-
How narrow is the sQGP transition?A simple non-perturbative
approach to hot gluodynamics
compared to lattice data
Chris Korthals Altes
Centre Physique Théorique au CNRSLuminy, F-13288, Marseille
2NIKHEF, theory groupAmsterdam
April 2011
With A. Dumitru, Y. Guo, Y. Hidaka, R. Pisarski,
arXiv:1011.3820, Phys.Rev D
How narrow is the sQGP transition?
-
We all want to understand the groundstate in
RHICexperiments...In this modest building housing the RHIC
VACUUMFACILITY the decision is made on a day to day basiswhether to
present the groundstate as AdS/CFT, monopolecondensate, ...whatever
the theorist likes.
How narrow is the sQGP transition?
-
We all want to understand the groundstate in
RHICexperiments...In this modest building housing the RHIC
VACUUMFACILITY the decision is made on a day to day basiswhether to
present the groundstate as AdS/CFT, monopolecondensate, ...whatever
the theorist likes.
How narrow is the sQGP transition?
-
We all want to understand the groundstate in
RHICexperiments...In this modest building housing the RHIC
VACUUMFACILITY the decision is made on a day to day basiswhether to
present the groundstate as AdS/CFT, monopolecondensate, ...whatever
the theorist likes.
How narrow is the sQGP transition?
-
T
µ
early universe
ALICE
> 0
SPS
quark-gluon plasma
hadronic fluid
nuclear mattervacuum
RHICTc ~ 170 MeV
µ ∼ o
> 0
n = 0
∼ 0
n > 0
922 MeV
phases ?
quark matter
neutron star cores
crossover
CFLB B
superfluid/superconducting
2SC
crossover
Figure: Proposed phase diagram for QCD. 2SC and CFL refer to
thediquark condensates .
How narrow is the sQGP transition?
-
Facts and fancy, connecting the facts
Facts from the lattice: EOS and flux loops
Fancy: determining an Ansatz for the effective potentialfrom
EOS
Predictions from effective potential.
Discussion
How narrow is the sQGP transition?
-
Facts and fancy, connecting the facts
Facts from the lattice: EOS and flux loops
Fancy: determining an Ansatz for the effective potentialfrom
EOS
Predictions from effective potential.
Discussion
How narrow is the sQGP transition?
-
Facts and fancy, connecting the facts
Facts from the lattice: EOS and flux loops
Fancy: determining an Ansatz for the effective potentialfrom
EOS
Predictions from effective potential.
Discussion
How narrow is the sQGP transition?
-
Facts and fancy, connecting the facts
Facts from the lattice: EOS and flux loops
Fancy: determining an Ansatz for the effective potentialfrom
EOS
Predictions from effective potential.
Discussion
How narrow is the sQGP transition?
-
Pressure, energy density and interaction measure
0 200 400 600 800 10000
1
2
3
4
5
6
(ε−3p)/T4
3p/T4
ε/T4
S.B. limit
T [MeV]
β=6.1, aσ/aτ=4
L=20aσ ~1.9fm
a
Fixed scale data by T.Umeda et al.., arXiv0809.2842.
energy density much steeper than pressure, so is theinteraction
measure, with peak at ∼ 1.2Tc.interaction measure falls off like
1/T 2 beyond T = 1.2Tc ,not like (1/ log T )2
How narrow is the sQGP transition?
-
Pressure, energy density and interaction measure
0 200 400 600 800 10000
1
2
3
4
5
6
(ε−3p)/T4
3p/T4
ε/T4
S.B. limit
T [MeV]
β=6.1, aσ/aτ=4
L=20aσ ~1.9fm
a
Fixed scale data by T.Umeda et al.., arXiv0809.2842.
energy density much steeper than pressure, so is theinteraction
measure, with peak at ∼ 1.2Tc.interaction measure falls off like
1/T 2 beyond T = 1.2Tc ,not like (1/ log T )2
How narrow is the sQGP transition?
-
Pressure, energy density and interaction measure
0 200 400 600 800 10000
1
2
3
4
5
6
(ε−3p)/T4
3p/T4
ε/T4
S.B. limit
T [MeV]
β=6.1, aσ/aτ=4
L=20aσ ~1.9fm
a
Fixed scale data by T.Umeda et al.., arXiv0809.2842.
energy density much steeper than pressure, so is theinteraction
measure, with peak at ∼ 1.2Tc.interaction measure falls off like
1/T 2 beyond T = 1.2Tc ,not like (1/ log T )2
How narrow is the sQGP transition?
-
1 1.5 2 2.5 3 3.5 4T/Tc
0
0.1
0.2
0.3
0.4
Del
ta/(
N^2
-1)
SU(3)SU(4)SU(6)
Figure: Interaction measure scaled by N2 − 1, Panero 2009. Note
thesmall reduced discontinuity at Tc
How narrow is the sQGP transition?
-
Pressure of the SU(3) plasma and perturbation theory
1 10 100 1000
T/ΛMS_
0.0
0.5
1.0
1.5
p/p 0
g2
g3
g4
g5
g6(ln(1/g)+0.7)
4d lattice
How narrow is the sQGP transition?
-
Pressure of the plasma
1 10 100 1000
T/ΛMS_
0.0
0.5
1.0
1.5
p/p 0
g2
g3
g4
g5
g6(ln(1/g)+0.7)
4d lattice
Comparison of perturbative results.The O(g3) has thewrong sign.
Electric quasiparticles not good enough!The pressure gets
contribution from magnetic sectorstarting from g6. What is in this
magnetic sector??
How narrow is the sQGP transition?
-
Pressure of the plasma
1 10 100 1000
T/ΛMS_
0.0
0.5
1.0
1.5
p/p 0
g2
g3
g4
g5
g6(ln(1/g)+0.7)
4d lattice
Comparison of perturbative results.The O(g3) has thewrong sign.
Electric quasiparticles not good enough!The pressure gets
contribution from magnetic sectorstarting from g6. What is in this
magnetic sector??
How narrow is the sQGP transition?
-
Pressure of the plasma
1 10 100 1000
T/ΛMS_
0.0
0.5
1.0
1.5
p/p 0
g2
g3
g4
g5
g6(ln(1/g)+0.7)
4d lattice
Comparison of perturbative results.The O(g3) has thewrong sign.
