Estimates of Scaling Violations for Pure SU(2) LGT

We investigate the approach of pure SU(2) lattice gauge theory with the Wilson action to its continuum limit using the deconfining transition, Luescher's gradient flow, and the cooling flow to set the scale. Of those, the cooling flow turns out to be computationally most efficient. We explore systematic errors due to use of three different energy observables and two distinct reference values for the flow time, the latter obtained by matching initial scaling behavior of some energy observables to that of the deconfining transition. Another important source of systematic errors are distinct fitting forms for the approach to the continuum limit. Besides relying in the conventional way on ratios of masses, we elaborate on a form introduced by Allton, which incorporates asymptotic scaling behavior. Ultimately we find that, though still small, our systematic errors are considerably larger than our statistical errors. ~


Introduction
Tests of the Standard Model, and New Physics searches require precise predictions of physical observables and accurate estimates of systematic uncertainty. Since scale setting is a source of uncertainty, it is useful to investigate scales that can achieve small statistical error bars, and to estimate the systematic error picked up from the choice of scale and the choice of fitting form for continuum limit extrapolations. Pure SU(2) gauge theory is a good testing ground for new scales because one can generate large statistics with relatively modest CPU resources.
Here we summarize the results [5] of a study of pure SU (2) LGT with the Wilson action where g is the bare coupling constant and the summation runs over all plaquettes. We parameterize lattice expectation values of plaquette matrices by With this parameterization we use three definitions of the energy density: a ul i + a ur i + a dl i + a dr i where E 4 is Lüscher's energy density, which averages over the four plaquettes attached to each site x in a fixed µν, µ ν plane; the superscripts of a i stand for up (u), left (l), right (r), and down (d) with respect to x (drawn in figure 1 of [1]). All definitions become ∼ F αβ F αβ in the continuum limit. Configurations are obtained by Markov chain Monte Carlo (MCMC) simulations with Monte Carlo plus Overrelaxation (MCOR) updating. One MCOR sweep updates each link once in a systematic order with the Fabricius-Haan-Kennedy-Pendleton [6,7] heatbath algorithm and, in the same systematic order, twice by overrelaxation [8]. Using checkerboard coding [9] and MPI Fortran, parallel updating of sublattices is implemented.

Deconfinement scale
The first scale investigated is the deconfinement scale. It is used to guide our choice of target values for the other scales. To calculate this scale we perform simulations on N 3 s N τ lattices and estimate pseudocritical coupling constants β c (N s , N τ ) by reweighting the Polyakov loop susceptibility to nearby β values and finding the maximum. Critical coupling constants β c (N τ ) = β c (∞, N τ ) are then extrapolated using three-parameter fits Inverting the results of these fits defines the deconfining length scale N τ (β c ). Configurations were generated using 2 19 − 2 25 MCOR sweeps, which were partitioned into 32 or more bins. Error bars are then calculated using the jackknife procedure on these bins. The reweighting curve for the susceptibility is rather flat near the maximum and has large error bars, as shown in figure 1 (left). Nevertheless the error bar on the position of the pseudocritical beta is relatively small. This is explained by the fact that the error bars of figure 1 (left) are strongly correlated, since they are reweighted from the same simulation. Dividing out the maximum susceptibility in each jackknife bin gives us figure 1 (right), which makes the rather small error bar of β max plausible. 17

Gradient scale
The gradient flow equation [ is an evolution equation in the fictitious flow time t that decreases the action as t increases. The SU(2) link derivatives are defined by and the initial condition is U µ (x, 0) = U µ (x). The functions are used to define gradient scales by choosing appropriate fixed target values y i and integrating the gradient flow equation until eq. (8) is satisfied. As a function of β, the observable scales like a length provided that 2. the values of β are large enough to be in the SU(2) scaling region; and 3. the target values are large enough so that √ 8s i ≫ 1 for the smallest used flow time.
One can choose target values so that initial estimates of the scales s i agree with the deconfinement scale for low values of β. For example figure 2 (left) plots the ratio of s i (N = 12, β = 2.43) to s i (N = 8, β = 2.30) against y i . Due to scaling, one expects this ratio to agree with the ratio of the deconfinement scales, N τ (2.43)/N τ (2.3) = 12/8 = 1.5. This leads to two intersections, one coming from the E 4 curve and another coming from the E 0 and E 1 curves, which practically agree.  (right) plots the function t 2 E i against the flow time. Picking initially y 1 4 = 0.030, the target value corresponding to the aforementioned E 4 intersection, defines a flow time, indicated by the vertical dotted blue line at t = 1.85. This flow time is then used to define two more target values y 1 0 and y 1 1 using that figure. Similarly, picking initially y 2 0 = 0.0755 (or equivalently y 1 2 = 0.0748) delivers three more targets. Hence, we consider altogether six target values:

Cooling scale
The cooling algorithm was introduced as part of an investigation of topological charge in the 2D O(3) sigma model [2]. Since then it has found many applications [10]. Bonati and D'Elia showed that using cooling as a smoothing technique produces similar results for topological observables as the gradient flow [3]. An SU(2) cooling step updates a link variable by where n c is the number of cooling steps and The update (12) minimizes the local contribution to the action, so that the "cooling flow" decreases the action. A cooling scale is then defined in the same way as a gradient scale through eq. (9) by iterating eq. (12) until a specified target value is reached. In 4D, n c cooling steps correspond to a gradient flow time t c = n c /3 [3]; therefore the cooling flow attains its target value at least 34 times faster than the gradient flow, assuming eq. (6) is integrated using the Runge-Kutta scheme with ǫ = 0.01. Matching cooling scale ratios to deconfinement scale ratios, the target values are y 2 0 = 0.0822, y 2 1 = 0.0812, y 2 4 = 0.0656.

