Using&Higher&Moments&of&Fluctua5ons&and&their& Ra5os&in ... · Using&Higher&Moments&of&Fluctua5ons&and&their& Ra5os&in&the&Search&for&the&QCD&Cri5cal&Point& Chrisana& Athanasiou,&MIT&

Using Higher Moments of Fluctua5ons and their Ra5os in the Search for the QCD Cri5cal Point

Chris5ana Athanasiou, MIT 4

work with: Krishna Rajagopal (MIT) Misha Stephanov (University of Illinois)

Outline

•  Introduc)on •  Cri)cal Contribu)on to Par)cle Mul)plicity Fluctua)ons

•  Ra)os of Fluctua)on Observables •  Summary

QCD Phase Diagram

vacuum

Quark-Gluon Plasma

Critical point

μB / MeV

T / MeV

~ 170

~ 940

nuclear matter

0

Hadron gas Color Superconductor

crossover

Models

LaNce simula5ons

Heavy-­‐Ion Collision Experiments

•  Loca)ng the cri)cal point from first-­‐principles – hard   Heavy-­‐Ion Collision Experiments

•  RHIC: Au-­‐Au collisions at √smax = 200 GeV

•  Momentum asymmetry collec)ve flow   strongly-­‐coupled QGP

~ 940

vacuum

Quark-Gluon Plasma

Critical point

μB / MeV

T / MeV

~ 170

nuclear matter

0

Hadron gas Color Superconductor

crossover

QCD Phase Diagram

√s

Heavy-­‐Ion Collision Experiments -­‐ con5nued

•  As QGP expands and cools, it follows trajectories with approx. nB/s = const.

Quark-Gluon Plasma

μB / MeV

T / MeV

~ 170

~ 940

nuclear matter

0

Color Superconductor

QCD Phase Diagram

√s

Critical point

Hadron gas

crossover

vacuum

Heavy-­‐Ion Collision Experiments 2

•  As QGP expands and cools, it follows trajectories with approx.

•  Chemical freeze-­‐out: system dilute enough that par)cle numbers freeze

nB/s = const.

•  To maximize cri)cal point (CP) effects vary to get freeze-­‐out point near CP

√s

Event-­‐by-­‐Event fluctua)ons

•  Detector “sees” par)cle mul)plici)es from freeze-­‐out condi)ons

•  Find observables that are sensi)ve to proximity to the CP

Outline

•  Introduc)on •  Cri)cal Contribu)on to Par)cle Mul)plicity Fluctua)ons

•  Ra)os of Fluctua)on Observables •  Summary

•  Cri)cal mode -­‐ σ : order parameter of the chiral phase transi)on

•  Correla)on length diverges at the CP ξ = m−1σ

•  Develops long wavelength correla)ons at the CP

Ω(σ) =

d3x

12(∇σ)2 +

m2σ

2σ2 +

λ3

3σ3 +

λ4

4σ4 + ...

.

•  Effec)ve ac)on

Cri5cal Mode Fluctua5ons Cri5cal Mode

•  Near the CP: with dimensionless and known in the

Ising universality class

λ3 = λ3 T (T ξ)−3/2, λ4 = λ4 (T ξ)−1

4 λ4 200 λ3 8,

Cri5cal Mode Fluctua5ons

•  at CP in the thermodynamic limit ξ →∞

•  Finite system life)me compared to away from the CP (Berdnikov, Rajagopal 00)

ξmax ∼ 2 fm

∼ 0.5 fm

•  Cri)cal mode fluctua)ons affect   Par)cle mul)plicity fluctua)ons

  Momentum distribu)ons   Ra)os, etc…

of these par)cles.

