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The Wroblewski parameter from lattice QCD
Workshop on Field Theories Near EquilibriumRajiv V. Gavai
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 1
The Wroblewski parameter from lattice QCD
Workshop on Field Theories Near EquilibriumRajiv V. Gavai
Introdution
λs from Quark Number Susceptibility
Pressure for small baryon density
Summary
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 1
Introduction
• Quark-Gluon Plasma in Heavy Ion Collisions.
• Reliable signals needed to establish it.
• Enhancement of strangeness production as a promising signal of QGP(Rafelski-Muller, Phys. Rev. Lett ’82, Phys. Rept ’86..).
• A variety of aspects studied and many different variations proposed.
• Most signal considerations based on Simple Models.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 2
Introduction
• Quark-Gluon Plasma in Heavy Ion Collisions.
• Reliable signals needed to establish it.
• Enhancement of strangeness production as a promising signal of QGP(Rafelski-Muller, Phys. Rev. Lett ’82, Phys. Rept ’86..).
• A variety of aspects studied and many different variations proposed.
• Most signal considerations based on Simple Models.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 2
Introduction
• Quark-Gluon Plasma in Heavy Ion Collisions.
• Reliable signals needed to establish it.
• Enhancement of strangeness production as a promising signal of QGP(Rafelski-Muller, Phys. Rev. Lett ’82, Phys. Rept ’86..).
• A variety of aspects studied and many different variations proposed.
• Most signal considerations based on Simple Models.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 2
Introduction
• Quark-Gluon Plasma in Heavy Ion Collisions.
• Reliable signals needed to establish it.
• Enhancement of strangeness production as a promising signal of QGP(Rafelski-Muller, Phys. Rev. Lett ’82, Phys. Rept ’86..).
• A variety of aspects studied and many different variations proposed.
• Most signal considerations based on Simple Models.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 2
FromSTAR
Webpage
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 3
Ratio of newly createdstrange quarks to lightquarks :
λs =2〈ss〉〈uu+ dd〉
(1)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 4
Wroblewski Parameter
• Hadron gas fireball model(Becattini-Heinz ’97).
• 3 Free parameters : T , V , andNss.
• Fit many hadron abundunces.
• Obtain λs from data.
• Find λs ∼ 0.4 (0.2) for AA(pp).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 5
Wroblewski Parameter
• Hadron gas fireball model(Becattini-Heinz ’97).
• 3 Free parameters : T , V , andNss.
• Fit many hadron abundunces.
• Obtain λs from data.
• Find λs ∼ 0.4 (0.2) for AA(pp).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 5
Wroblewski Parameter
• Hadron gas fireball model(Becattini-Heinz ’97).
• 3 Free parameters : T , V , andNss.
• Fit many hadron abundunces.
• Obtain λs from data.
• Find λs ∼ 0.4 (0.2) for AA(pp).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 5
Wroblewski Parameter
• Hadron gas fireball model(Becattini-Heinz ’97).
• 3 Free parameters : T , V , andNss.
• Fit many hadron abundunces.
• Obtain λs from data.
• Find λs ∼ 0.4 (0.2) for AA(pp).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 5
Wroblewski Parameter
• Hadron gas fireball model(Becattini-Heinz ’97).
• 3 Free parameters : T , V , andNss.
• Fit many hadron abundunces.
• Obtain λs from data.
• Find λs ∼ 0.4 (0.2) for AA(pp).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 5
Quark Number Susceptibility
♠ We have argued that
λs =2χs
χu + χd. (2)
(Gavai & Gupta, PR D ’02 )
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 6
Quark Number Susceptibility
♠ We have argued that
λs =2χs
χu + χd. (2)
(Gavai & Gupta, PR D ’02 )
♠ Quark Number Susceptibilities also crucial for other QGP Signatures : Q, BFluctuations
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 6
Quark Number Susceptibility
♠ We have argued that
λs =2χs
χu + χd. (2)
(Gavai & Gupta, PR D ’02 )
♠ Quark Number Susceptibilities also crucial for other QGP Signatures : Q, BFluctuations
♠ Finite Density Results by Taylor Expansion in µ
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 6
Quark Number Susceptibility
♠ We have argued that
λs =2χs
χu + χd. (2)
(Gavai & Gupta, PR D ’02 )
♠ Quark Number Susceptibilities also crucial for other QGP Signatures : Q, BFluctuations
♠ Finite Density Results by Taylor Expansion in µ
♠ Theoretical Checks : Resummed Perturbation expansions, DimensionalReduction..
