Pushing dimensional reduction of QCD to lower temperatures Philippe - - PowerPoint PPT Presentation

university-logo Intro DimRed Center symm. SU(2) Outlook

Pushing dimensional reduction of QCD to lower temperatures

Philippe de Forcrand ETH Zürich and CERN

arXiv:0801.1566 with A. Kurkela and A. Vuorinen Really: hep-ph/0604100, A. Vuorinen and L. Yaffe

GGI, Florence, June 2008

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Motivation

QCD thermodynamics well understood at low T: hadron resonance gas

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

T/Tc

ε/T4

Karsch et al., hep-ph/0303108

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Motivation

... and at asymptotically high T: gas of free quarks and gluons What about T ∼ a few Tc, ie. experimental range?

0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 T/Tc p/pSB 3 flavour 2 flavour 2+1 flavour

Karsch et al., hep-lat/9602007 Still far from non-interacting gas

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Motivation

... and at asymptotically high T: gas of free quarks and gluons What about T ∼ a few Tc, ie. experimental range? Pisarski, hep-ph/0612191 Far from leading order perturbation theory (e − 3p)/T 4 ∼ logT

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Perturbative expansion

IR divergences → non-perturbative (Linde)

1 10 100 1000

T/ΛMS

_

0.0 0.5 1.0 1.5

p/pSB

g

2

g

3

g

4

g

5

g

6(ln(1/g)+0.7)

4d lattice

Kajantie et al., hep-ph/0211321 Spatial area law ↔ non-perturbative ∀ T

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Brute force

Solution: (3+ 1)d lattice simulations However:

• Nτ must be large (O (10)) to control a → 0 limit

Tc ? Fodor et al. ↔ Karsch et al.

• Finite density ??

Alternative approach?

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo 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 φ0(
• x)

φ(

• x,τ) = T ∑+∞

n=−∞ exp(iωnτ)φn(

• x)
• Effective action: integrate out non-static modes

Z = D φ0D φn exp(−S0(φ0)− Sn(φ0,φn))

=

D φ0 exp(−S0(φ0)− Seff(φ0)) with exp(−Seff(φ0)) ≡ D φn exp(−Sn(φ0,φn))

• In practice?

Goal is to reproduce Green’s fncts φ0(

• 0)φ0(
• x) for |
• x| ≫ β

T is UV cutoff for Seff

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo 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 A0(0)A0(x)
• ultrasoft modes, O (g2T): magnetic masses Ai(0)Ai(x)
• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Pert. Results

Perturbative approach

Asymptotic freedom allows/enforces evaluation of Seff by perturbation theory

• Degrees of freedom are static Ai,A0, ie. 3d YM with adjoint Higgs
• Adjust couplings of Seff to match Green’s fncts in perturbation theory

→ after integrating out hard modes: EQCD

SEQCD =

• d3x

1

2F 2 ij +TrDiA0DiA0 + m2 EA2 0 +λEA4

• with Fij = ∂iAj −∂jAi − ig3[Ai,Aj], and g2

3 = g2T

mE(T),λE(T) fixed by perturbative matching

→ after integrating out soft modes: MQCD

SMQCD =

• d3x 1

2F 2 ij

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Pert. Results

Successes and limitations of EQCD

Screening masses down to ∼ 2Tc

0+

++ 0-

• +

0+

++ 0-

• +

0-

+- JR PC

2 3 4 5 6 7 8

M/T SU(2) SU(3)

Laermann & Philipsen, hep-ph/0303042

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Pert. Results

Successes and limitations of EQCD

Screening masses at finite density (T = 2Tc)

1 2 3 4 |µ|/T 5 10 M/T µ real, J

P=0 + + + + + − −

1 2 3 4 |µ|/T 5 10 M/T µ real, J

P=0 − − + − − − +

Hart et al., hep-ph/0004060

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Pert. Results

Successes and limitations of EQCD

Spatial string tension down to ∼ Tc ?

0.4 0.5 0.6 0.7 0.8 5 4.5 4 3.5 3 2.5 2 1.5 1 1 T/T0 r0 T T/σs

1/2(T)

2 0.5 Nτ=4 Nτ=6 Nτ=8

Karsch et al., 0806.3264

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Pert. Results

Successes and limitations of EQCD

Wrong phase diagram ⇒ must fail near Tc

0.00 0.10 0.20 0.30 0.40

x

−0.02 0.00 0.02 0.04 0.06

xy

phase diagram βG = 12

tricritical point 2nd order 1st order

• pert. theory

broken symmetry phase symmetric phase

• pert. theory

4d matching line

SEQCD =

• d3x

1

2F 2 ij +TrDiA0DiA0 + m2 EA2 0 +λEA4

• Symmetry is A0 ↔ −A0, ie. Z2

Matching line is in the wrong phase

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Root of the problem

• Perturbation theory ⇒ small fluctuations around one vacuum A0 = 0
• YM vacuum is Nc-degenerate:

center symmetry (spontaneously broken for T > Tc) Aµ(x) → s(x)(Aµ(x)+ i∂µ)s(x)†, with s(x +βˆ eτ) = exp(i 2π