•  σ couples to pions and protons: Lσππ,σpp = 2 G σ π+π− + g σ p p

(t, V →∞)

3 2 1 0 1 2 3

10

20

30

40

Number

of events

Number

of protons

Measuring fluctua5ons in par5cle mul5plici5es

measure the mean, variance, skewness, etc…

•  Can repeat these calcula)ons for pions, net protons, etc •  Want to obtain the cri)cal contribu)on to these quan))es

•  We will use cumulants, e.g.:

κ2 = N2, κ3 = N3, κ4 ≡ N4 = N4 − 3N22

mσ = ξ−1, γk =

k2 + m2/m, v2k = nk(1± nk),

gπ = G/mπ, d = 2, λ3 = λ3 T (T ξ)−3/2, λ4 = λ4 (T ξ)−1

δnk1δnk2σ = d2 1m2

σV

g2

T

v2k1

γk1

v2k2

γk2

Cri5cal contribu5on to pion/proton correlators

(Rajagopal, Shuryak, Stephanov 99, Stephanov 08)

ξ2

ξ7

ξ9/2

δnk1δnk2δnk3σ = d3 2λ3

V 2T

g

m2σ

3 v2k1

γk1

v2k2

γk2

v2k3

γk3

δnk1δnk2δnk3δnk4σ = d4 6V 3T

2

λ3

mσ

2

− λ4

×

g

m2σ

4 v2k1

γk1

v2k2

γk2

v2k3

γk3

v2k4

γk4

+ …

Net protons and mixed correlators

•  Note: correlators depend on 5 parameters:

which have large uncertain)es G, g, ξ, λ3, λ4

•  Net protons: Adapt previous expressions by replacing:

Np −Np

•  Can also calculate mixed correlators, e.g. 2 pion – 2 proton:

δnπp1

δnπp2

δnpp3

δnpp4σ = d2

pd2π

6V 3T

2

λ3

mσ

2

− λ4

gp gπ

m4σ

2 vπ 2p1

γπp1

vπ 2p2

γπp2

vp 2p3

γpp3

vp 2p4

γpp4

vp 2k → vp 2

k − vp 2k

Calcula5ng mul5plicity cumulants

•  Second cumulant – variance:

Poisson -­‐ Bose-­‐Einstein effects -­‐ Other interac)ons -­‐ Etc..

ignore

ω2p, σ =κ2p,σ

Np= d2

pg2ξ2

T

k

v2k

γk

2

knk

−1

•  Normalizing:

ωipjπ =κipjπ

Ni

i+jp N

ji+jπ

•  For mixed cumulants with i protons and j pions:

•  Non-­‐cri)cal contribu)on to ωipjπ = δi,i+j + δj,i+j + (few %)

κ2p,σ = (δNp)2σ =

k1

k2

δnk1δnk2σ ∝ V 1

Mul5plicity cumulants – cri5cal point signature

•  Higher cumulants depend stronger on ξ: ω2 ∝ ξ2,

ω3 ∝ ξ9/2,

ω4 ∝ ξ7

•  As we approach the CP ξ increases and then decreases as we move away from it

•  CP signature: Non-­‐monotonic behavior, as a func5on of collision energy, of mul5plicity cumulants

ξ(µB) =2 fm

(1 + (µB − 400)2/W 2)1/3

0.1 0.2 0.3 0.4 0.5 0.6ΜBGeV

0.5

1.0

1.5

2.0

Ξfm0.1 GeV

0.2 GeV

!

•  E.g. toy example

0.0 0.1 0.2 0.3 0.4 0.5ΜBGeV

2

4

6

8

10

Ω

Ω1 p2Π

Ω3 Π

Ω2 p1Π

Mul5plicity cumulants – example plots

0.0 0.1 0.2 0.3 0.4 0.5ΜBGeV0

100

200

300

400

Ω4 p

g7, 0.2 GeVg7, 0.1 GeVg5, 0.1 GeV

0 0.1 0.2 0.30

1

2

0.0 0.1 0.2 0.3 0.4 0.5ΜBGeV

10

20

30

40

50

60

Ω

Ω4 Π

Ω2 p2Π

Ω3 p

Parametriza)on (Cleymans et al 05):