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 6
Quark Number Susceptibility
♠ We have argued that
λs =2χs
χu + χd. (2)
(Gavai & Gupta, PR D ’02 )
♠ Quark Number Susceptibilities also crucial for other QGP Signatures : Q, BFluctuations
♠ Finite Density Results by Taylor Expansion in µ
♠ Theoretical Checks : Resummed Perturbation expansions, DimensionalReduction..
♠ Our improvement: Fixed mq/Tc, Continuum limit...
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 6
Assuming three flavours, u, d, and s quarks, and denoting by µf thecorresponding chemical potentials, the QCD partition function is
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 7
Assuming three flavours, u, d, and s quarks, and denoting by µf thecorresponding chemical potentials, the QCD partition function is
Z =∫
DU exp(−SG)∏f=u,d,s Det M(mf,µf ) . (3)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 7
Assuming three flavours, u, d, and s quarks, and denoting by µf thecorresponding chemical potentials, the QCD partition function is
Z =∫
DU exp(−SG)∏f=u,d,s Det M(mf,µf ) . (3)
Defining µ0 = µu + µd + µs and µ3 = µu − µd, baryon and isospindensity/susceptibilities can be obtained as :(Gottlieb et al. ’87, ’96, ’97, Gavai et al. ’89)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 7
Assuming three flavours, u, d, and s quarks, and denoting by µf thecorresponding chemical potentials, the QCD partition function is
Z =∫
DU exp(−SG)∏f=u,d,s Det M(mf,µf ) . (3)
Defining µ0 = µu + µd + µs and µ3 = µu − µd, baryon and isospindensity/susceptibilities can be obtained as :(Gottlieb et al. ’87, ’96, ’97, Gavai et al. ’89)
ni = TV∂ lnZ∂µi
, χij = TV∂2 lnZ∂µi∂µj
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 7
Assuming three flavours, u, d, and s quarks, and denoting by µf thecorresponding chemical potentials, the QCD partition function is
Z =∫
DU exp(−SG)∏f=u,d,s Det M(mf,µf ) . (3)
Defining µ0 = µu + µd + µs and µ3 = µu − µd, baryon and isospindensity/susceptibilities can be obtained as :(Gottlieb et al. ’87, ’96, ’97, Gavai et al. ’89)
ni = TV∂ lnZ∂µi
, χij = TV∂2 lnZ∂µi∂µj
Higher order susceptibilities are defined by
χfg··· =T
V
∂n logZ∂µf∂µg · · ·
=∂nP
∂µf∂µg · · ·. (4)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 7
Assuming three flavours, u, d, and s quarks, and denoting by µf thecorresponding chemical potentials, the QCD partition function is
Z =∫
DU exp(−SG)∏f=u,d,s Det M(mf,µf ) . (3)
Defining µ0 = µu + µd + µs and µ3 = µu − µd, baryon and isospindensity/susceptibilities can be obtained as :(Gottlieb et al. ’87, ’96, ’97, Gavai et al. ’89)
ni = TV∂ lnZ∂µi
, χij = TV∂2 lnZ∂µi∂µj
Higher order susceptibilities are defined by
χfg··· =T
V
∂n logZ∂µf∂µg · · ·
=∂nP
∂µf∂µg · · ·. (4)
These are Taylor coefficients of the pressure P in its expansion in µ.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 7
All of these can be written as traces of products of M−1 and various derivatives ofM .
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 8
All of these can be written as traces of products of M−1 and various derivatives ofM .
Setting µi = 0, ni =0 but χ are nontrivial. Diagonal χii’s are
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 8
All of these can be written as traces of products of M−1 and various derivatives ofM .
