We determine the critical couplings for the deconfinement phase transition in
$SU(2)$ gauge theory on $N_\tau \times N_\sigma^3$ lattices with $N_\tau = 8$
and 16 and $N_\sigma$ varying between 16 and 48. A comparison with string
tension data shows scaling of the ratio $T_c / \sqrt{\sigma}$ in the entire
coupling regime $\beta =2.30-2.75$, while the individual quantities still
exhibit large scaling violations. We find $T_c / \sqrt{\sigma}=0.69(2)$. We
also discuss in detail the extrapolation of $T_c / Lambda_{\rm{\bar{M}
\bar{S}}}$ and $\sqrt{\sigma} / Lambda_{\rm{\bar{M}\bar{S}}}$ to the continuum
limit. Our result, which is consistent with the above ratio, is $T_c /
Lambda_{\rm{\bar{M}\bar{S}}} = 1.23(11)$ and $\sqrt{\sigma} /
Lambda_{\rm{\bar{M}\bar{S}}} = 1.79(12)$. We also comment upon corresponding
results for $SU(3)$ gauge theory and four flavour QCD.Comment: 27 pages with 9 postscript figures included. Plain TeX file (needed
  macros are included). BI-TP 92-26, FSU-SCRI-92-103, HLRZ-92-39 (Quote of
  UKQCD string tension, and accordingly Figs. 5 and 7a, plus a few typo's
  corrected.

Attig

Baker

Barber

Barkai

Bhanot

Binder

Booth

Born

Bowler

Brown

Christ

Creutz

Dashen

Decker

Ding

El-Khadra

Engels

F. Karsch

Falconi

Ferrenberg

Fredenhagen

Gavai

Gottlieb

Green

Gupta

Gutbrod

Hasenfratz

Heller

Iwasaki

J. Fingberg

Karsch

Kennedy

Lautrup

Lepage

Lüscher

Makeenko

Marinari

Michael

Parisi

Perantonis

Privman

Samuel

U. Heller

van Baal

Wolff

English

arXiv

Fingberg J, Heller UM, Karsch F. Scaling and asymptotic scaling in the SU(2) gauge theory. Nuclear Physics, B: Proceedings Supplements. 1993;30:343-346.We present results from a large volume study of the finite temperature deconfinement phase transition. The questions of scaling and asymptotic scaling are addressed. We find that an extrapolation of the critical temperature and the string tension to the continuum limit is possible when an effectivecoupling defined in terms of the plaquette operator is used

Fingberg, Jochen

Heller, U. M.

