Recent Developments in THERMUS

“The Wonders of Z”

Spencer WheatonDept of PhysicsUniversity of Cape Town

Statistical-Thermal Model

Fireball resulting from high-energy heavy-ion collision treated as an ideal

gas of hadrons

At freeze-out these hadrons are assumed to be described by local

thermal distributions

Chemical Freeze-out:

multiplicities fixed

Thermal Freeze-out:

momenta fixed

THERMUS – Statistical-Thermal Model Analysis within ROOT

SW, J. Cleymans and M. Hauer, Comput. Phys. Commun. 180 (2009) 84.

SW, PhD Thesis, UCT (2005)

17 152 lines of codeDecays & properties of 361 hadrons 24 C++ classes

… and growing!

Grand-Canonical Ensemble: B, S, Q & E conserved on

average

Strangeness Canonical Ensemble: S conserved exactly;

B, Q & E conserved on average

Fully Canonical Ensemble: B, S & Q conserved exactly;E conserved on average

THERMUS performs calculations within 3 commonly applied statistical

ensembles:T, B, S, Q, V

T, B, S, Q, V

T, B, S, Q, V

THERMUS has proved extremely successful in describing hadron

multiplicities and ratios… but like all statistical models it battles to reproduce

the K+/+ “horn”:

(J. Cleymans, H. Oeschler, K. Redlich and SW,

Phys.Lett.B615:50-54, 2005)

Estimate of rest of Resonance Spectrum

Including high-mass resonances (and meson) improves the situation:

(Andronic, Braun-Munzinger, Stachel Phys.Lett.B673:142,2009)

Baryons and Mesons with u, d and s quarks

up to 2.6 GeV( meson included) …

2005 Particle Set:

2010 Particle Set:Dawit Worku (UCT) has since updated

the THERMUS particle set to

include also c and b quarks …

Extended THERMUS Particle Set

With extended particle set comes need for extension in ensembles. So, coming soon:

B, S, Q, C and b GCES, C and b CE

Fully B, S, Q, C and b CE

But what about calculations beyond particle multiplicities and mean values?

Work on fluctuations and correlations with Michael Hauer (Frankfurt) has all been

done within THERMUS …

Statistical Thermal Model: Ensembles and Partition

Functions Micro-canonical ensemble: Fixed E, P and B, S, Q, C, b …

Canonical ensemble: Fixed B, S, Q, C, b …

Grand-canonical ensemble: Nothing Fixed Exactly

, , , exp , ,GCE ll

Z V u V u

1

3

33

ln 1

2

jjl l

jjl l

p u q

ll

p u q

d p eg

d p e

3

32states

Vd p

Grand Canonical Ensemble:

Canonical Ensemble (Traditional Approach):

, , , , ,2 2 2

exp , ,

i B S QQ B S QQ B S Q SBCE

ll

dddZ V u e

V i u

Quantum Stats

Boltzmann Stats

3

32

p ui ii

g Vz d p e

Chemical potentials gone!

include flow here

, , ,2

exp , ,

, ,

jj

ii

JiQQ

J

ll

Q QCE

dV u e

V i u

e Z V u

Z

Retain the chemical potentials & project out the GCE partition function:

, , ,

, , ,

, , ,

B S Q

B S Q

Q B S Q

Canonical Ensemble (Alternative Approach):ala Michael Hauer

Example 1: Static Pion Gas (conserved Q)

Q

5 integrandCE

QZ 5 integrandQZ

as in GCE

Q

T = 150 MeVR = 6 fm

Boltzmann Approximation:

50 integrandCE

QZ

Chemical potential in Z is a free parameter, but

choose well and oscillations cease or at

least are reduced!

Q

/ /, , exp2

exp exp 2 cosh cos2

cos cos 2 sinh sin

sin sin 2 sinh sin

Q Q Q Q QiQ T i T iQQQ

Q QQ

QQ Q

QQ Q

dV e ze ze z

dz z

T

Q zT

Q zT

Z

Little bit of work required to get integrand into a manageable form:

de Klerk, Hauer, SW

Example 2: Full HR Gas

iQ QCE

QCE

Z

Z

Canonical Correction for hadron with charge

content Bi, Si, Qi :

0

iQ Qi iCECE GCEQ

CE

ZN N

Z

BSQ ensemble implemented in THERMUSuses result of Becattini and Keranen to

calculate correction factors …

a) Becattini & Keranen - Static Boltzmann:

, , , exp 22 2

cos arg

exp 2 cos

QQ B S Q SCE n B

Neutrals n

S Q

m m S m QMesons m

ddZ V z I

S Q B

z S Q

3D 2D integration,

but still oscillatory

expb b S b QBaryons b

V z i S Q Q

S Works well for THERMUS BSQ ensemble

1

2

1

2

, , ,

, , ,

ii iQ QQ Q Q QCE

Q Q QCE

Z V Ve

Z V Ve

Z

Z

Choose chemical potentials to smooth out integrands in both numerator and

denominator- much easier to integrate

A Carbon-Carbon

Correction Factor

Symmetry of before

disappears, so 3D integration

needed

Hauer, de Klerk, SWb) Approach using :

, , exp2 2 2

exp 2 cosh cos

cos cos 2 sinh sin

sin sin 2 sinh sin

QB S Q SBn

Neutrals n

j j jParticles j

j j jParticles j

j j jParticles j

dddz

z

Q z Q Q

Q z Q Q

Z

Only 13 distinct quantum content

combinations

Q

S

So, on the cards is the application of this numerical technique to the

B,S,Q,C,b ensemble in THERMUS…

An analytic result has recently been derived by Beutler et. al.

arXiv:0910.1697

Beutler et. al. arXiv:0910.1697: 

Canonical treatment of B, S, Q, C and b: Quantum statistics for lightest bosons…

5D integration 3D integration Bessel functions

Fluctuations and Correlations

, ,

/ , ,

/

, ,

1

number of states with , and ,

number of states with

,

( , )

, ( , , )

( , , ) ( , )

, ,

A B

CE

CE

Q A B

CE

Q

CE

A B

A Bce A B

Q N N

Q

Q T Q N N

GCEQ T Q

GCE

Q N NGCE

QGCE

gce A B gce

Q N NP N N

Q

Z V

Z V

e Z V Z V

Z V e Z V

Z

Z

P Q N N P Q

ZZ

Starting point for fluctuations & correlations is again the partition

function: E.g.

