Symanzik flow on HISQ ensembles
We report on a scale determination with gradient-flow techniques on the HISQ ensembles generated by the MILC collaboration. The lattice scale , originally proposed by the BMW collaboration, is computed using Symanzik flow at four lattice spacings ranging from 0.15 to 0.06 fm. With a Taylor series ansatz, the results are simultaneously extrapolated to the continuum and interpolated to physical quark masses. We give a preliminary determination of the scale in physical units, along with associated systematic errors, and compare with results from other groups. We also present a first estimate of autocorrelation lengths as a function of flowtime for these ensembles.
Symanzik flow on HISQ ensembles
A. Bazavov, C. Bernard, N. Brown††thanks: Speaker. , C. DeTar, J. Foley, Steven Gottlieb, U.M. Heller, J.E. Hetrick, J. Laiho ††thanks: Present address: Department of Physics, Syracuse University, Syracuse, New York, USA, L. Levkova, M. Oktay, R.L. Sugar, D. Toussaint, R.S. Van de Water, R. Zhou ††thanks: Present address: Fermi National Accelerator Laboratory, Batavia, IL 60510, USA (MILC Collaboration)
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Department of Physics, Washington University, St. Louis, MO 63130, USA
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA
Department of Physics, Indiana University, Bloomington, IN 47405, USA
American Physical Society, One Research Road, Ridge, NY 11961, USA
Physics Department, University of the Pacific, Stockton, CA 95211, USA
SUPA, Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, United Kingdom
Physics Department, University of California, Santa Barbara, CA 93106, USA
Physics Department, University of Arizona Tucson, AZ 85721, USA
Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia 60510, USA
Scale setting holds central importance in lattice QCD. The precision with which any dimensionful quantity can be computed is limited by the precision with which the scale is determined in physical units. Furthermore, continuum extrapolation of any quantity, dimensionful or dimensionless, requires precise determination of the relative scale between ensembles with different bare couplings. Any dimensionful quantity that is finite in the continuum limit may be used for scale-setting, but if it is not experimentally measured its value in physical units needs to be determined on the lattice by comparison to an experimentally determined quantity. An ideal scale-setting quantity should be easy and cheap to compute on the lattice, have small statistical errors, and be relatively insensitive to systematic issues, such as finite-volume effects or differences between simulated and physical quark-mass values. This has led to the consideration of quantities such as and , defined from the static quark potential, and, more recently,  and  from gradient flow . Gradient flow has received a particularly high interest level over the past year, as evident from the two plenary talks dealing extensively with the flow and its applications [4, 5]. This interest stems from the particular ease with which a scale from the flow can be computed and the resulting small statistical errors. Computing the gradient flow requires no costly quark propagator computations and avoids fits, such as to the asymptotic time behavior of correlation functions or Wilson loops.
Here, we present our computation of on the MILC HISQ ensembles, and perform a continuum extrapolation to determine in physical units. We also investigate the autocorrelation length of the energy density as a function of flow time on these ensembles.
2 Gradient Flow and the Scale
Gradient flow  is a smoothing of the original gauge fields towards stationary points of the gauge action . Successive links are updated according to the diffusion equation,
where the equation in square brackets is the corresponding continuum flow equation. The flow time increases as the gauge fields are smoothed. By repeatedly integrating infinitesimal smearing steps, the diffusion equation can be solved numerically. Note that the action is not necessarily the same as the action used to generate the original gauge fields. As the gauge fields diffuse, high momentum artifacts of the lattice spacing are removed. Thus, statistical fluctuations and discretization effects are suppressed while preserving low-energy physics.
To extract the scale using the gradient flow one needs a dimensionful quantity. One of the simplest is the flow time which has units of . Pick a reference timescale at which a dimensionless quantity reaches a predefined value. If this quantity is finite in the continuum limit, then will be independent of lattice spacing up to discretization corrections in powers of . Lüscher  has shown the energy density is finite to next-to-leading order (when expressed in terms of renormalized quantities), where and is the clover-leaf definition of the field strength tensor at flow time . The reference timescale is chosen to satisfy . Based on empirical evidence, the discretization effects can be reduced by considering the slope and defining the new reference timescale where . In both cases, the cutoff of is chosen to minimize discretization effects from small flow times and finite volume effects at large flow times.
3 HISQ Ensemble Results
We have computed the scale on the MILC HISQ ensembles . The tree-level Symanzik improved action is used in the flow, partially for comparison to earlier work with Wilson flow on the same ensembles by HPQCD . Note that the energy density is computed with the clover-leaf definition of the field strength tensor; thus, the final scale is not Symanzik improved. Numerical integration is performed with the fourth-order Runga-Kutta scheme originally proposed in Ref. . We found a step size of sufficient to render integration step-size errors negligible. Results for each ensemble are compiled in Table 1.
As can be seen in Table 1, the statistical error of is below 0.1% for almost all ensembles. Furthermore, the gradient flow has been run on only a fraction of the configurations in most ensembles. Each ensemble has or will have approximately 1000 configurations, which leaves plenty of room for reduction of statistical errors. Note, the reduction would be particularly large for the finer and fm ensembles where is still small. As a comparison, Fig. 1 (Left) plots the percent error of and other scale-setting quantities against for the four physical quark mass ensembles. Only , the pseudoscalar decay constant at the fiducial point with valence-quark masses 0.4 times the strange quark mass , has lower percent errors than . However, was computed over the entirety of each ensemble. Using conservative estimates of the autocorrelation length, the statistical error of should be comparable to or better than the error of if computed on the same number of configurations.