T (µB) = a− bµ2B − cµ4

B

a = 0.166 GeV, b = 0.139 GeV−1, c = 0.053 GeV−3

and using λ3 = 4, λ4 = 12, G = 300 MeV, g = 7

Data on net proton cumulants

κ σ2 ≡ κ4

κ2where

(STAR Collabora)on 2010)

Cri5cal contribu5on to proton ω4

0.0 0.1 0.2 0.3 0.4 0.5ΜBGeV0

100

200

300

400

Ω4 p

g7, 0.2 GeVg7, 0.1 GeVg5, 0.1 GeV

0 0.1 0.2 0.30

1

2

Mul5plicity cumulants – movie

Changing the cri)cal μB – the loca)on of the CP:

Outline

•  Introduc)on •  Cri)cal Contribu)on to Par)cle Mul)plicity Fluctua)ons

•  Ra)os of Fluctua)on Observables •  Summary

Uncertain5es of parameters

•  Cumulants depend on 5 non-­‐universal parameters: κ2π ∼ V T−1G2ξ2N2

π ,

κ3π ∼ V T−3/2G3λ3ξ9/2N3

π ,

κ4π ∼ V T−2G4(2λ23 − λ4)ξ7N4

π

•  have large uncertain)es hard to predict the cri)cal contribu)on to cumulants

G, g, ξ, λ3, λ4

•  By taking ra)os of cumulants can cancel some parameter dependence

minimize observable uncertain)es

Ra5os of mul5plicity cumulants

No parameter dependence

Ra)os taken aher subtrac)ng Poisson and defined λ4 ≡ 2λ2

3 − λ4

r = i + j

r1 =skewnesspskewnessπ

, r2 =kurtosispkurtosisπ

,

Parameter independent ra5os

•  Parameter and energy independent ra)os:

where skewness =κ3

κ3/22

, kurtosis =κ4

κ22

•  All equal to 1 if CP contribu)on dominates

•  How to use these ra)os: •  If one sees peaks in the measured cumulants at some μB •  Calculate these ra)os around the peak •  If equal to 1 Parameter independent way of verifying

that the fluctua)ons you see are due to the CP

r4 =κ2

2p2π

κ4πκ4p

r1 = (Nπ/Np)1/2 , r2 = Nπ/Np,•  Poisson contribu)on: r3 = r4 = 0

r3 =κ2p1π

κ2/33p κ1/3

3π

,

Constraining parameters

•  If CP found, can constrain parameters by measuring cumulant ra)os near the CP

•  Parameters appear in certain combina)ons in the cumulants can only constraint 4 independent (but not unique) combina)ons

κ4pN2π/κ4πκ2

2p,

•  For example, some choices are:

1.  using or 2.  using or 3.  using 4.  using

G ξ

κ3p1πNp/κ4pNπ,G/g

λ4/λ2

3 κ2pκ4p/κ23p,

κ2p2πNπ/κ4πκ2p

κ2p2πN2p /κ4pN

2π

κ3pN3/2p /κ9/4

2p N1/4π .λ2

3/g3

Outline

•  Introduc)on •  Cri)cal Contribu)on to Par)cle Mul)plicity Fluctua)ons

•  Ra)os of Fluctua)on Observables •  Summary

Summary

•  We used par)cle mul)plicity fluctua)ons as a probe to the loca)on of the CP

•  Higher cumulants of event-­‐by-­‐event distribu)ons are more sensi)ve to cri)cal fluctua)ons

•  Constructed cumulant ra)os to iden)fy the CP loca)on with reduced parameter uncertain)es

•  CP signature: Non-­‐monotonic behavior, as a func)on of collision energy, of mul)plicity cumulants

•  If CP is found, showed how to use cumulant ra)os to constraint the values of the non-­‐universal parameters

Thank you!