Electric quasiparticles not good enough!The pressure gets
contribution from magnetic sectorstarting from g6. What is in this
magnetic sector??
How narrow is the sQGP transition?
-
EQCD prediction for pressure relates ultrahigh T points
(arXiv:0710.4197)data hep-lat/9602007. For HTL improvement see
arXiv:1005.1603.
How narrow is the sQGP transition?
-
The 1/T 2 law for the interaction measure
RDP at Kyoto 2006 (also Meisinger et al.hep-phys/ 0108009);data
hep-lat/9602007; arxiv.org/abs/0810.1570, /0809.2842 .
How narrow is the sQGP transition?
-
What is the plasma consisting of, as seen by fluxloops?
electric colour flux as seen by a spatial ’t Hooft
loop("e-loop")
magnetic colour flux as seen by a spatial Wilson
loop(’m-loop")
How narrow is the sQGP transition?
-
What is the plasma consisting of, as seen by fluxloops?
electric colour flux as seen by a spatial ’t Hooft
loop("e-loop")
magnetic colour flux as seen by a spatial Wilson
loop(’m-loop")
How narrow is the sQGP transition?
-
What is the plasma consisting of, as seen by fluxloops?
electric colour flux as seen by a spatial ’t Hooft
loop("e-loop")
magnetic colour flux as seen by a spatial Wilson
loop(’m-loop")
How narrow is the sQGP transition?
-
Flux of Debye screened glue
1/2
(a)(b)
(c)
z
x
y
x
z
l_D
l_Dl_D
Phi _1
Spatial ’t Hooft loop Vk = exp(i 4πg∫
Tr ~EYk .d~S) in x-yplane.Yk generalized hypercharge exp(i2πYk)
= exp(ik2π/N)1.There are k(N-k) gluons with Yk charge 1, etc...Thin
slab of thickness lD defining the effective flux volumeHow narrow
is the sQGP transition?
-
Flux of Debye screened glue
1/2
(a)(b)
(c)
z
x
y
x
z
l_D
l_Dl_D
Phi _1
Spatial ’t Hooft loop Vk = exp(i 4πg∫
Tr ~EYk .d~S) in x-yplane.Yk generalized hypercharge exp(i2πYk)
= exp(ik2π/N)1.There are k(N-k) gluons with Yk charge 1, etc...Thin
slab of thickness lD defining the effective flux volumeHow narrow
is the sQGP transition?
-
Flux of Debye screened glue
1/2
(a)(b)
(c)
z
x
y
x
z
l_D
l_Dl_D
Phi _1
Spatial ’t Hooft loop Vk = exp(i 4πg∫
Tr ~EYk .d~S) in x-yplane.Yk generalized hypercharge exp(i2πYk)
= exp(ik2π/N)1.There are k(N-k) gluons with Yk charge 1, etc...Thin
slab of thickness lD defining the effective flux volumeHow narrow
is the sQGP transition?
-
At T >> Tc a gas of Debye screened gluons:lD ∼ 1gT
>> 1TAny gluon species with charge ±1: contributesexp(i2π/2)
= −1.In the slab are on average l̄ = n(T )lD.Area gluons of
hatspecies. Poisson distribution for average due to a
chargedspecies:
< Vk >one cs=∑
ll̄ ll!(−1)l exp(−l̄) = exp(−2̄l)
All 2k(N − k) charged gluon species (supposedindependent):
< Vk >= exp(−4k(N − k)lDn(T ).Area)
Casimir scaling: ρk (T ) ∼ k(N − k)lDn(T )
How narrow is the sQGP transition?
-
At T >> Tc a gas of Debye screened gluons:lD ∼ 1gT
>> 1TAny gluon species with charge ±1: contributesexp(i2π/2)
= −1.In the slab are on average l̄ = n(T )lD.Area gluons of
hatspecies. Poisson distribution for average due to a
chargedspecies:
< Vk >one cs=∑
ll̄ ll!(−1)l exp(−l̄) = exp(−2̄l)
All 2k(N − k) charged gluon species (supposedindependent):
< Vk >= exp(−4k(N − k)lDn(T ).Area)
Casimir scaling: ρk (T ) ∼ k(N − k)lDn(T )
How narrow is the sQGP transition?
-
At T >> Tc a gas of Debye screened gluons:lD ∼ 1gT
>> 1TAny gluon species with charge ±1: contributesexp(i2π/2)
= −1.In the slab are on average l̄ = n(T )lD.Area gluons of
hatspecies. Poisson distribution for average due to a
chargedspecies:
< Vk >one cs=∑
ll̄ ll!(−1)l exp(−l̄) = exp(−2̄l)
All 2k(N − k) charged gluon species (supposedindependent):
< Vk >= exp(−4k(N − k)lDn(T ).Area)
Casimir scaling: ρk (T ) ∼ k(N − k)lDn(T )
How narrow is the sQGP transition?
-
At T >> Tc a gas of Debye screened gluons:lD ∼ 1gT
>> 1TAny gluon species with charge ±1: contributesexp(i2π/2)
= −1.In the slab are on average l̄ = n(T )lD.Area gluons of
hatspecies. Poisson distribution for average due to a
chargedspecies:
< Vk >one cs=∑
ll̄ ll!(−1)l exp(−l̄) = exp(−2̄l)
All 2k(N − k) charged gluon species (supposedindependent):
< Vk >= exp(−4k(N − k)lDn(T ).Area)
Casimir scaling: ρk (T ) ∼ k(N − k)lDn(T )
How narrow is the sQGP transition?
-
At T >> Tc a gas of Debye screened gluons:lD ∼ 1gT
>> 1TAny gluon species with charge ±1: contributesexp(i2π/2)
= −1.In the slab are on average l̄ = n(T )lD.Area gluons of
hatspecies. Poisson distribution for average due to a
chargedspecies:
< Vk >one cs=∑
ll̄ ll!(−1)l exp(−l̄) = exp(−2̄l)
All 2k(N − k) charged gluon species (supposedindependent):
< Vk >= exp(−4k(N − k)lDn(T ).Area)
Casimir scaling: ρk (T ) ∼ k(N − k)lDn(T )
How narrow is the sQGP transition?
