Status of the Lambda Lattice Scale for the SU(3) Wilson gauge action
aa r X i v : . [ h e p - l a t ] J a n Status of the lambda lattice scale for the SU(3)Wilson gauge action
Bernd A. Berg ∗ Department of Physics, Florida State University, Tallahassee, FL 32306, USAE-mail: [email protected]
With the emergence of the Yang-Mills gradient flow technique there is renewed interest in theissue of scale setting in lattice gauge theory. Here I compare for the SU(3) Wilson gauge actionnon-perturbative scale functions of Edwards, Heller and Klassen (EHK), Necco and Sommer(NS), both relying on Sommer’s method using the quark potential, and the scale function derivedby Bazavov, Berg and Velytsky (BBV) from a deconfining phase transition investigation by theBielefeld group. It turns out that the scale functions are based on mutually inconsistent data,though the BBV scale function is consistent with the EHK data when their low b ( b = . ±
2% of their values, but clearly visible within thestatistical accuracy.
The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
U(3) lambda lattice scale
Bernd A. Berg
1. Introduction
With the emergence of the Yang-Mills gradient flow technique [1] there is renewed interestinto the issue of scale setting in lattice gauge theory. For a review see [2]. Therefore, it appearsto be worthwhile to analyze the status of the lambda scale for the SU(3) Wilson gauge actionfrom previous literature. Based on the Sommer scale [3] there are two estimates (parametrizations)of the SU(3) scaling function, a paper by Edwards, Heller and Klassen [4] (EHK) and anotherby Necco and Sommer [5] (NS). Independently an estimate of the SU(3) scaling function waslater extracted by Bazavov, Berg and Velytsky [6] (BBV) using deconfining transition coupling b t estimates and other information from a paper by the Bielefeld group [7] ( b = / g , where g is thebare coupling of the SU(3) Wilson gauge action). In addition to the data points on which thesescale function estimates are based, we include three data points calculated by Lüscher [1] with thegradient method and two recent large lattice estimates of b t [8]. Summary and conclusions followin the final section 3.
2. Definition and comparison of the scales
Sommer [3] proposed to set a hadronic scale r i / a through the force F ( r ) between static quarksat intermediate distances r by r i F ( r i ) = c i (Sommer scale). For their SU(3) investigations NS [5]use the values r F ( r ) = .
65 and r c F ( r c ) = . . (2.1)The r value was suggested in the original paper by Sommer. It is used by NS for their smallerlattices and also by EHK, who employ also larger values for c i , which we do not discuss here.The r c definition is used by NS for their set of large lattices. While a number of choices haveto be made when calculating r i / a (for details see the EHK and NS papers), estimations of thedeconfining transition temperatures T t = / [ a ( b t ) N t ] are in essence free of ambiguities when oneuses maxima of the Polyakov loop susceptibility on N N t lattices to determine b t ( N t ) for the limit N → ¥ . In particular, when refining the lattice a switch of a reference value, like from r to r c (2.1),is unwarranted when T t is used.In the following we compile the analytical expressions of the three scaling functions. TheEHK scaling function, the second of Eqs. (4.4) in their paper [4] with ˆ a defined by their Eq. (4.1),is given by [ a L L ] EHK = f EHK l ( b ) = l EHK ( g ) f as l ( g ) , (2.2)and derived from data in the range 5 . ≤ b ≤ .
