QCD critical point and event-by-event fluctuations M. Stephanov U. - - PowerPoint PPT Presentation

QCD critical point and event-by-event fluctuations

• M. Stephanov
• U. of Illinois at Chicago

QCD critical point and event-by-event fluctuations – p. 1/15

QCD phase diagram (a sketch)

crossover 1 0.1 T, GeV µB, GeV point critical quark(yonic) matter crystals, ..? phases: c.s.c., CFL nuclear matter vacuum hadron gas QGP

Models (and lattice) suggest the transition becomes 1st order at some µB. Can we observe the critical point in heavy ion collisions, and how?

QCD critical point and event-by-event fluctuations – p. 2/15

Critical point(s) in known liquids

Most liquids have a critical point (seen, e.g., by critical opalescence). Water: Does QCD “perfect liquid” have one?

QCD critical point and event-by-event fluctuations – p. 3/15

What do we need to discover the critical point?

LTE03 LR01 LR04 LTE08 LTE04 11 5 19 2 7.7 130 50 100 150 200 400 800 600 200

R H I C s c a n T, MeV µB, MeV

Experiments: RHIC, NA61(SHINE), FAIR/GSI, NICA Better lattice predictions, with known systematic uncertainties. Good experimental signatures.

QCD critical point and event-by-event fluctuations – p. 4/15

Critical fluctuations: theory

1 2 3 2 σ2

V ∼ (Ω′′)−1

Ω(σ0) µ = µCP (Ω′′)−1 → ∞ large equilibrium fluctuations Einstein, 1910: µ > µCP µ T 1 3 i.e., eS, or e−Ω/T

Consider an observable such as, e.g., σV =

• V σ, where σ ∼ ¯

ψψ.

P(σV ) ∼ number

• f states with that σV

σV µ < µCP

Why does CP seem to defy the central limit theorem? Because, correlation length ξ → ∞. This is a collective phenomenon. Indeed, the magnitude of fluctuations σ2

V ∼ ξ2.

QCD critical point and event-by-event fluctuations – p. 5/15

Fluctuation signatures

Experiments measure multiplicities Nπ, Np, . . . , mean pT , etc. These quantities fluctuate event-by-event. Fluctuation magnitude is quantified by e.g., (δN)2,(δpT )2. What is the magnitude of these fluctuations near the QCD C.P .? (Rajagopal-Shuryak-MS, 1998)

)

p

N ∆ Net Proton (

• 20
• 10

10 20

Number of Events

1 10

2

10

3

10

4

10

5

10

6

10 0-5% 30-40% 70-80% Au+Au 200 GeV

<0.8 (GeV/c)

T

0.4<p |y|<0.5

Universality tells us how it grows at the critical point: (δN)2 ∼ ξ2. Magnitude of ξ is limited < O(2–3 fm)

(Berdnikov-Rajagopal).

“Shape” of the fluctuations can be measured: non-Gaussian moments. As ξ → ∞ fluctuations become less Gaussian. Higher cumulants show even stronger dependence on ξ

(PRL 102:032301,2009):

(δN)3 ∼ ξ4.5, (δN)4 − 3(δN)22 ∼ ξ7 which makes them more sensitive signatures of the critical point.

QCD critical point and event-by-event fluctuations – p. 6/15

Higher moments (cumulants) and ξ

Consider probability distribution for the order-parameter field: P[σ] ∼ exp {−Ω[σ]/T} , Ω = Z d3x »1 2(∇σ)2 + m2

σ

2 σ2 + λ3 3 σ3 + λ4 4 σ4 + . . . – . ⇒ ξ = m−1

σ

Moments (connected) of q = 0 mode σV ≡ R d3x σ(x): κ2 = σ2

V = V T ξ2 ;

κ3 = σ3

V = 2V T 2 λ3 ξ6 ;

κ4 = σ4

V c ≡ σ4 V − 3σ2 V 2 = 6V T 3 [ 2(λ3ξ)2 − λ4 ] ξ8 .

