Download - SU(N) Gauge Theory Thermodynamics from Lattice · 2009. 7. 31. · SU(N) Gauge Theory at Large N SU(N) gauge theory conﬁnes for all N. Nonperturbative physics for q

SU(N) Gauge Theory Thermodynamics from

Lattice

Saumen Datta(in collaboration with Sourendu Gupta)

July 21, 2009

Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice

Introduction

The theory of Strong interactions: SU(3) gauge theoryQuark fields ψi : triplet of SU(3), i = 1, 2, 3 color

Gluon fields: traceless Hermitian matrices Aµij

αS (Q2) = g2(Q2)4π = 4π

(11−2Nf3

) lnQ2/Λ2in leading order

The interactions become strong at hadron physics scales:no natural small expansion parameter

G. ’t Hooft 1974 Consider SU(N), use 1/N as expansion parameterLeads to some simplifications and qualitative understandings ofsome properties of hadrons


kl

k

k

i

i

i

j

j j l

l

i j

jj

i i

j

i

i

i

ij

kj

j

k

g NO( )2

O(g4N )2

O(g2)

g ∼ 1/√

N for nontriviallarge N limitλtH = g2N

Quark loops ignored inleading orderunless Nf ∼ Nc

λtH(Q2) ∼ 48π2

11 lnQ2/Λ2


SU(N) Gauge Theory at Large N

◮ SU(N) gauge theory confines for all N.Nonperturbative physics for q < ΛSU(N)

◮ Glueballs, mass ∼ O(N0c ), decay widths ∼ O(1/Nc )

Mesons, mass ∼ O(N0c ), scattering ∼ O(1/Nc )

Lattice study: more expensive. String tension, Glueballspectrum, etc. studied.

Lucini & Teper; Narayanan & Neuberger; etc.

◮ At large temperatures, O(N2c ) gluon degrees of freedom

◮ 1st order phase transition at Tc ∼ ΛSU(N) with latent heatO(N2

c )

SU(3): 1st order phase transition at Tc ∼ 270 MeV


Equilibrium Thermodynamics on Lattice

Z (T ) =∫

dU exp(−β∫ 1/T

0,pbcdτ

∫

d3xL(U))

1/T

L(x) As lattice spacing a → 0,βL → TrF 2

µν withβ = 2N/g2 = 2N2/λtH

◮ Aµ(~x , 0) = Aµ(~x , 1/T )ZN group of aperiodic gauge transformationUµ(~x , 1/T ) = e2πi/NUµ(~x , 0)

◮ Order parameter for deconfinement transition:

L =1

NTr

Nτ∏

x0=1

U(x0,x),0̂→ 1/N Tr P e

R

1/T

0dτ A0(x̃,τ)

ZN : L =1

V

∑

x

L(~x) → e2πi/NL


ZN Symmetry and Deconfinement

◮ | < L > | ∼ e−F , F free energy of a static quark source

◮ < L > 6= 0 → ZN broken and deconfinement

◮ SU(2): 2nd order transition in Z2 universality class

Engels et al. 1982-90

◮ For N > 2: ZN : 1st order transition expected

◮ The QCD transition Nf = 2 + 1 is a crossover.Expected to be first order for all finite mass at SU(∞)

◮ Chiral limit may be more involved.

◮ Similarly, complicated phase structures have been predictedfor µB ∼ O(Nc).

McLerran & Pisarski ’07


Study of SU(N) theory at finite T

Interesting predictions at finite temperature, which may help ourunderstanding of SU(3)

◮ Symmetry of Polyakov loop → strong 1st order transition?

◮ How does Tc/ΛMS scale?

◮ For N > 3 the latent heat ∼ N2

◮ Does one reach the asymptotic state soon after the transition?

◮ Deviation from conformality in the plasma phase?

Deconfinement transition for SU(4)

Gavai; Ohta & Wingate; 2001

Deconfinement transition for SU(N), and pressure of the hightemperature phase from coarse lattices.

