Difference between revisions of "Lattice spacing"
(→Lattice spacing for Iwasaki action) |
(→Lattice spacing for Iwasaki action) |
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The relevant studies are: | The relevant studies are: | ||
| − | * Nonperturbatively: $$\beta$$ obtained from Necco and Sommer ([http:// | + | * Nonperturbatively: $$\beta$$ obtained from Necco and Sommer ([http://arxiv.org/abs/hep-lat?papernum=0108008 hep-lat/0108008], (2.6), valid up to 6.92) |
* Nonperturbatively: $$\beta$$ as determined by Alpha collaboration ([http://arxiv.org/pdf/hep-lat/9806005v2 hep-lat/9806005v2], (2.18), valid for $5.7\leq\beta\leq6.57$) | * Nonperturbatively: $$\beta$$ as determined by Alpha collaboration ([http://arxiv.org/pdf/hep-lat/9806005v2 hep-lat/9806005v2], (2.18), valid for $5.7\leq\beta\leq6.57$) | ||
| − | * Perturbatively: perturbative <math>\beta</math> ((3.6) in [http:// | + | * Perturbatively: perturbative <math>\beta</math> ((3.6) in [http://arxiv.org/abs/hep-lat?papernum=9609025 hep-lat/9609025]), matched with Necco-Sommer at 6.92 |
We are going to use the non-perturbative parametrization determined by Alpha collaboration: | We are going to use the non-perturbative parametrization determined by Alpha collaboration: | ||
| Line 24: | Line 24: | ||
chosen properly. | chosen properly. | ||
| − | * <math>n_{or}</math>: this is computed using the formula from page 3 of [http:// | + | * <math>n_{or}</math>: this is computed using the formula from page 3 of [http://arxiv.org/abs/hep-lat?papernum=9806005 hep-lat/9806005]: <math>n_{or} \approx 1.5 r_0/a</math>. |
* <math>n_{skip}</math>: You should also skip <math>20\times (n_{or}+1)</math> steps between two measurements. | * <math>n_{skip}</math>: You should also skip <math>20\times (n_{or}+1)</math> steps between two measurements. | ||
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The relevant papers are | 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, [http:// | + | * 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, [http://arxiv.org/abs/hep-lat/9905005 hep-lat/9905005] |
The pure gauge partition function for Iwasaki action is given by | The pure gauge partition function for Iwasaki action is given by | ||
\[ | \[ | ||
| − | Z = \int {\ | + | Z = \int {\rm D}U\, e^{\beta S_g(U)}\,,\qquad S_g(U) = c_0 \sum_p U_p + c_1 \sum_{dp} U_{dp}\,. |
\] | \] | ||
| − | 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 $$c_1=-0.331$$. | + | 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: | + | 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: |
\[ | \[ | ||
a\sqrt{\sigma}\vert_\beta = f(\beta) \frac{1+c_2 \hat{a}(\beta)^2 + c_4 \hat{a}(\beta)^4}{c_0}\,,\qquad \hat{a}(\beta) \equiv \frac{f(\beta)}{f(\beta_1)}\,, | a\sqrt{\sigma}\vert_\beta = f(\beta) \frac{1+c_2 \hat{a}(\beta)^2 + c_4 \hat{a}(\beta)^4}{c_0}\,,\qquad \hat{a}(\beta) \equiv \frac{f(\beta)}{f(\beta_1)}\,, | ||
\] | \] | ||
| − | where $f(\beta)$ is the two loop scaling function of $SU(3)$ (a perturbative result): | + | 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} \,. | 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} \,. | ||
Latest revision as of 14:08, 14 March 2023
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 \[ Z = \int {\rm D}U\, e^{\beta S_g(U)}\,,\qquad S_g(U) = c_0 \sum_p U_p + c_1 \sum_{dp} U_{dp}\,. \] 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.
