Wilson Discretization Research Articles

Non-perturbative definition of the QCD energy-momentum tensor on the lattice

We present a strategy to define non-perturbatively the energy-momentum tensor in Quantum Chromodynamics (QCD) which satisfies the appropriate Ward identities and has the right trace anomaly. The tensor is defined by regularizing the theory on a lattice, and by fixing its renormalization constants non-perturbatively by suitable Ward identities associated to the Poincaré invariance of the continuum theory. The latter are derived in thermal QCD with a non-zero imaginary chemical potential formulated in a moving reference frame. A renormalization group analysis leads to simple renormalization- group-invariant definitions of the gluonic and fermionic contributions to either the singlet or the non-singlet components of the tensor, and therefore of their form factors among physical states. The lattice discussion focuses on the Wilson discretization of quark fields but the strategy is general. Specific to that case, we also carry out the analysis for the on-shell O(a)-improvement of the energy-momentum tensor. The renormalization and improvement programs profit from the fact that, as shown here, the thermal theory enjoys de-facto automatic O(a)-improvement at finite temperature. The validity of the proposal is scrutinized analytically by a study to 1-loop order in lattice perturbation theory with shifted and twisted (for quarks only) boundary conditions. The latter provides also additional useful insight for a precise non-perturbative calculation of the renormalization constants. The strategy proposed here is accessible to Monte Carlo computations, and in this sense it provides a practical way to define non-perturbatively the energy-momentum tensor in QCD.

SU(3) sextet model with Wilson fermions

We investigate the spectrum and IR properties of the SU(3) "sextet" model with two Dirac fermions in the two-index symmetric representation via lattice simulations. This model is a prime candidate for a realization of Walking Technicolor, which features a minimal matter content and it is expected to be inside or very close to the lower boundary of the conformal window. We use the Wilson discretization for the fermions and map the phase structure of the lattice model. We study several spectral and gradient flow observables both in the bulk and the weak coupling phases. While in the bulk phase we find clear signs of chiral symmetry breaking, in the weak coupling phase there is no clear indication for it, and instead the chiral limit of the model seems compatible with an IR-conformal behavior.

Supersymmetric QCD on the lattice: An exploratory study

We perform a pilot study of the perturbative renormalization of a Supersymmetric gauge theory with matter fields on the lattice. As a specific example, we consider Supersymmetric ${\cal N}{=}1$ QCD (SQCD). We study the self-energies of all particles which appear in this theory, as well as the renormalization of the coupling constant. To this end we compute, perturbatively to one-loop, the relevant two-point and three-point Green's functions using both dimensional and lattice regularizations. Our lattice formulation involves the Wilson discretization for the gluino and quark fields; for gluons we employ the Wilson gauge action; for scalar fields (squarks) we use na\"ive discretization. The gauge group that we consider is $SU(N_c)$, while the number of colors, $N_c$, the number of flavors, $N_f$, and the gauge parameter, $\alpha$, are left unspecified. We obtain analytic expressions for the renormalization factors of the coupling constant ($Z_g$) and of the quark ($Z_\psi$), gluon ($Z_u$), gluino ($Z_\lambda$), squark ($Z_{A_\pm}$), and ghost ($Z_c$) fields on the lattice. We also compute the critical values of the gluino, quark and squark masses. Finally, we address the mixing which occurs among squark degrees of freedom beyond tree level: we calculate the corresponding mixing matrix which is necessary in order to disentangle the components of the squark field via an additional finite renormalization.

Bootstrap Algebraic Multigrid for the 2D Wilson Dirac system

We develop an algebraic multigrid method for solving the non-Hermitian Wilson discretization of the two-dimensional Dirac equation. The proposed approach uses a bootstrap setup algorithm based on a multigrid eigensolver. It computes test vectors which define the least squares interpolation operators by working mainly on coarse grids, leading to an efficient and integrated self-learning process for defining algebraic multigrid interpolation. The algorithm is motivated by the $\gamma_5$-symmetry of the Dirac equation, which carries over to the Wilson discretization. This discrete $\gamma_5$-symmetry is used to reduce a general Petrov Galerkin bootstrap setup algorithm to a Galerkin method for the Hermitian and indefinite formulation of the Wilson matrix. Kaczmarz relaxation is used as the multigrid smoothing scheme in both the setup and solve phases of the resulting Galerkin algorithm. The overall method is applied to the odd-even reduced Wilson matrix, which also fulfills the discrete $\gamma_5$-symmetry. ...