Setting µi = 0, ni =0 but χ are nontrivial. Diagonal χii’s are
χ0 =T
2V[〈O2(mu) +
12O11(mu)〉] (5)
χ3 =T
2V〈O2(mu)〉 (6)
χs =T
4V[〈O2(ms) +
14O11(ms)〉] (7)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 8
All of these can be written as traces of products of M−1 and various derivatives ofM .
Setting µi = 0, ni =0 but χ are nontrivial. Diagonal χii’s are
χ0 =T
2V[〈O2(mu) +
12O11(mu)〉] (5)
χ3 =T
2V〈O2(mu)〉 (6)
χs =T
4V[〈O2(ms) +
14O11(ms)〉] (7)
Here O2 = Tr M−1u M ′′u − Tr M−1
u M ′uM−1u M ′u, and O11(mu) = (Tr M−1
u M ′u)2,and the traces are estimated by a stochastic method:Tr A =
∑Nvi=1R
†iARi/2Nv , and (Tr A)2 = 2
∑Li>j=1(Tr A)i(Tr A)j/L(L− 1) ,
where Ri is a complex vector from a set of Nv subdivided in L independent sets.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 8
Gavai & Gupta PR D ’01; Gavai, Gupta & Majumdar, PR D 2002
χFFT — Ideal gas results for same Lattice.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 9
Gavai & Gupta PR D ’01; Gavai, Gupta & Majumdar, PR D 2002
χFFT — Ideal gas results for same Lattice.
0
0.2
0.4
0.6
0.8
0.9
1
0.5 1 1.5 2 2.5 3
χ /χ�
FF
T3
T/Tc
mv
0.1
0.3
0.75
1.03
12 x 4 LatticeN = 2; m /T = 0.1
f s cN = 0; Quenched
f
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 9
Gavai & Gupta PR D ’01; Gavai, Gupta & Majumdar, PR D 2002
χFFT — Ideal gas results for same Lattice.
0
0.2
0.4
0.6
0.8
0.9
1
0.5 1 1.5 2 2.5 3
χ /χ�
FF
T3
T/Tc
mv
0.1
0.3
0.75
1.03
12 x 4 LatticeN = 2; m /T = 0.1
f s cN = 0; Quenched
f
Note that PDG values forstrange quark mass =⇒mstrangev /Tc' 0.3-0.7 (Nf=0);
0.45-1.0(Nf=2).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 9
Perturbation Theory
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 10
Perturbation Theory
Weak coupling expansion gives:χ
χFFT= 1− 2(αsπ ) + 8
√(1 + 0.167Nf)(αsπ )
32
(Kapusta 1989).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 10
Perturbation Theory
Weak coupling expansion gives:χ
χFFT= 1− 2(αsπ ) + 8
√(1 + 0.167Nf)(αsπ )
32
(Kapusta 1989).
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
0 0.02 0.04 0.06 0.08 0.1
χ/χ F
FT
αs/π
| || |3Tc 3Tc1.5Tc 1.5Tc
LO Plasmon Nf=0 Plasmon Nf=2 Plasmon Nf=4
♣ Minm 0.981 (0.986) at0.03 (0.02)for Nf = 0 (2).♣ For 1.5 ≤ T/Tc ≤ 3pert. theory −→ 0.99-0.98(1.08=1.03) for Nf = 0 (2).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 10
Resummed Perturbation Theory
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 11
Resummed Perturbation Theory
Hard Thermal Loop & Self-consistent resummation give :(Blaizot, Iancu & Rebhan, PLB ’01; Chakraborty, Mustafa & Thoma, EPJC ’02).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 11
Resummed Perturbation Theory
Hard Thermal Loop & Self-consistent resummation give :(Blaizot, Iancu & Rebhan, PLB ’01; Chakraborty, Mustafa & Thoma, EPJC ’02).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 11
Resummed Perturbation Theory
Hard Thermal Loop & Self-consistent resummation give :(Blaizot, Iancu & Rebhan, PLB ’01; Chakraborty, Mustafa & Thoma, EPJC ’02).
