Search for the critical point of QCDsgupta/courses/qgp2016/lec4.pdfBasics Experiment Statistics...

Basics Experiment Statistics Summary

Search for the critical point of QCD

Sourendu Gupta

SERC School in EHEPDelhi University28 April, 2016

Sourendu Gupta QGP@SERC-EHEP: Lecture 4


Thermodynamics and statistical mechanics

Experiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point

Error analysis for ratios

Summary



Outline




Summary



Can experiments see the critical point?

T

µ

freezeout



Thermodynamic fluctuations

Thermodynamics deals with matter in bulk: Avogadro number ofparticles (NA = 0.602× 1024). Short range interactions, so energyis additive. Particle number, charge, etc are all additive. All giverise to thermodynamic extensive quantities.

Equilibrium system: all extensive quantities have a precise andmeasurable value. Disturb the system a little and let it settle toequilibrium: new precise and measurable values.

Thermal fluctuations deduced by Boltzmann, Maxwell.Observation (1827) by Robert Brown of position fluctuation oftracer particles in water explained in 1905 by Einstein as due tothermal fluctuations of molecules.



Grand canonical ensemble (GCE)

To study fluctuations, divide material into small part which isobserved (system) and large unobserved part (heat bath).Extensive thermodyamic variables of system fixed; but microscopicvariables can still fluctuate. Need a collection (ensemble) ofdifferent realizations of the microscopic states in thermodynamicequilibrium.

Grand canonical ensemble: all conserved quantities allowed tofluctuate by exchanging energy and material with the heat bath.Possible to realize by observing a small portion of the fireball byputting angular cuts.

Which ensemble is created in an experiment? Different realizationsof a system may be same system at different times: normal intabletop experiments. More natural in colliders to usemeasurements in different collisions to have many realizations ofthe same system.



Fluctuations and correlations



Fluctuations and correlations

ξ

For almost ideal gases ξ ≃ λ.



Gaussian fluctuations

When Vobs ≫ ξ3 then distribution of any extensive quantity isGaussian: central limit theorem. Mean value of Gaussian is theextensive thermodynamic equilibrium value. Thermodynamicquantities alone have little connection to microscopic properties.

Study of microscopic properties therefore requires study offluctuations. Width of Gaussian related to specific heat. Widthproportional to

√NA, therefore usually negligible, with respect to

mean.

When system is small enough then deviations from Gaussian arevisible. These are reflections of microscopic physics.

Central limit theorem fails when ξ ≃ L, i.e., ξ3 ≃ Vobs . Then nodifference between microscopic and macroscopic.



Non-linear susceptibilities

Taylor expansion of the pressure in µB

P(T , µB +∆µB)/T4 =

∑

n

1

n!

[

χ(n)(T , µB)Tn−4]

(

∆µBT

)n

has Taylor coefficients called non-linear susceptibilities (NLS).When µB = 0 they can be computed directly on the lattice,otherwise reconstructed from such computations.Gavai, SG: 2003, 2010

Cumulants of the event-to-event distribution of baryon number aredirectly related to the NLS:

[B2] = T 3V

(

χ(2)

T 2

)

, [B3] = T 3V

(

χ(3)

T

)

, [B4] = T 3Vχ(4).

V unknown, can be removed by taking ratios.(SG: 2009)



What are the cumulants?

[B] = 〈B〉[B2] = 〈(B − 〈B〉)2〉[B3] = 〈(B − 〈B〉)3〉

[B4] = 〈(B − 〈B〉)4〉 − 3[B2]2

[B5] = 〈(B − 〈B〉)5〉 − 10[B2] [B3][B6] = 〈(B − 〈B〉)6〉 − 15[B2] [B4]− 10[B3]2 + 30[B2]3



Tests and assumptions

m1 :[B3]

[B2]=

χ(3)(T , µB)/T

χ(2)(T , µB)/T 2

m2 :[B4]

[B2]=

χ(4)(T , µB)

χ(2)(T , µB)/T 2

m3 :[B4]

[B3]=

χ(4)(T , µB)

χ(3)(T , µB)/T

Also for cumulants of electric charge, Q, and strangeness, S .

1. Two sides of the equation equal if there is thermal equilibriumand no other sources of fluctuations.

2. Right hand side computed in the grand canonical ensemble(GCE). Can observations simulate a grand canonicalensemble? What T and µB?

3. Why should hydrodynamics and diffusion be neglected?



Thermodynamics and fluctuations

Observations

In a single heavy-ion collision, each conserved quantity (B , Q, S) isexactly constant when the full fireball is observed. In a small partof the fireball they fluctuate: from part to part and event to event.




Observations


Thermodynamics

If ξ3 ≪ Vobs ≪ Vfireball , then fluctuations can be explained in thegrand canonical ensemble: energy and B, Q, S allowed to fluctuatein one part by exchange with rest of fireball (diffusion: transport).




