzy Sphere

We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to the scalar model. Then we comment on the results recently obtained from Monte Carlo simulations, and show a preview of new numerical data, which are consistent with the expected transition between two phases characterised by the topology of the support of a matrix eigenvalue distribution.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Casimir scaling and renormalization of Polyakov loops in large-N gauge theories

We study Casimir scaling and renormalization properties of Polyakov loops in different irreducible representations in SU(N) gauge theories; in particular, we investigate the approach to the large-N limit, by performing lattice simulations of Yang-Mills theories with an increasing number of colors, from 2 to 6. We consider the twelve lowest irreducible representations for each gauge group, and find strong numerical evidence for nearly perfect Casimir scaling of the bare Polyakov loops in the deconfined phase. Then we discuss the temperature dependence of renormalized loops, which is found to be qualitatively and quantitatively very similar for the various gauge groups. In particular, close to the deconfinement transition, the renormalized Polyakov loop increases with the temperature, and its logarithm reveals a characteristic dependence on the inverse of the square of the temperature. At higher temperatures, the renormalized Polyakov loop overshoots one, reaches a maximum, and then starts decreasing, in agreement with weak-coupling predictions. The implications of these findings are discussed.Comment: 1+33 pages, 14 figures; v2: expanded discussion in sections 2 and 3, added references: version published in JHE

Momentum broadening of partons on the light cone from the lattice

The jet quenching parameter describes the momentum broadening of a high-energy parton moving through the quark-gluon plasma. Following an approach originally proposed by Caron-Huot, we discuss how one can extract information on the collision kernel associated with the parton momentum broadening, from the analysis of certain gauge-invariant operators in dimensionally reduced effective theories, and present numerical results from a lattice study.Comment: 7 pages, 3 pdf figures, talk presented at the 31st International Symposium on Lattice Field Theory "Lattice 2013" (29 July - 3 August 2013, Mainz, Germany

Investigating jet quenching on the lattice

Due to the dynamical, real-time, nature of the phenomenon, the study of jet quenching via lattice QCD simulations is not straightforward. In this contribution, however, we show how one can extract information about the momentum broadening of a hard parton moving in the quark-gluon plasma, from lattice calculations. After discussing the basic idea (originally proposed by Caron-Huot), we present a recent study, in which we estimated the jet quenching parameter non-perturbatively, from the lattice evaluation of a particular set of gauge-invariant operators.Comment: 4 pages, 2 figures, talk presented at the 6th International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions "Hard Probes 2013" (4-8 November 2013, Cape Town, South Africa

A numerical study of a confined Q anti-Q system in compact U(1) lattice gauge theory in 4D

We present a numerical study about the confining regime of compact U(1) lattice gauge theory in 4D. To address the problem, we exploit the duality properties of the theory. The main features of this method are presented, and its possible advantages and limits with respect to alternative techniques are briefly discussed. In Monte Carlo simulations, we focus our attention onto the case when a pair of static external charges is present. Some results are shown, concerning different observables which are of interest in order to understand the confinement mechanism, like the profile of the electric field induced by the static charges, and the ratios between Polyakov loop correlation functions at different distances

Comparing the Nambu-Goto string with LGT results

We discuss a way to evaluate the full prediction for the interquark potential which is expected from the effective Nambu-Goto string model. We check the correctness of the prescription reproducing the results obtained with the zeta function regularization for the first two perturbative orders. We compare the predictions with existing Monte Carlo data for the (2+1) dimensional Z(2), SU(2) and SU(3) gauge theories: in the low temperature regime, we find good agreement for large enough interquark distances, but an increasing mismatch between theoretical predictions and numerical results is observed as shorter and shorter distances are investigated. On the contrary, at high temperatures (approaching the deconfinement transition from below) a remarkable agreement between Monte Carlo data and the expectations from the Nambu-Goto effective string is observed for a wide range of interquark distances.Comment: 25 pages, 4 eps figures; added a reference, included remarks, corrected a typo; version accepted for publication in JHE