Our results for Nt = 4 Lattice artifacts ?Check for larger Nt and improved actions.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 11
χud
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 12
χud
Off-diagonal Susceptibility : χud = 〈 TV Tr M−1u M ′uTr M−1
d M ′d〉
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 12
χud
Off-diagonal Susceptibility : χud = 〈 TV Tr M−1u M ′uTr M−1
d M ′d〉
♥ Zero within 1–σ ∼ O(10−6) for T > Tc.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 12
χud
Off-diagonal Susceptibility : χud = 〈 TV Tr M−1u M ′uTr M−1
d M ′d〉
♥ Zero within 1–σ ∼ O(10−6) for T > Tc.
♥ Identically zero for Ideal gas but O(α3s) in P.T.
Using the same scale and αs as for χ3 −→ χud ∼ O(10−4) !!
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 12
χud
Off-diagonal Susceptibility : χud = 〈 TV Tr M−1u M ′uTr M−1
d M ′d〉
♥ Zero within 1–σ ∼ O(10−6) for T > Tc.
♥ Identically zero for Ideal gas but O(α3s) in P.T.
Using the same scale and αs as for χ3 −→ χud ∼ O(10−4) !!
♥ NONZERO for T < Tc and ∝M−2π .
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 12
χud
Off-diagonal Susceptibility : χud = 〈 TV Tr M−1u M ′uTr M−1
d M ′d〉
♥ Zero within 1–σ ∼ O(10−6) for T > Tc.
♥ Identically zero for Ideal gas but O(α3s) in P.T.
Using the same scale and αs as for χ3 −→ χud ∼ O(10−4) !!
♥ NONZERO for T < Tc and ∝M−2π .
0
1
2
3
4
5
6
7
8
2 2.5 3 3.5 4 4.5 5
M /
T c4
ud
�
2χ
π10
5
M /Tcπ
♣ 123 × 4 Lattice; Quenched.♣ T = 0.75Tc♣ Gavai, Gupta & Majumdar,
PR D 2002
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 12
Taking Continuum Limit
(Gavai & Gupta, PR D ’02 and PR D ’03)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 13
Taking Continuum Limit
(Gavai & Gupta, PR D ’02 and PR D ’03)
♠ Investigate larger Nt : 6, 8, 10, 12 and 14.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 13
Taking Continuum Limit
(Gavai & Gupta, PR D ’02 and PR D ’03)
♠ Investigate larger Nt : 6, 8, 10, 12 and 14.
♠ Naik action : Improved by O(a) compared to Staggered.Introduction of µ nontrivial but straightforward.(Naik, NP B 1989; Gavai, NP B ’03)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 13
Taking Continuum Limit
(Gavai & Gupta, PR D ’02 and PR D ’03)
♠ Investigate larger Nt : 6, 8, 10, 12 and 14.
♠ Naik action : Improved by O(a) compared to Staggered.Introduction of µ nontrivial but straightforward.(Naik, NP B 1989; Gavai, NP B ’03)
0
0.5
1
1.5
2
2.5
2 4 6 8 10 12 14
χ/Τ2
Nt
Aspect ratio 3
naive naik
p4
♠ Does improve theNt-dependence of the freefermions.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 13
Results at 2Tc :
0.8
1
1.2
1.4
1.6
1.8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
/T2
χ 3
1/N t2
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 14
Results at 2Tc :
0.8
1
1.2
1.4
1.6
1.8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
/T2
χ 3
1/N t2
♦ N−2t ∼ a2 extrapolation works and leads to same results within errors for both
staggered and Naik fermions.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 14
Results at 2Tc :
0.8
1
1.2
1.4
1.6
1.8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
/T2
χ 3
1/N t2
♦ N−2t ∼ a2 extrapolation works and leads to same results within errors for both
staggered and Naik fermions.
