Computation of the string tension in three dimensional Yang-Mills theory using large N reduction Joe Kiskis UC Davis Rajamani Narayanan Florida International University 1
Outline Quick result Introduction Details Conclusion 2
Quick result arxiv:0807.1315 5 3 lattice N = 47 b = 1 g 2 N = 0.6 to 0.8 Wilson loops 1x1 to 7x7 σb = 0.1964 ± 0.0009 (continuum extrapolation) 3
Introduction Large N Large N reduction Phase structure Project description 4
Large N Expansion parameters α(q 2 ) or 1/N N simplifications Planar graphs Factorization Non-interacting mesons OZI rule 1/3 1/ 5
Large N reduction Reduction to a one point (Eguchi-Kawai) 1 d lattice Z d N center symmetry But broken at weak coupling 6
Work-arounds Quenched E-K But Bringoltz and Sharpe Twisted E-K But Teper and Vairinhos Continuum or partial reduction i.e. reduction to finite physical size l > 1/T c 7
Center symmetry breaking at physical scale Z d N Zd 1 N Zd 2 N... 11 10 9 4D 2!loop %&function for L c (b) Tadpole Improved L 0 I =0.275 L 0 I =0.260 L 0 I =0.245 0c <!> 1c (stable) 0c <!> 1c (metastable) 1c(metastable)!> 0h(stable) 8 L c (b) 7 6 5 4 0.34 0.35 0.36 0.37 b=1/(g 2 0 N) =1/$ Kiskis, Narayanan, and Neuberger 8
Phase structure 3 dimensions Lattice size L L=3 L=1 0h-phase: No gap in the plaquette distribution Phase transition in plaquette distribution Possibly third order 0c-phase: Plaquette distribution opens up a gap around!=". QCD in the confined phase. No volume dependence Durhuus-Olesen transition in Wilson loops L 1 (b I )=5.90(47) b I Deconfining transition L 3 (b I )=2.14(26)b I L 2 (b I )=3.85(43)b I Continuum limit 1c-phase: U(1) one direction is broken. QCD in the deconfined phase. 2c-phase: U(1) in two directions are broken. QCD in a small box at zero temperature. 3c-phase: U(1) in all three directions are broken. QCD in a small box at high temperatures. Continuum limit b B I =0.26 Tadpole improved t Hooft Coupling b I Figure 8: Summary of large N QCD in d = 3 on L 3 lattice Narayanan and Neuberger 9
Project Context Karabali, Kim, and Nair σb = 1 8π 0.1995 Bringoltz and Teper Large lattices N up to 8 Polyakov loops σb = 0.1975 ± 0.0002 0.0005 10
This work 5 3 lattice N = 47 b = 0.6 to 0.8 Smear space-like links with staples in the same time slice Wilson loops 1x1 to 7x7 Fit to get quark-antiquark potential and string tension 11
Details Wilson gauge field action with bare coupling g b = 1 Tadpole improved to g 2 b N I = e(b)b with e(b) the average plaquette Space-like and time-like separations K, T in lattice units. Physical units k = K/b I and t = T/b I 12
Smearing 2 S - U S + 1 U = P SU(N) [(1 f)u + f 2 S + + f 2 S ] Iterate n times τ = fn f = 0.1 n = 25 τ = 2.5 13
Compute all Wilson loops 1x1 to 7x7 Fit to W (k, t) = e a m(k)t 0-1 -2 ln W(k,t) -3-4 -5-6 -7 2 4 6 8 10 12 t 14
Fit m(k) to m(k) = σb 2 Ik + c 0 b I + c 1 k 0.5 b=0.6 0.0296(15)k + 0.129(4) - 0.160(4)/k 0.0296k+0.116 b=0.8 0.0342(11)k + 0.140(3) - 0.101(2)/k 0.0342k+0.132 0.4 m(k) 0.3 0.2 0.1 0 2 4 6 8 10 12 k 15
Extrapolate: b I σb I 0.1964 ± 0.0009 0.2 0.195 0.19 0.185 σ 1/2 b I 0.18 0.175 0.17 0.165 0.1964(9)-0.00403(22)/b I 2 0.16 0 1 2 3 4 5 6 7 2 1/b I 16
Are N and L large enough? m(k) 0.55 0.5 0.45 0.4 0.35 N=47 0.0342(11)k +0.140(3)-0.101(2)/k N=41 0.0341(14)k+0.140(3)-0.100(3)/k N=37 0.0339(14)k+0.139(4)-0.099(3)/k N=31 0.0330(17)k+0.143(4)-0.102(3)/k N=23 0.0323(22)k+0.141(6)-0.097(4)/k 0.3 0.25 0.2 0.15 0.1 2 4 6 8 10 12 k 17
0.5 4 3 0.0316(23)k+0.108(5)-0.117(6)/k 5 3 0.0296(15)k+0.129(4)-0.160(4)/k 0.4 m(k) 0.3 0.2 0.1 2 4 6 8 10 12 k 18
Are the results sensitive to smearing? 0.5 τ=1.25 0.0354(12)k + 0.134(3) - 0.093(3)/k τ=2.5 0.0342(11)k + 0.140(3) - 0.101(2)/k 0.4 m(k) 0.3 0.2 0.1 2 4 6 8 10 12 k 19
Do Creutz ratios work well? 0.22 [χ(k,k)] 1/2 b I 0.2 0.18 0.16 0.14 2 4 6 8 10 12 k 20
Conclusion 5 3 lattice N = 47 b = 1 g 2 N = 0.6 to 0.8 Wilson loops 1x1 to 7x7 σb = 0.1964 ± 0.0009 (continuum extrapolation) 21