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.

Meson and baryon dispersion relations with Brillouin fermions

We study the dispersion relations of mesons and baryons built from Brillouin quarks on one N_f=2 gauge ensemble provided by QCDSF. For quark masses up to the physical strange quark mass, there is hardly any improvement over the Wilson discretization, if either action is link-smeared and tree-level clover improved. For quark masses in the range of the physical charm quark mass, the Brillouin action still shows a perfect relativistic behavior, while the Wilson action induces severe cut-off effects. As an application we determine the masses of the \Omega_c^0, \Omega_{cc}^+ and \Omega_{ccc}^{++} baryons on that ensemble.

Least-Squares Finite Element Methods for Quantum Electrodynamics

A significant amount of the computational time in large Monte Carlo simulations of lattice field theory is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical parameters, the discretized operator is large and ill-conditioned and has random coefficients. More recently, adaptive algebraic multigrid (AMG) methods have been shown to be effective preconditioners for Wilson's discretization [J. Brannick et al., Lect. Notes Comput. Sci. Eng., 55 (2006), pp. 499–506], [J. Brannick et al., Phys. Rev. Lett., 100 (2008), pp. 041601–041604] of the Dirac equation. This paper presents an alternate discretization of the two-dimensional Dirac operator of quantum electrodynamics (QED) based on least-squares finite elements. The discretization is systematically developed and, physical properties of the resulting matrix system are discussed. Finally, numerical experiments are presented that demonstrate the effectiveness of adaptive smoothed aggregation ($\alpha$SA) multigrid as a preconditioner for the discrete field equations.

The low-lying mass spectrum of the N = 1 SU(2) SUSY Yang-Mills theory with Wilson fermions

We analyze the low energy spectrum of bound states of the N = 1 SU(2) SUSY Yang-Mills Theory (SYM). This work continues the investigation of the non-perturbative properties of SYM by Monte Carlo simulations in the Wilson discretization with dynamical gluinos. The dynamics of the gluinos is included by the Two-Step Multi-Bosonic Algorithm (TSMB) for dynamical fermions. A new set of configurations has been generated on a \(16^3\cdot 32\) lattice at \(\beta = 2.3\) and \(\kappa = 0.194\). The analysis also includes sets of configurations previously generated on a smaller (\(12^3\cdot 24\)) lattice at \(\kappa = 0.1925, 0.194\) and 0.1955. Guided by predictions from low energy Lagrangians, we consider spin-1/2, scalar and pseudoscalar particles. The spectrum of SYM is a challenging subject of investigation because of the extremely noisy correlators. In particular, meson-like correlators contain disconnected contributions. The larger time-extension of the \(16^3\cdot 32\) lattice allows to observe two-state signals in the effective mass. Finite-volume effects are monitored by comparing results from the two lattice sizes.

Towards the lattice study of M-theory

We propose the Wilson discretization of the supersymmetric Yang-Mills Quantum Mechanics as a lattice version of the matrix model of M-theory. An SU(2) model is studied numerically in the quenched approximation for D=4. A clear signal for the existence of two different phases is found and the continuum pseudocritical temperature is determined. We have also extracted the continuum limit of the total size of the system in both phases and for different temperatures.

Chiral symmetry and charge density in mesons for Kogut-Susskind and Wilson fermions

We investigate the sea quark distribution around static quarks in the framework of lattice QCD by calculating the correlations of Polyakov loops with the local chiral condensate ψψ(r) and the color charge density ψ †ψ(r) . A comparison of dynamical quarks in Kogut-Susskind and Wilson discretization is performed. We observe a restoration of chiral symmetry in the vicinity of color sources for both fermion regularization schemes as a consequence of gluon string formation. It turns out that virtual antiquarks screen static quarks as expected from QED.