♦ Milder N−2t ∼ a2-dependence for Naik fermions.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 14
The continuum susceptibility vs. T therefore is :
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 15
The continuum susceptibility vs. T therefore is :
0.6
0.7
0.8
0.9
1.0
1 2 3
χ /Τ
23
T/Tc
NL
HTL
Naik action (Squares) and Staggered action (circles)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 15
The continuum susceptibility vs. T therefore is :
0.6
0.7
0.8
0.9
1.0
1 2 3
χ /Τ
23
T/Tc
NL
HTL
Naik action (Squares) and Staggered action (circles)
♥ Also reproduced in dimensional reduction (1 free parameter). Vuorinen, PR D ’03.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 15
The continuum susceptibility vs. T therefore is :
0.6
0.7
0.8
0.9
1.0
1 2 3
χ /Τ
23
T/Tc
NL
HTL
Naik action (Squares) and Staggered action (circles)
♥ Also reproduced in dimensional reduction (1 free parameter). Vuorinen, PR D ’03.
♥ Note that χud behaves the same way for ALL Nt and both fermions, leading to
the same O(10−6) values in continuum too.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 15
Wroblewski Parameter
Using our continuum QNS, ratio χs/χu can be obtained.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 16
Wroblewski Parameter
Using our continuum QNS, ratio χs/χu can be obtained.
m/Tc = 0.03 for u, d and m/Tc = 1 for s quark → λs(T ). Extrapolate to Tc.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 16
Wroblewski Parameter
Using our continuum QNS, ratio χs/χu can be obtained.
m/Tc = 0.03 for u, d and m/Tc = 1 for s quark → λs(T ). Extrapolate to Tc.
0.2 0.4 0.6 0.8 1λs
AGS Si-Au
AGS Au-Au
SpS Pb-Pb
SpS S-Ag
SpS S-S
RHIC Au-Au
Quenched QCD (T )c
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 16
Caveats
• Quenched approximation – Expect a shift of 5-10 % in full QCD.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 17
Caveats
• Quenched approximation – Expect a shift of 5-10 % in full QCD.
• Extrapolation to Tc – Straightforward but better to do it for full QCD
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 17
Caveats
• Quenched approximation – Expect a shift of 5-10 % in full QCD.
• Extrapolation to Tc – Straightforward but better to do it for full QCD .
• Preliminary results for Full 2-flavour QCD (Gavai & Gupta):
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 17
Caveats
• Quenched approximation – Expect a shift of 5-10 % in full QCD.
• Extrapolation to Tc – Straightforward but better to do it for full QCD .
• Preliminary results for Full 2-flavour QCD (Gavai & Gupta):
0.38
0.4
0.42
0.44
0.46
0.48
0.8 0.85 0.9 0.95 1 1.05 1.1
sλ
T/Tc
Nt=4, 2 flavour QCD, mu/Tc=0.1, ms/Tc=1.0
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 17
Caveats
• Quenched approximation – Expect a shift of 5-10 % in full QCD.
• Extrapolation to Tc – Straightforward but better to do it for full QCD .
• Preliminary results for Full 2-flavour QCD (Gavai & Gupta):
0.38
0.4
0.42
0.44
0.46
0.48
0.8 0.85 0.9 0.95 1 1.05 1.1
sλ
T/Tc
Nt=4, 2 flavour QCD, mu/Tc=0.1, ms/Tc=1.0
♣ Large finite volumeeffects below Tc♣ Up to 123 Lattices used.♣ Strong dependence onms expected.♣ Large finite a effects.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 17
• At SPS and RHIC, µB 6= 0 ; But observed λs is insensitive to it.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 18
• At SPS and RHIC, µB 6= 0 ; But observed λs is insensitive to it. .
– Theoretically, Screening mass- Susceptibility correlation and µ-dependenceresults of QCD-TARO on screening masses too suggest such an insensitivity.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 18
• At SPS and RHIC, µB 6= 0 ; But observed λs is insensitive to it. .
– Theoretically, Screening mass- Susceptibility correlation and µ-dependenceresults of QCD-TARO on screening masses too suggest such an insensitivity.
– Needs to be checked explicitly.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 18
• At SPS and RHIC, µB 6= 0 ; But observed λs is insensitive to it. .
– Theoretically, Screening mass- Susceptibility correlation and µ-dependenceresults of QCD-TARO on screening masses too suggest such an insensitivity.
– Needs to be checked explicitly.
• Assumed : characteristic time scale of plasma are far from the energy scales ofstrange or light quark production.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 18
• At SPS and RHIC, µB 6= 0 ; But observed λs is insensitive to it. .
