Michele Brambilla

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 nf=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.

Numerical Stochastic Perturbation Theory in the Schrödinger Functional

The Schrodinger functional (SF) is a powerful and widely used tool for the treatment of a variety of problems in renormalization and related areas. Albeit offering many conceptual advantages, one major downside of the SF scheme is the fact that perturbative calculations quickly become cumbersome with the inclusion of higher orders in the gauge coupling and hence the use of an automated perturbation theory framework is desirable. We present the implementation of the SF in numerical stochastic perturbation theory (NSPT) and compare first results for the running coupling at two loops in pure SU(3) Yang-Mills theory with the literature.

Efficiency on multi-core CPUs: the Wilson Dirac operator on Aurora

An optimized code has to be tuned to the CPU architecture: a cu rrent trend in modern CPUs is the increasing number of cores per socket, with different level s of cache. It turns out to be natural to have different parallelization “granularities” (multith reading and multiprocessing) characterized by completely different bandwidth and latencies. We present different strategies for the implementation of t he Wilson Dirac operator which aim at maximizing the performance on the Aurora architecture.

Three loops renormalization constants in Numerical Stochastic Perturbation Theory

We present three loops renormalization constants for Wilson fermion bilinears (vector, scalar, axial, pseudoscalar currents). Two gluonic regularizations are considered: tree level Symanzik improved action (with N f = 2) and Iwasaki action (with N f = 4). Both cases are amenable for comparisons with non-perturbative results. We discuss the issue of taming both finite lattice spacing and finite volume artifacts. As a byproduct, we comment on two loops matching of lattice and continuum couplings.

Finite size effects in lattice RI-MOM

RI-MOM (or its RI’-MOM variant) is one of the most polular renormalization schemes for Lattice QCD; being regulator independent, it can be effectively adopted in a lattice regularization. RI-MOM is defined in infinite volume. This is in principle a fundamental problem for the lattice, since any simulation is performed in a finite volume. From a practical point of view, one most often verifies a posteriori (by performing computations on different physical volumes) the expectation that renormalization constants, determined in the RI-MOM scheme at large momenta, should not be affected by significant finite size effects. In the context of Numerical Stochastic Perturbation Theory, we have in recent years devised a novel method to explicitly look and correct for finite size effects (in a convenient window). We review this method, discussing how it can be applied in a non-perturbative formulation as well.

An update on the status of NSPT computations

We will discuss final results obtained by the Parma group for the computation of renormalization constants of quark bilinears for the regularizations defined by nf=2 Wilson fermions/tree level Symanzik improved gauge and nf=4 Wilson fermions/Iwasaki improved gauge actions. Computation has been performed in the framework of Numerical Stochastic Perturbation Theory (NSPT) up to three loop. Perturbative results will be compared with the ones coming from non perturbative determinations.

Renormalization constants for Iwasaki action

By numerically integrating the differential equations of S tochastic Perturbation Theory, Numerical Stochastic Perturbation Theory can perform high order perturbative calculations in lattice gauge theory. We report on the computation of renormalization constants for Iwasaki gauge action and Wilson fermions. We generated configurations at dif ferent lattice volumes V=12 4 , 16 4 , 20 4 , 24 4 , and 32 4 . To remove the effect of finite time step in the integration of stochastic differential equations, for each volume we generate configuration at different time step τ=0.010, 0.02, and 0.030. Renormalization constants are defined in the RI’- MOM scheme. We extrapolate them to the continuum limit and also correct for finite volume effe cts. Here we present one loop results, checking that they are consistant with analytical results.

Perturbative vs non-perturbative renormalization: The case of the quark mass

Comparing perturbative and non-perturbative results for renormalization constants has been an issue for a while. The quark mass renormalization constant is a prototype example: discrepancies between different results have been several times ascribed to this issue. Given the logarithmic nature of the divergence, there is no theoretical obstruction to a perturbative computation. The problem, as it is obvious, is how to perform the computation at high loops. Truncation errors should in turn be compared to a variety of errors (e.g. irrelevant effects, chiral extrapolation, finite size) which should be carefully assessed as well. We discuss the status of our computations in Numerical Stochastic Perturbation Theory, in particular for the tree level Symanzik improved gauge action at n f = 2. The emphasis is on main goal: how to take all the systematic effects under control at three loop level.

The Dirac operator spectrum: a perturbative approach

By computing the Dirac operator spectrum by means of Numerical Stochastic Perturbation Theory, we aim at throwing some light on the widely accepted picture for the mechanism which is behind the Bank-Casher relation. The latter relates the chiral condensate to an accumulation of eigenvalues in the low end of the spectrum. This can be in turn ascribed to the usual mechanism of repulsion among eigenvalues which is typical of quantum interactions. First results appear to confirm that NSPT can indeed enable us to inspect a huge reshuffling of eigenvalues due to quantum repulsion.