Lucini & Teper; Lucini, Teper & Wenger; Bringholtz & Teper 2003–

Need to be careful about discretization errorsSaumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice

Our study

Numerical investigation of SU(4) and SU(6) gauge theoriesStudy large N theory using results for N=3,4,6

S. Datta & S. Gupta, arXiv:0906.3929 and in preparation

◮ Study of the cutoff dependenceIs the lattice spacing small enough for 2-loop running?

◮ Study of the phase transition in SU(4) and SU(6)Finite volume study

◮ Direct estimation of Tc/ΛMS

◮ Equation of state for N=4 and 6 theoriesAnalyse SU(3) with existing data.

Engels et al., Nucl. Phys. B 469 (1996) 419

◮ ǫ− 3p and conformal symmetry breaking in the deconfinedphase


Scaling

Tc(g2,Nt) = 1

a(g2)Nt

Use Tc(Nt) to predict Tc(N′t)

S. Gupta, PRD 64(2001) 034507

we use for coupling VQ,Q̄(q2) = −N2−12N

g2

q2 (V-scheme)

1

2

3

4

24.5 25 25.5 26

T/T

c(N

t=6)

β

SU(6), 2-loop

Tc(Nt=4)Tc(Nt=6)Tc(Nt=8)


Scaling

Tc(g2,Nt) = 1

a(g2)Nt

Use Tc(Nt) to predict Tc(N′t)

S. Gupta, PRD 64(2001) 034507

we use for coupling VQ,Q̄(q2) = −N2−12N

g2

q2 (V-scheme)

0.6

1

1.4

1.8

10.4 10.6 10.8 11 11.2 11.4

T/T

c(N

t=6)

β

SU(4), 2-loop

Tc(Nt=4)

Tc(Nt=6)Tc(Nt=8)


Bulk and Deconfinement Transitions for SU(6)

Pµ,ν(x) =1

NTrUµ(x)Uν(x + m̂u)U†

µ(x + ν)U†ν(x)

P =1

6VT

∑

µ,ν

∑

x

Pµ,ν(x)

0.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

24 24.2 24.4 24.6 24.8 25

<P

>

β

SU(6), 163xNt Lattice

Nt=4Nt=6

Nt=16 0

0.1

0.2

24 24.2 24.4 24.6 24.8 25

<|L

|>

β

SU(6), 163xNt Lattice

Nt=4Nt=6


Bulk and Deconfinement Transition for SU(4)

0.5

0.52

0.54

0.56

0.58

10.4 10.6 10.8 11

<P

>

β

SU(4)

164

163x4183x6

0

0.04

0.08

0.12

10.4 10.6 10.8 11 11.2 11.4<

|L|>

β

SU(4), L3x6 lattice

L=18L=24

〈Pt − Ps〉 associated with deconfinement:ǫ

T 4 = 6N2N4τ

Pt−Ps

λtH+ corrections


Bulk and Deconfinement Transition for SU(4)

0.5

0.52

0.54

0.56

0.58

10.4 10.6 10.8 11

<P

>

β

SU(4)

164

163x4183x6 0

0.0001

0.0002

0.0003

10.4 10.6 10.8 11 11.2 11.4<

Pt-

-Ps>

β

SU(4), L3x6 lattice

L=18L=24

〈Pt − Ps〉 associated with deconfinement:ǫ

T 4 = 6N2N4τ

Pt−Ps

λtH+ corrections


Degenerate Vacuua at Tc

-0.08

-0.04

0

0.04

0.08

-0.08 -0.04 0 0.04 0.08

Im L

Re L

SU(4), Nt=6, β=10.79, Hot run, every 400th

Ns=16Ns=18Ns=20Ns=22


Finite Size Analysis

Susceptibility χL = V (〈|L|2〉 − 〈|L|〉2) ∼ V for 1st order.

