Björn Leder

The strange quark mass and Lambda parameter of two flavor QCD

We complete the non-perturbative calculations of the strange quark mass and the Lambda parameter in two flavor QCD by the ALPHA collaboration. The missing lattice scale is determined via the kaon decay constant, for whose chiral extrapolation complementary strategies are compared. We also give a value for the scale r0 in physical units as well as an improved determination of the renormalization constant ZA.

Adaptive Aggregation-Based Domain Decomposition Multigrid for the Lattice Wilson-Dirac Operator

In lattice quantum chromodynamics (QCD) computations a substantial amount of work is spent in solving discretized versions of the Dirac equation. Conventional Krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions. We present a domain decomposition adaptive algebraic multigrid method used as a preconditioner to solve the “clover improved” Wilson discretization of the Dirac equation. This approach combines and improves two approaches, namely domain decomposition and adaptive algebraic multigrid, that have been used separately in lattice QCD before. We show in extensive numerical tests conducted with a parallel production code implementation that considerable speedup can be achieved compared to conventional Krylov subspace methods, domain decomposition methods, and other hierarchical approaches for realistic system sizes.

Determination of the static potential with dynamical fermions

Abstract We present in detail a technique to extract the potential between a static quark and anti-quark pair from Wilson loops measured on dynamical configurations. This technique is based on HYP smearing and leads to an exponential improvement of the noise-to-signal ratio of Wilson loops. We explain why the correct continuum potential is obtained and show numerical evidence that the cut-off effects are small. We present precise results for the non-perturbative potential. As applications, we determine the scale r 0 / a and study the shape of the static potential in the range of distances around r 0 , where it can be compared with phenomenological potential models.

Multigrid preconditioning for the overlap operator in lattice QCD

The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics (QCD), the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations, it preserves the important physical property of chiral symmetry, at the expense of requiring much more effort when solving systems posed with this operator. We present a preconditioning technique based on another lattice discretization, the Wilson-Dirac operator. The mathematical analysis precisely describes the effect of this preconditioning strategy in the case that the Wilson-Dirac operator is normal. Although this is not exactly the case in realistic settings, we show that current smearing techniques indeed drive the Wilson-Dirac operator towards normality, thus providing motivation for why our preconditioner works well in practice. Results of numerical experiments in physically relevant settings show that our preconditioning yields accelerations of more than an order of magnitude compared to unpreconditioned solvers.

One flavor mass reweighting in lattice QCD

Abstract One flavor mass reweighting can be used in lattice QCD computations to fine tune the quark masses to their physical values. We present a new method that utilizes an unbiased stochastic estimation of the one flavor determinant. The stochastic estimation is based on the integral representation of the determinant of a complex matrix, which we prove. In contrast to other methods it can also be applied in situations where the determinant has a complex phase. The stochastic error is controlled by determinant factorizations based on mass interpolation and Schur decomposition. As an example of an application we demonstrate how the method can be used to tune the up–down quark mass difference.

Fermions as Global Correction: the QCD Case

Abstract It is widely believed that the fermion determinant cannot be treated in global acceptance–rejection steps of gauge link configurations that differ in a large fraction of the links. However, for exact factorizations of the determinant that separate the ultraviolet from the infrared modes of the Dirac operator it is known that, the latter show less variation under changes of the gauge field compared to the former. Using a factorization based on recursive domain decomposition allows for a hierarchical algorithm that starts with pure gauge updates of the links within the domains and ends after a number of filters with a global acceptance–rejection step. Ratios of determinants have to be treated stochastically and we construct techniques to reduce the noise. We find that the global acceptance rate is high on moderate lattice sizes and demonstrate the effectiveness of the hierarchical filter.

The Schrödinger functional for Gross-Neveu models

Gross-Neveu type models with a finite number of fermion flavours are studied on a two-dimensional Euclidean space-time lattice. The models are asymptotically free and are invariant under a chiral symmetry. These similarities to QCD make them perfect benchmark systems for fermion actions used in large scale lattice QCD computations. The Schroedinger functional for the Gross-Neveu models is defined for both, Wilson and Ginsparg-Wilson fermions, and shown to be renormalisable in 1-loop lattice perturbation theory. In two dimensions four fermion interactions of the Gross-Neveu models have dimensionless coupling constants. The symmetry properties of the four fermion interaction terms and the relations among them are discussed. For Wilson fermions chiral symmetry is explicitly broken and additional terms must be included in the action. Chiral symmetry is restored up to cut-off effects by tuning the bare mass and one of the couplings. The critical mass and the symmetry restoring coupling are computed to second order in lattice perturbation theory. This result is used in the 1-loop computation of the renormalised couplings and the associated beta-functions. The renormalised couplings are defined in terms of suitable boundary-to-boundary correlation functions. In the computation the known first order coefficients of the beta-functions are reproduced. One of the couplings is found to have a vanishing beta-function. The calculation is repeated for the recently proposed Schroedinger functional with exact chiral symmetry, i.e. Ginsparg-Wilson fermions. The renormalisation pattern is found to be the same as in the Wilson case. Using the regularisation dependent finite part of the renormalised couplings, the ratio of the Lambda-parameters is computed.

Application of Domain Decomposition to the Evaluation of Fermion Determinant Ratios

We analyze the fluctuations in the case of mass reweighting for N f = 2 Wilson fermions. We use a domain decomposition factorization of the fermion determinant. Ratios of determinants are estimated stochastically. We study the stochastic and the ensemble fluctuations as a function of the volume V and the mass shift ∆m. With our result it is possible to estimate the cost and the effectiveness of mass reweighting. In addition we introduce a stochastic estimation for the one flavor case without using the square root.

Lattice and string worldsheet in AdS/CFT: a numerical study

We consider a possible discretization for the gauge-fixed Green-Schwarz (two-dimensional) sigma-model action for the Type IIB superstring and use it for measuring the cusp anomalous dimension of planar

Testing universality and automatic O(a) improvement in massless lattice QCD with Wilson quarks

\mathcal{N}=4