Lattice spacing

Lattice spacing for Wilson action

This page describes the connection between the coupling parameter \(\beta\) that defines the Wilson gauge action and the lattice spacing as determined from measuring the interquark potential.

The relevant studies are:

• Nonperturbatively: $$\beta$$ obtained from Necco and Sommer (hep-lat/0108008, (2.6), valid up to 6.92)
• Nonperturbatively: $$\beta$$ as determined by Alpha collaboration (hep-lat/9806005v2, (2.18), valid for $5.7\leq\beta\leq6.57$)
• Perturbatively: perturbative \(\beta\) ((3.6) in hep-lat/9609025), matched with Necco-Sommer at 6.92

We are going to use the non-perturbative parametrization determined by Alpha collaboration:

\[ \ln(a/r_0)=-1.6805-1.7139(\beta-6)+0.8155(\beta-6)^2- 0.6667(\beta-6)^3 \]

Since we use a heatbath algorithm together with over-relaxation steps we need to also insure that the parameters used in generating the configuration are chosen properly.

• \(n_{or}\): this is computed using the formula from page 3 of hep-lat/9806005\[n_{or} \approx 1.5 r_0/a\].
• \(n_{skip}\): You should also skip \(20\times (n_{or}+1)\) steps between two measurements.

A Mathematica notebook implementing these formulas can be downloaded from here.

Lattice spacing for Iwasaki action

This note describes the connection between the coupling parameter \(\beta\) that defines the Iwasaki gauge action and the lattice spacing as determined from measuring the interquark potential.

The relevant papers are

• CP-PACS Collaboration Collaboration, M. Okamoto et. al., Equation of state for pure SU(3) gauge theory with renormalization group improved action, Phys.Rev. D60 (1999) 094510, hep-lat/9905005

The pure gauge partition function for Iwasaki action is given by $$ Z = \int {\cal D}U\, e^{\beta S_g(U)}\,,\qquad S_g(U) = c_0 \sum_p U_p + c_1 \sum_{dp} U_{dp}\,. $$ Here $p$ is the sum over plaquettes and $dp$ indicates the sum over double plaquettes ($1\times 2$ plaquettes). The coefficients are set to $c_0=1-8 c_1$ and $c_1=-0.331$.

The connection between $\beta$ and the lattice spacing is determined non-perturbatively: lattice measurements for the string tension, $\sigma$, are fitted to the following ansatz: $$ a\sqrt{\sigma}\vert_\beta = f(\beta) \frac{1+c_2 \hat{a}(\beta)^2 + c_4 \hat{a}(\beta)^4}{c_0}\,,\qquad \hat{a}(\beta) \equiv \frac{f(\beta)}{f(\beta_1)}\,, $$ where $f(\beta)$ is the two loop scaling function of $SU(3)$ (a perturbative result): $$ f(\beta=6/g^2)\equiv (b_0 g^2)^{-b_1/2b_0^2} e^{-\frac1{2b_0 g^2}} \,, \qquad b_0 = \frac{11}{(4\pi)^2}\,\quad b_1=\frac{102}{(4\pi)^4} \,. $$ The fit uses $\beta_1=2.40$ and the coefficients are $$ c_0=0.524(15)\,, \quad c_2=0.274(76)\,, \quad c_4=0.105(36)\,. $$ The physical value of $\sigma$ is usually set to $(440\,{\tt MeV})^2$. A Mathematica notebook implementing these formulas can be downloaded from here.

As in the case of Wilson action, if we are using heat-bath with over-relaxation, we should set the number of quasi-heat-bath hits to $5$ and the number of over-relation steps to $n_{or}=1.5 r_0/a$ and skip $20\times(n_{or}+1)$ between measurements.