0

0.0002

0.0004

0.0006

10.77 10.78 10.79 10.8

χL/V

β

L=16L=18L=20L=22L=24

0.00034

0.00036

0.00038

0.0004

6e-05 0.00012 0.00018 0.00024

χL/V

1/V

SU(4), Nt=6


Deconfinement Transition for SU(6)

0

0.03

0.06

0.09

24.4 24.6 24.8 25 25.2 25.4 25.6

<|L

|>

β

SU(6), L3x6 Lattice

L=16L=20L=24 0

0.0001

0.0002

24.4 24.8 25.2 25.6

<P

t--P

s>

β

SU(6), L3x6 Lattice

L=16L=20L=24


N-dependence of βc

β = 2N2/λtH

0.66

0.68

0.7

0.72

3 4 5 6 7 8 9 10

β c/N

2

N

SU(N), Nt=6

βc

N2 = 0.7008(6) − 0.413(6)N2

0.66

0.68

0.7

0.72

0.74

0.76

3 4 5 6 7 8 9 10

β c/N

2

N

SU(N), Nt=8

βc

N2 = 0.7186(5) − 0.406(5)N2


Tc/ΛMS

◮ Use 2-loop running to directly calculate Tc/ΛMS

◮ Definitions of g2 through different schemesg2E = 8N

N2−1(1 − P) (E-scheme)

besides g2V and g2

MS

◮ For each N, continuum limit of g2scheme

◮ Scheme-dependence observed to be stronger than cutoffdependence

◮ N-dependence using c0 + c1N2

Tc

ΛMS

|N→∞ = 1.17(5)


Equation of State from Lattice

Define thermodynamic quantitiesF (T ,V ) = T lnZ (T ,V ) Z: grand canonical partition function

ǫ = − 1V

∂F (T ,V )/T

∂(1/T )

p = ∂F∂V

= F/V for homogeneous




ǫ = − 1V

∂F (T ,V )/T

∂(1/T )

p = ∂F∂V


Trace of energy momentum tensor θµµ = ∆ = ǫ−3pT 4

measure of breaking of conformal invariance

∆/T 4 = T ∂∂T

(p/T 4)




ǫ = − 1V

∂F (T ,V )/T

∂(1/T )

p = ∂F∂V




∆/T 4 = T ∂∂T

(p/T 4)

Convenient to calculate this on lattice

T ∂∂T

→ −a ∂∂a

p/T 4 → N3t

N3s

lnZ




ǫ = − 1V

∂F (T ,V )/T

∂(1/T )

p = ∂F∂V




∆/T 4 = T ∂∂T

(p/T 4)

Convenient to calculate this on lattice

T ∂∂T

→ −a ∂∂a

p/T 4 → N3t

N3s

lnZ ∆/T 4 = ∂∂aβ N3

t

N3s〈 dSdβ 〉


Eqn. of State (Contd.)

p(T )T 4 − p(T0)

T 40

=∫ ββ0

dβ (P(β) − P(β0))

ǫT 4 = 3 p

T 4 + ∆T 4

sT 3 = ǫ

T 4 + pT 4

For free gas, Stefan-Boltzmann limitǫ

T 4 = 3 pT 4 = (N2 − 1)π2

15R(Nτ )Here R(Nτ ) discretization error= 1 + 10

21( πNτ

)2 + ...

Boyd et al., Nucl.Phys. B 469(’96) 419;Engels et al., Nucl.Phys. B 205(’82) 545.


Cutoff and Volume Dependence of EOS for SU(4)

0

4

8

12

1 2 3 4T/Tc

ε/T4

p/T4

SU(4), Nt=6

Ns=24Ns=18

0

4

8

12

1 2 3 4T/Tc

p/T4

ε/T4

SU(4)

Nt=6Nt=8


Thermodynamics for SU(4)

0

4

8

12

1 2 3 4T/Tc

p/T4

ε/T4

s/T3SU(4)


Thermodynamics for SU(6)

0

10

20

30

1 2 3T/Tc

p/T4

ε/T4

s/T3

SU(6)


Scaling of Thermodynamic Quantities with N

0

0.25

0.5

0.75

1

1 2 3 4T/Tc

SU(3)

SU(4)

SU(6)

p/pSB

ε/εSB


Scaling of Thermodynamic Quantities with N

0

0.25

0.5

0.75

1

1 2 3 4T/Tc

SU(3)

SU(4)

SU(6)

p/pSB

s/sSB


Interaction Measure ǫ − 3p

SU(N) gauge theory classically scale invariant → Tµµ = 0Quantum theory may break this symmetryǫ− 3p measure of breaking of scale invariance