-
At T >> Tc a gas of Debye screened gluons:lD ∼ 1gT
>> 1TAny gluon species with charge ±1: contributesexp(i2π/2)
= −1.In the slab are on average l̄ = n(T )lD.Area gluons of
hatspecies. Poisson distribution for average due to a
chargedspecies:
< Vk >one cs=∑
ll̄ ll!(−1)l exp(−l̄) = exp(−2̄l)
All 2k(N − k) charged gluon species (supposedindependent):
< Vk >= exp(−4k(N − k)lDn(T ).Area)
Casimir scaling: ρk (T ) ∼ k(N − k)lDn(T )
How narrow is the sQGP transition?
-
At T >> Tc a gas of Debye screened gluons:lD ∼ 1gT
>> 1TAny gluon species with charge ±1: contributesexp(i2π/2)
= −1.In the slab are on average l̄ = n(T )lD.Area gluons of
hatspecies. Poisson distribution for average due to a
chargedspecies:
< Vk >one cs=∑
ll̄ ll!(−1)l exp(−l̄) = exp(−2̄l)
All 2k(N − k) charged gluon species (supposedindependent):
< Vk >= exp(−4k(N − k)lDn(T ).Area)
Casimir scaling: ρk (T ) ∼ k(N − k)lDn(T )
How narrow is the sQGP transition?
-
Reduced electric flux tension in deconfined phase
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
1 1.5 2 2.5 3 3.5 4 4.5
σ k/T
2 / (
k (N
-k))
T/Tc
SU(3)SU(4), k=1SU(4), k=2SU(6), k=1SU(6), k=2SU(6), k=3SU(8),
k=1SU(8), k=2SU(8), k=3SU(8), k=4
GKA T/ΛMSbar=1.35
Nc ≤ 8, de Forcrand et al.,hep-lat/0510081, Bursa/Teper,
hep-lat/0505025
GKA: field theory calculation to two loop order
hep-ph/0102022,cubic order in hep-ph0412322. BGKAP: PRL66, 998,
1991.
How narrow is the sQGP transition?
-
Electric flux tension in the deconfined phase
e-tension for SU(Nc), Nc ≤ 8, PdF et al.,hep-lat/051008
How narrow is the sQGP transition?
-
Electric flux tension
Casimir scaling good for ANY T above 1.15 Tc indeconfined
phase
Two loop reduced tension does not match the
latticecalculation
Warranted: 3 or more loop calculation (Yannis Burnier,‘CPKA,
York Schroeder, Aleksi Vuorinen).
This talk: perhaps a more insightful way-beyondperturbation
theory- to understand the Casimir scalingdown to ≥ Tc
How narrow is the sQGP transition?
-
Casimir scaling of spatial Wilson loops
GKA (hep-ph0102022): at high T a dilute gas of adjointmonopoles
causes Casimir scaling for Wilson loops.Lucini Teper (2001....) and
hep-lat/051008 :
How narrow is the sQGP transition?
-
Casimir scaling of spatial Wilson loops
GKA (hep-ph0102022): at high T a dilute gas of adjointmonopoles
causes Casimir scaling for Wilson loops.Lucini Teper (2001....) and
hep-lat/051008 :
How narrow is the sQGP transition?
-
Casimir scaling of spatial Wilson loops
GKA (hep-ph0102022): at high T a dilute gas of adjointmonopoles
causes Casimir scaling for Wilson loops.Lucini Teper (2001....) and
hep-lat/051008 :
How narrow is the sQGP transition?
-
m-flux tension at asymptotic T. Lattice results
0
1
2
3
4
5
6
7
8
9
0 0.05 0.1 0.15 0.2 0.25
(k-σ
k/σ 1
)N
1/N
Binding energy of k-strings
k=2k=3
Casimir scalingSine formula
flux-tension for SU(Nc), Nc ≤ 8, Meyer, hep-lat/0412021)
How narrow is the sQGP transition?
-
m-flux tension at asymptotic T
0
0.2
0.4
0.6
0.8
1
0 0.05 0.1 0.15 0.2 0.25
2σk=
N/2
/ (N
σ 1)
1/N
The k=N/2 string tension
Casimir scalingSine formula
m-tension for SU(Nc), Nc ≤ 8, Meyer, hep-lat/0412021
How narrow is the sQGP transition?
-
3d results propagate in ALL of the deconfined phasethrough the
running coupling
1.0 2.0 3.0 4.0 5.0T / T
c
0.6
0.8
1.0
1.2
T/σ
s1/2
4d lattice, Nτ 1 = 8
Tc / Λ
MS = 1.10...1.35_
2-loop
1-loop
m-tension, Nc = 3, hep-lat/0503003σ(T ) = c3d g4M(T ) = c3d
g
4E(T )(1 + small) = c3d g
4(T )(1 + ..+ 3loop)T 2
How narrow is the sQGP transition?
-
How do magnetic and electric flux compare?
��������
��������
������
������
��������
��������
��������
��������
��������
���
���
���
���
����
������
������
������
������
����
���
���
������
������
����������������
����
���
���
���
���
1.0
l l
0.6
0.8
0.4
1.2
1−4
1.6
1.8
2.0
l1 2 3 4
T/T c
SU(3) colour electric flux versus SU(3) colour magnetic flux
Note: equality inside the peak of the interaction measure T/Tc ∼
1.10.So peak might be due to a correlation of electric and magnetic
quasi-particles
How narrow is the sQGP transition?
-
Correlations between loops
Measure correlation on the lattice between nearby,
almostcontingent ’t Hooft and Wilson loop as function
oftemperature.
For very high T: magnetic and electric populations
areuncorrelaled, so expect no correlation between loops.
For T in critical region around the peak of the conformalenergy
the correlation may become quite strong.
The correlation is a key quantity for understanding thebehaviour
of the plasma components.
Unfortunately it is subleading in in large N limit, so
simplestAdS/CFT is not enought to access it.
How narrow is the sQGP transition?
-
The ratio δ as function of T. SU(3) case
\delta=\sigma_s/(m_0^++)^2, colours as in previous figure.
