Computation of the string tension in three dimensional Yang-Mills theory using large N reduction

Transcription

1 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

2 Outline Quick result Introduction Details Conclusion 2

3 Quick result arxiv: lattice N = 47 b = 1 g 2 N = 0.6 to 0.8 Wilson loops 1x1 to 7x7 σb = ± (continuum extrapolation) 3

4 Introduction Large N Large N reduction Phase structure Project description 4

5 Large N Expansion parameters α(q 2 ) or 1/N N simplifications Planar graphs Factorization Non-interacting mesons OZI rule 1/3 1/ 5

6 Large N reduction Reduction to a one point (Eguchi-Kawai) 1 d lattice Z d N center symmetry But broken at weak coupling 6

7 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

8 Center symmetry breaking at physical scale Z d N Zd 1 N Zd 2 N D 2!loop %&function for L c (b) Tadpole Improved L 0 I =0.275 L 0 I =0.260 L 0 I = c <!> 1c (stable) 0c <!> 1c (metastable) 1c(metastable)!> 0h(stable) 8 L c (b) b=1/(g 2 0 N) =1/$ Kiskis, Narayanan, and Neuberger 8

9 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

10 Project Context Karabali, Kim, and Nair σb = 1 8π Bringoltz and Teper Large lattices N up to 8 Polyakov loops σb = ±

11 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

12 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

13 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 =

14 Compute all Wilson loops 1x1 to 7x7 Fit to W (k, t) = e a m(k)t ln W(k,t) t 14

15 Fit m(k) to m(k) = σb 2 Ik + c 0 b I + c 1 k 0.5 b= (15)k (4) (4)/k k b= (11)k (3) (2)/k k m(k) k 15

16 Extrapolate: b I σb I ± σ 1/2 b I (9) (22)/b I /b I 16

17 Are N and L large enough? m(k) N= (11)k (3)-0.101(2)/k N= (14)k+0.140(3)-0.100(3)/k N= (14)k+0.139(4)-0.099(3)/k N= (17)k+0.143(4)-0.102(3)/k N= (22)k+0.141(6)-0.097(4)/k k 17

18 (23)k+0.108(5)-0.117(6)/k (15)k+0.129(4)-0.160(4)/k 0.4 m(k) k 18

19 Are the results sensitive to smearing? 0.5 τ= (12)k (3) (3)/k τ= (11)k (3) (2)/k 0.4 m(k) k 19

20 Do Creutz ratios work well? 0.22 [χ(k,k)] 1/2 b I k 20

21 Conclusion 5 3 lattice N = 47 b = 1 g 2 N = 0.6 to 0.8 Wilson loops 1x1 to 7x7 σb = ± (continuum extrapolation) 21