Observations


Thermodynamics

If ξ3 ≪ Vobs ≪ Vfireball , then fluctuations can be explained in thegrand canonical ensemble: energy and B, Q, S allowed to fluctuatein one part by exchange with rest of fireball (diffusion: transport).

Comparison

When Vobs ≪ Vfireball , Gaussian as Vobs/ξ3 → ∞. Finite sizeeffects are mainly controlled by NLS. Otherwise system is in thecritical regime.



Ensuring control

Ensembles used in tabletop experiments keep intensive quantitiesfixed by controlling the heatbath: fixed temperature, fixedcomposition etc.

Collider experiments find this harder. Events may be classified byimpact parameter, but there may still be fluctuations in the totalenergy content of the fireball. Similarly, the chemical compositionof the fireball may need to be controlled. Compositional controlmay be harder to achieve at low densities.

LHC will play an important role in ensuring that these controlmechanisms are implemented properly.



Why thermodynamics and not dynamics?

Chemical species may diffuse on the expanding background of thefireball, so why should we neglect diffusion and expansion?

First check whether the system size, L, is large enough comparedto the correlation length ξ: Knudsen’s number K = L/ξ. IfK ≫ 1, ie, L ≫ ξ then central limit theorem will apply.Next, compare the relative importance of diffusion and flowthrough a dimensionless number (Peclet’s number):

W = L2

tD =Lvflow

D =Kξvflowξcs

=Kvflow

cs= MK ,

where M is the Mach number. When W ≪ 1 diffusion dominates.Near chemical freeze-out K ≃ 1 and M ≃ 1, so flow dominates:fluctuations are frozen in. Detector observes thermodynamicfluctuations at chemical freeze out.(Bhalerao, SG: 2009)



Peclet phenomenology

leng

ths

time

V

V1/3

1/3obs

ξ

fireball

λ p

Define Peclet length λp ≃ ξ/M where W = 1. λp remains fairlyconstant until time R0/cs then falls rapidly as flow becomes fully3d. So freezeout time is not very strongly dependent on rapiditywindow.


Basics Experiment Statistics Summary Thermodynamics Thermalization Criticality

Outline




Summary



Event by event fluctuations

)pN∆Net Proton (-20 -10 0 10 20

Num

ber

of E

vent

s

1

10

210

310

410

510

610 0-5%30-40%70-80%

Au+Au 200 GeV


Grand canonical thermodynamics?

When Vobs/ξ3 → ∞ and Vfireball/Vobs → ∞, then thermodynamics

in the grand canonical ensemble works; all distributions ofconserved quantities are Gaussian. For a Gaussian the onlynon-vanishing cumulants are the mean, [B], and the variance [B2].

Observation of any other non-vanishing cumulant [Bn] means thefluctuations are non-Gaussian. Is it because the system is not largeenough for thermodyamics to be applied? If it is a finite size effect,we are still ok.

If finite size effect, then trivial volume dependence of cumulants,i.e., all cumulants scale as V .



Is the fluctuation Gaussian?

STAR: QM 2009, Knoxville

◮ Higher cumulantsscale down withlarger powers of V .

◮ Npart is a proxy forV .

◮ Cumulants observedto scale correctly asNpart .

◮ Can one connect toQCD?



Same data, different plot

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400

Cum

ulan

ts

N part

from STAR data [naive error propagation, no systematic errors]200 GeV Au Au

[B ]1

[B ]2

[B ]3

[B ]4

Central limit theorem follows. Scaling implies ξ3 ≪ Vobs at some√S .




0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400

Cum

ulan

ts

N part

from STAR data [naive error propagation, no systematic errors]62.4 GeV Au Au

[B ]1

[B ]2

[B ]3

[B ]4





0

2

4

6

8

10

12

0 50 100 150 200 250 300 350 400

Cum

ulan

ts

N part

from STAR data [naive error propagation, no systematic errors]19.6 GeV Au Au

[B ]1

[B ]3

[B ]2

[B ]4




Very important deduction

The cumulants are linear in the volume, so the higher orderfluctuations will scale away in the limit when the volume becomesinfinite. The fluctuations will then become Gaussian.

This happens at almost all√S . This is consistent with

thermodynamics at an ordinary point.

This means that departure from Gaussian at one√S could be

significant, and should be investigated as a signal of a criticalpoint.



