We review results from lattice QCD calculations on the thermodynamics of strong-interaction matter with emphasis on input these calculations can provide to the exploration of the phase diagram and properties of hot and dense matter created in heavy ion experiments. This review is organized in sections as follows: (1) Introduction, (2) QCD thermodynamics on the lattice, (3) QCD phase diagram at high temperature, (4) Bulk thermodynamics, (5) Fluctuations of conserved charges, (6) Transport properties, (7) Open heavy flavors and heavy quarkonia, (8) QCD in external magnetic fields, (9) Summary.

Thermodynamics of strong-interaction matter from Lattice QCD

Properties of hot and dense matter from relativistic heavy ion collisions

A primer on resurgent transseries and their asymptotics

The $XYZ$ states: experimental and theoretical status and perspectives

theta dependence of SU(N) gauge theories in the presence of a topological term

It is sometimes speculated that the sign problem that afflicts many quantum field theories might be reduced or even eliminated by choosing an alternative domain of integration within a complexified extension of the path integral (in the spirit of the stationary phase integration method). In this paper we start to explore this possibility somewhat systematically. A first inspection reveals the presence of many difficulties but---quite surprisingly---most of them have an interesting solution. In particular, it is possible to regularize the lattice theory on a Lefschetz thimble, where the imaginary part of the action is constant and disappears from all observables. This regularization can be justified in terms of symmetries and perturbation theory. Moreover, it is possible to design a Monte Carlo algorithm that samples the configurations in the thimble. This is done by simulating, effectively, a five-dimensional system. We describe the algorithm in detail and analyze its expected cost and stability. Unfortunately, the measure term also produces a phase which is not constant and it is currently very expensive to compute. This residual sign problem is expected to be much milder, as the dominant part of the integral is not affected, but we have still no convincing evidence of this. However, the main goal of this paper is to introduce a new approach to the sign problem, that seems to offer much room for improvements. An appealing feature of this approach is its generality. It is illustrated first in the simple case of a scalar field theory with chemical potential, and then extended to the more challenging case of QCD at finite baryonic density.

New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble

We present the first practical Monte Carlo calculations of the recently proposed Lefschetz thimble formulation of quantum field theories. Our results provide strong evidence that the numerical sign problem that afflicts Monte Carlo calculations of models with complex actions can be softened significantly by changing the domain of integration to the Lefschetz thimble or approximations thereof. We study the interacting complex scalar field theory (relativistic Bose gas) in lattices of size up to ${8}^{4}$ using a computationally inexpensive approximation of the Lefschetz thimble. Our results are in excellent agreement with known results. We show that---at least in the case of the relativistic Bose gas---the thimble can be systematically approached and the remaining residual phase leads to a much more tractable sign problem (if at all) than the original formulation. This is especially encouraging in view of the wide applicability---in principle---of our method to quantum field theories with a sign problem. We believe that this opens up new possibilities for accurate Monte Carlo calculations in strongly interacting systems of sizes much larger that previously possible.

Monte Carlo simulations on the Lefschetz thimble: Taming the sign problem

We explore the highly non-perturbative hot region of the QCD phase diagram
close to Tc by use of an imaginary chemical potential mu which avoids the sign
problem. The number density and the quark number susceptibility are consistent
with a critical behaviour associated with the transition line in the negative
mu^2 half-plane. We compare the analytic continuation of these results with
various phenomenological models, none of which provides a satisfactory
description of data, a failure on which we make some comments. These results
complement and extend the information obtained via the analysis of the
susceptibilities evaluated at zero mu, yielding a simple description of the
candidate strongly interacting QGP phase. As a byproduct of our analysis we
investigate the Polyakov loop and its hermitian conjugate. Our data offer a
vivid evidence of the importance of the complex nature of the functional
integral measure, which results in L (mu) ne \bar L(mu) for a real chemical
potential.

Strongly interacting quark-gluon plasma, and the critical behavior of QCD at imaginary μ

We compute the perturbative expansion of the Lattice SU(3) plaquette to order β−10 on 84 and 244 volumes. The result is found to be consistent both with the expected renormalon behaviour and with finite size effects on top of that. Complete control over the latter effects was the only loose end still to be tied up from previous works of our group.

A consistency check for renormalons in lattice gauge theory: β−10 contributions to the SU(3) plaquette

We determine to order α3 in the quenched approximation the so-called residual mass in the lattice regularisation of the Heavy Quark Effective Theory. We follow a gauge-invariant strategy which exploits the fact that this mass term dominates the exponential decrease of perturbative Wilson loops at large perimeters. Our computational tool is Numerical Stochastic Perturbation Theory. The new coefficient we compute is crucial to improve the determination of the (MSbar) mass of the b-quark from lattice simulations of the Heavy Quark Effective Theory.

Francesco Di Renzo

Papers

New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble

Monte Carlo simulations on the Lefschetz thimble: Taming the sign problem

Strongly interacting quark-gluon plasma, and the critical behavior of QCD at imaginary μ

A consistency check for renormalons in lattice gauge theory: β−10 contributions to the SU(3) plaquette

The residual mass in Lattice Heavy Quark Effective Theory to α 3 order