Pushing dimensional reduction of QCD to lower temperatures

Transcription

1 Intro DimRed Center symm. SU(2) Outlook Pushing dimensional reduction of QCD to lower temperatures Philippe de Forcrand ETH Zürich and CERN arxiv: with A. Kurkela and A. Vuorinen Really: hep-ph/ , A. Vuorinen and L. Yaffe GGI, Florence, June 2008

2 Motivation Intro DimRed Center symm. SU(2) Outlook QCD thermodynamics well understood at low T : hadron resonance gas ε/t T/T c Karsch et al., hep-ph/

3 Motivation Intro DimRed Center symm. SU(2) Outlook... and at asymptotically high T : gas of free quarks and gluons What about T a few T c, ie. experimental range? p/p SB flavour 2 flavour 2+1 flavour T/T c Still far from non-interacting gas Karsch et al., hep-lat/

4 Motivation Intro DimRed Center symm. SU(2) Outlook... and at asymptotically high T : gas of free quarks and gluons What about T a few T c, ie. experimental range? Pisarski, hep-ph/ Far from leading order perturbation theory (e 3p)/T 4 logt

5 Intro DimRed Center symm. SU(2) Outlook Perturbative expansion IR divergences non-perturbative (Linde) p/p SB 0.5 g 6 (ln(1/g)+0.7) 4d lattice T/Λ_ MS g 2 g 3 g 4 g 5 Kajantie et al., hep-ph/ Spatial area law non-perturbative T

6 Brute force Intro DimRed Center symm. SU(2) Outlook Solution: (3 + 1)d lattice simulations However: N τ must be large (O (10)) to control a 0 limit T c? Fodor et al. Karsch et al. Finite density?? Alternative approach?

7 Intro DimRed Center symm. SU(2) Outlook Pert. Results Dimensional reduction dim (d + 1) system with one compact dimension: looks like dim d at distances β = 1 T β degrees of freedom are static modes φ 0 ( x) φ( x,τ) = T + n= exp(iω nτ)φ n ( x) Effective action: integrate out non-static modes with Z = D φ 0 D φ n exp( S 0 (φ 0 ) S n (φ 0,φ n )) = D φ 0 exp( S 0 (φ 0 ) S eff (φ 0 )) exp( S eff (φ 0 )) D φ n exp( S n (φ 0,φ n )) In practice? Goal is to reproduce Green s fncts φ 0 ( 0)φ0 ( x) for x β T is UV cutoff for S eff

8 Intro DimRed Center symm. SU(2) Outlook Pert. Results Dimensional reduction for QCD Asymptotic freedom: g(t) 1/logT causes separation of scales at high T : hard modes, energy O (T): non-static, esp. fermions (odd Matsubara) soft modes,o (gt): Debye mass A 0 (0)A 0 (x) ultrasoft modes,o (g 2 T): magnetic masses A i (0)A i (x)

9 Intro DimRed Center symm. SU(2) Outlook Pert. Results Perturbative approach Asymptotic freedom allows/enforces evaluation of S eff by perturbation theory Degrees of freedom are static A i,a 0, ie. 3d YM with adjoint Higgs Adjust couplings of S eff to match Green s fncts in perturbation theory after integrating out hard modes: EQCD S EQCD = ( d 3 1 x 2 F ij 2 + TrD i A 0 D i A 0 + me 2 A2 0 + λ ) EA 4 0 with F ij = i A j j A i ig 3 [A i,a j ], and g 2 3 = g2 T m E (T),λ E (T) fixed by perturbative matching after integrating out soft modes: MQCD S MQCD = d 3 x 1 2 F 2 ij

10 Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Screening masses down to 2T c 8 7 SU(2) SU(3) 6 M/T PC J R Laermann & Philipsen, hep-ph/

11 Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Screening masses at finite density (T = 2T c ) µ real, J P =0 + µ real, J P = M/T 5 M/T µ /T µ /T Hart et al., hep-ph/

12 Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Spatial string tension down to T c? 0.5 r 0 T T/σ s 1/2 (T) N τ =4 N τ =6 N τ = T/T 0 Karsch et al.,