_
_
_
_
_
_
_
| | |
0.3
0.6
0.9
1.2
1.5
1.8
2.1
0 1 2 3 00T/T_c
\delta
x
X
x
x
x
x
xx
x
x
x
x
The ratio σ1/m2++, SU(3), Datta, Gupta,hep-lat/0208001
How narrow is the sQGP transition?
-
At T ∼ 1.2Tc the ratio has risen with a factor 10From large T to
Tc the ratio increases with a factor 40! !
SU(3) weakly first order, may explain the large ratio.
m−− is probably the inverse radius of the adjoint magneticquasi
particle, determines a much smaller ratio whichwould be the
diluteness l3
−−nM , but is not yet available for
all T.
How narrow is the sQGP transition?
-
Perturbation theory and the flux loops
Once the non-perturbative 3d part of the magnetic loops
isdetemined on lattice, perturbation theory works, and theyhave
Casimir scaling.
Although the magnetic free energy scales as a gas ofadjoint
quasi-particles, no classical adjoint monopoles areknown in
QCD.
The electric loops have Casimir scaling according to onetwo and
two loop order. To three loop order the preliminaryresults suggest
the same.
"Precocious" QGP behaviour (see below) may be analternative
explanation
How narrow is the sQGP transition?
-
Perturbation theory and the flux loops
Once the non-perturbative 3d part of the magnetic loops
isdetemined on lattice, perturbation theory works, and theyhave
Casimir scaling.
Although the magnetic free energy scales as a gas ofadjoint
quasi-particles, no classical adjoint monopoles areknown in
QCD.
The electric loops have Casimir scaling according to onetwo and
two loop order. To three loop order the preliminaryresults suggest
the same.
"Precocious" QGP behaviour (see below) may be analternative
explanation
How narrow is the sQGP transition?
-
Perturbation theory and the flux loops
Once the non-perturbative 3d part of the magnetic loops
isdetemined on lattice, perturbation theory works, and theyhave
Casimir scaling.
Although the magnetic free energy scales as a gas ofadjoint
quasi-particles, no classical adjoint monopoles areknown in
QCD.
The electric loops have Casimir scaling according to onetwo and
two loop order. To three loop order the preliminaryresults suggest
the same.
"Precocious" QGP behaviour (see below) may be analternative
explanation
How narrow is the sQGP transition?
-
Perturbation theory and the flux loops
Once the non-perturbative 3d part of the magnetic loops
isdetemined on lattice, perturbation theory works, and theyhave
Casimir scaling.
Although the magnetic free energy scales as a gas ofadjoint
quasi-particles, no classical adjoint monopoles areknown in
QCD.
The electric loops have Casimir scaling according to onetwo and
two loop order. To three loop order the preliminaryresults suggest
the same.
"Precocious" QGP behaviour (see below) may be analternative
explanation
How narrow is the sQGP transition?
-
Perturbation theory and the flux loops
Once the non-perturbative 3d part of the magnetic loops
isdetemined on lattice, perturbation theory works, and theyhave
Casimir scaling.
Although the magnetic free energy scales as a gas ofadjoint
quasi-particles, no classical adjoint monopoles areknown in
QCD.
The electric loops have Casimir scaling according to onetwo and
two loop order. To three loop order the preliminaryresults suggest
the same.
"Precocious" QGP behaviour (see below) may be analternative
explanation
How narrow is the sQGP transition?
-
Field theory calculation of loop average
〈Vk (L)〉 = TrphysVk (L)exp(−H/T )/Trphys exp(−H/T )
Bytranslation into path integral language: Domainwall at z=0between
domains where Polyakov loop takes different Z(N)values has energy
ρk (T ).
k
z
P=1
P=exp(ik2π/Ν)
z=0 >
0
N−1
numbered by k
Mimima of effective potential
Periodic time direction and z−direction orthogonal
1
2
3
k
x
t
to (x,y) plane. Loop L x L at z=0.x y
qY
Polyakov loop profile along z-direction, and Z(N) vacua.
Effective action: U = K (q)q′2 + V (q) in loop
expansion.tunnneling along qYk between the minima gives ρkenergy of
wall/per unit length=ρk(T ).
How narrow is the sQGP transition?
-
Field theory calculation of loop average
〈Vk (L)〉 = TrphysVk (L)exp(−H/T )/Trphys exp(−H/T )
Bytranslation into path integral language: Domainwall at z=0between
domains where Polyakov loop takes different Z(N)values has energy
ρk (T ).
k
z
P=1
P=exp(ik2π/Ν)
z=0 >
0
N−1
numbered by k
Mimima of effective potential
Periodic time direction and z−direction orthogonal
1
2
3
k
x
t
to (x,y) plane. Loop L x L at z=0.x y
qY
Polyakov loop profile along z-direction, and Z(N) vacua.
Effective action: U = K (q)q′2 + V (q) in loop
expansion.tunnneling along qYk between the minima gives ρkenergy of
wall/per unit length=ρk(T ).
How narrow is the sQGP transition?
-
Field theory calculation of loop average
〈Vk (L)〉 = TrphysVk (L)exp(−H/T )/Trphys exp(−H/T )
Bytranslation into path integral language: Domainwall at z=0between
domains where Polyakov loop takes different Z(N)values has energy
ρk (T ).
k
z
P=1
P=exp(ik2π/Ν)
z=0 >
0
N−1
numbered by k
Mimima of effective potential
Periodic time direction and z−direction orthogonal
1
2
3
k
x
t
to (x,y) plane. Loop L x L at z=0.x y
qY
Polyakov loop profile along z-direction, and Z(N) vacua.
Effective action: U = K (q)q′2 + V (q) in loop
expansion.tunnneling along qYk between the minima gives ρkenergy of
wall/per unit length=ρk(T ).
How narrow is the sQGP transition?
-
Field theory calculation of loop average
〈Vk (L)〉 = TrphysVk (L)exp(−H/T )/Trphys exp(−H/T )
Bytranslation into path integral language: Domainwall at z=0between
domains where Polyakov loop takes different Z(N)values has energy
ρk (T ).
k
z
P=1
P=exp(ik2π/Ν)
z=0 >
0
N−1
numbered by k
Mimima of effective potential
Periodic time direction and z−direction orthogonal
1
2
3
k
x
t
to (x,y) plane. Loop L x L at z=0.x y
qY
Polyakov loop profile along z-direction, and Z(N) vacua.
