t Hooft limit (1, 2, 3, ···∞) - Theoretical Physics (TIFR)

Outline T c
Latent heat EOS Summary
The gluoN c plasma and its ’t Hooft limit
(1, 2, 3, · · · ∞)
Saumen Datta and Sourendu Gupta
ILGTI: TIFR
Extreme QCD 2010
Physikzentrum Bad Honnef, Germany
June 21, 2010
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Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
RG scaling and ’t Hooft scaling
Latent heat
Equation of state
Summary
SG
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
Limiting procedures
◮ ’t Hooft scaling limit: take N c → ∞ and g 2 → 0 keeping
λ = g 2 N c fixed. Correlation functions can be computed
non-perturbatively but diagrammatically by resumming a
well-defined class of diagrams. Simple caricature of hadron
physics in this limit.
’t Hooft, Coleman; tested by Teper and collaborators
Extended to many other classes of theories. AdS/CFT
correspondence works in the ’t Hooft limit of N c → ∞.
◮ Strong scaling limit: work at (fixed) long distance scales and
take N c → ∞. Non-perturbative content of the theory may be
explored on the lattice and the limit taken.
Teper, Lucini and collaborators
Corrections organized in power of 1/N c
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Flag diagram of gluoN c plasmas
line of chiral transitions
0
m
N c
3
2+ε
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Flag diagram of gluoN c plasmas
line of chiral transitions
N c
3
0
~10 MeV
m
2+ε
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Flag diagram of gluoN c plasmas
line of chiral transitions
0
m
Τ
Deconfined
Baryonic
Α
Quarkyonic
µ
N c
3
2+ε
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Flag diagram of gluoN c plasmas
line of chiral transitions
Τ
Baryonic
Deconfined
Quarkyonic?
µ
N c
Β
3
0
m
2+ε
SG
Large N c Thermodynamics

