The critical end point of QCD
R. V. Gavai and Sourendu Gupta
ILGTI: TIFR

ATHIC 2008
Tsukuba, Japan
October 14, 2008

SG (ILGTI: TIFR)

The critical end point of QCD

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Crawling towards the continuumµB 6= 0 through Taylor expansions of thermodynamic quantities.
Before this year: state of the art lattice computations of physics forµB 6= 0 used lattice cutoff Λ = 4T ≃ 800 MeV near Tc .
Our earlier computation used mπ ≃ 230 MeV and spatial sizes with
LT = 2, 3, 4 and 6. This enabled extrapolation to thethermodynamic limit, i.e., L → ∞.
Now: new computations with Λ = 6T ≃ 1200 MeV near Tc (Gavaiand SG, arXiv:0806.2233 [hep-lat]).
mπ remains unchanged (230 MeV), but spatial volumes are somewhat
smaller (LT = 2, 3 and 4). No extrapolation to L → ∞ yet.
20000–50000 configurations (≥ 100 independent) at each coupling;
stochastic determination of traces with 500 random vectors on eachconfiguration. (Gavai and SG, Phys. Rev. D 68, 2003, 034506.)
SG (ILGTI: TIFR)

The critical end point of QCD

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1

Quark Number Susceptibilities

2The Critical End Point

3Series sums and Padé resummations

4Experiment

SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities

Outline1Quark Number Susceptibilities

2The Critical End Point

3Series sums and Padé resummations

4Experiment

SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities

What is a QNS?
Taylor coefficient of the pressure in Nf = 2 QCD isX
1P(T , µu , µd ) =χnu ,nd (T )µnuu µndd ,n!n!
u dn ,nu

dand, since the two quark flavours are degenerate, χnu ,nd = χnd ,nu . Tradeµu,d for µB,Q . ThenχB =2(χ20 + χ11 ) = 2χBQ
9χQ =1(5χ20 − 4χ11 ) .
9Transforming to µB,I3 , one has
χBI3 = 0,O 11
SG (ILGTI: TIFR)

O2

χI3 =1(χ20 − χ11 ) .
2O 22
The critical end point of QCD

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Quark Number Susceptibilities

Off-diagonal QNS

χ11 /Τ2-0.02

Nt=4

Nt=60-0.04-0.06-0.08-0.10
0.6

0.8

11.2

1.4 1.6
T/Tc

1.82

2.2

2.4

Sees only hO11 i. No evidence for lattice spacing effects.
SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities

Diagonal QNS
2.0
Nt=4

1.8
1.6

χ20 /Τ2

1.4
Nt=6

1.2
1.0
0.8
0.6
0.4
0.2
0
0.6

0.8

1

1.2

1.4 1.6
T/Tc

1.8

2

2.2

2.4

Sees hO11 i and hO2 i. Second expectation value is cutoff dependent. Also,
has a cross over. We look at its susceptibility hO22 ic to identify Tc .
SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities“Susceptibility” of QNS: hO22 ic — 4th order QNS
2.0
Nt=4(T/V)<O22 >

1.6

1.2
Nt=6
0.8

0.4

0
0.5

1

1.5
T/Tc

2

2.5

Peak at the same coupling as peak of χL . Within the 1% precision ofT /Tc , the two quantities peak at the same coupling. See Gavai andGupta, PR D72 (2005) 054006.
SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities

Diagonal fourth-order QNS
87

Nt=46χ40543

Nt=62
100.5

11.5
T/Tc

22.5

Non-zero for T > Tc . Has contribution from hO4 i, which has
non-vanishing value for the ideal gas.
SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities

The operator O4
2.0
Nt=4(T/V)<O4 >

1.6

1.2
Nt=60.8

0.4

00.5

1

1.5
T/Tc

22.5

Rapid cross over from a small value in the hadronic phase to a
non-vanishing value for the ideal gas.
SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities“Susceptibility” of O4 : hO44 ic — 8th order QNS
800
700(T/V) <O44>