Data generation for gradient and cooling scales
Lattice sizes and β values used are given in Table 1. In each run, 128 configurations were generated, and on each configuration the gradient and cooling flows were performed. We allocated approximately equal amounts of CPU time to configuration generation and to smoothing flow. Subsequent configurations are separated by 2 11 − 2 13 MCOR sweeps, depending on how many sweeps are needed to reach the target value. Error bars are calculated using the jackknife method with respect to the 128 configurations. Integrated autocorrelation times τ int for the series of configurations are estimated using the software of reference [11] and are found to be statistically compatible with 1. Additionally we calculated for each configuration the cooling flow of the topological charge defined as in reference [3]. An example of cooling trajectories for the 128 configurations is given in figure 3. Topological correlations are then estimated by calculating τ int for the series of topological charges, which are also found to be statistically compatible with 1. Thus we treat our scale measurements as statistically independent.  4 2.300 28 4 2.430, 2.510 40 4 2.574, 2.620, 2.670, 2.710, 2.751 44 4 2.816 52 4 2.875

Scaling and asymptotic scaling analysis
We analyze 13 length scales, the deconfinement scale L 0 , the gradient scales L 1 − L 6 and the cooling scales L 7 − L 12 . Our goals are to determine whether using cooling scales over gradient scales leads to a significant loss of accuracy; estimate scaling violations; investigate whether the choice of target value leads to seriously distinct scaling behavior; compare scaling and asymptotic scaling fit forms; and finally give an overall estimate of systematic error picked up from reference scale choice and choice of fitting form for continuum limit extrapolations. We first examine O(a 2 ) scaling corrections to ratios of lengths. These are fitted using where r i, j is the continuum limit value and k i, j and c i, j are constants. In the following we fix L j = L 10 and plot the normalized ratio R i,10 r i,10 = 1 + c ′ i,10 1 L 10 2 (17) in figure 4 (left). We find no discernible loss of accuracy using the cooling scale over the gradient scale, which one can see from relative sizes of error bars in this plot. There is clear overlap between   cooling and gradient scales, for example cooling scales L 10 − L 12 fall within the spread of gradient scales L 1 − L 6 , which shows that cooling scales do not suffer significant scaling violations compared to gradient scales. At (1/L 10 ) 2 ≈ 0.30, which corresponds to β = 2.300, we read off scaling violations of about 10%. Deeper in the scaling region at (1/L 10 ) 2 ≈ 0.05, which corresponds to β = 2.574, the violations are less than 2%. Results for all our L i /L j continuum estimates are compiled in Table 2.
The three-loop asymptotic scaling relation for SU (2) is where b 0 = 11/24π 2 [12,13] and b 1 = 17/96π 4 [14,15] are the regularization scheme independent coefficients and q 1 = 0.08324 [16] using lattice regularization. The superscript of f 1 as indicates that only the O(g 2 ) correction to the asymptotic scaling relation is included. Allton suggested [4] using eq. (18) to fit the approach to the continuum limit by expanding in powers of a. For the inverse length 1/L i this reads Plugging eq. (18) into eq. (19) and truncating the series after 3 terms, which is essentially the minimum number needed to get acceptable q values for asymptotic scaling fits, one finds Due to the α i,1 term in eq. (20) corrections to ratios would in general be of order a in the lattice spacing. This may be the main reason why Allton's approach never became popular. In [17] this problem was avoided by combining the scales discussed there into a single fit, which is only possible if the relative scaling violations are so weak that they can be neglected within the statistical errors. In [5] we relaxed this to the requirement that the α i,1 coefficients have to agree for all scales, i.e. α i,1 ≡ α 1 , which provides a general solution to the problem. The master coefficient was then determined by the maximum likelihood approach, varying α 1 and repeating all fits for each value. Figure 5 plots eq. (20) against β, with the asymptotic scaling behavior divided out. With this normalization the curves approach 1 in the continuum limit. At β = 2.300 asymptotic scaling violations range from 28% to 37%. Differences of ratios at this β value reach 14% (from 0.72/0.63 ≈ 1.14), in agreement with 12% (from 1.04/0.93 ≈ 1.12) of the scaling fits at the same β value, shown in figure 4 (left). For a more direct comparison with scaling, we compute ratios of gradient and cooling lengths using asymptotic scaling. Since all scales have the same first order coefficient, they cancel in the ratio, leading to normalized ratio functions R i,10 r i, 10 = 1 + 1 r i,10 In the figure 4 (right), normalized ratio functions using scaling and asymptotic scaling fits are both plotted against the squared lattice spacing. Straight line fits are scaling fits, while slightly curved fits are asymptotic scaling fits. Systematic errors due to choice of fitting form alone seems not to exceed about 0.6%. At (1/L 10 ) 2 = 0.05, the combined systematic error due to choice of scale, target value, and continuum limit fitting form is read off to be around 2%.

Summary and Conclusions
A physical scale, here the deconfinement scale, is well suited for determining initial target values for the cooling and gradient flow. The cooling scale is found to reach its target values much faster than the gradient scale without losing accuracy in scale setting. Systematic errors due to using distinct operators, within the same scale, gradient or cooling, are considerably larger than the systematic error encountered by switching from the gradient to the cooling scale or vice versa. For the gradient and cooling scales six target values were used, and the choice of target value gives the largest contribution to the total systematic error, which is approximately 2% at about β = 2.6. This is small, but larger than the statistical errors of Table 2. Fits relying on asymptotic scaling were compared with the conventionally used scaling fit, and the systematic error between the two fit forms contributes only about 0.6% to the total systematic error. Our results show that one must probe rather deep in the scaling region to achieve systematic errors of the discussed variety below 1%.