Effective action: U = K (q)q′2 + V (q) in loop
expansion.tunnneling along qYk between the minima gives ρkenergy of
wall/per unit length=ρk(T ).
How narrow is the sQGP transition?
-
A non-perturbative approach
I_3
ω !/3 TrP=1
ω *
0
domain at infinite T
domain at T_c
Y
Domain of the SU(3) effective potential in Cartan space
Infinite T see perturbative potentialT ≥ Tc see
histogramThermodynamic functions live on the C invariant minima(red
lines, Z(3) related copies)we want a model for the potential V in
between thesetemperatures.we can compute the tunneling between Z(3)
related vacua(e flux tension) or the tunneling from V at T to TrP
0,
How narrow is the sQGP transition?
-
A non-perturbative approach
I_3
ω !/3 TrP=1
ω *
0
domain at infinite T
domain at T_c
Y
Domain of the SU(3) effective potential in Cartan space
Infinite T see perturbative potentialT ≥ Tc see
histogramThermodynamic functions live on the C invariant minima(red
lines, Z(3) related copies)we want a model for the potential V in
between thesetemperatures.we can compute the tunneling between Z(3)
related vacua(e flux tension) or the tunneling from V at T to TrP
0,
How narrow is the sQGP transition?
-
A non-perturbative approach
I_3
ω !/3 TrP=1
ω *
0
domain at infinite T
domain at T_c
Y
Domain of the SU(3) effective potential in Cartan space
Infinite T see perturbative potentialT ≥ Tc see
histogramThermodynamic functions live on the C invariant minima(red
lines, Z(3) related copies)we want a model for the potential V in
between thesetemperatures.we can compute the tunneling between Z(3)
related vacua(e flux tension) or the tunneling from V at T to TrP
0,
How narrow is the sQGP transition?
-
A non-perturbative approach
I_3
ω !/3 TrP=1
ω *
0
domain at infinite T
domain at T_c
Y
Domain of the SU(3) effective potential in Cartan space
Infinite T see perturbative potentialT ≥ Tc see
histogramThermodynamic functions live on the C invariant minima(red
lines, Z(3) related copies)we want a model for the potential V in
between thesetemperatures.we can compute the tunneling between Z(3)
related vacua(e flux tension) or the tunneling from V at T to TrP
0,
How narrow is the sQGP transition?
-
0.5
-1.0
-0.5
1.00.5
0.0
0.0
-0.5
Re[p]
Im[p]
1.00V
eff
0.2
0.4
0.6
0.8
SU(3), lowest order perturbative effective potential.Z(3)
minima: gulleys of least action along the boundary of the domain
(hypercharge
How narrow is the sQGP transition?
-
-0.15 0
0.15
0.15
0
-0.15
Re Ω
Im Ω
Histogram of the Polyakov loop P in SU(3). It equals exp−(Vol)V
(P).The Z(3) minima have moved in towards the symmetric point.
the Z(3) symmetric point a new minimum is developing. Tc when
all degenerate
How narrow is the sQGP transition?
-
A non perturbative approach to the effective potential
Intiated by Meisinger, Miller and Ogilvie hep-phys/0109009and
0108026Motivated by a remark of RDP at Kyoto 2006Idea is to make an
Ansatsz for V that consists of Z(N)symmetric "trial functions":
Bp(P) =
∑
l wlTradjP(A0)l .
Simplest are those with wl = 1/lp,p = 4,2. Arecorresponding to
the fluctuation determinant, resp tadpoleof the gluon. Correspond
to simple Bernoulli polynomials:
P = diag(
exp(i2πq1), ..........,exp(i2πqN))
B2p(P) ∼∑
i ,j
|qi − qj |p(1 − |qi − qj |)p,
Perturbative answer is B4, minima at qi − qj == 0 mod1.To
destabilize those minima: need linear term in qi − qj ,and the
unique candidate is B2 with a negative coefficient.
How narrow is the sQGP transition?
-
A non perturbative approach to the effective potential
Intiated by Meisinger, Miller and Ogilvie hep-phys/0109009and
0108026Motivated by a remark of RDP at Kyoto 2006Idea is to make an
Ansatsz for V that consists of Z(N)symmetric "trial functions":
Bp(P) =
∑
l wlTradjP(A0)l .
Simplest are those with wl = 1/lp,p = 4,2. Arecorresponding to
the fluctuation determinant, resp tadpoleof the gluon. Correspond
to simple Bernoulli polynomials:
P = diag(
exp(i2πq1), ..........,exp(i2πqN))
B2p(P) ∼∑
i ,j
|qi − qj |p(1 − |qi − qj |)p,
Perturbative answer is B4, minima at qi − qj == 0 mod1.To
destabilize those minima: need linear term in qi − qj ,and the
unique candidate is B2 with a negative coefficient.
How narrow is the sQGP transition?
-
A non perturbative approach to the effective potential
Intiated by Meisinger, Miller and Ogilvie hep-phys/0109009and
0108026Motivated by a remark of RDP at Kyoto 2006Idea is to make an
Ansatsz for V that consists of Z(N)symmetric "trial functions":
Bp(P) =
∑
l wlTradjP(A0)l .
Simplest are those with wl = 1/lp,p = 4,2. Arecorresponding to
the fluctuation determinant, resp tadpoleof the gluon. Correspond
to simple Bernoulli polynomials:
P = diag(
exp(i2πq1), ..........,exp(i2πqN))
B2p(P) ∼∑
i ,j
|qi − qj |p(1 − |qi − qj |)p,
Perturbative answer is B4, minima at qi − qj == 0 mod1.To
destabilize those minima: need linear term in qi − qj ,and the
unique candidate is B2 with a negative coefficient.
How narrow is the sQGP transition?
-
A non perturbative approach to the effective potential
Intiated by Meisinger, Miller and Ogilvie hep-phys/0109009and
0108026Motivated by a remark of RDP at Kyoto 2006Idea is to make an
Ansatsz for V that consists of Z(N)symmetric "trial functions":
Bp(P) =
∑
l wlTradjP(A0)l .