5. Here f as ( g ) is the universal two-loop scalingfunction of SU(3) gauge theory, f as ( g ) = (cid:0) b g (cid:1) − b / ( b ) e − / ( b g ) with b =
113 316 p , b = (cid:18) p (cid:19) . (2.3)Higher perturbative and non-perturbative corrections are parametrized by l EHK ( g ) = ( + a ˆ a + a ˆ a ) / a with ˆ a = ˆ a ( g ) = f as ( g ) / f as ( ) (2.4)2 U(3) lambda lattice scale
Bernd A. Berg b EHK r / a b NS r / a b NS r c / a b t Bielefeld ( aT t ) − ∗ √ t / a − − ( aT t ) − − − Table 1:
Data used. ∗ The statistical error bar of this data point has been increased, so that it does notdominate the whole T c set, when the overall constant is adjusted to fit to the NS or EHK scale function. and the coefficients are given by a = . a = . a = . / a , the asymptotic scale f as ( g ) is approached for b → ¥ . In contrast to that NS presenttheir scale in form of a polynomial fit, Eq. (2.6) in their paper [5], which is supposed to be valid inthe region 5 . ≤ b ≤ . [ a L L ] NS = f NS l ( b ) with f NS l ( b ) = exp (cid:2) − . − . ( b − ) + . ( b − ) − . ( b − ) (cid:3) . (2.5)The BBV scaling function, Eq. (19) in their paper [6], is given by [ a L L ] BBV = f BBV l ( b ) = × l BBV ( g ) f as ( g ) , (2.6)where f as is again the asymptotic scaling function (2.3) and higher perturbative and non-perturbativecorrections are parametrized by l BBV ( g ) = + e ln a e − a / g + a g + a g (2.7)with the coefficients ln a = . a = . a = − . a = . . As theEHK scale, the BBV scale approaches up to a constant factor f as ( g ) for b → ¥ .In table 1 data are compiled on which the scales rely. As usual error bars are given in paren-thesis and apply to the last digits. The EHK data are from table 4 of their paper [4], which includesalso results from other groups. Thus several data point exists at some b , which are here combinedinto one estimate per b value. Their b = .
54 data point is omitted, because it is not used for thedetermination of their r / a scaling function (2.2). The NS data are from table 1 of their paper [5].The Bielefeld data are from table 2 of their paper [7]. We also list the three gradient flow datapoints from Lüscher [1] and two recent large-lattice b t estimates from Francis et al. [8]. As thesedata are not used for the determination of the scaling functions they provide independent tests.The statistical errors for estimates of deconfining transition transition temperatures are in b t with N t fixed. To allow for direct comparison with the statistical accuracy of the Sommer method, weattach to ( aT t ) − error bars by means of the equation △ ( aT t ) − = N t f BBV l ( b t ) (cid:2) f BBV l ( b t ) − f BBV l ( b t − △ b t ) (cid:3) . (2.8) To get convenient constants in the upcoming table 2, our definition (2.6) differs by a factor 10 from the one in [6]. U(3) lambda lattice scale
Bernd A. Berg
EHK r EHK r − r NS r c Bielefeld + LüscherE 0.9994 (14) 0.9996 (15) 0.99204 (97) 0.5172 (17) 1.35102 (81) 0.94272 (62)N 1.0055 (14) 1.0031 (15) 0.99995 (98) 0.5140 (17) 1.36108 (81) 0.94420 (62)B 0.21566 (28) 0.21646 (31) 0.21415 (21) 0.11024 (35) 0.29146 (17) 0.20388 (13)
Table 2:
Scale constants c from fitting Eq. (2.10) to the data (E for EHK, N for NS and B for BBV). EHK r EHK r − r NS r c Bielefeld + LüscherEHK 0.83 0.66 10 − − − NS 10 − − − BBV 10 − Table 3:
Probabilities Q that the discrepancy between scale and data set is due to chance. Zero indicates apositive number smaller than 10 − . For each of the the three scaling functions we perform one-parameter fits of the form c / f l ( b ) (2.9)to altogether six data sets: EKH r data, EHK r data with the data point for b = . b entering the determination of their scaling function) and denoted EHK r −
1, NS r data, NS r c data, combined Bielefeld and Francis et al. data denoted Bielefeld + and Lüscher’sdata points. The NS data are split, because their r and r data require independent determinationsof the over all constant in (2.9), while the Bielefeld and Francis et al. data are combined by theopposite reason. The results for the twelve constants are compiled in table 2.Even more interesting than the constants are the thus obtained goodness of fit values Q , whichare given in table 3. We see that the EHK r data are only consistent with the EHK scale, similarlythe NS r data are only consistent with the NS scale and the Bielefeld + data only with the BBVscale. The NS r c data from large lattices are rather inaccurate. They are consistent with the NS andBBV scales and almost consistent with the EHK scale. Leaving the b = . b value, the EHK r − Q values, Fig. 1 is obtained forthe differences between the data and the BBV scale function divided by this function (relative de-viation). Correspondingly, the relative deviations to the EHK and NS scale functions are calculatedand shown in the figure. Rotating the scale functions around, the relative deviations from the NSand EHK scales are found in the same way and shown in Figs. 2 and 3.The ratio between the NS data sets r and r c changes when different scale functions are used.From the constants of table 2 one finds ( r c / r ) BBV = . ( ) / . ( ) = . ( ) , (2.10) ( r c / r ) NS = . ( ) / . ( ) = . ( ) , (2.11) ( r c / r ) EHK = . ( ) / . ( ) = . ( ) . (2.12)4 U(3) lambda lattice scale
Bernd A. Berg −0.03−0.02−0.01 0 0.01 0.02 5.6 5.8 6 6.2 6.4 6.6 6.8 7 D a t a b NS r NS r EHK r LüscherBBV BielefeldFrancis et al.