600
500
400
300
200
100
0
0.5

1

1.5
T/Tc

22.5

This quantity peaks at the same coupling as χL and hO22 ic . Within theprecision of our measurement there is no dependence of the cross overcoupling on these observables.
SG (ILGTI: TIFR)

The critical end point of QCD

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Quark Number Susceptibilities

The operator O6 — 6th order QNS
2.0

1.6(T/V)<O6 >Nt=4
1.20.8
Nt=60.4

00.5

1

1.5
T/Tc

2

2.5

The operator expectation value hO6 i has structure below Tc and hence its“susceptibility” cannot be used to probe the cross over coupling. Similar
observation for hO8 i.
SG (ILGTI: TIFR)

The critical end point of QCD

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The Critical End Point

Outline1

Quark Number Susceptibilities

2

The Critical End Point

3Series sums and Padé resummations

4Experiment

SG (ILGTI: TIFR)

The critical end point of QCD

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The Critical End Point

Finite size effects
At critical point correlation length becomes infinite, appropriate
susceptibilities diverge and free energy becomes singular ... in theinfinite volume limit (van Hove’s theorem).
No numerical computation ever performed on infinite volumes.
Deduce the existence of a critical point through extrapolations: finite
size scaling (FSS) well developed for direct simulations.
Example: peak of susceptibility scales as power of volume. Smaller
effect: position of peak shifts from its infinite volume position by adifferent power of volume—χmax (L) ∝ Lp ,Tc (L) = Tc − a/Lq ,(p, q > 0).

FSS not well developed for series expansions; some aspects are known.
SG (ILGTI: TIFR)

The critical end point of QCD

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The Critical End Pointln χ (n)Series expansionsn (L3)*n (L2)*n (L1)*L1 < L 2< L 3n

For the divergent quantity: χB (T , µB ) =volume effects in the series coefficients.
SG (ILGTI: TIFR)(n)nn χ (T )µB , the leading finiteP

The critical end point of QCD

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The Critical End Point

Nt = 46 24mπ Ns
68105µB/T43210 51015Ns

2025

At fixed T /Tc ≃ 0.95. Circles: ratio of order 0 and 2; boxes: ratio oforder 2 and 4. Gavai and SG, Phys. Rev. D 71, 2005, 114014.
SG (ILGTI: TIFR)

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The Critical End Point

Nt = 6: Radius of convergence
1.4
1.2µ/(3Τ)

10.8
0.6
0.40.2LT=4; T/Tc=0.99
001234n5

Filled symbols: (χ(0) /χ(n) )1/n . Open symbols:
SG (ILGTI: TIFR)

The critical end point of QCD

678p
χ(n−1) /χ(n+1) .
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The Critical End Point

Nt = 6: Radius of convergence
1.4
1.2µ/(3Τ)

1
0.8
0.6
0.4
0.2
LT=4; T/Tc=0.97
001

234n5

Filled symbols: (χ(0) /χ(n) )1/n . Open symbols:
SG (ILGTI: TIFR)

The critical end point of QCD

678p
χ(n−1) /χ(n+1) .
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The Critical End Point

Nt = 6: Radius of convergence
1.4
1.2µ/(3Τ)

10.8
0.6
0.4
0.2
LT=4; T/Tc=0.94
001234n5

Filled symbols: (χ(0) /χ(n) )1/n . Open symbols:
SG (ILGTI: TIFR)

The critical end point of QCD

678p
χ(n−1) /χ(n+1) .
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The Critical End Point

Nt = 6: Finite size scaling
1.4
1.2µ/(3Τ)

10.8
0.6
0.4
0.2
LT=2, T/Tc=0.94 Preliminary
001234n5Filled symbols: (χ(0) /χ(n) )1/n . Open symbols:
SG (ILGTI: TIFR)

The critical end point of QCD

678p
χ(n−1) /χ(n+1) .
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The Critical End Point

Nt = 6: Finite size scaling
1.4
1.2µ/(3Τ)