– Theoretically, Screening mass- Susceptibility correlation and µ-dependenceresults of QCD-TARO on screening masses too suggest such an insensitivity.
– Needs to be checked explicitly.
• Assumed : characteristic time scale of plasma are far from the energy scales ofstrange or light quark production.
– Observation of spikes in photon production may falsify this.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 18
• At SPS and RHIC, µB 6= 0 ; But observed λs is insensitive to it. .
– Theoretically, Screening mass- Susceptibility correlation and µ-dependenceresults of QCD-TARO on screening masses too suggest such an insensitivity.
– Needs to be checked explicitly.
• Assumed : characteristic time scale of plasma are far from the energy scales ofstrange or light quark production.
– Observation of spikes in photon production may falsify this.
• Assumed : Chemical equilibration in the plasma.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 18
EoS for nonzero baryon density
Recall,
χfg··· =T
V
∂n logZ∂µf∂µg · · ·
=∂nP
∂µf∂µg · · ·. (8)
Thus χuuuu involves terms having fourth derivative w. r. to µ while χuudd onlysecond derivatives.
In continuum, f(aµ) = 1 + aµ→ f ′′(0) = 0.On lattice, in general, all derivatives exist and depend on the nature of function :prescription dependence !
Fodor-Katz used fHK and got µE = 725 MeV for Nt = 4. If they were to usefBG, then µE = 692 MeV.
Easy to show that f ′′(0) = 1 always but all higher derivatives depend on choice of
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 19
f . Thus, one can write
χuuuu = χHKuuuu + ∆f (3)(χuuT 2
)( 4N2t
), (9)
where ∆f (3) = f (3) − 1 is 2 for fBG.
Prescription dependence must go away for small a or large enough Nt.How large an Nt needed ? Nt ≥ 10, see below.
Defining
µ∗T
=
√12χuu/T 2
|χuuuu|, (10)
and ∆P = P (µ)− P (µ = 0), the Taylor series expansion for Pressure P for 2flavours can be re-organized as,
∆PT 4
=(χuuT 2
)(µT
)2[
1 +(µ/T
µ∗/T
)2
+O(µ4
µ4∗
)]. (11)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 20
Note that
• Each term in ∆P is prescription dependent, except the 1st. Physical ∆P maybe best obtained by evaluating each in continuum limit, as we do below. Moreimportant for larger µ.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 21
Note that
• Each term in ∆P is prescription dependent, except the 1st. Physical ∆P maybe best obtained by evaluating each in continuum limit, as we do below. Moreimportant for larger µ.
• The above is true for all physical quantities.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 21
Note that
• Each term in ∆P is prescription dependent, except the 1st. Physical ∆P maybe best obtained by evaluating each in continuum limit, as we do below. Moreimportant for larger µ.
• The above is true for all physical quantities.
• µ� µ∗ for prescription independence, provided still higher susceptibilities≤ χuuuu.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 21
Note that
• Each term in ∆P is prescription dependent, except the 1st. Physical ∆P maybe best obtained by evaluating each in continuum limit, as we do below. Moreimportant for larger µ.
• The above is true for all physical quantities.
• µ� µ∗ for prescription independence, provided still higher susceptibilities≤ χuuuu.
• (TE, µE) may be identified from the radius of convergence using many highersusceptibilities obtained in continuum limit term by term. What about series onfinite lattice and estimate of (TE, µE) as done presently ?