A simple continuum extrapolation can be performed by including only the physical quark-mass ensembles. With just these ensembles, quark mass mistuning effects cannot be accounted for, and the statistical error will be larger than from a fit to the complete data set. Nevertheless, this extrapolation provides a double check on the final value from the more complicated fit. The results of a linear and quadratic fit to in powers of are shown in Fig. 1 (Right). Taking from the quadratic fit as the central value, and the difference between the two fits as extrapolation error, this fit gives fm.
4 Fitting and Extrapolation
The value of may be simultaneously interpolated to physical quark masses and extrapolated to the continuum by fits to all of our data. All fit functions used are power series in or , , and where the latter two are proxies for and , respectively. Starting at linear order, fit functions up to cubic order in each variable (including cross terms) are considered. The lowest order discretization term is always , and higher orders are powers of either or . Including the non-physical ensembles gives us the most data to work with and helps correct mistuning errors. However, those ensembles cover a wide range of values, so we also consider fits that drop one or more of those ensembles with the lightest strange-quark masses. Five different cutoffs on were considered. In total, there are five choices for parameterizing discretization effects (powers of or ), three possible orders of and , and five different datasets to choose from (corresponding to different cutoffs on ), yielding 215 different fits. Note that 10 such fits would have more parameters than degrees of freedom and are therefore excluded from the count.
A fit is considered acceptable if it has a p-value greater than 0.01. An acceptable fit is considered to be “preferred” if it displays at least two of the following three attributes: a p-value greater than 0.1, at least twice as many degrees of freedom as fit parameters, or a deviation smaller than one sigma from the physical 0.06 fm ensemble, which is the most important ensemble since it has the finest lattice spacing and physical masses. A representative preferred fit is plotted in Fig. 2 (Left). The values of for all acceptable fits are binned and placed in the histogram in Fig. 2 (Right). From among the preferred fits, a central fit is chosen close to the median of the preferred and acceptable fits. The error from this procedure is conservatively estimated as the width of the histogram. The chosen central fit has the functional form
with 20 points fit (all non-physical ensembles included). The fit has , , and is from the re-tuned physical 0.06 fm ensemble.
To help pick the jackknife bin size and control correlations, we compute the autocorrelation function of as a function of the flow time and the number, , of unit trajectories separating configurations. Autocorrelation functions are notoriously difficult to estimate precisely, and for many of the ensembles the between the configurations used to determine is not small enough to see correlations reliably. In particular, at 0.09 and 0.06 fm, running over the entire ensembles to compute autocorrelation lengths is computationally expensive. Instead, we run on only 50 and 25 equilibrated configurations separated by 24 unit trajectories from the , and 0.06 fm ensembles, respectively. For the ensembles at 0.15 and 0.12 fm where the full ensembles have been run, we have a much better estimate of the autocorrelation function. A comparison of the integrated autocorrelation length as a function of flow time for different lattice spacings and light sea quark masses is plotted in Fig. 3 (Left).
To estimate the statistical error on the integrated autocorrelation length, we jackknife the results for and compute the average autocorrelation function across all flow times. We integrate the the average autocorrelation function up to a cutoff value of where the autocorrelations become indistinguishable from noise. There is additional error from the choice of the cutoff, which is not completely separable from the estimate of the statistical error. We attempt to account for this error by measuring the difference between the largest and smallest integrated autocorrelation lengths for which the statistical errors are not completely uncontrolled. As a function of increasing cutoff, the errors are considered to have become “uncontrolled” when the deviation between the result on the full dataset and those on the first or second halves is more than 3 sigma. Fig. 3 (Right) illustrates the integration error estimation process for the 0.12 fm ensemble. The integration region yielding the largest integrated autocorrelation length is chosen for each ensemble in Fig. 3 (Left) in order to estimate the worse-case scenario.
6 Results and Conclusion
For each HISQ ensemble, we jackknife with a bin size larger than the estimated autocorrelation length. Then we compute through a simultaneous extrapolation to the continuum and physical quark masses, and use the experimental value of to yield in physical units. This yields the final (but still preliminary) value of
where the errors are statistical, systematic error from the continuum extrapolation and chiral interpolation, residual finite volume error in , and experimental error in , respectively. The result is in agreement with simple continuum extrapolation through the physical mass ensembles, which gives fm (statistical and rough continuum extrapolation errors only).
HPQCD has also calculated on a the 0.15 to 0.09 fm, physical strange quark mass MILC HISQ ensembles , a subset of the ensembles considered here. They used the Wilson action in the flow, and also set the absolute scale with , as we have done here. They report a final value of fm, well within one standard deviation of our result. At the moment, there are no other complete computations of with fermions. ETM reported a preliminary value of fm on their twisted mass ensembles, but the error analysis at that time was not complete . BMW reports fm on 2-HEX smeared Wilson-clover ensembles, a value about 2.1 (joint) sigma from ours. It is conceivable that the difference between their result and ours is due to the different numbers of flavors in the sea. However, as reported in Ref. , there is no obvious trend in as increases.
This work was supported by the U.S. Department of Energy and the National Science Foundation. Computations for the calculation of Symanzik flow were carried out with resources provided by the Texas Advanced Computing Center (TACC). The HISQ gauge configurations were generated with resources provided by the Argonne Leadership Computing Facility, the Blue Waters Project at the National Center for Supercomputing Applications (NCSA), the National Energy Resources Supercomputing Center (NERSC), the National Institute for Computational Sciences (NICS), TACC, the National Center for Atmospheric Research (UCAR), and the USQCD facilities at Fermilab, under grants from the DOE and NSF.
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