0

0.2

0.4

0.6

1 2 3 4

(ε-3

p)/T

4 /(N

2 -1)

T/Tc

SU(3)SU(4)SU(6)

Substantial conformal symmetry breaking in plasmaPeak moves from ∼ 1.09Tc at N=3 to ∼ 1.025Tc at N=6∼ 1/T 2 fall-off at intermediate temperature

R. Pisarski, hep-ph/0612191Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice

Conformal vs. Weak Coupling Limit

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p/p S

B

ε/εSB

SU(3)

Weak coupling result from Laine et al., Phys.Rev. D 73 (2006) 085009



0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p/p S

B

ε/εSB

SU(3)

SU(4)




0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p/p S

B

ε/εSB

SU(3)

SU(4)

SU(6)



Summary and Conclusions

◮ SU(N) gauge theories studied at finite temperature forN=3,4,6

◮ Important to check for discretization errors.Under control for Nτ ∼

> 6.

◮ First order deconfinement transition confirmed for N=4,6

◮ Tc/ΛMS = 1.17(5) with small N dependence

◮ Nice N2 scaling for thermodynamic quantities for T/Tc ∼> 1.2.

◮ Large deviation from Stefan-Boltzmann limit

◮ Considerable conformal symmetry breaking till deep in theplasmaNo evidence of a window for strongly coupled conformal theory


Extra: (ǫ − 3p)/T 4 scaling

0

0.2

0.4

0.6

1 2 3 4

(ε-3

p)/T

2 Tc2 /(

N2 -1

)

T/Tc

SU(3)SU(4)SU(6)


Extra: Tc/ΛMS

1.1

1.2

1.3

0 0.01 0.02 0.03

Tc/

ΛM

S

1/Nt2

SU(4)

E-schemeV-scheme

MS-scheme


Extra: Tc/ΛMS

1.1

1.2

1.3

0 0.01 0.02 0.03

Tc/

ΛM

S

1/Nt2

SU(6)E-schemeV-scheme

MS-scheme


Extra: Tc/ΛMS

1.1

1.2

1.3

0 0.01 0.02 0.03

Tc/

ΛM

S

1/Nt2

SU(6)E-schemeV-scheme

MS-scheme

1.05

1.1

1.15

1.2

1.25

1.3

0 0.02 0.04 0.06 0.08 0.1 0.12

Tc/

ΛM

S

1/N2

SU(N)

E-schemeV-scheme

MS-scheme


Extra: βc for Nτ = 8

0

0.01

0.02

0.03

11.02 11.05 11.08 11.11

<|L

|>

β

SU(4), L3x8 lattices

L=22L=24L=28L=30

0

2e-05

4e-05

6e-05

8e-05

0.0001

11 11.04 11.08 11.12 11.16

χL/V

β

SU(4),L3x8 lattices, Cold start

L=22L=24L=28L=30


Extra: Integration Method in Pressure

-0.5

0

0.5

1

1.5

2

2.5

10.6 10.8 11 11.2 11.4

P/T

4

β

SU(4), 183x6 Lattice

Quadratic fn.Linear fn.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

10.9 11 11.1 11.2 11.3 11.4 11.5

P/T

4

β

SU(4), 243x8 Lattice

Quadratic fn.Linear fn.


Extra: Volume Dependence of ǫ − 3p

0

5

10

15

20

0.9 1 1.1 1.2 1.3 1.4

(ε-3

p)/

T4

T/Tc

SU(6),Nt=6 Lattice

L=24L=16


Extra: Volume Dependence of ǫ − 3p

0

0.1

0.2

0.3

0.4

0.5

0.6

1 1.2 1.4 1.6 1.8 2

(ε-3

p)/

T4/N

2

T/Tc

SU(4)SU(6)


Extra: Latent Heat for Nt=5

0.6

0.8

1

1.2

1Tc

{

Lh

N2

}14

r

r

rr


Download - SU(N) Gauge Theory Thermodynamics from Lattice · 2009. 7. 31. · SU(N) Gauge Theory at Large N SU(N) gauge theory conﬁnes for all N. Nonperturbative physics for q

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