Simplest are those with wl = 1/lp,p = 4,2. Arecorresponding to
the fluctuation determinant, resp tadpoleof the gluon. Correspond
to simple Bernoulli polynomials:
P = diag(
exp(i2πq1), ..........,exp(i2πqN))
B2p(P) ∼∑
i ,j
|qi − qj |p(1 − |qi − qj |)p,
Perturbative answer is B4, minima at qi − qj == 0 mod1.To
destabilize those minima: need linear term in qi − qj ,and the
unique candidate is B2 with a negative coefficient.
How narrow is the sQGP transition?
-
A non perturbative approach to the effective potential
Intiated by Meisinger, Miller and Ogilvie hep-phys/0109009and
0108026Motivated by a remark of RDP at Kyoto 2006Idea is to make an
Ansatsz for V that consists of Z(N)symmetric "trial functions":
Bp(P) =
∑
l wlTradjP(A0)l .
Simplest are those with wl = 1/lp,p = 4,2. Arecorresponding to
the fluctuation determinant, resp tadpoleof the gluon. Correspond
to simple Bernoulli polynomials:
P = diag(
exp(i2πq1), ..........,exp(i2πqN))
B2p(P) ∼∑
i ,j
|qi − qj |p(1 − |qi − qj |)p,
Perturbative answer is B4, minima at qi − qj == 0 mod1.To
destabilize those minima: need linear term in qi − qj ,and the
unique candidate is B2 with a negative coefficient.
How narrow is the sQGP transition?
-
A non perturbative approach to the effective potential
Intiated by Meisinger, Miller and Ogilvie hep-phys/0109009and
0108026Motivated by a remark of RDP at Kyoto 2006Idea is to make an
Ansatsz for V that consists of Z(N)symmetric "trial functions":
Bp(P) =
∑
l wlTradjP(A0)l .
Simplest are those with wl = 1/lp,p = 4,2. Arecorresponding to
the fluctuation determinant, resp tadpoleof the gluon. Correspond
to simple Bernoulli polynomials:
P = diag(
exp(i2πq1), ..........,exp(i2πqN))
B2p(P) ∼∑
i ,j
|qi − qj |p(1 − |qi − qj |)p,
Perturbative answer is B4, minima at qi − qj == 0 mod1.To
destabilize those minima: need linear term in qi − qj ,and the
unique candidate is B2 with a negative coefficient.
How narrow is the sQGP transition?
-
Repulsive and attractive eigenvalues of the Wilson line
SU(2): P = diagonal(exp(iφ/2,exp(−iφ/2))At high T in
perturbation theory: phases cluster atcentergroup values φ = 2πq =
0, πT 4Vpert = T 4(π2/15) + 2π
2
3 q2(1 − |q|)2) minima in q=0,1
Adding a term Vnonpert = −M2T 2|q|(1 − |q|) inducestendency to
repulsion: minima are φ = ±πSo our mean field like Ansatz is:
T 4V = T 4(Vpert + Vnonpert)
= T 4(
π2
15+
23π2q2(1 − |q|)2 − (M
T)2(|q|(1 − |q|) + d)
)
(1)
At high T: the perturbative determinant term dominates:e.v.’s
cluster in Z(N)As T ∼ M: the "non-perturbative" Ansatz starts to
kick in:the linear term destabilizes the perturbative vacuum,
e.v’srepel, equal spacing,Tr P=0, and d fixes pressure=0 at Tc.
How narrow is the sQGP transition?
-
Repulsive and attractive eigenvalues of the Wilson line
SU(2): P = diagonal(exp(iφ/2,exp(−iφ/2))At high T in
perturbation theory: phases cluster atcentergroup values φ = 2πq =
0, πT 4Vpert = T 4(π2/15) + 2π
2
3 q2(1 − |q|)2) minima in q=0,1
Adding a term Vnonpert = −M2T 2|q|(1 − |q|) inducestendency to
repulsion: minima are φ = ±πSo our mean field like Ansatz is:
T 4V = T 4(Vpert + Vnonpert)
= T 4(
π2
15+
23π2q2(1 − |q|)2 − (M
T)2(|q|(1 − |q|) + d)
)
(1)
At high T: the perturbative determinant term dominates:e.v.’s
cluster in Z(N)As T ∼ M: the "non-perturbative" Ansatz starts to
kick in:the linear term destabilizes the perturbative vacuum,
e.v’srepel, equal spacing,Tr P=0, and d fixes pressure=0 at Tc.
How narrow is the sQGP transition?
-
Repulsive and attractive eigenvalues of the Wilson line
SU(2): P = diagonal(exp(iφ/2,exp(−iφ/2))At high T in
perturbation theory: phases cluster atcentergroup values φ = 2πq =
0, πT 4Vpert = T 4(π2/15) + 2π
2
3 q2(1 − |q|)2) minima in q=0,1
Adding a term Vnonpert = −M2T 2|q|(1 − |q|) inducestendency to
repulsion: minima are φ = ±πSo our mean field like Ansatz is:
T 4V = T 4(Vpert + Vnonpert)
= T 4(
π2
15+
23π2q2(1 − |q|)2 − (M
T)2(|q|(1 − |q|) + d)
)
(1)
At high T: the perturbative determinant term dominates:e.v.’s
cluster in Z(N)As T ∼ M: the "non-perturbative" Ansatz starts to
kick in:the linear term destabilizes the perturbative vacuum,
e.v’srepel, equal spacing,Tr P=0, and d fixes pressure=0 at Tc.
How narrow is the sQGP transition?
-
Repulsive and attractive eigenvalues of the Wilson line
SU(2): P = diagonal(exp(iφ/2,exp(−iφ/2))At high T in
perturbation theory: phases cluster atcentergroup values φ = 2πq =
0, πT 4Vpert = T 4(π2/15) + 2π
2
3 q2(1 − |q|)2) minima in q=0,1
Adding a term Vnonpert = −M2T 2|q|(1 − |q|) inducestendency to
repulsion: minima are φ = ±πSo our mean field like Ansatz is:
T 4V = T 4(Vpert + Vnonpert)
= T 4(
π2
15+
23π2q2(1 − |q|)2 − (M
T)2(|q|(1 − |q|) + d)
)
(1)
At high T: the perturbative determinant term dominates:e.v.’s
cluster in Z(N)As T ∼ M: the "non-perturbative" Ansatz starts to
kick in:the linear term destabilizes the perturbative vacuum,
e.v’srepel, equal spacing,Tr P=0, and d fixes pressure=0 at Tc.
