Lattice spacing guide

Theory

The standard way to set the lattice spacing is to compare a dimensionful observable measured on the lattice with its physical value, and then define the lattice spacing as the value which makes the lattice value match the physical one. There are various observables chosen for this role. A natural observable would be one closely related to the means by which QCD sets its own scale by the running of the coupling constant and the onset of confinement. However, \( \Lambda_{\mathrm{QCD}} \) has no precise definition; in order to compute the lattice spacing we need an observable with a precise definition.

Such an observable is a Sommer scale, defined in terms of the static quark potential. The static quark potential is simply the potential between two infinitely-heavy quarks separated by a spatial vector . This potential is related in turn to the value of the Wilson loop with spatial separation \( \vec r\) (along any path from the origin to \( \vec r\)) and temporal separation \(t\); for \(t\) sufficiently large, this should have the form \(A \exp(-V(\vec r)t)\).

Then the Sommer scale \(r_{\gamma}\) is defined as that scale for which \(r^2 \frac{dF}{dr} = \gamma\), with any particular choice of \(\gamma\). Common choices are \(\gamma=1\) and \(\gamma=1.65\); the resulting scales are called \(r_1\) and \(r_0\) respectively.

Empirically, one observes that the static quark potential takes the form \(V(\vec r)\) = C + \frac{\alpha}{r} + \sigma r</math>: it is Coulombic at short distance and constant-force at long distance. Then the Sommer scale \(r_{\gamma}\) is given by \(r_{\gamma}=\sqrt{\frac{\gamma - \alpha}{\sigma}}\).

Thus, we have the following procedure for computing the lattice spacing:

• Measure the amplitudes of Wilson loops of various separations \(r\)
• Determine the resulting WORK IN PROGRESS

Nucl. Phys. B 411 2 (1994) 839-854