Abstract
The use of the Yang-Mills gradient flow in step-scaling studies of lattice QCD is expected to lead to results of unprecedented precision. Step scaling is usually based on the Schr\"odinger functional, where time ranges over an interval [0,T] and all fields satisfy Dirichlet boundary conditions at time 0 and T. In these calculations, potentially important sources of systematic errors are boundary lattice effects and the infamous topology-freezing problem. The latter is here shown to be absent if Neumann instead of Dirichlet boundary conditions are imposed on the gauge field at time 0. Moreover, the expectation values of gauge-invariant local fields at positive flow time (and of other well localized observables) that reside in the center of the space-time volume are found to be largely insensitive to the boundary lattice effects.
Highlights
In numerical lattice field theory, step scaling refers to a finite-size scaling technique that allows the physics at high energies to be related to the characteristic low energy scales of the theory [1]
Step scaling is usually based on the Schrodinger functional, where time ranges over an interval [0, T ] and all fields satisfy Dirichlet boundary conditions at time 0 and T
The combination of observables and boundary conditions advertised in this paper provides a technically attractive framework for step-scaling studies
Summary
In numerical lattice field theory, step scaling refers to a finite-size scaling technique that allows the physics at high energies to be related to the characteristic low energy scales of the theory [1]. A modification of the Schrodinger functional is considered, where open (Neumann) boundary conditions are imposed on the gauge field at one of the space-time boundaries and Dirichlet boundary conditions at the opposite boundary With this choice, the renormalizability of the theory is preserved, the perturbation expansion in powers of the gauge coupling remains regular and the topology-freezing problem is avoided, because the topological charge can freely flow in and out of the volume through the open boundary [19]. The volume-dependence of the expectation values of observables localized in the center of the space-time volume is examined and shown to be small in the kinematical situations of interest Such quantities are practically unaffected by boundary lattice effects. Both limitations of the Schrodinger functional setup can be overcome with the suggested change of boundary conditions and if suitable local observables (such as the ones obtained with the gradient flow) are used
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