How narrow is the sQGP transition?
-
Repulsive and attractive eigenvalues of the Wilson line
SU(2): P = diagonal(exp(iφ/2,exp(−iφ/2))At high T in
perturbation theory: phases cluster atcentergroup values φ = 2πq =
0, πT 4Vpert = T 4(π2/15) + 2π
2
3 q2(1 − |q|)2) minima in q=0,1
Adding a term Vnonpert = −M2T 2|q|(1 − |q|) inducestendency to
repulsion: minima are φ = ±πSo our mean field like Ansatz is:
T 4V = T 4(Vpert + Vnonpert)
= T 4(
π2
15+
23π2q2(1 − |q|)2 − (M
T)2(|q|(1 − |q|) + d)
)
(1)
At high T: the perturbative determinant term dominates:e.v.’s
cluster in Z(N)As T ∼ M: the "non-perturbative" Ansatz starts to
kick in:the linear term destabilizes the perturbative vacuum,
e.v’srepel, equal spacing,Tr P=0, and d fixes pressure=0 at Tc.
How narrow is the sQGP transition?
-
REPULSION or SYMMETRY IS RESTOREDN=2 N=3 N=4
ATTRACTION or SYMMETRY IS BROKEN
As T goes down the eigenvalues start to decluster andmove out to
the equal spacing positions. In all but SU(2)the transition is
first order, so the eigenvalues stop short ofthe equal spacing
positions.
How narrow is the sQGP transition?
-
REPULSION or SYMMETRY IS RESTOREDN=2 N=3 N=4
ATTRACTION or SYMMETRY IS BROKEN
As T goes down the eigenvalues start to decluster andmove out to
the equal spacing positions. In all but SU(2)the transition is
first order, so the eigenvalues stop short ofthe equal spacing
positions.
How narrow is the sQGP transition?
-
Simple relations
To determine pressure from V (q):find the extrema q0 of V (q), V
′(q0) = 0:
p = −V (q0). (2)
Now the relation of ∆ to V is immediate:
∆
T 4= T
∂
∂T
(
p/T 4)
= −∂V (q0)∂T
= −T ∂q0∂T
V ′(q0) + 2M2/T 2Vnonpert(q0) (3)
So ∆ relates only (not unexpected) to the
non-perturbativepotential.:
∆
T 4= 2M2/T 2Vnonpert(q0). (4)
How narrow is the sQGP transition?
-
-0.5
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3
T / TC
orig. model Aorig. model A
p/T4
Latt. p/T4e/3T4
Latt. e/3T4∆/T4
Latt. ∆/T4
How narrow is the sQGP transition?
-
This Ansatz is good, but not good enough!The interaction measure
not rising steep enough: themaximum is displaced to much too high
TWe need another parameter to fix this:
Vnonpert → Vnonpert − c(M/T )2(|q|2(1 − |q|)2
-0.5
0
0.5
1
1.5
1 1.5 2 2.5 3
T / TC
Ext. model A
p/T4
Latt. p/T4e/3T4
Latt. e/3T4∆/T4
Latt. ∆/T4
How narrow is the sQGP transition?
-
-0.5
0
0.5
1
1.5
1 1.5 2 2.5 3
T / TC
p/T4
Latt. p/T4e/3T4
Latt. e/3T4∆/T4
Latt. ∆/T4
How narrow is the sQGP transition?
-
-0.5
0
0.5
1
1.5
1 1.5 2 2.5 3
T / TC
p/T4
Latt. p/T4e/3T4
Latt. e/3T4∆/T4
Latt. ∆/T4
SU(2) thermodynamic functions, c=2.
How narrow is the sQGP transition?
-
SU(3),thermodynamic functions c=1 .
How narrow is the sQGP transition?
-
Predictions
The effective potential is now fixed. There are fourpredictions
to be checked by lattice.
Interface tension for T ≥ TcFor SU(3) and higher Nc : tension at
Tc for coexistingphases.
Polyakov loop average
How narrow is the sQGP transition?
-
Figure: Potential at Tc , showing a a VERY weak first order
transition,as function of 1-q. q=1 is the confined state. You see a
very smallmaximum at 1 − q = 0.16, i.e. 1 − qc = 0.33 is the
minimumdegenerate with the minimum at 1 − q = 0, the confining
vacuum.
How narrow is the sQGP transition?
-
Figure: Vertical blow up of the first graph and now you see the
firstorder transition, i.e. the degeneracy at 1 − q = 0 and 1 − q =
0.33
How narrow is the sQGP transition?
-
Figure: Potential at 0.99 Tc , as function of 1-q. The
metastableminimum at non-zero 1-q has almost gone away.
How narrow is the sQGP transition?
-
For coexisting phases the tension is (SU(3))= 0.0258012T 2c
/
√
g2(T ) (smaller than Bursa-Teperresult). Large latent heat means
very broad flat potential atTcSU(2) interface tension= 4π
2T 2
3√
6g2(T )(1 − (Tc/T )2)3/2 if c=0,
and kinetic energy is taken classical.
SU(2): the fit is done with (obviously) c=1.5 (SU(2) and justthe
two loop corrections from the complete QGP. The latterare not the
whole story. We have to include loopcorrections from fluctuations
around the SQGP minima.This is being done.
For SU(3) the tunneling path is only for the QGP along theλ8.
Away from QGP there is a λ3 component takennumerically into
account, to obtain the minimal action.
How narrow is the sQGP transition?
-
For coexisting phases the tension is (SU(3))= 0.0258012T 2c
/
√
g2(T ) (smaller than Bursa-Teperresult). Large latent heat means
very broad flat potential atTcSU(2) interface tension= 4π
2T 2
3√
6g2(T )(1 − (Tc/T )2)3/2 if c=0,
and kinetic energy is taken classical.
SU(2): the fit is done with (obviously) c=1.5 (SU(2) and justthe
two loop corrections from the complete QGP. The latterare not the
whole story. We have to include loopcorrections from fluctuations
around the SQGP minima.This is being done.