Tree graphs. Each propagator gives ξ2.

+

Scaling requires “running”: λ3 = ˜ λ3T(Tξ)−3/2 and λ4 = ˜ λ4(Tξ)−1, i.e., κ3 = σ3

V = 2V T 3/2 ˜

λ3 ξ4.5 ; κ4 = 6V T 2 [ 2(˜ λ3)2 − ˜ λ4 ] ξ7 .

QCD critical point and event-by-event fluctuations – p. 7/15

Moments of observables

Example: Fluctuation of multiplicity is the fluctuation of occup. numbers, δN = X

p δnp.

Any moment of the multiplicity distribution is related to a correlator of δnp: κ3π = (δN)3 = X

p1

X

p2

X

p3δnp1δnp2δnp3 ,

where P

p = V

R

d3p (2π)3 .

np fluctuates around ¯ np(m), which also fluctuates: δm = gδσ, i.e., δnp = δn0

p + ∂¯ np ∂m g δσ .

δnp1δnp2δnp3σ = 2λ3 V 2T „ g m2

σ

«3 v2

p1

γp1 v2

p2

γp2 v2

p3

γp3

v2

p = ¯

np(1 ± ¯ np), γp = (dEp/dm)−1

Similarly for (δN)4c. Since (δN)3 scales as V 1 we suggest ω3(N) ≡ (δN)3 ¯ N which is V 0.

QCD critical point and event-by-event fluctuations – p. 8/15

Energy scan and fluctuation signatures: notes

crossover (˜ λ3 = 0) 1st order µB T contours of equal ξ with max ξ vs √s freeze-out point freeze-out points critical point

Higher moments provide more sensitive signatures. However, this comes at a price: Harder to predict – more theoretical uncertainties. Signal/noise lower for higher moments. Sensitivity could mean narrower critical region: if it rises faster with rising ξ, it also falls faster with falling ξ. One could, e.g., try combining various higher moments to eliminate uncer- tainties.

QCD critical point and event-by-event fluctuations – p. 9/15

Using ratios and mixed moments

Athanasiou, Rajagopal, MS (2010)

The dominant dependence on µB (i.e., on √s) is from two sources ξ and np, e.g., κ3p ∼ ˜ λ3 g3

p ξ4.5n3 p.

ξ(µB) has a peak at µB = µcritical

B

; nB ∼ eµcritical

B

/T determines the height of the peak;

• ther factors: g3

p and ˜

λ3 depend on µB weaker. Leading dependence on µcritical

B

can be cancelled in ratios. E.g., κ3p Np „Nπ Np «2 ∼ ˜ λ3 g3

p ξ4.5

Unknown/poorly known coupling parameters gp or gπ can be also cancelled in ratios. E.g., κ4p κ2

2p

κ2

2π

κ4π ,

• r

κ3

4p

κ4

3p

κ4

3π

κ3

4π

. These ratios are 1 when critical point fluctuations dominate. Mixed moments allow more possibilities. E.g., κ2

2p2π

κ4pκ4π . Mixed moments have no trivial Poisson contribution.

QCD critical point and event-by-event fluctuations – p. 10/15

Using ratios and mixed moments

Athanasiou, Rajagopal, MS (2010)

The dominant dependence on µB (i.e., on √s) is from two sources ξ and np, e.g., κ3p ∼ ˜ λ3 g3

p ξ4.5n3 p.

ξ(µB) has a peak at µB = µcritical

B

; nB ∼ eµcritical

B

/T determines the height of the peak;

• ther factors: g3

p and ˜

λ3 depend on µB weaker. Leading dependence on µcritical

B

can be cancelled in ratios. E.g., κ3p Np „Nπ Np «2 ∼ ˜ λ3 g3

p ξ4.5

Unknown/poorly known coupling parameters gp or gπ can be also cancelled in ratios. E.g., κ4p κ2

2p

κ2

2π

κ4π ,

• r

κ3

4p

κ4

3p

κ4

3π

κ3

4π

. These ratios are 1 when critical point fluctuations dominate. Mixed moments allow more possibilities. E.g., κ2

2p2π

κ4pκ4π . Mixed moments have no trivial Poisson contribution.