Karsch, Frithjof

Publications at Bielefeld University

Nuclear Physics B (Proc. Suppl.) 30 (1993) 343,--346 North-Holland I ~Lll[ai l l~ l l i l  Bib'&'! [ I k l  "! PROCEEDINGS SUPPLEMENTS Scal ing and Asymptot i c  Scal ing in the SU(2)  Gauge Theory  J. FINGBERG ffi , U. M. HELLER b and F. KARSCH a, c ~HLRZ, c/o KFA Jiilich, D-5170 Jiilich, Germany bSCRI, The Florida State University, Tallahassee, FL 32306-44052 U.S.A. CFakult~t fiir Physik, Universit~t Bielefeld, D-4800 Bielefeld 1, Germany We present results from a large volume study of the finite temperature deconfinement phase transition. The questions of scaling and asymptotic s aling are addressed. We find that an extrapolation of the critical temperature and the string tension to the continuum limit is possible when an effective coupling defined in terms of the plaquette operator is used. 1. INTRODUCTION First results for the deconfinement tempera- ture [1], all in the region around 200 MeV, were promising but very soon it became clear that larger lattices are required. Even advanced stud- ies of the heavy quark potential on large lattices [2-4] showed that asymptotic scaling is not yet re- stored at the values of the bare coupling constant presently accessible. Recent investigations of the finite temperature deconfinement transition using finite-size scaling techniques showed that finite spatial size correc- tions are well under control and that reliable methods to determine the critical coupling [5-7] are available. We have extended these studies to smaller lattice spacings using lattices of size N a x N~ with Nr = 8 and 16. This makes it pos- sible to study scaling of the dimensionless ratio Tc/v~ and the approach to asymptotic scaling in different extrapolation schemes. 2. F IN ITE-S IZE  SCAL ING AND THE CONTINUUM L IMIT  Finite-size scaling techniques can be used to re- late the critical behaviour observed on finite lat- tices to the corresponding continuum field theory. The singular part of the free energy density for a continuum field theory with a simple critical point in a finite volume can be written in terms of lattice variables as the product of a scaling func- tion Qjo and a factor y which is the ratio of the spatial and temporal lattice size [8]. Q j, depends on the reduced temperature t and the external magnetic field strength h in form of a thermal and a magnetic scaling field g~ and gh. f,(t,h;N¢;N,)=Y-aQjo (g,Y~,ghY ~ ) (I) The Binder cumulant g, is defined through the free energy density as - 0h' Ih=o/( 0h2 h--o/ g, (2) = (L')/(L') 2 -  3 . (3) On a finite lattice, g, is measured by the second and fourth moment of the Polyakov loop opera- tor. The cumulant g, depends only on g~yl/v and for small values of the reduced temperature one can use a linear expansion. g4(gcytlv) = g,,o % g,,ltY 1/~ (4) For fixed Nr and varying Nq the critical coupling for infinite volume ~c,oo can be determined from the fixed point g,(t = 0). For a fixed value of y, varying Nr and Nq accordingly, the critical cou- pling can be extracted from the shift A~ which is necessary to overlay two curves of g4. g4(/3 -- A/3; No; N,) -- g4(/3; 2-N't,; 2N,) (5) = + (6) The continuum limit can be found by keeping ~/ fixed and increasing Nr. The numerical results for g4 are shown in Figs. 1 and 2. The continu- ous curves were obtained by the density of states 0920-5632/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved. 344 J. Fingberg et al. / Scaling aJut asymptotic s alb~g inthe SU(2) gauge theory g,i -0.8 -I .0 -1.2 -1.4 - I  .S -0.8 g4 I . . . .  I ' " " " I . . . .  I • " " " I | :. . N, N, _ , 24 s J "':k | t ._ ~t~ x 32 8 I • I A , I . * * • I • • • • I . , • • I 2.sos 2.s ,o  2.5,5 2.s2o =41g -1.0 -1.2 -1.4 -1 .6  %N, ~ . .  o 32 16 -1 ,S -1 .S J I I I I 2.500 2.72 2.725 2.73 2.735 2.74 2.745 2.75 B=41g"  Figure 1. The cumulant g4 for N~ = 8. Figure 2. The cumulant g4 for N~ = 16. method. Using the fixed point method for Nr = 8 and the shift for Nr = 16 we obtain the following values for the critical couplings 2.5115(40), N, = 8 /~c,oo = 2.7395(100), Nr = 16 . (7) 3. SCALING AND SCALING ASYMPTOTIC  The scaling behaviour near the continuum limit is controlled by the asymptotic freedom property of QCD which implies scaling of dimensionless ra- tios of physical quantities. For SU(N) gauge theories the ratio Vc(R)/V'~ scales as a function of Rye" for ~ _> 2.4, 6.0 for g - 2, 3 [2, 3]. A comparison with string tension data [8] shown in Fig. 3 reveals caling of Tc /v~ in the region 4 < Nr _< 16 for SU(2) and in the region 8 < Nr < 14 for SU(3). By averaging over the scaling region we obtain the continuum result Tc _ I 0.69 4- 0.02, SU(2) - .  0.50 4- 0.03, su(3) (s) The fact that we observe scaling but not asymp- totic scaling indicates the presence of universal scaling violating terms which can be incorporated by a redefinition of the bare coupling/~. Follow- ing Parisi [9], an effective coupling scheme which has been successfully applied [10] can be defined in terms of the plaquette operator. N 2 - 1 19E -- 4(1 - (Up)) (9) Various other definitions giving similar results are possible [11, 12]. In Figs. 4 and 5 we show that violations of asymptotic freedom are strongly re- duced in the effective coupling scheme. In the continuum limit scaling violating terms are ex- pected to be O(I/Ina). The data are not (yet) sensitive to this logarithmic correction. A linear extrapolation to a = 0 yields Tc = { 1.23 4- 0.11, SU(2) A~--~ _ 1.03±0.19, SU(3) . (10) The corresponding extrapolation for the string tension, shown in Fig. 5, results in v~ [ 1.79-4-0.12, SU(2) AIW ~- = ~. 1.75 -+- 0.20, SU(3) (11) J. Fingberg et aL ~Scaling and asymptotic s albzg in the SU(2) gauge theory 345 1.0  0.S  0 .6  0 .4  0,2  I • r=/o' /= 1 I I I I I o SU(2) .~-0 ,~ SU(3)  n , -O  X SU(3)  he-4 I 1 x z z T I , I = I * J , I . I , I , I , 4 6 S 10 12 14 16 18 N r Figure 3. The critical temperature in units of the square root of the string tension. We stress once more that the linear extrapolation eventually should turn to a logarithmic one at even smaller values of the lattice spacing. 4. Conclusion The ratio Tc/v~ scales for /~ _> 2.3 over a range, where the lattice spacing varies by a factor of 4 and the smallest lattice spacing we reached was a = 0.09/V~. A comparison between the bare and an effective coupling scheme shows that violations of asymp- totic scaling are strongly reduced in the latter case. Converting our continuum result for the string tension to a value of the A-parameter and comparing with other recent results we see that the numbers do not differ significantly. The last value in Eq. 12 has been obtained by using the perturbative form of the running coupling [13]. 0.559(37) A~I-~S-(2)/v~ - 0.617(55), Ref.[2] 0.478, Ref.[12] (12) 0.571(37) Asu(s)/.~_ 0 558 +°'°lz Ref.[3] • -0.00r, (13) • -~--~ ,vv -  0.581(45), Ref.[14] 0.532, Ref.[11] These results give further confidence that an ex- trapolation to the continuum limit a - 0 is within reach for the pure gauge theory. For the fu- ture systematic errors possibly present in the ap- plied methods have to be further educed• Then a remaining challenging task is the extension of these methods to full QCD including dynamical fermions. REFERENCES 1 J. Kuti, J. Pol6nyi and K. Szlach~nyi, Phys. Lett. 98B (1980) 199, J. Engels, F. Karsch and H. Satz, Phys. Left. 101B (1981) 89, K. Kajantie, C. Montonen and E. Pietarinen, Z. Phys. C9 (1981) 253. 2 S.P. Booth et al. (the UKQCD Collabora- tion), Phys. Left. 275B (1992) 424, Liverpool preprint LTH284. 3 G. S. Bali and K. Schilling, Wuppertal preprint WUB 92-02 and WUB 92-29. 4 C. Michael, Phys. Left. 283B (1992) 103. 5 J. Engels, J. Fingberg, and M. Weber, NucL Phys. B332 (1990) 737. 6 J. Engels, J. Fingberg and D. Miller, to ap- pear in Nuc]. Phys. B (1992). 7 J. Engels, J. Fingberg and V. K. Mitrjushkin, Bielefeld preprint BI-TP 92/22. 8 J. Fingberg, U. Heller and F. Karsch, Bide- feld preprint BI-TP 92/26 and references within. 9 G. Parisi, Proceedings of the XX zh Confer- ence on High Energy Physics, Madison 1980. 10 F. Karsch and R. Petronzio, Phys. Left. 139B (1984) 403. 11 A. X. E1-Khadra et al., FERMILAB-PUB- 91/354-T, G. P. Lepage and P. B. Mackenzie, FERMILAB-PUB-91/355-T. 12 M. Liischer, R. Sommer, P. Weisz and U. Wolff, Cern preprint CERN-TH 6566/92. 13 A. Billoire, Phys. Lett. 104B (1981) 472. 14 S. P. Booth et al. (the UKQCD Collabora- tion), Liverpool preprint LTH285. 346 J. Fingberg et aL ~Scaling and asymptotic s aling in the SU(2) gauge theory I I I I I I T J^  e Su(2) _ r t f  =0 o ~ 2..'$ 2.0 1.5 1.0 O.S 0 I I I I I I 0 0.1 0.2 0.3 0.4 0.5 aT= 3.5 | I oT/~/^ m 2.S A 3 -  a*"  o.o~ ° 0 0.1 ! su(2) nr=O 0 m I 0.2 aAi I 0.3 3.0 1 I I I I 3.0~ x T J^  m suo) I 4 • r ) f=  0 2.5 ~ -"  o J A ~,o ° jo 2 .0 , "  0 .5 0 I I 0 0.1 0.2 0.3 0.4 0.5 aT© 4 3.5 3 2.5 2 1.5 I • j al/2/Am I i ° / e / O su(3) nf  zO - 1 I I 0 0.1 0.2 0.3 aA t Figure 4. Extrapolation of the critical tempera- ture in the bare (triangles) and the effective (cir- cles) coupling scheme. Figure 5. Extrapolation of the string tension in the bare (triangles) and the effective (circles) cou- pling scheme. 