For SU(3) the tunneling path is only for the QGP along theλ8.
Away from QGP there is a λ3 component takennumerically into
account, to obtain the minimal action.
How narrow is the sQGP transition?
-
For coexisting phases the tension is (SU(3))= 0.0258012T 2c
/
√
g2(T ) (smaller than Bursa-Teperresult). Large latent heat means
very broad flat potential atTcSU(2) interface tension= 4π
2T 2
3√
6g2(T )(1 − (Tc/T )2)3/2 if c=0,
and kinetic energy is taken classical.
SU(2): the fit is done with (obviously) c=1.5 (SU(2) and justthe
two loop corrections from the complete QGP. The latterare not the
whole story. We have to include loopcorrections from fluctuations
around the SQGP minima.This is being done.
For SU(3) the tunneling path is only for the QGP along theλ8.
Away from QGP there is a λ3 component takennumerically into
account, to obtain the minimal action.
How narrow is the sQGP transition?
-
For coexisting phases the tension is (SU(3))= 0.0258012T 2c
/
√
g2(T ) (smaller than Bursa-Teperresult). Large latent heat means
very broad flat potential atTcSU(2) interface tension= 4π
2T 2
3√
6g2(T )(1 − (Tc/T )2)3/2 if c=0,
and kinetic energy is taken classical.
SU(2): the fit is done with (obviously) c=1.5 (SU(2) and justthe
two loop corrections from the complete QGP. The latterare not the
whole story. We have to include loopcorrections from fluctuations
around the SQGP minima.This is being done.
For SU(3) the tunneling path is only for the QGP along theλ8.
Away from QGP there is a λ3 component takennumerically into
account, to obtain the minimal action.
How narrow is the sQGP transition?
-
For coexisting phases the tension is (SU(3))= 0.0258012T 2c
/
√
g2(T ) (smaller than Bursa-Teperresult). Large latent heat means
very broad flat potential atTcSU(2) interface tension= 4π
2T 2
3√
6g2(T )(1 − (Tc/T )2)3/2 if c=0,
and kinetic energy is taken classical.
SU(2): the fit is done with (obviously) c=1.5 (SU(2) and justthe
two loop corrections from the complete QGP. The latterare not the
whole story. We have to include loopcorrections from fluctuations
around the SQGP minima.This is being done.
For SU(3) the tunneling path is only for the QGP along theλ8.
Away from QGP there is a λ3 component takennumerically into
account, to obtain the minimal action.
How narrow is the sQGP transition?
-
Interface SU(2), SU(3), de Forcrand, Noth, hep-lat/0506005
Interface tension ∼ ((T − Tc)/Tc)3/2, i.e. critical exponent
=1.5instead of universal 1.26.
How narrow is the sQGP transition?
-
SU(3), Polyakov loop average, Gupta et al., arXiv:0711.2251
Narrowness of the sQGP (T/Tc = 1 to 1.2)) is closelyrelated to
narrowness of interaction measureOur result does contradict the
data. O(g2) correctionsunlikely to produce agreement. Data without
fuzzing theloop.
How narrow is the sQGP transition?
-
SU(3), Polyakov loop average, Gupta et al., arXiv:0711.2251
Narrowness of the sQGP (T/Tc = 1 to 1.2)) is closelyrelated to
narrowness of interaction measureOur result does contradict the
data. O(g2) correctionsunlikely to produce agreement. Data without
fuzzing theloop.
How narrow is the sQGP transition?
-
SU(3), Polyakov loop average, Gupta et al., arXiv:0711.2251
Narrowness of the sQGP (T/Tc = 1 to 1.2)) is closelyrelated to
narrowness of interaction measureOur result does contradict the
data. O(g2) correctionsunlikely to produce agreement. Data without
fuzzing theloop.
How narrow is the sQGP transition?
-
1 1.5 2 2.5 3 3.5 4T/Tc
0
0.1
0.2
0.3
0.4
Del
ta/(
N^2
-1)
SU(3)SU(4)SU(6)
Figure: Interaction measure scaled by N2 − 1, Panero
2009.Reduced discontinuity looks very small, like we find.
How narrow is the sQGP transition?
-
4 5 6 7 8 9 10
0.06
0.08
0.10
0.12
0.14
Figure: Latent heat in units of (N2 − 1)T 4c , Teper/Bursa:0.744
− 0.34/N2
How narrow is the sQGP transition?
-
4 6 8 10
0.005
0.010
0.015
0.020
Figure: Order-disorder interface between coexisting phases, in
unitsof (N2 − 1)T 2c /
√
g2(Tc)N , as function of N
How narrow is the sQGP transition?
-
4 5 6 7 8 9 10
0.50
0.55
0.60
Figure: Jump of the normalized Polyakov loop at Tc , as function
of N
How narrow is the sQGP transition?
-
Masses induced by the presence of the loop
The presence of the loop induces a shift in the timederivatives,
or equivalently in the Matsubara frequencies ofoff diagonal
fluctuations:
p0 → p0 + 2πT (qi − qj)So the corresponding inverse propagator
is corrected notonly by m2(q) a q dependent Debye mass (O(gT)), but
alsoby an O(1) shift: ~p2 + m2D(q) + (2πT (qi − qj))2
How narrow is the sQGP transition?
-
Illustration of the behaviour of the masses for SU(3).
How narrow is the sQGP transition?
-
10-4
10-3
10-2
10-1
100
101
0 2 4 6 8 10 12 14 16
|F1(
R)-
F 1(∞
)|/Τ
R/a = 4RT
β=4.0760, T/Tc=1.005fit, large R
fit, interm. R
SU(3),
How narrow is the sQGP transition?
-
Conclusions
Model: EOS fully fixes effective potential
Predicts surface tensions (o-o, o-d),Polyakov loop
average,latent heat,
Our model finds precocious QGP at T = 1.20Tc, beyondwhich
P=1
The precociousness is persisting for more coloursNc = 4,5,
....
If so: the Casimir scaling of the e-tension down toT ∼ 1.2Tc may
be understandable, and should becompared to the Teper/Bursa lattice
data.
Conspicuously absent is prediction for magnetic tension
Introduce quarks!
How narrow is the sQGP transition?
-
How narrow is the sQGP transition?
