Spatial string tension revisited

Transcription

1 Spatial string tension revisited (dimensional reduction at work) York Schröder work together with: M. Laine 1

2 Motivation RHIC QCD at T > (a few) 100 MeV asymptotic freedom weak coupling expansion slow convergence, non-trivial structure problematic dof's are identified soft modes p gt odd powers in g ultrasoft modes p g 2 T non-pert coeffs general picture perturbation theory OK for parametrically hard scales p 2πT soft and ultrasoft scales need improved analytic schemes, or non-pert treatment starting point: dim red eff. theory, or HTL eff. theory quantitative evidence: pick some simple observables compare 4d lattice vs soft/ultrasoft eff. theory e.g. static correlation lengths agreement down to T 2T c 2

3 Thermal pressure p(t ): 4d vs 3d p QCD (T ) lim V T V ln D[A a µ, ψ, ψ] ( exp 1 h h/t 0 dτ d 3 2ɛ x L QCD ) L QCD = 1 4 F a µνf a µν + ψγ µ D µ ψ + L GF + L FP p/p 0 asymptotically, expect ideal gas: p QCD (T ) p 0 = 0.5 g 6 (log(1/g)+0.9) g 6 (log(1/g)+0.7) g 6 (log(1/g)+0.5) 4d lattice T/Λ_ MS N π 2 T 4 f 90 3

4 Spatial string tension σ s study an observable allowing an unambiguous comparison take rectangular Wilson loop W s (R 1, R 2 ) in (x 1, x 2 ) plane def potential V s (R 1 ) = lim R2 1 R 2 ln W s (R 1, R 2 ) def spatial string tension σ s lim R1 V s(r 1 ) R 1 σ s has been measured in SU(3) on the (4d) lattice as e.g. ( σ s T = φ ) T T c [Boyd et al, 96] aim: get the eff. theory prediction for σ s effective theory setup σ s from 3d lattice perturbative matching to 4d Λ MS vs T c 4

5 Spatial string tension σ s : 4d vs 3d (anticipating all that follows) 1.2 T c / Λ MS _ = loop 1.0 T/σ s 1/ loop 0.6 4d lattice, N τ 1 = T / T c [4d lattice data from Boyd et al, 96] (cave: , no cont. extrapolation: Nτ = 8, T = 1/aNτ ) 5

6 Effective theory setup: QCD EQCD high T: QCD dynamics contained in 3d EQCD L E = 1 2 T r F 2 kl + T r [D k, A 0 ] 2 + m 2 ET r A λ (1) E (T r A 2 0) 2 + λ (2) E T r A matching coefficients [E. Braaten, A. Nieto, 95; M. Laine, YS, 05] m 2 E = T 2 { #g 2 + #g } λ (1/2) E = T { #g 4 + #g } g 2 E = T { g 2 + #g 4 + #g } higher order operators do not (yet) contribute [S. Chapman, 94; Kajantie et al, 97, 02] δl E g 2 D kd l (2πT ) 2L E g 2 (g2 T ) 2 (2πT ) 2L E (i.e. g 8 for g 2 E ) 6

7 Digression: g 2 E numerically in practice, need to renormalize: g 2 = g 2 ( µ) is MS coupling from soln of RGE at 2-loop level, define as usual [ ] Λ MS lim µ b1 b 0 g 2 /2b 2 [ 0 1 ] ( µ) exp µ 2b 0 g 2 ( µ) hence g 2 E = g 2 E ( µ, Λ MS, T ) = T φ ( µ T, T Λ MS )

8 Digression: g 2 E numerically in practice, need to renormalize: g 2 = g 2 ( µ) is MS coupling from soln of RGE at 2-loop level, define as usual [ ] Λ MS lim µ b1 b 0 g 2 /2b 2 [ 0 1 ] ( µ) exp µ 2b 0 g 2 ( µ) hence g 2 E = g 2 E ( µ, Λ MS, T ) = T φ ( µ T, T Λ MS ) ren. scale dependence formally, µ dependence is of higher order numerically, there is µ dependence free to choose some optimisation procedure, e.g. PMS choose µ opt at extremum of 1-loop vary scale within µ = ( ) µ opt 7

9 3.5 Choosing µ opt via PMS _ N f = 0, T = 2 Λ MS loop / T g E loop _ µ / T 8

10 3.5 Choosing µ opt via PMS _ N f = 2, T = 2 Λ MS loop / T g E loop _ µ / T 9

11 3.5 Choosing µ opt via PMS _ N f = 3, T = 2 Λ MS loop / T g E loop _ µ / T 10

12 Digression: g 2 E numerically 3.5 N f = / T g E loop loop _ T / Λ MS 11

13 Digression: g 2 E numerically 3.5 N f = / T g E loop loop _ T / Λ MS 12

14 Digression: g 2 E numerically 3.5 N f = / T g E loop 1-loop _ T / Λ MS 13

15 Digression: g 2 E numerically 0.40 [Laine, Schroder,.. hep-ph/ ] 0.30 α s eff = g E 2 / 4πT 0.20 N f = 3 N f = 0 N f = _ T / Λ MS 14

16 Effective theory setup: QCD EQCD MQCD 3d EQCD is contained in 3d MQCD L M = 1 2 T r F 2 kl +... matching coefficient [P. Giovannangeli, 04; M. Laine, YS, 05] g 2 M = g 2 E { 1 + # g2 E m E + # g4 E m 2 E + # g2 E λ (1/2) E m 2 E +... } expansion converges extremely well, even close to T c can safely ignore higher loop corrections for g 2 M higher order operators could contribute δl M g 2 E D k D l m 3 L M g 2 E E (g 2 T ) 2 m 3 E L M 15

17 Effective theory prediction for σ s observable σ s exists in 3d SU(3) gauge theory (MQCD) dimensionful gauge coupling σ s = #g 4 M most recent lattice data σs g 2 M = 0.553(1) [M. Teper, B. Lucini, 02]

18 Effective theory prediction for σ s observable σ s exists in 3d SU(3) gauge theory (MQCD) dimensionful gauge coupling σ s = #g 4 M most recent lattice data σs g 2 M = 0.553(1) [M. Teper, B. Lucini, 02] to compare with 4d lattice, need to relate g 2 M σs T = 0.553(1) g2 M g 2 E g 2 E T = φ «T Λ MS and T

19 Effective theory prediction for σ s observable σ s exists in 3d SU(3) gauge theory (MQCD) dimensionful gauge coupling σ s = #g 4 M most recent lattice data σs g 2 M = 0.553(1) [M. Teper, B. Lucini, 02] to compare with 4d lattice, need to relate g 2 M σs T = 0.553(1) g2 M g 2 E g 2 E T = φ «T Λ MS and T finally, need to relate Λ MS e.g. via T = 0 string tension e.g. via Sommer scale e.g. via scaling at crit. point and T c T c Λ = T c/ σ Λ / σ MS MS = 1.16(4) [Teper et al, 03; Bali, Schilling, 92] T c Λ = r 0 T c r MS 0 Λ = 1.25(10) [S. Necco, 03; ALPHA coll, 98] MS to be conservative, consider the interval Tc Λ MS = 1.15(5) [S. Gupta, 00] 16

20 Spatial string tension σ s : 4d vs 3d 1.2 T c / Λ MS _ = loop 1.0 T/σ s 1/ loop 0.6 4d lattice, N τ 1 = T / T c [4d lattice data from Boyd et al, 96] (cave: , no cont. extrapolation: Nτ = 8, T = 1/aNτ ) parameter-free comparison! support for hard/soft+ultrasoft picture of thermal QCD 17