Outline
Outline T c
Latent heat EOS Summary
RG scaling and ’t Hooft scaling
Latent heat
Equation of state
Summary
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Unphysical strong–weak coupling crossover
0.65
0.60
0.55
0.50
0.45
0.40
8 terms
7 terms
0.35
0.30
0.25
8 9 10 11 12
β
Wilson action. Earlier generation of simulations limited by this
bulk transition; solution now: move to smaller lattice spacing.
Teper and collaborators, Panero
SG
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
· · · no longer a problem
0.6
SU(4)
0.55
N s =18
N
0.5
s =24
10.4 10.6 10.8 11 11.2 11.4
Clear 1st order transition signal in L but not in plaquette: therefore
thermal transition, not bulk. Similarly for N c = 6, 8 and 10.
Computations for N t = 6, 8, 10 and (sometimes) 12.
SG
β
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
· · · no longer a problem
0.12
SU(4)
0.08
0.04
N s =18
N
0
s =24
10.4 10.6 10.8 11 11.2 11.4
Clear 1st order transition signal in L but not in plaquette: therefore
thermal transition, not bulk. Similarly for N c = 6, 8 and 10.
Computations for N t = 6, 8, 10 and (sometimes) 12.
SG
β
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
· · · no longer a problem
0.08
SU(4),β=10.79
0.04
N s =22
Im L
0
-0.04
-0.08
-0.08 -0.04 0 0.04 0.08
Re L
Clear 1st order transition signal in L but not in plaquette: therefore
thermal transition, not bulk. Similarly for N c = 6, 8 and 10.
Computations for N t = 6, 8, 10 and (sometimes) 12.
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Precision test of RG
Nt
10
Τ
c
8
Τ
c
6
Τ
c
β 10.788 11.078 11.339
β c determined to one part in 10 4 . Strong coupling determined
non-perturbatively. 2-loop RG works with precision of two parts in
10 3 for a ≤ 1/(8T c ). For larger a: non-perturbative RG.
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Precision test of RG
Nt
10
8
(4/5)
Τ c
Τ
c
Τ
c
6
Τ
c
(4/3) Τ
c
β 10.788 11.078 11.339
β c determined to one part in 10 4 . Strong coupling determined
non-perturbatively. 2-loop RG works with precision of two parts in
10 3 for a ≤ 1/(8T c ). For larger a: non-perturbative RG.
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Precision test of RG
Nt
10
(4/5)
Τ c
0.804(3)
Τ
c
8
Τ
c
6
Τ
c
(4/3) Τ
c
1.308 (2)
β 10.788 11.078 11.339
β c determined to one part in 10 4 . Strong coupling determined
non-perturbatively. 2-loop RG works with precision of two parts in
10 3 for a ≤ 1/(8T c ). For larger a: non-perturbative RG.
SG
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
Precision test of RG
2.5
SU(4)
T/T c (N t =6)
2
1.5
1
N t =5
N t =6
N t =8
N t =10
10.8 11 11.2 11.4 11.6 11.8
β c determined to one part in 10 4 . Strong coupling determined
non-perturbatively. 2-loop RG works with precision of two parts in
10 3 for a ≤ 1/(8T c ). For larger a: non-perturbative RG.
SG
β
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
Precision test of RG
4
3
SU(4)
T/T c (N t =6)
2
1
N t =5
N t =6
N t =8
N t =10
10.4 11 11.6 12.2
β c determined to one part in 10 4 . Strong coupling determined
non-perturbatively. 2-loop RG works with precision of two parts in
10 3 for a ≤ 1/(8T c ). For larger a: non-perturbative RG.
SG
β
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
Step-scaling functions
0.6
∆β=β(a)-β(2a)
0.5
0.4
V-scheme
E-scheme
Nonperturbative
Boyd et al
SU(3)
0.3
6 6.5 7
β(a)
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Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Step-scaling functions
1
V-scheme
E-scheme
∆β=β(a)-β(2a)
0.6
Nonperturbative
0.2
SU(4)
10.6 11 11.4 11.8 12.2
β(a)
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
’t Hooft scaling: fixed λ, fixed physics
10.5
10
λ c (2πT c )
9.5
9
8.5
8
0 0.04 0.08 0.12
2
1/N c
λ c = g 2 (T c )N c . Clearly non-linear (even ignoring N t = 8 and 10).
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Very large corrections
Best-fit function:
λ c =
{
9.8771(4) −
14.2562(2)
N 2 c
9.9904(6) + 1.2081(3)
N 2 c
+ 54.7830(2)
N 4 c
− 23.5709(3)
N 4 c
(non-perturbative),
(2-loop).
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Very large corrections
Best-fit function:
λ c =
{
9.8771(4) −
14.2562(2)
N 2 c
9.9904(6) + 1.2081(3)
N 2 c
+ 54.7830(2)
N 4 c
− 23.5709(3)
N 4 c
(non-perturbative),
(2-loop).
For N c = 3
λ c =
{
9.8771(4) − 1.584 + 0.676 (non-perturbative),
9.9904(6) + 0.134 − 0.291 (2-loop).
1/N 2 c and 1/N 4 c corrections nearly equal to each other.
SG
Large N c Thermodynamics

Outline
Outline T c
Latent heat EOS Summary
RG scaling and ’t Hooft scaling
Latent heat
Equation of state
Summary
SG
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
Multiple peaks?
50
SU(4), N t =6, β=10.79
40
Prob(|L|)
30
20
10
N s =16
0
N s =24
0 0.02 0.04 0.06 0.08
|L|
Clear multiple peaks for L but not for plaquette. Entropy surface
not flat? Not a true first order transition?
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Multiple peaks?
800
700
600
SU(4), N t =6
β=10.79
N s =24
Prob(δP)
500
400
300
200
100
N s =16
0
-0.002 -0.001 0 0.001 0.002 0.003
δP
Clear multiple peaks for L but not for plaquette. Entropy surface
not flat? Not a true first order transition?
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Isolate phases using order parameter
800
β=10.78
800
β=10.78
600
Low
High
600
Low
High
Prob(δP)
Prob(δP)
Prob(δP)
400
200
8000
600
400
200
8000
600
400
200
Low
Low
β=10.79
High
β=10.80
High
Prob(δP)
Prob(δP)
Prob(δP)
400
200
8000
600
400
200
8000
600
400
200
Low
Low
β=10.79
High
β=10.80
High
Stable against change in V and a
0
-0.002 -0.001 0 0.001 0.002 0.003
0
-0.002 -0.001 0 0.001 0.002 0.003
δP
δP
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Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Is N c = 3 special?
N t N s ∆E/Tc 4 ∆E/∆ max
4 16 2.06(1)(3)
24 1.93(1)(3)
32 1.90(2)(2)
6 16 1.79(2)(4) 0.65(2)
32 1.54(2)(5)
48 1.44(4)(3)
N c ≥ 4 results stable for N s /N t ≃ 3. But large finite size effect for
N c = 3.
SG
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
Scaling with N c
Latent heat depends on N c even after scaling by number of gluons:
T A = N 2 c − 1. Best fit result—
∆ǫ
T A T 4 c
= 0.388(3) − 1.61(4)
Nc
2 .
N c = 3 may have larger finite volume effects than larger N c .
For N c = 2
∆ǫ
T A Tc
4 = 0.388(3) − 0.40(1) = 0!
Good news? Bad news?
SG
Large N c Thermodynamics

