raghunath(sahoo( indian(instute(of(technology(indore,(india(

Raghunath Sahoo Indian Ins/tute of Technology Indore, India

arXiv:1507.08434 +

arXiv:1511. +

arXiv:1511.

Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 3

The Mo'va'on Hagedorn: !  StaGsGcal descripGon of momentum spectra !  ExponenGal decay of differenGal cross-‐secGon: E

d

3�

dp

3= A.exp(

�pT

T

)

Experimental Data:

High pT-‐tail deviates from exponenGal (equilibrium Boltzmann staGsGcs).

Hagedorn’s proposal:

QCD-‐inspired empirical formula, that describes the spectra over wide pT-‐range:

Ed3�

d3p= A.

✓1 +

pTp0

◆�n

�! exp

✓�npT

p0

◆for pT ! 0,

�!✓p0pT

◆n

for pT ! 1.

A, n and p0 are fiZng parameters.


The Mo'va'on "  In an equilibrated staGsGcal ensemble T is associated with <pT>.

" When the system shows deviaGon from equilibrium, the above may not be correct.

"  Either T fluctuates event-‐by-‐event or within the same event. (See presentaGon by me on 7th Nov.)

"  Such a system could be described by non-‐extensive generalizaGon of staGsGcal mechanics:

f(pT ) = Cq.h1 + (q � 1)

pTT

i� 1q�1

q is the non-‐extensive parameter and a measure of degree of deviaGon from Boltzmann staGsGcs. In the limit q # 1, we get back to Boltzmann staGsGcs.

The Tsallis non-‐extensive staGsGcs #

For n =

1

q � 1

, and p0 =

T

q � 1

the QCD-‐inspired formula and Tsallis distribuGon funcGon coincide with each other. M. Rybczynski, Z. Wlodarczyk and G. Wilk, JPG 39 (2012) 095004


Makes sense to use Nonextensive Sta's'cs at High-‐energies

PHENIX: Phys. Rev. 83, 064903 (2011) Experiments at RHIC and LHC use non-‐extensive staGsGcs to describe parGcle spectra: STAR: Phys. Rev. 75, 064901 (2007)-‐ Strange parGcle yields: pp @ 200 GeV PHENIX: Phys. Rev. 83, 064903 (2011) –IdenGfied parGcles pp @ 200 GeV CMS: Eur. Phys. J. C 72, 2164 (2012)–IdenGfied parGcles pp @ 0.9, 2.76, and 7 TeV ALICE: Eur. Phys. J. C 71, 1655 (2011)–IdenGfied parGcles pp @ 900 GeV Physics LeFers B 712, 309 (2012)-‐MulG-‐strange parGcles pp@ 7 TeV Heavy-‐Ion data: TBW or Tsallis with radial flow in a relaGvisGc approach: (PRC 79 (2009) 051901(R) & 1507.08434).


The Tsallis Distribu'on

Pros and Cons:

$  Experimental data at high energies follow Tsallis distribuGon.

$  A secGon of theoreGcians believe it as “SuperstaGsGcs”.

$  Another secGon of theoreGcians say:

!  No convincing, rigorous derivaGon of Tsallis distribuGon from fundamental microscopic physics.

!  No theoreGcally derived or even physically moGvated transport equaGon for which the Tsallis distribuGon is a soluGon.

We need more theoreGcal helps in this direcGon……..

$  Analysis of identified particle spectra with nonextensive staGsGcs. $  Taylor expansion in (q-‐1) and study of degree of deviaGon from non-‐extensivity.

$  AnalyGc inclusion of radial flow in Tsallis distribuGon and heavy-‐ion spectra. $  Mass ordering of T, V and β for idenGfied parGcles: Indica/on of differen/al freeze-‐out. $  Speed of sound for physical hadron gas in nonextensive staGsGcs.


I am going to discuss on…..

The thermodynamical quanGGes are:

where “V” is the volume and “g” is the degeneracy factor. The q-‐logarithm is defined as ,

These equaGons are thermodynamically consistent.