13 Intro DimRed Center symm. SU(2) Outlook Pert. Results Successes and limitations of EQCD Wrong phase diagram must fail near T c phase diagram β G = symmetric phase 1st order xy pert. theory 4d matching line pert. theory 2nd order tricritical point 0.00 broken symmetry phase x S EQCD = ( d 3 1 x 2 F ij 2 + TrD i A 0 D i A 0 + me 2 A2 0 + λ EA 4 0 Symmetry is A 0 A 0, ie. Z 2 Matching line is in the wrong phase )

14 Intro DimRed Center symm. SU(2) Outlook Root of the problem Perturbation theory small fluctuations around one vacuum A 0 = 0 YM vacuum is N c -degenerate: center symmetry (spontaneously broken for T > T c ) A µ (x) s(x)(a µ (x)+i µ )s(x), with s(x + βê τ ) = exp(i 2π N c k)s(x) P(x) : exp(i β 0 dτa 0(x,τ)) : Effective action should respect symmetries of original action

15 Intro DimRed Center symm. SU(2) Outlook Polyakov loop vs coarse-grained Polyakov loop Use Polyakov loop P(x) instead of A 0 in S eff Pisarski But: P(x) SU(N) non-renormalizable (non-linear σ-model) cf. PNJL Here: degree of freedom is coarse-grained Polyakov loop Z (x) T V d 3 y block U(x,y)P(y)U(y,x) Yaffe renormalizability preserved: easy T matching with perturbation theory

16 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Simplest: SU(2) Yang-Mills sum of SU(2) matrices is multiple of SU(2) matrix Z = λω, Ω SU(2), λ > 0 Parametrization:Z = 1 2 (Σ1+iΠ aσ a ) L eff = g 2 3 [ 1 2 TrF 2 ij + Tr(D i Z D i Z )+V(Z ) ] Include all Z 2 -symmetric super-renormalizable terms: V(Z ) = b 1 Σ 2 + b 2 Π 2 a + c 1 Σ 4 + c 2 (Π 2 a) 2 + c 3 Σ 2 Π 2 a Local gauge invariancez (x) Ω(x)Z (x)ω 1 (x) Global Z 2 symmetry:z Z (actually Σ Σ,Π Π indep.)

17 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Perturbative matching Determine [almost] parameters {b 1,b 2,c 1,c 2,c 3 } using perturbation theory Split potential into hard and soft pieces: V(Z ) = V h + g3 2 V s Hard potential scales T (magnitude of coarse-grained Pol.) V h = h 1 TrZ Z + h 2 (TrZ Z ) 2 O(4) symmetric Soft potential EQCD at high T V s = s 1 TrΠ 2 + s 2 (TrΠ 2 ) 2 + s 3 Σ 4

18 Leading order Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Classical solution: Σ(x) = Σ, Π(x) = Πδ a,3 which minimize V class = g 2 3 ( Σ 2 + Π 2 )(2h h 2 ( Σ 2 + Π 2 )) Two possible cases (h 2 > 0 for stability): h 1 < 0 deconfined h 1 > 0 confined Σ = Π = h 1 h 2,TrZ v Σ = Π = TrZ = 0

19 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results 1-loop effective potential At 1-loop, U(1) symmetry of potential is broken: Σ = v cos(πα), Π = v sin(πα) V eff = s 1v 2 2 sin2 (πα)+ s 2v 4 4 sin4 (πα)+s 3 v 4 cos 4 (πα) v 3 3π sin(πα) 3 +O (g 2 3 )

20 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Matching to EQCD Decompose fluctuations into radial + angular: Z = ± [ 1 2 v1+g 3( 1 2 φ1+iχ)] ] ] L = 1 2 TrF ij 2 + [( 1 2 i φ) 2 + mφ 2φ2 + Tr [(D i χ) 2 + mχχ V s (φ,χ) m 2 φ = 8v 2 c 1 = 2h 1, heavy m 2 χ = 2(b 2 + v 2 c 3 ) = g 2 3 (s 1 4v 2 s 3 ), light L EQCD = 1 2 F 2 ij + TrD i A 0 D i A 0 + m 2 E A2 0 + λ EA 4 0 χ is A 0 2 equations m 2 χ = m 2 E, λχ 4 = λ E A 4 0

21 Domain wall Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Two more equations from matching domain wallz (x = ± ) = ± v 2 1 tension and width at 1-loop: F( z) π 2 T 4 /6 1 F F YM z L SU(2) andl eff wall profiles z relative difference

