su(n) gauge theory thermodynamics from lattice · 2009. 7. 31. · su(n) gauge theory at large n...
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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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Trace of energy momentum tensor θµµ = ∆ = ǫ−3pT 4
measure of breaking of conformal invariance
∆/T 4 = T ∂∂T
(p/T 4)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Trace of energy momentum tensor θµµ = ∆ = ǫ−3pT 4
measure of breaking of conformal invariance
∆/T 4 = T ∂∂T
(p/T 4)
Convenient to calculate this on lattice
T ∂∂T
→ −a ∂∂a
p/T 4 → N3t
N3s
lnZ
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Trace of energy momentum tensor θµµ = ∆ = ǫ−3pT 4
measure of breaking of conformal invariance
∆/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 〉
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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.
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
Thermodynamics for SU(4)
0
4
8
12
1 2 3 4T/Tc
p/T4
ε/T4
s/T3SU(4)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
Thermodynamics for SU(6)
0
10
20
30
1 2 3T/Tc
p/T4
ε/T4
s/T3
SU(6)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen 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)
SU(4)
Weak coupling result from Laine et al., Phys.Rev. D 73 (2006) 085009
Saumen 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)
SU(4)
SU(6)
Weak coupling result from Laine et al., Phys.Rev. D 73 (2006) 085009
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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.
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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)
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
Extra: Latent Heat for Nt=5
0.6
0.8
1
1.2
1Tc
{
Lh
N2
}14
r
r
rr
Saumen Datta (in collaboration with Sourendu Gupta) SU(N) Gauge Theory Thermodynamics from Lattice
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