Outline




Summary



We are so lucky

0.7

0.8

0.9

1

1.1

0 1 2 3 4 5

T/T

c

/TBµ

Freezeout curve

10 GeV

20 GeV30 GeV Mumbai Nt=4

Mumbai Nt=6Mumbai Nt=8

Look at m1, m2 and m3 along the freeze-out curve.Sourendu Gupta QGP@SERC-EHEP: Lecture 4


m1 = S/σ

0.001

0.01

0.1

1

10

1 10 100 1000 10000

m1

S (GeV)NN

Nt=4Nt=6

Gavai, SG: 2010



m1 = S/σ

0.001

0.01

0.1

1

10

1 10 100 1000 10000

m1

S (GeV)NN

HRG

power law

Gavai, SG: 2010



m2 = Kσ2

-1

0

1

2

3

4

5

1 10 100 1000 10000

m2

S (GeV)NN

Nt=4Nt=6

Gavai, SG: 2010



m2 = Kσ2

-1

0

1

2

3

4

5

1 10 100 1000 10000

m2

S (GeV)NN

HRG

Gavai, SG: 2010



m3 = Sσ/K

0.1

1

10

100

1000

1 10 100 1000 10000

m3

S (GeV)NN

Nt=4Nt=6

Filled circles: negative values

Gavai, SG: 2010



m3 = Sσ/K

0.1

1

10

100

1000

1 10 100 1000 10000

m3

S (GeV)NN

power lawFilled circles: negative values

HRG

Gavai, SG: 2010



Comparison with lattice predictions

STAR

Collaboration,1004.4959(2010)

Surprising agreementwith lattice QCD:

◮ impliesnon-thermalsources offluctuations arevery small

◮ T does not varyacross thefreezeout surface.

◮ tests QCD innon-perturbativethermal region

Gavai, SG, 1001.3796



Experiment vs lattice QCD



Outline




Summary



Finite size effects damp divergences

1. System size limits correlation lengths near the critical point:ℓ ≃ ξ. The Knudsen number is never small near the CEP, socentral limit theorem will stop working. Check the scaling ofσ2, S and K and see whether there are violations of thecentral limit theorem. (SG: 2009)

2. As a result, the Peclet number need not be large, anddiffusion may play an important role even close to kineticfreeze-out. Then fluctuations of conserved quantities may notbe comparable to thermal equilibrium values at chemicalfreeze-out!

3. Another way of saying this is: critical divergences are limiteddue to finite size effects: no singularities, hence no directmeasurement of the critical exponents. System drops out ofequilibrium due to finite lifetime. (Stephanov: 2008; Berdnikov,Rajagopal: 1998)



Outline




Summary



The meaning of errors and error propagation

One usually uses the central limit theorem to say that the error ina measurement, ∆x , is related to the variance of the measurement,σ2(x), and the number of measurements, N, by the formula

(∆x)2 =σ2(x)

N.

This works if the distribution of the estimates of the measurementsis Gaussian. The error is the 65% confidence limit.

It is very tempting to use standard error analysis for products andratios of numbers. If t = x/y and then one usually writes

(

∆t

t

)2

=

(

∆x

x

)2

+

(

∆y

y

)2

.

What does this mean?



The error formula for ratios is meaningless

One can prove that the ratio (or product) of two Gaussiandistributed numbers is not Gaussian. In fact the ratio has infinitevariance! So the formula for error propagation gives some ∆twhich has no meaning as a confidence limit. In fact, since thevariance is infinite, such a ∆t is perfectly meaningless. (Geary: 1930)





You do not believe this?






Do a Monte Carlo experiment. Draw two numbers x and y eachfrom a Gaussian distribution. Take the ratio t. Repeat this processN times so you have N values of x , y and t. Now compute thevariance of each quantity. As you increase N you find σ2(x) andσ2(y) are finite numbers but σ2(t) grows without bound.






Do a Monte Carlo experiment. Draw two numbers x and y eachfrom a Gaussian distribution. Take the ratio t. Repeat this processN times so you have N values of x , y and t. Now compute thevariance of each quantity. As you increase N you find σ2(x) andσ2(y) are finite numbers but σ2(t) grows without bound.

Do not use the error propagation formula for ratios. Othermethods are possible.



The variance of a ratio grows with statistics

10

100

1000

100 1000 10000

σ (ra

tio)/

N1/

2

N

Result of a simple Monte Carlo test.Sourendu Gupta QGP@SERC-EHEP: Lecture 4


Alternatives to variance

half−widthmodehalf−width t

Measures of error exist even when variance is infinite.Phase transitions and statistics are very closely linked!



Alternatives to variance

33% of area 33% of area

mean t

Measures of error exist even when variance is infinite.Phase transitions and statistics are very closely linked!



Outline




Summary



Summary

1. The ensemble of collider events can be used to simulate thegrand canonical ensemble provided

◮ the fireball is thermalized◮ one observes only a small part of the fireball◮ this part is much larger than the correlation volume

GCE fluctuations can then be measured through event byevent fluctuations.

2. Crucial check: cumulants of B , Q, and S , must scale linearlywith system volume (proxy: Npart).

3. If the ratios of cumulants equal the results of the lattice, thenthe fireball is in thermal equilibirum.

4. Near a critical point the fireball cannot come to thermalequilibrium.


Thermodynamics and statistical mechanicsExperiment sees thermodynamicsObserving nearly Gaussian fluctuationsDetailed test of thermalizationWhat could happen near the critical point

Error analysis for ratiosSummary

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