Nc k)s(x)

P(x) ≡: exp(i β

0 dτA0(x,τ)) :

−0.1 0.1 −0.1 −0.05 0.05 0.1 −0.1 0.1 −0.1 −0.05 0.05 0.1 −0.1 0.1 −0.1 −0.05 0.05 0.1 −0.1 0.1 −0.1 −0.05 0.05 0.1

Effective action should respect symmetries of original action

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Polyakov loop vs coarse-grained Polyakov loop

• Use Polyakov loop P(x) instead of A0 in Seff

Pisarski But: P(x) ∈ SU(N) → non-renormalizable (non-linear σ-model)

• cf. PNJL
• Here: degree of freedom is coarse-grained Polyakov loop

Yaffe

Z (x) ≡

T Vblock

• d3y U(x,y)P(y)U(y,x)

renormalizability preserved: easy T → ∞ matching with perturbation theory

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo 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+ iΠaσa)

Leff = g−2

3

1

2TrF 2 ij +Tr(DiZ †DiZ )+ V(Z )

• Include all Z2-symmetric super-renormalizable terms:

V(Z ) = b1Σ2 + b2Π2

a + c1Σ4 + c2(Π2 a)2 + c3Σ2Π2 a

Local gauge invariance Z (x) → Ω(x)Z (x)Ω−1(x) Global Z2 symmetry: Z → −Z (actually Σ → −Σ,Π → −Π indep.)

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Perturbative matching

• Determine [almost] parameters {b1,b2,c1,c2,c3} using perturbation

theory

• Split potential into hard and soft pieces: V(Z ) = Vh + g2

3Vs

• Hard potential →

scales ∼ T (magnitude of coarse-grained Pol.) Vh = h1TrZ †Z + h2(TrZ †Z )2 O(4) symmetric

• Soft potential →

EQCD at high T Vs = s1TrΠ2 + s2(TrΠ2)2 + s3Σ4

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Leading order

• Classical solution: Σ(x) = ¯

Σ, Π(x) = ¯ Πδa,3 which minimize

Vclass =

g−2

3

4 (¯

Σ2 + ¯ Π2)(2h1 + h2(¯ Σ2 + ¯ Π2))

Two possible cases (h2 > 0 for stability): h1 < 0 → deconfined h1 > 0 → confined

¯ Σ = ¯ Π = − h1

h2 ,TrZ ≡ v

¯ Σ = ¯ Π = TrZ = 0

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo 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)

Veff = s1v2

2

sin2(πα)+ s2v4

4

sin4(πα)+ s3v4 cos4(πα)− v3

3π|sin(πα)|3 +O (g2 3)

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Matching to EQCD

Decompose fluctuations into radial + angular:

Z = ±

1

2v1+ g3( 1 2φ1+ iχ)

• L = 1

2TrF 2 ij + 1 2

• (∂iφ)2 + m2

φφ2

+Tr

• (Diχ)2 + m2

χχ2

+ Vs(φ,χ)

m2

φ = 8v2c1 = −2h1, heavy

m2

χ = 2(b2 + v2c3) = g2

3(s1 − 4v2s3), light

LEQCD = 1

2F 2 ij +TrDiA0DiA0 + m2 EA2 0 +λEA4

χ is A0 →

2 equations m2

χ = m2

E,

˜ λχ4 = λEA4

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Domain wall

• Two more equations from matching domain wall Z (x = ±∞) = ± v

21

tension and width at 1-loop:

1 2 3 4 5 0.2 0.4 0.6 0.8 1

¯ z

F(¯ z) π2T 4/6 2 4 6 8

• 0.01

0.01 0.02

¯ z

∆F FYM

L SU(2) and Leff wall profiles

relative difference

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Finally...

• 5 couplings, 4 equations → 1 free parameter r

Leff = g−2

3

1

2TrF 2 ij +Tr(DiZ †DiZ )+ V(Z )

• V(Z ) = b1Σ2 + b2Π2

a + c1Σ4 + c2(Π2 a)2 + c3Σ2Π2 a

g2

3 = g2T

b1 = − 1

4r 2T 2,

b2 = − 1

4r 2T 2 + 0.441841g2T 2

c1 = 0.0311994r 2 + 0.0135415g2 c2 = 0.0311994r 2 + 0.008443432g2 c3 = 0.0623987r 2

• Couplings determined by (g,T) and r = mφ/T
• r fixes mass of coarse-grained Pol. loop magnitude

→ any r ∼ O (1) should give similar IR physics

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Phase diagram?