4,

4, , ,2 2

exp , ,

jj

JiQ iPQ P

J

ll

d dV u e e

V i u u i

Z

1 2

1 2

, ,...,

0

...!

n

n

nj j j

n j j jn

i

n

, ,Q P =Q

(Taylor expansion)

0

eq

ZQ

QQ

Approximation good if chemical potentials and four-

temperature chosen such that:

Micro-Canonical Ensemble:

/2

1 1/21

1 1/22

, , , 1 1 1exp

, , , det 22j

GCE jJGCE

jj k k

k

V uP

Z V u V

V V

k

Z Q

Q

Q

Large Volume (Thermodynamic) Limit:

22

2 22 2

X X

XX

A B A BAB

A A B B

N N

N

N N N N

N N N N

GCE mean

Multi-variate normal distribution

To get joint particle multiplicity

distributions in various ensembles need to consider

slices through GCE distribution

ala Michael Hauer

Neutral Pion Gas (+,-,0) in MCE (large V)

Correlation coefficient within bin…..

Correlation coefficient between bins : no dynamics!

- at T = 160 MeV

– & at T = 160 MeV

M. Hauer, G. Torrierri & S.W.

Correlation between disconnected momentum space bins (no dynamical

effects)

M. Hauer, G. Torrierri & SW

V1

Monte Carlo Particle Generator

V2

V1: observed sub-system

V2: unobserved

Vg = V1 + V2 : total system

Constraints placed on system are imposed

only on the total volume

1

g

V

V

M. Hauer & SW published

The crux is the following:

1

1 1

1 1 1

1 1 1 1 1 1

, ; ,1 1 1 1

, ,, , , ,

, ,

; , , , , , ,

i

j jg g

j jg g gj i j

Njg g g

P Q P Q j ig j gce j

Z V V P P Q QP P Q N Z V P Q

Z V P Q

V V u P P Q N u

W

1 1

1 1

, ; ,1 1

,1,

,

/21

; , , , , ,

, ,

, , ,

1 1 1exp

2 11

j jg g

jjg g

jg g

P Q P Qg j GCE j

P P Q Qg j

P Qg j

llL

V V u Z V u

V V u

V u

V

W

Z

Z

Strategy of Monte Carlo Generator:

1)Sample sub-system V1 Grand-Canonically in Boltzmann Approximation

2)For each particle of type i, generate a momentum magnitude following a

Boltzmann Distribution:

3)Assign direction to particle assuming isotropic particle emission…

4)Allow 2 and 3 body decays

5) Reweight

System Considered

Neutral , Static , T = 160 GeV and V1 = 2000 fm3

B, S, Q, E, Pz considered for reweighting

= 0.00

= 0.75

= 0.25

= 0.50

= 0.00

= 0.25

= 0.50

= 0.75

= 0

S(s) = -1 Q(s) = -1/3

Prim

ordi

al N

E [GeV] E [GeV]

Str

ange

ness

Con

tent

Strangeness ContentBaryon Content

p Z [

GeV

/c]

Cha

rge

Con

tent

= 0.25

Prim

ordi

al N

E [GeV] E [GeV]

Str

ange

ness

Con

tent

Strangeness ContentBaryon Content

p Z [

GeV

/c]

Cha

rge

Con

tent

= 0.50

Prim

ordi

al N

E [GeV] E [GeV]

Str

ange

ness

Con

tent

Strangeness ContentBaryon Content

p Z [

GeV

/c]

Cha

rge

Con

tent

= 0.75

Prim

ordi

al N

E [GeV] E [GeV]

Str

ange

ness

Con

tent

Strangeness ContentBaryon Content

p Z [

GeV

/c]

Cha

rge

Con

tent

Fully-Phase Space Integrated Results

Averages Variances

Co-Variances Correlation Coefficients

20 runs of 2.5 × 104 events

Linear extrapolation to MCE

Momentum Space Dependence

Divide into 5 bins such that each

bin contains 1/5th of positives

Momentum Bin Selectionpositives

positives

GCE Correlation Coefficients

20 runs of 105 events

Extrapolating Variances to

MCE20 runs of 105 events

Linear extrapolation to MCE

primordial

primordial

Largest baryon and strangeness content in pT,5

Linear extrapolation

to MCE

Non-Linear extrapolation

to MCE

Extrapolating Covariances and Correlation Coefficients to MCEprimordial

primordial

20 runs of 105 events

Multiplicity Fluctuations and

Correlations

GCE Scaled Variance of Positives

20 runs of 2 × 105 events

GCE Correlation Coefficient +-

Scaled Variance of Primordial Positives

20 runs of 2 × 105 events

Primordial Correlation Coefficient

Scaled Variance of Final State Positives

Final State Correlation Coefficient

Concluding Remarks

THERMUS proven itself in analysis of mean valuesNew extended particle set should allow better description of data.

Recent work aimed at extending functionality to include fluctuation and correlation analysis A Monte Carlo Particle Generator has been developed

In the process a new technique for the calculation of canonical correction factors has been developed