The Tsallis-‐Boltzmann distribuGon funcGon is given by,

f =

1 + (q � 1)E�µ

T

�� 1q�1

(q = 1) e�E�µT

s = �gvR d3p

(2⇡)3

fqlnqf � f

�

N = gvR d3p

(2⇡)3 fq

✏ = gRE d3p

(2⇡)3 fq

P = gR d3p

(2⇡)3p2

3E fq


The Non-‐extensive Sta's'cs

T =@✏

@s

��n

µ =@✏

@n

��s

n =@P

@µ

��T

s =@P

@T

��µ

Meaning:

s = S/V (entropy density), n = N/V (parGcle number density)

Assuming q ≈ 1, as is to be the case in high energy physics (1 < q < 1.2), the Tsallis distribuGon, appearing in the expressions for the thermodynamic quanGGes can be expanded in a Taylor series:

The first term is a Boltzmann distribu'on and other terms are higher order in (q-‐1).

1 + (q � 1)

E � µ

T

�� qq�1

' e�E�µT

(1 + (q � 1)

1

2

E � µ

T(�2 +

E � µ

T)

�

+O⇥(q � 1)2

⇤+O

⇥(q � 1)3

⇤+ ....

)


Taylor expansion in (q-‐1)

Using Taylor expansion, transverse momentum distribution is given by,

where (q-‐1) ≅ x and (E-‐μ)/T = Φ


The Transverse Momentum Distribu'on

For details on derivaGon: arXiv: 1507.08434

0.01

0.1

1

d2 N/d

p Tdy (

GeV

/c)-1

0 0.5 1 1.5 2 2.5 3pT (GeV/c)

-0.2

0

0.2

(M- E

)/M

Tsallis distribu'on without Taylor expansion

Fits to the normalized differenGal charged parGcle yields, measured by the ALICE experiment in p+p collisions at √s = 0.9 TeV.


pT-‐Distribu'on (ALICE p+p@ 0.9 TeV)

The final expression for transverse momentum distribuGon up to first order in (q-‐1) is given by,

1

pT

dN

dpT dy=

gV

(2⇡)2

⇢2T [rI0(s)K1(r)� sI1(s)K0(r)]� (q � 1)Tr2I0(s)[K0(r) +K2(r)]

+4(q � 1) TrsI1(s)K1(r)� (q � 1)Ts2K0(r)[I0(s) + I2(s)] +(q � 1)

4Tr3I0(s)[K3(r) + 3K1(r)]

�3(q � 1)

2Tr2s[K2(r) +K0(r)]I1(s) +

3(q � 1)

2Ts2r[I0(s) + I2(s)]K1(r)

� (q � 1)

4Ts3[I3(s) + 3I1(s)]K0(r)

�

p

µ = (mT coshy, pT cos�, pT sin�,mT sinhy)

To include flow in Tsallis Boltzmann distribuGon funcGon with Taylor expansion, we have taken the cylindrical symmetry in which it has a explicit dependence on flow parameter v. Here, we have replaced f(E) by f(pμuμ), where pμuμ is a Lorentz invariant quanGty and

uµ = (�cosh⇠, �vcos↵, �vsin↵, �sinh⇠)


Inclusion of Radial Flow to order (q-‐1)

0.01

0.1

1

10

100

1000

1/N

1/2πp

T d2 N

/dp Tdy

(G

eV/c

)-20 0.5 1 1.5 2 2.5 3

pT (GeV/c)

-0.4-0.2

00.2

(M-E

)/M

0.1

1

10

100

1000

10000

1/N

1/2πp

T d2 N

/dp Tdy

(G

eV/c

)-2

0 0.5 1 1.5 2 2.5 3 3.5pT (GeV/c)

-1-0.5

00.5

(M-E

)/M

where

Without flow With flow

FiZng parameters: T = 146 MeV, q = 1.030, β = 0.609, r = 29.8 fm

In(s) and Kn(r) are the modified Bessel funcGons of the first and second kind.


pT-‐Distribu'on (ALICE Pb+Pb@ 2.76 TeV)