Outline
Outline T c
Latent heat EOS Summary
RG scaling and ’t Hooft scaling
Latent heat
Equation of state
Summary
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Conformal symmetry breaking
0.4
SU(3)
SU(4)
SU(6)
∆/(T A T 4 )
0.2
0
1 2 3 4
T/T c
Good scaling of ∆/T 4 = (E − 3P)/T 4 with N c at fixed T/T c
(strong N c scaling).
∆ 1/4 ≃ T even at T ≃ 2T c for N c = 3: conformal symmetry
broken. SG Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Evidence for mass?
0.4
∆/(T A T 2 T c
2 )
0.2
0
SU(3)
SU(4)
SU(6)
1 2 3 4
T/T c
∆/T 2 ≃ constant is generic evidence for mass scales persisting in
the high temperature phase.
Meisinger, Miller, Ogilvie, 2002; Pisarski, 2007
SG
Large N c Thermodynamics

Outline T c
Latent heat EOS Summary
Cutoff dependence of pressure
1
0.8
SU(4)
p/p SB
0.6
0.4
0.2
0
N t =5(Panero)
N t =6
N t =8
1 2 3 4
T/T c
SG
Large N c Thermodynamics

Outline T c
Main results
Latent heat EOS Summary
4
SU(4)
Ts/p SB
3
ε/p SB
2
1
p/p SB
0
1 2 3 4
T/T c
Far from ideal gas.
Good scaling of EOS with N c at fixed T/T c (strong N c scaling).
SG
Large N c Thermodynamics

Outline T c
Main results
Latent heat EOS Summary
1
0.8
0.2
0
ε/ε SB
p/p SB
SU(3)
SU(4)
SU(6)
1 2 3 4
T/T c
0.6
0.4
Far from ideal gas.
Good scaling of EOS with N c at fixed T/T c (strong N c scaling).
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
’t Hooft scaling
1
0.8
s/s SB
0.6
0.4
0.2
0
SU(3)
SU(4)
SU(6)
6 7 8 9 10
λ(2πT)
’t Hooft scaling fails for T ≤ 2T c .
N = 4 SYM does not describe pure gauge theory.
SG
Large N c Thermodynamics

Outline T c Latent heat EOS Summary
Conformal theory?
1
0.8
SU(3)
SU(4)
SU(6)
p/p SB
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
ε/ε SB
SG
Large N c Thermodynamics

Outline
Outline T c
Latent heat EOS Summary
RG scaling and ’t Hooft scaling
Latent heat
Equation of state
Summary
SG
Large N c Thermodynamics

Outline T c
Main results
Latent heat EOS Summary
◮ Location of deconfining first order transition measured with
high accuracy. Yields high precision test of RG scaling and ’t
Hooft scaling. ’t Hooft coupling at T c has large 1/N c
corrections near N c = 3. Signs of breakdown of the ’t Hooft
procedure.
◮ New method developed for determination of latent heat in
gluoN c plasmas. Find
∆ǫ
T A T 4 c
= 0.388(3) − 1.61(4)
Nc
2 .
◮ Strong N c scaling works very well for EOS. Conformal
symmetry strongly broken up to T ≃ 3T c . Resummed weak
coupling theory works much better in description of lattice
data.
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Large N c Thermodynamics