• To determine phase diagram, need [non-perturbative] lattice

simulations Slat = SW + SZ + V(ˆ

Σ, ˆ Π),

SW = β∑x,i<j

• 1− 1

2Tr[Uij]

• , Wilson action

SZ = 2

• 4

β

• ∑x,i Tr
• ˆ

Π2 − ˆ Π(x)Ui(x)ˆ Π(x +ˆ

i)U†

i (x)

• +
• 4

β

• ∑x,i
• ˆ

Σ2(x)− ˆ Σ(x)ˆ Σ(x +ˆ

i)

• , kinetic term

V =

• 4

β

3 ∑x

• ˆ

b1 ˆ

Σ2 + ˆ

b2 ˆ

Π2

a +ˆ

c1 ˆ

Σ4 +ˆ

c2

• ˆ

Π2

a

2 +ˆ

c3 ˆ

Σ2 ˆ Π2

a

• with β =

4 ag2

3 , a lattice spacing

• Explicit Z(2) symmetry ˆ

Π → −ˆ Π, ˆ Σ → −ˆ Σ

• Lattice couplings?
• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Matching lattice and MS continuum couplings

Perturbative 2-loop lattice calculation

• A. Kurkela, 0704.1416

Σ = g3 ˆ Σ+O (a), Π = g3ˆ Π+O (a),

ci = ˆ ci +O (a)

ˆ

b1 = b1/g4

3 − 2.38193365 4π

(2ˆ

c1 +ˆ c3)β

+

1 16π2

• (48ˆ

c2

1 + 12ˆ

c2

3 − 12ˆ

c3)[log1.5β+ 0.08849]− 6.9537ˆ c3

• +O (a),

ˆ

b2 = b2/g4

3 − 0.7939779 4π

(10ˆ

c2 +ˆ c3 + 2)β

+

1 16π2

• (80ˆ

c2

2 + 4ˆ

c2

3 − 40ˆ

c2)[log1.5β+ 0.08849]− 23.17895ˆ c2 − 8.66687

• +O (a)

Matching exact in g3, but O (a) error → continuum extrapolation

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Simulation results

0.1 0.2 0.3 0.4 0.5 <|Σ|> 0.1 0.15 0.2 0.25 0.3 1/g

2

0.002 0.004 0.006 <|Σ|

2>−<|Σ|> 2

0.1 0.2 0.3 0.4 0.5 <|Σ|> 0.1 0.15 0.2 0.25 0.3 1/g

2

0.001 0.002 0.003 <|Σ|

2>−<|Σ|> 2

r 2 = 5,643,β = 12 r 2 = 5,643,β = 6 Z2-restoring phase transition

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Universality class

• 100

100 (1/g

2-1/gc 2)N 1/ν

1 1.5 2 2.5 3 B4 N=64 N=96 N=128 1.604

• 100
• 50

50 100 (1/g

2-1/gc 2)N 1/ν

1 1.5 2 2.5 3 3.5 B4 N=64 N=96 N=128 1.604

r 2 = 5 r 2 = 10 3d Ising universality class: B4 ≡ Σ4

Σ22 = 1.604... at criticality

ν ≈ 0.63

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Thermodynamic + continuum extrapolations

50 100 150 N 0.2 0.22 0.24 0.26 0.28 1/g

2

r

2=10, β=12

r

2=5, β=12

0.05 0.1 0.15 1/β 0.3 0.35 0.4 0.45 1/g

2

r

2=10

r

2=5

V → ∞ a → 0 r small → large correlation length → large volume r large → large cutoff effects → fine lattice r = 0 : Σ decouples → λφ4 already done X.P . Sun, hep-lat/0209144

• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook Setup Matching Lattice Results

Phase diagram

2 4 6 8 10 r

2

0.05 0.1 0.15 0.2 0.25 0.3 1/g

2

g

2=5.1

Confined Deconfined

• Phase transition is robust ∀ r

r → ∞: no radial fluctuations, but still domain wall and transition

• g2

crit depends mildly on r for r > 1

• Fixing g2(Tc) = 5.1 gives r ∼ 2.6:
• dim. red. action completely defined
• Ph. de Forcrand

GGI, June 2008 DimRed

university-logo Intro DimRed Center symm. SU(2) Outlook

Prospects

• To do with SU(2):
• determine r(T) non-perturbatively
• check accuracy of effective theory at T near Tc
• 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 ) = Vh(Z )+ g2

3Vs(Z );

M ≡ Z − 1

31TrZ

Vh(Z ) = c1TrZ †Z + c2(detZ + detZ †)+ c3Tr(Z †Z )2 Vs(Z ) = d1TrM†M + d2Tr(M3 + M3†)+ d3Tr(M†M)2 6 couplings, 4 equations → 2 heavy modes to tune

• A. Kurkela, 0704.1416
• Ph. de Forcrand

GGI, June 2008 DimRed