22 Finally... Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results 5 couplings, 4 equations 1 free parameter r [ L eff = g TrF ij 2 + Tr(D i Z D i Z )+V(Z ) ] V(Z ) = b 1 Σ 2 + b 2 Π 2 a + c 1 Σ 4 + c 2 (Π 2 a) 2 + c 3 Σ 2 Π 2 a g 2 3 = g2 T b 1 = 1 4 r 2 T 2, b 2 = 1 4 r 2 T g 2 T 2 c 1 = r g 2 c 2 = r g 2 c 3 = r 2 Couplings determined by (g,t) and r = m φ /T r fixes mass of coarse-grained Pol. loop magnitude any r O (1) should give similar IR physics

23 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Phase diagram? To determine phase diagram, need [non-perturbative] lattice simulations S lat = S W + S Z + V(ˆΣ, ˆΠ), [ S W = β x,i<j Tr[U ij] ], Wilson action ( ) ] 4 S Z = 2 β x,i Tr[ˆΠ2 ˆΠ(x)U i (x)ˆπ(x + î)u i (x) ( ) ) 4 + β x,i (ˆΣ 2 (x) ˆΣ(x)ˆΣ(x + î), kinetic term ( ) [ 3 2 V = x ˆb 1ˆΣ 2 +ˆb2ˆΠ 2 a + ĉ 1ˆΣ 4 + ĉ 2 (ˆΠ a) 2 + ĉ3ˆσ 2ˆΠ 2 a 4 β with β = 4, a lattice spacing ag3 2 Explicit Z(2) symmetry ˆΠ ˆΠ, ˆΣ ˆΣ ] Lattice couplings?

24 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Matching lattice and MS continuum couplings Perturbative 2-loop lattice calculation A. Kurkela, Σ = g 3ˆΣ+O (a), Π = g 3ˆΠ+O (a), c i = ĉ i +O (a) ˆb 1 = b 1 /g π (2ĉ 1 + ĉ 3 )β π 2 { (48ĉ ĉ ĉ 3)[log1.5β ] ĉ 3 } +O (a), ˆb 2 = b 2 /g π (10ĉ 2 + ĉ 3 + 2)β + 1 {(80ĉ 2 16π ĉ2 3 40ĉ 2)[log1.5β ] ĉ 2 } O (a) Matching exact in g 3, buto (a) error continuum extrapolation

25 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Simulation results < Σ 2 > < Σ > 2 < Σ > /g 2 < Σ 2 > < Σ > < Σ > /g 2 r 2 = 5,64 3,β = 12 r 2 = 5,64 3,β = 6 Z 2 -restoring phase transition

26 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Universality class N=64 N=96 N= N=64 N=96 N=128 B B (1/g 2 2 1/ν -1/g c )N (1/g 2 2 1/ν -1/g c )N r 2 = 5 r 2 = 10 3d Ising universality class: B 4 Σ4 Σ 2 2 = at criticality ν 0.63

27 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Thermodynamic + continuum extrapolations 1/g r 2 =10, β=12 r 2 =5, β=12 1/g r 2 =10 r 2 = N /β V a 0 r small large correlation length large volume r large large cutoff effects fine lattice r = 0 : Σ decouples λφ 4 already done X.P. Sun, hep-lat/

28 Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results Phase diagram 1/g Deconfined g 2 =5.1 Confined r 2 Phase transition is robust r r : no radial fluctuations, but still domain wall and transition gcrit 2 depends mildly on r for r > 1 Fixing g 2 (T c ) = 5.1 gives r 2.6: dim. red. action completely defined

29 Prospects Intro DimRed Center symm. SU(2) Outlook To do with SU(2): determine r(t) non-perturbatively check accuracy of effective theory at T near T c - domain wall tension - spatial string tension - screening masses make predictions using effective theory - heavy fermions - chemical potential Also SU(3):Z GL(3,C) V(Z ) = V h (Z )+g3 2 V s(z ); M Z 1 1TrZ 3 V h (Z ) = c 1 TrZ Z + c 2 (detz + detz )+c 3 Tr(Z Z ) 2 V s (Z ) = d 1 TrM M + d 2 Tr(M 3 + M 3 )+d 3 Tr(M M) 2 6 couplings, 4 equations 2 heavy modes to tune A. Kurkela,