Square Symanzik action to one-loop order

TL;DR

This paper calculates the one-loop coefficients for a new Symanzik improved lattice gauge action applicable to SU(2) and SU(3) groups, enhancing the precision of lattice gauge theory simulations.

Contribution

It introduces and computes the one-loop coefficients for an alternative Symanzik improved action for SU(2) and SU(3) gauge groups, providing tools for more accurate lattice calculations.

Findings

Derived one-loop coefficients for the new action

Applicable to SU(2) and SU(3) gauge groups

Improves lattice gauge theory accuracy

Abstract

We present the one-loop coefficients for an alternative Symanzik improved lattice action with gauge groups SU(2) or SU(3).

Tables

Table 1. Table 1: One-loop improvement coefficients c i ′ superscript subscript 𝑐 𝑖 ′ c_{i}^{\prime} , defined by c i ​ ( g 0 2 ) = c i + c i ′ ​ g 0 2 + 𝒪 ​ ( g 0 4 ) subscript 𝑐 𝑖 superscript subscript 𝑔 0 2 subscript 𝑐 𝑖 superscript subscript 𝑐 𝑖 ′ superscript subscript 𝑔 0 2 𝒪 superscript subscript 𝑔 0 4 c_{i}(g_{0}^{2})=c_{i}+c_{i}^{\prime}\,g_{0}^{2}+\mbox{$\cal O$}(g_{0}^{4}) , for the Lüscher-Weisz  and square Symanzik actions. N 𝑁 N is the number of colors, while c ~ 0 ′ superscript subscript ~ 𝑐 0 ′ \tilde{c}_{0}^{\prime} and c ~ 1 ′ superscript subscript ~ 𝑐 1 ′ \tilde{c}_{1}^{\prime} stand for c 0 ′ − 16 ​ c 4 ′ superscript subscript 𝑐 0 ′ 16 superscript subscript 𝑐 4 ′ c_{0}^{\prime}-16c_{4}^{\prime} and c 1 ′ + 4 ​ c 4 ′ superscript subscript 𝑐 1 ′ 4 superscript subscript 𝑐 4 ′ c_{1}^{\prime}+4c_{4}^{\prime} respectively. For completeness we include the Lambda parameter ratios and the expectation values of a few a × b 𝑎 𝑏 a\times b Wilson loops, ⟨ N − 1 ​ Re Tr  ​ U ​ ( a × b ) ⟩ ≡ 1 − 1 4 ​ g 0 2 ​ ( N − N − 1 ) ​ w ​ ( a , b ) + 𝒪 ​ ( g 0 4 ) delimited-⟨⟩ superscript 𝑁 1 Re Tr  𝑈 𝑎 𝑏 1 1 4 superscript subscript 𝑔 0 2 𝑁 superscript 𝑁 1 𝑤 𝑎 𝑏 𝒪 superscript subscript 𝑔 0 4 \langle N^{-1}\mbox{Re Tr }U(a\times b)\rangle\equiv 1-\raisebox{0.6458pt}{\scriptsize$\frac{1}{4}$}g_{0}^{2}(N-N^{-1})w(a,b)+\mbox{$\cal O$}(g_{0}^{4}) . By convention the tadpole parameter u 0 subscript 𝑢 0 u_{0} equals ⟨ N − 1 ​ Re Tr  ​ U ​ ( 1 × 1 ) ⟩ 1 / 4 superscript delimited-⟨⟩ superscript 𝑁 1 Re Tr  𝑈 1 1 1 4 \langle N^{-1}\mbox{Re Tr }U(1\times 1)\rangle^{1/4} .

	Lüscher-Weisz			square
${\tilde{c}}_0^{\prime }$	0.	135160(13)	$\left(N=2\right)$	0.	113417(11)	$\left(N=2\right)$
	0.	23709(6)	$\left(N=3\right)$	0.	19320(4)	$\left(N=3\right)$
${\tilde{c}}_1^{\prime }$	-0.	0139519(8)	$\left(N=2\right)$	-0.	0112766(7)	$\left(N=2\right)$
	-0.	025218(4)	$\left(N=3\right)$	-0.	019799(2)	$\left(N=3\right)$
$c_2^{\prime }$	-0.	0029431(8)	$\left(N=2\right)$	-0.	0029005(7)	$\left(N=2\right)$
	-0.	004418(4)	$\left(N=3\right)$	-0.	004351(2)	$\left(N=3\right)$
$\Lambda /{\Lambda }_{\text{Wilson}}$	4.	1308935(3)	$\left(N=2\right)$	4.	0919901(2)	$\left(N=2\right)$
	5.	2921038(3)	$\left(N=3\right)$	5.	2089503(2)	$\left(N=3\right)$
$w(1,1)$	0.366262680(2)			0.3587838551(1)
$w(1,2)$	0.662626785(2)			0.6542934512(1)
$w(2,2)$	1.098143594(2)			1.0887235337(1)

Equations

$$S(\{c_i\})$$

$$c_0=5/3,c_1=-1/12,c_2=0,c_4=0.$$

$$c_0=16/9,c_1=-1/9,c_2=0,c_4=1/144,$$

$$\frac{\tilde{c}_{1}(g_{0}^{2})}{\tilde{c}_{0}(g_{0}^{2})}=-\frac{1}{20}(1+0.1217g_{0}^{2}+\mbox{$\cal O$}(g_{0}^{4}))$$

$$\frac{c_{1}u_{0}^{-2}+4c_{4}u_{0}^{-4}}{c_{0}-16c_{4}u_{0}^{-4}}=-\frac{1}{20}(1+0.0957g_{0}^{2}+\mbox{$\cal O$}(g_{0}^{4})).$$

Full text

INLO-PUB-17/96

Square Symanzik action to one-loop order

Jeroen Snippe

Instituut-Lorentz for Theoretical Physics,

University of Leiden, PO Box 9506,

NL-2300 RA Leiden, The Netherlands.