1
0.8
0.6
0.4
0.2
LT=4, T/Tc=0.94
0012

34n5

Filled symbols: (χ(0) /χ(n) )1/n . Open symbols:
SG (ILGTI: TIFR)

The critical end point of QCD

678p
χ(n−1) /χ(n+1) .
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The Critical End Point“Kurtosis” and radius of convergence
5
Nt=6
Nt=44µ(2)/T
*32

10
0.6

0.8

1.0

1.2

1.4 1.6T/Tc

1.82

2.2

2.4

p
Plot of 12χ(2) /χ(4) . Lattice spacing dependence quantifies possible
systematic errors. Related to Kurtosis: experimental measurement
possible.
SG (ILGTI: TIFR)

The critical end point of QCD

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The Critical End Point

Critical end point
Multiple criteria agree:
Small window in T where all the coefficients are positive.
Stability of radius of convergence with order and estimator
Finite size effects follow correct trend; more planned for the future.
Pinching of the radius of convergence with T : (experimentalmeasurement of Kurtosis?)

This gives
TE= 0.94 ± 0.01
TcandµEB= 1.8 ± 0.1
TEwith Nf = 2 when mπ /mρ ≃ 0.3 at a finite volume with LT = 4 andlattice cutoff of a = 1/6T E .
For a lattice cutoff of a = 1/4T E at the same renormalized quark
mass and on the same volume we had found a similar value for
T E /Tc and µEB /T E = 1.3 ± 0.3. Extrapolation to L → ∞ reduced
this to 1.1 ± 0.1.
SG (ILGTI: TIFR)

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Series sums and Padé resummations

Outline1Quark Number Susceptibilities

2

The Critical End Point

3

Series sums and Padé resummations

4Experiment

SG (ILGTI: TIFR)

The critical end point of QCD

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Series sums and Padé resummations

Fluctuations of Baryon number
Suggestion by Stephanov, Rajagopal, Shuryak; Asakawa, Heinz, Muller;
Jeon, Koch(∆B)2P(∆B) = exp −.
2VT χBExtrapolate χB to finite chemical potential: peak at Tc ?
1Series coefficients

0.8

0.6

0.4

0.2

00.75
SG (ILGTI: TIFR)

11.25

1.5

The critical end T/Tc
point of QCD

1.75

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Series sums and Padé resummations

Sum the series5χ Β /Τ 2

4n=4

3
2
10n=2
0

0.5

1

1.5µ/Τ

2

2.5

3Summing the series never shows critical behaviour: sum is a polynomial
and smoothly behaved. The sum peaks at Tc : incorrect (see SG, SEWM
2006).
SG (ILGTI: TIFR)

The critical end point of QCD

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Series sums and Padé resummations

Critical fluctuations
5P01χ Β /Τ 24P1132
1000.511.5µ/Τ

22.5

3Use Padé approximants for the extrapolations: divergence only at thecritical end point. Error propagation requires care: see arXiv:0806.2233[hep-lat].
SG (ILGTI: TIFR)

The critical end point of QCD

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Experiment

Outline1

Quark Number Susceptibilities

2The Critical End Point

3

Series sums and Padé resummations

4Experiment

SG (ILGTI: TIFR)

The critical end point of QCD

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Experiment

Experimental measurement of critical index
Near the critical point
χ(2) ∝ |µ − µEB |−γ ,χ(4) ∝ |µ − µEB |−γ−2(γ > 0).

The Kurtosis diverges:
K = −1 +χ(4)∝ |µ − µc |γ−2 ;3[χ(2) ]2(recall that P = P0 + p|µ − µc |−γ+2 is non-analytic but non-divergent).
Along freezeout trajectory in an energy scan, the Kurtosis isnon-monotonic. Should be visible in experiment. Look for non-Gaussian(E-to-E) fluctuations. Big question: what can hide non-Gaussian
fluctuations?
Exciting possibility: can one measure a critical index in experiments?
SG (ILGTI: TIFR)

The critical end point of QCD

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