(the UKQCD Collaboration), Liverpool preprint LTH285. 346 J. Fingberg et aL ~Scaling and asymptotic scaling in the SU(2) gauge theory

424, Liverpool preprint LTH284.

E1-Khadra et al., FERMILAB-PUB-

NucL Phys.

Scaling and asymptotic scaling in the SU(2) gauge theory

Crossref

U.M. Heller

Kuti

Kajantie

Billoire

Fingberg J, Heller UM, Karsch F. Scaling and asymptotic scaling in the SU(2) gauge theory. Nuclear Physics, B. 1993;392(2):493-517.We determine the critical couplings for the deconfinement phase transition in SU(2) gauge theory on N(tau) x N(sigma)3 lattices with N(tau) = 8 and 16 and N(sigma) varying between 16 and 48. A comparison with string tension data shows scaling of the ratio T(c)/square-root sigma in the entire coupling regime beta = 2.30-2.75, while the individual quantities still exhibit large scaling violations. We find T(c)/square-root sigma = 0.69(2). We also discuss in detail the extrapolation of T(c)/LAMBDA(MSBAR) and square-root sigma/LAMBDA(MSBAR) to the continuum limit. Our result, which is consistent with the above ratio, is T(c)/LAMBDA(MSBAR) = 1.23(11) and square-root sigma/LAMBDA(MSBAR) = 1.79(12). We also comment upon corresponding results for SU(3) gauge theory and four-flavour QCD

119;

331; E i Marinari,

3343; Y. Iwasaki et al.,

347; C,

and LTH284

in Finite-size scaling and numerical simulations of statistical systems (World ScientifiC,

L715 * Here we consider the system as a (d + D-dimensional model at &quot;invers~ temperature./3, which is not to he confused with the PhYSical temperature T. r 516 J. Fingberg et al. / Scaling #t SU(2)

Three-loop results on the lattice,

http://pub.uni-bielefeld.de/publication/1784454

Scaling and Asymptotic Scaling in the SU(2) Gauge Theory

Scaling and Asymptotic Scaling in the SU(2) Gauge Theory

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