QCD critical point and event-by-event fluctuations – p. 10/15

Experiment (pre-QM)

(GeV)

NN

s 4 5 10 20 100 200

2

σ κ

1 2 3

AMPT AMPT (SM) Hijing UrQMD

Au+Au Collisions

Critical Point Search

720 420 210 54 20

(MeV)

B

µ

10

Therminator <0.8 (GeV/c)

T

0.4<p STAR Data |y|<0.5

Lattice QCD

STAR Preliminary

LTE03 LR01 LR04 LTE08 LTE04 50 100 150 200 400 800 600 200

R H I C s c a n T, MeV µB, MeV

(κσ2 = κ4/κ2 ≈ ω4 if κ2 ≈ N). No critical signatures seen at those values of µB. Consistent with expectations that µcritical

B

> 200 MeV. What is happening at √s = 19.6 GeV? Low statistics. Large positive contribution to Poisson is excluded, but large negative — is not.

QCD critical point and event-by-event fluctuations – p. 11/15

Negative kurtosis?

Could the critical contribution to kurtosis be negative? (MS, arxiv:1104.1627) (δN)4c = N + σ4

V c

„ g T Z

p

v2

p

γp «4 + . . . , σ4

V c = 6V T 2 [ 2˜

λ2

3 − ˜

λ4 ] ξ7 . On the crossover line ˜ λ3 = 0 by symmetry, while ˜ λ4 ≈ 4. > 0. P(σV ): → Thus σ4

V c < 0 and ω4(N) < 1 on the crossover line. And around it.

Universal Ising eq. of state: M = Rβθ, t = R(1 − θ2), H = Rβδh(θ)

here κ4 is κ4(M) ≡ M4c

0.4 0.2 0.0 0.2 0.4 0.6 20 20 40 60 80 100 120 t Κ4

QCD critical point and event-by-event fluctuations – p. 12/15

Implications for the energy scan

µB, GeV , GeV T 1 0.1 critical point freezeout curve nuclear matter QGP hadron gas CFL+

QCD critical point and event-by-event fluctuations – p. 13/15

Implications for the energy scan

H t µB, GeV , GeV T 1 0.1 critical point nuclear matter QGP hadron gas CFL+

QCD critical point and event-by-event fluctuations – p. 13/15

Implications for the energy scan

H t µB, GeV , GeV T 1 0.1 critical point freezeout curve nuclear matter QGP hadron gas CFL+

19

QCD critical point and event-by-event fluctuations – p. 13/15

Implications for the energy scan

H t µB, GeV , GeV T 1 0.1 critical point freezeout curve nuclear matter QGP hadron gas CFL+

?

19 11

If the kurtosis stays significantly below Poisson value in 19 GeV data, the logical place to take a closer look is between 19 and 11 GeV.

QCD critical point and event-by-event fluctuations – p. 13/15

Implications for the energy scan

ω4 √s

baseline

If the kurtosis stays significantly below Poisson value in 19 GeV data, the logical place to take a closer look is between 19 and 11 GeV.

QCD critical point and event-by-event fluctuations – p. 13/15

Concluding remarks

Critical point is a special singular point on the phase diagram, with unique

• signatures. This makes its experimental discovery possible.

Locating the point is still a challenge for theory. The search for the critical point is on. We are waiting for RHIC results at µB > 200 MeV are just in! If kurtosis stays significantly below Poisson value at √s = 19 GeV, then the critical point could be close, to the right, on the phase diagram. Then: √s = 15 GeV?

QCD critical point and event-by-event fluctuations – p. 14/15

QM notes

Potential sources of negative contribution at high baryon density: Fermi statistics: ω4 ≈ 1 − 7npp (small, but increases with µB). Baryon number conservation (talk by M. Nahrgang).

QCD critical point and event-by-event fluctuations – p. 15/15