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 21
Our Results
Our results for χuuuu and ∆P : Gavai and Gupta, PR D68, ’03
0
0.2
0.4
0.6
0.8
1
1 1.5 2 2.5 3 3.5
χ uuu
u
�
T/Tc
Nt = 8 Nt = 10 Nt = 12 Nt = 14
Continuum limitIdeal Gas
0
0.1
0.2
0.3
0.4
0.5
1 1.5 2 2.5 3 P
/T
�
4∆
T/T c
0.15
0.44
0.73
1.03
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 22
Our Results
Our results for χuuuu and ∆P : Gavai and Gupta, PR D68, ’03
0
0.2
0.4
0.6
0.8
1
1 1.5 2 2.5 3 3.5
χ uuu
u
�
T/Tc
Nt = 8 Nt = 10 Nt = 12 Nt = 14
Continuum limitIdeal Gas
0
0.1
0.2
0.3
0.4
0.5
1 1.5 2 2.5 3 P
/T
�
4∆
T/T c
0.15
0.44
0.73
1.03
♥ Both reproduced in dimensional reduction (1 free parameter). Vuorinen, PR D68, ’03
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 22
Our Results
Our results for χuuuu and ∆P : Gavai and Gupta, PR D68, ’03
0
0.2
0.4
0.6
0.8
1
1 1.5 2 2.5 3 3.5
χ uuu
u
�
T/Tc
Nt = 8 Nt = 10 Nt = 12 Nt = 14
Continuum limitIdeal Gas
0
0.1
0.2
0.3
0.4
0.5
1 1.5 2 2.5 3 P
/T
�
4∆
T/T c
0.15
0.44
0.73
1.03
♥ Both reproduced in dimensional reduction (1 free parameter). Vuorinen, PR D68, ’03
♥ Our results for P agree with Fodor-Katz (PL B568, ’03) and the recentBielefeld results (PR D68, ’03).
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 22
Defining µ∗i to extend the definition of µ∗2 (ith term =(i+ 2)th term), the Taylorseries expansion for Pressure ∆P for 2 flavours can be re-organized as,
∆PT 4
=(χuuT 2
)(µT
)2[
1 +(µ
µ∗2
)2[
1 +(µ
µ∗4
)2 [1 + . . .
]]]. (12)
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 23
Defining µ∗i to extend the definition of µ∗2 (ith term =(i+ 2)th term), the Taylorseries expansion for Pressure ∆P for 2 flavours can be re-organized as,
∆PT 4
=(χuuT 2
)(µT
)2[
1 +(µ
µ∗2
)2[
1 +(µ
µ∗4
)2 [1 + . . .
]]]. (12)
130
140
150
160
170
600 800 1000 1200 1400 1600
T
�
µ
6/8 order4/6 order2/4 order
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 23
Summary
• Quark number susceptibilities −→ RHIC signal physics.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 24
Summary
• Quark number susceptibilities −→ RHIC signal physics.
• Continuum limit of χuu and χuuuu obtained in Quenched QCD. Broadly inagreement with BIR resummation and dimensional reduction. Still scope forimprovement in them ?
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 24
Summary
• Quark number susceptibilities −→ RHIC signal physics.
• Continuum limit of χuu and χuuuu obtained in Quenched QCD. Broadly inagreement with BIR resummation and dimensional reduction. Still scope forimprovement in them ?
• Continuum limit of χuu yields λs in agreement with RHIC and SPS results afterextrapolation to Tc. First full QCD investigations show encouraging trend.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 24
Summary
• Quark number susceptibilities −→ RHIC signal physics.
• Continuum limit of χuu and χuuuu obtained in Quenched QCD. Broadly inagreement with BIR resummation and dimensional reduction. Still scope forimprovement in them ?
• Continuum limit of χuu yields λs in agreement with RHIC and SPS results afterextrapolation to Tc. First full QCD investigations show encouraging trend.
• Pressure for nonzero µ obtained. At both SPS and RHIC, χuu is the majorcontribution.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 24
Summary
• Quark number susceptibilities −→ RHIC signal physics.
• Continuum limit of χuu and χuuuu obtained in Quenched QCD. Broadly inagreement with BIR resummation and dimensional reduction. Still scope forimprovement in them ?
• Continuum limit of χuu yields λs in agreement with RHIC and SPS results afterextrapolation to Tc. First full QCD investigations show encouraging trend.
• Pressure for nonzero µ obtained. At both SPS and RHIC, χuu is the majorcontribution.
• Phase diagram in T − µ on small Nt = 4 has begun to emerge: Differentmethods, same (TE, µE). Beware of prescription dependence and lookforward to larger Nt.
NEFT ’03, TIFR, December 18, 2003 R. V. Gavai Top 24
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