Simulating lattice field theories on multiple thimbles

Simulating thimble regularization of lattice field theory can be tricky when more than one thimble is to be taken into account. A couple of years ago we proposed a solution for this problem. More recently this solution proved to be effective in the case of 0+1 dimensional QCD. A few lessons we can learnt, including the role of symmetries and general hints on algorithmic solutions.Comment: 8 pages, 2 figures; Proceedings of the 35th International Symposium on Lattice Field Theory, Granada, Spai

Renormalization constants for Lattice QCD: new results from Numerical Stochastic Perturbation Theory

By making use of Numerical Stochastic Perturbation Theory (NSPT) we can compute renormalization constants for Lattice QCD to high orders, e.g. three or four loops for quark bilinears. We report on the status of our computations, which provide several results for Wilson quarks and in particular (values and/or ratios of) Z_V, Z_A, Z_S, Z_P. Results are given for various number of flavors (n_f = 0, 2, 3, 4). While we recall the care which is due for the computation of quantities for which an anomalous dimension is in place, we point out that our computational framework is well suited to a variety of other calculations and we briefly discuss the application of NSPT to other regularizations (in particular the Clover action).Comment: 7 pages, talk given at Lattice 2006 (Quark Masses, Gauge Couplings, and Renormalization

High-loop perturbative renormalization constants for Lattice QCD (II): three-loop quark currents for tree-level Symanzik improved gauge action and n_f=2 Wilson fermions

Numerical Stochastic Perturbation Theory was able to get three- (and even four-) loop results for finite Lattice QCD renormalization constants. More recently, a conceptual and technical framework has been devised to tame finite size effects, which had been reported to be significant for (logarithmically) divergent renormalization constants. In this work we present three-loop results for fermion bilinears in the Lattice QCD regularization defined by tree-level Symanzik improved gauge action and n_f=2 Wilson fermions. We discuss both finite and divergent renormalization constants in the RI'-MOM scheme. Since renormalization conditions are defined in the chiral limit, our results also apply to Twisted Mass QCD, for which non-perturbative computations of the same quantities are available. We emphasize the importance of carefully accounting for both finite lattice space and finite volume effects. In our opinion the latter have in general not attracted the attention they would deserve.Comment: 23 pages, 7 figures, pdflate

Thimble regularization at work: from toy models to chiral random matrix theories

We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional phi^4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very crude nature. The deviation of results from phase quenched results highlights that in a certain region of parameter space there is a quite important sign problem. In view of this, the success of our thimble approach is quite a significant one.Comment: 33 pages, 8 figures. Some extra references have been added and subsection 3.1 has been substantially expanded. Some extra comments on numerics have also been added in subsection 4.4. Appendix A and appendix B.1 now features some more detail

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 strategy to tackle finite size effects which can be quite sizeable in the computation of logarithmically divergent renormalization constants. Our first high loop determination of quark bilinears for Wilson fermions was limited to finite constants and finite ratios. A precise determination of Z_P and Z_S (and hence of Z_m) 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 computations for the Lattice QCD regularization defined by tree level Symanzik improved gauge action and Wilson 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.Comment: 7 pages, poster presented at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, German

Non perturbative physics from NSPT: renormalons, the gluon condensate and all that

Numerical Stochastic Perturbation Theory (NSPT) enables very high order computations in Lattice Gauge Theories. We report on the determination of the gluon condensate from lattice QCD measurements of the basic plaquette. This is a long standing problem, which was eventually solved a few years ago in pure gauge. In this context NSPT is crucial: it is actually the only tool enabling the subtraction of the power divergent contribution associated to the identity operator in the OPE for the plaquette. This subtraction is actually a delicate issue, since the perturbative expansion of the plaquette is on general ground expected to be an asymptotic one, due to renormalons. This in turn results in ambiguities and the separation of scales in the OPE does not correspond to a separation of perturbative and non-perturbative contributions. All in all, one needs to absorb the ambiguities attached to the perturbative series into the definition of the condensate itself, i.e. one needs a prescription. A possible one amounts to summing the perturbative series up to its minimal term, which means computing up to orders which only NSPT can aim at. Our computation is the first one in QCD, with massless staggered fermions. In order to remove the zero-mode of the gauge field, twisted boundary conditions are adopted for the latter, consistently coupled to fermions in the fundamental representation supplemented with smell degrees of freedom.Comment: 7 pages, 4 figures. Talk given at the 36th Annual International Symposium on Lattice Field Theory, 22-28 July 2018, Michigan State University, East Lansing, Michigan, US

QCD at High Temperature : Results from Lattice Simulations with an Imaginary mu

We summarize our results on the phase diagram of QCD with emphasis on the high temperature regime. For $T \ge 1.5 T_c$ the results are compatible with a free field behavior, while for $T \simeq 1.1 T_c$ this is not the case, clearly exposing the strongly interacting nature of QCD in this regionComment: 7 pages, 2 figures; To appear in the proceedings of QCD@Work 2005,International Workshop on Quantum Chromodynamics, Conversano, Bari, Italy, 16-20 Jun 200

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

We determine to order alpha^3 the so-called residual mass in the lattice regularisation of the Heavy Quark Effective Theory for Nf=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 (MS bar) mass of the b-quark from lattice simulations of the Heavy Quark Effective Theory.Comment: 11 pages 3 figures. Inconsistent notation correcte