Figure 1:
Relative deviations after the best fit of each data set to the BBV scale function. −0.03−0.02−0.01 0 0.01 0.02 5.6 5.8 6 6.2 6.4 6.6 6.8 7 D a t a b BBV BielefeldFrancis et al.NS r NS r EHK r Lüscher
Figure 2:
Relative deviations after the best fit of each data set to the NS scale.
The first values for ( r c / r ) BBV and ( r c / r ) NS are in statistical agreement with one another as well aswith the ratio r c / r = . ( ) , which is given in Eq. (2.5) of the NS paper and used to determinethe NS scale function. For ( r c / r ) NS this is obvious in Fig. 2, where the NS scale function (i.e., thezero-line) fits both NS data sets well. All other data are in disagreement with this scale. The BBVreference scale of Fig. 1 fits the EHK r − , the T c , the NS r c and Lüscher’s data well and The fit drawn is for the EHK r data. It becomes good for the EHK r − b = . U(3) lambda lattice scale
Bernd A. Berg −0.03−0.02−0.01 0 0.01 0.02 5.6 5.8 6 6.2 6.4 6.6 6.8 7 D a t a b NS r NS r EHK r LüscherFrancis et al.BBV Bielefeld
Figure 3:
Relative deviations after the best fit of each data set to the EHK scale function. is in disagreement with the NS r data and the b = . r data should in Fig. 1 be slightly higherthan the NS scale on its r c data. As this stays within statistical errors, we have just averaged thetwo curves, but use distinct colors, red for the r and blue for the r c range. Such averaging is notpossible when plotting the NS data versus the EHK scale, because the ratio (2.12) is incompatiblewith the other two ratios. It amounts to the difference between the red and blue curves in Fig. 3.
3. Summary and conclusions
Table 3 shows that the three scale functions (EHK, NS and BBV) are derived from data sets,given in table 1, which are mutually inconsistent in the range up to b = .
4, while the NS r c datafor the range 6 . ≤ b ≤ .
92 are not very restrictive. Only the BBV scaling function is consistentwith Lüscher’s accurate data (see also Figs. 1 to 3).In the range 5 . ≤ b ≤ .
92 the relative discrepancy between the scales is never largerthan ±
2% as is shown in the upper part of Fig. 4 for ratios of the form const f
EHK l / f BBV l and const ′ f NS l / f BBV l (the upper abscissa and the right ordinate apply and the constants (2.9) used fromtable 2 are the same as those for Fig. 1). Note that the previous figures, which exhibit relativedeviations from the scales, cover a corresponding [-0.03:0.02] range.The lower part of Fig. 4 shows that the EHK and BBV scales approach the universal asymptoticscale (2.3) in rather distinct ways, whereas such a parametrization is not attempted by NS (this partof the figure uses a normalization in which all scales agree at b = b values,which could come from calculations of the SU(3) deconfining temperature for N t >
12. This may6
U(3) lambda lattice scale
Bernd A. Berg R a t i o s b EHK/As scaleBBV/As scaleNS/As scaleEHK/BBV scaleBBV/BBV scaleNS/BBV scale
Figure 4:
Ratios with respect to the BBV scale (upper part, top abscissa and right ordinate) and asymptoticbehavior of the scales (lower part, bottom abscissa and left ordinate). need some innovative techniques as the N t =
14 and 16 data from Francis et al. are seen to exhibitsimilar inaccuracies as the large lattice NS data. Most promising may be calculations with thegradient method at larger b values. That this will work is also not obvious. For instance, thesensitivity of the gradient method to topological excitations [1, 9] on periodic lattices turns into adisadvantage when it comes to accurate scale calculations. Acknowledgments:
This work has in part been supported by DOE grant DE-FG02-97ER-41022and by NERSC ERCAP 86977 and ERCAP 86979. I like to thank Urs Heller for useful commentson the manuscript.
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