Francesco Di Renzo

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

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.

Numerical stochastic perturbation theory for full QCD

We give a full account of the Numerical Stochastic Perturbation Theory method for Lattice Gauge Theories. Particular relevance is given to the inclusion of dynamical fermions, which turns out to be surprisingly cheap in this context. We analyse the underlying stochastic process and discuss the convergence properties. We perform some benchmark calculations and?as a byproduct?we present original results for Wilson loops and the 3-loop critical mass for Wilson fermions.

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

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.

The strongly interacting Quark Gluon Plasma, and the critical behaviour of QCD at imaginary chemical potential

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.

The residual mass in Lattice Heavy Quark Effective Theory to α3 order

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.

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

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.

The sign problem and the Lefschetz thimble

Recently, we have introduced a novel approach to deal with the sign problem that prevents the Monte Carlo simulations of a class of quantum field theories (QFTs). The idea is to formulate the QFT on a Lefschetz thimble. Here we review the formulation of our approach and describe the Aurora Monte Carlo algorithm that we are currently testing on a scalar field theory with a sign problem.

3-d lattice Yang-Mills free energy to four loops

We compute the expansion of the 3-d Lattice Yang-Mills free energy to four-loop order by means of Numerical Stochastic Perturbation Theory. The first and second order are already known and are correctly reproduced. The third- and fourth-order coefficients are new results. The known logarithmic divergence in the fourth order is correctly identified. We comment on the relevance of our computation in the context of dimensionally reduced finite temperature QCD.

High loop renormalization constants by NSPT: a status report

We present an update on Numerical Stochastic Perturbation Theory projects for Lattice QCD, which are by now run on apeNEXT. As a first issue, we discuss a st rategy to tackle finite size effects which can be quite sizeable in the computation of logarithmically divergent renormalization constants. Our first high loop determination of quar k bilinears for Wilson fermions was limited to finite constants and finite ratios. A precise deter mination of ZP and ZS (and hence of Zm) now becomes possible. We also give an account of computations for actions different from the standard regularization we have taken into account so far (Wilson gauge action and Wilson fermions). In particular, we present the status of computat ions for the Lattice QCD regularization defined by tree level Symanzik improved gauge action and Wils on fermions, which became quite popular in recent times. We also take the chance to discuss the related topic of the computation of the gluon and ghost propagators (which we undertook in collaboration with another group). This is relevant in order to better understand non-perturbative computations of propagators aiming at qualitative/quantitative understanding of confinement.

TheNf=2 residual mass in Perturbative Lattice-HQET for an improved determination of the (MS bar) b-quark mass

We determine to order α3 the so-called residual mass in the lattice regularisation of the Heavy Quark Effective Theory for N f = 2. Our (gauge-invariant) strategy makes use of Numerical Stochastic Perturbation Theory to compute the static interquark potential where the above mentioned mass term appears as an additive contribution. We discuss how the new coefficient we compute in the expansion of the residual mass can improve the determination of the () mass of the b-quark from lattice simulations of the Heavy Quark Effective Theory.