Abstract: We present the one-loop coefficients for an alternative Symanzik improved lattice action with gauge groups SU(2) or SU(3).

Recently a new improved lattice action, called the square Symanzik action, was introduced by adding a $2\times 2$ Wilson loop to the Lüscher-Weisz Symanzik action [1, 2]:

[TABLE]

where the $<>$ imply averaging over the two opposite directions for each of the links. The inclusion of the $2\times 2$ loop allows a simple diagonalization of the gauge field propagator, provided one takes $c_{0}\,c_{4}=c_{1}^{2}$ (where $c_{i}\equiv c_{i}(g_{0}^{2}=0)$). This simplifies certain analytic calculations, even if tadpole corrections [3] are incorporated. For details we refer to ref. [4].

The aim of Symanzik improvement is to cancel leading (\mbox{\cal O}(a^{2})) corrections in the lattice spacing $a$. The simplest choice for (on-shell) Symanzik improvement at tree-level amounts to [2]

[TABLE]

For the square action one takes instead [4]

[TABLE]

which satisfies $c_{0}\,c_{4}=c_{1}^{2}$. At tree-level many other Symanzik improved actions can be easily constructed. This freedom has been used, e.g. in ref. [5], to study the universality of improvement by comparing the effectiveness of alternative actions.

Up to now there has been only one choice of the improvement coefficients, eq. (2), for which a one-loop calculation was completed [2]. Here we present our results of a one-loop calculation belonging to the square Symanzik action, eq. (3). Details of our calculation, that is based on the methods of Lüscher, Weisz and Wohlert [2, 6], will be presented elsewhere. Introducing the notation c_{i}(g_{0}^{2})=c_{i}+c_{i}^{\prime}\,g_{0}^{2}+\mbox{\cal O}(g_{0}^{4}) we refer to table 1 for the coefficients $c_{i}^{\prime}$. At this point we stress that $c_{4}^{\prime}$ is a free parameter because in the expansion of the action to \mbox{\cal O}(a^{2}) only the combinations $\tilde{c}_{0}(g_{0}^{2})\equiv c_{0}(g_{0}^{2})-16c_{4}(g_{0}^{2})$ and $\tilde{c}_{1}(g_{0}^{2})\equiv c_{1}(g_{0}^{2})+4c_{4}(g_{0}^{2})$ contain $c_{4}(g_{0}^{2})$.

The following checks were performed to convince ourselves of the validity of the results in table 1.

• •

For the Lüscher-Weisz action all results of the original calculation [2, 6] were reproduced, in most cases to a slightly higher accuracy. Especially the agreement with ref. [2] is non-trivial because we used covariant, instead of coulomb, gauge fixing.

• •

Coefficients are extracted from physical quantities computed as a function of the lattice spacing. We checked that divergences cancel, the one-loop beta function is reproduced, continuum limits are independent of the action chosen, and $a^{2}\ln(a)$ terms do not appear for the Lüscher-Weisz and square actions—as expected from Symanzik’s analysis for $\varphi^{4}$ [1].

• •

The combination $\tilde{c}_{1}^{\prime}-c_{2}^{\prime}$ was computed both using the static quark potential method of ref. [6] and the twisted finite volume method of ref. [2]. The agreement is better than 0.003%.

• •

Using three completely different methods: (a) static quark potential; (b) three point vertex in a twisted finite volume; (c) (for SU(2)) a background field calculation in a periodic finite volume [4], the Lambda parameters extracted agree to at least six digits.

We conclude with testing how well the tadpole correction [3] to the SU(3) tree-level square Symanzik action predicts the one-loop correction. Since, to \mbox{\cal O}(a^{2}), $c_{4}(g_{0}^{2})$ can be freely chosen, the relevant test is comparing

[TABLE]

to

[TABLE]

Here $u_{0}=1-0.3588g_{0}^{2}/6$ was taken from table 1. It follows that the tadpole prediction captures 79% of the one-loop correction, a result similar to the 76% found for the Lüscher-Weisz Symanzik action. (For SU(2) one finds 80% for both actions).

Of course one may consider the ratios $c_{1}(g_{0}^{2})/c_{0}(g_{0}^{2})$ and $c_{4}(g_{0}^{2})/c_{0}(g_{0}^{2})$ separately. While for $c_{4}^{\prime}=0.003058$, satisfying $c_{4}(g_{0}^{2})c_{0}(g_{0}^{2})=(c_{1}(g_{0}^{2}))^{2}$ to one-loop order, the tadpole prediction is off by 21% in both ratios, for $c_{4}^{\prime}=0.002401$ the deviations are only 11%.

Acknowledgments

The author is grateful to Pierre van Baal and Margarita García Pérez for encouraging discussions.