Advanced Lattice QCD

TL;DR

The paper surveys a comprehensive program for non-perturbative lattice QCD, anchored by Symanzik's effective theory to control lattice artifacts and by O($a$) improvement (notably the clover term) to accelerate the $a\to0$ limit. It develops and tests non-perturbative methods to determine improvement and renormalization constants, including the Schrödinger functional in finite volume to tune $c_{\rm sw}$ and $Z_A$, and to define a non-perturbative running coupling via finite-volume schemes. It then connects low-energy hadronic physics to high-energy perturbation theory through a recursive finite-size renormalization group, yielding a non-perturbative determination of $\Lambda_{\overline{\rm MS}}$ and a controlled matching to MSbar. The framework demonstrates that with $a\lesssim 0.1$ fm and non-perturbative improvement, continuum QCD predictions can be obtained with controllable discretization errors, while sea quark effects remain the main frontier for fully realistic simulations. Altogether, the ALPHA program provides a coherent strategy to non-perturbatively renormalize QCD and to compute the running coupling and fundamental parameters from first principles.

Abstract

The topics covered by the lectures include Symanzik's effective continuum theory, O(a) improvement, chiral symmetry on the lattice and non-perturbative renormalization.

Figures (17)

• Figure 1: Calculated values of the vector meson mass (full circles) and linear extrapolation to the continuum limit (cross). Simulation data from Butler et al. (GF11 collab.) [\ref{['WeingartenI']}].
• Figure 2: Graphical representation of the products of gauge field variables contributing to the lattice field tensor [eq. (\ref{['fieldtensor']})]. The point $x$ is at the centre of the diagram where all loops start and end.
• Figure 3: Choice of the region $R$ when deriving the PCAC relation. The product of fields $\@fontswitch\mathcal{O}$ is assumed to be localized in the shaded area away from $R$.
• Figure 4: Assumed localization regions of the field products ${\@fontswitch\mathcal{O}}_{\rm int}$ and ${\@fontswitch\mathcal{O}}_{\rm ext}$ when deriving the integrated Ward identity eq. (\ref{['intwardid']}).
• Figure 5: Assumed shape of the region $R$ in eq. (\ref{['aav']}). The vector $n_{\mu}$ denotes the outward normal to the boundary $\partial R$.