Without radial-‐flow

μ≠0

E d3Ndp3 = gV mT

(2⇡)3

1 + (q � 1)mT�µ

T

�� qq�1

With radial-‐flow

μ=0

Radial-‐flow using Taylor expansion

E d3Ndp3 = gV mT

(2⇡)3

1 + (q � 1)mT

T

�� qq�1

1

pT

dN

dpT dy=

gV

(2⇡)2

⇢2T [rI0(s)K1(r)� sI1(s)K0(r)]

�(q � 1)Tr2I0(s)[K0(r) +K2(r)]

+4(q � 1) TrsI1(s)K1(r)� (q � 1)

Ts2K0(r)[I0(s) + I2(s)] +(q � 1)

4Tr3I0(s)

[K3(r) + 3K1(r)]�3(q � 1)

2Tr2s

[K2(r) +K0(r)]I1(s) +3(q � 1)

2Ts2r

[I0(s) + I2(s)]K1(r)�(q � 1)

4Ts3

[I3(s) + 3I1(s)]K0(r)

�

Tsallis StaGsGcs


pT-‐Spectra and Tsallis Func'on

" We have fixed Tsallis distribuGon to the pT-‐spectra for p+p and A+A collisions at RHIC and LHC energies.

"  It gives bexer χ2/ndf for peripheral heavy-‐ion

collisions than central collisions.

(GeV/c)T

p0 0.5 1 1.5 2 2.5 3

-2 (G

eV/c

)dy T

dp TpN2 d

Nπ21

5−10

4−10

3−10

2−10

1−101

10

210

310

410

510

610

Au+Au 200GeV(Most Central)

(GeV/c)T

p0.5 1 1.5 2 2.5 3

-2 (G

eV/c

)dy T

dp TpN2 d

Nπ21

5−10

4−10

3−10

2−10

1−101

10

210

310

410

510

610

Au+Au 200GeV(Most Peripheral)

90.5)×(+π 72.5)×(+ K

3.5)× p( 0.12)×(φ

Λ

Particle

/ndf

2 χ

0

1

2

3

4

+π +K p φ Λ

Au+Au Peripheral Particle Antiparticle

Particle

/ndf

2 χ

0

1

2

3

4

+π +K p φ Λ

Au+Au Central Particle Antiparticle

"  The χ2/ndf value is around 2.5 for K+ and for parGcles like π , p and Φ, the χ2/ndf ~ 1.


pT-‐Spectra & Tsallis+Radial Flow

Mass (MeV)200 400 600 800 1000 1200

3Vo

lum

e (fm

)

1−10

1

10

210

310

410

510

(most peripheral)Au+Au 200GeV

Mass (MeV)200 400 600 800 1000 1200

Tem

pera

ture

(GeV

)

0.2

0.4

0.6

0.8

p+p 200GeV

Mass (MeV)200 400 600 800 1000 1200

β

0

0.2

0.4

0.6

0.8

Mass (MeV)200 400 600 800 1000 1200

q

0.9

0.95

1

1.05

1.1

1.15

1.2

(GeV/c)T

p0 0.5 1 1.5 2 2.5 3

-2 (

GeV

/c)

dyT

dpTp

N2 d Nπ21

5−10

4−10

3−10

2−10

1−101

10

210

310

410

510

610

Au+Au 200GeV(Most Central)

(GeV/c)T

p0.5 1 1.5 2 2.5 3

-2 (

GeV

/c)

dyT

dpTp

N2 d

Nπ21

5−10

4−10

3−10

2−10

1−101

10

210

310

410

510

610

Au+Au 200GeV(Most Peripheral)

90.5)×(+π 72.5)×(+ K

3.2)× p( 0.12)×(φ

Λ

"  There is a mass ordering of T, β and V: picture of differenGal freeze-‐out. "  Trend of parameters in p+p and Au+Au

peripheral collisions are the same: formaGon of similar thermodynamic systems.


Possible mass ordering in T, β and V@RHIC

Mass (MeV)200 400 600 800 1000 1200

3Vo

lum

e (fm

)

10

210

310

410

510

(most peripheral)Pb+Pb 2.76TeV

Mass (MeV)200 400 600 800 1000 1200

Tem

pera

ture

(GeV

)

0.2

0.4

0.6

0.8

p+p 2.76TeV

Mass (MeV)200 400 600 800 1000 1200

β

0

0.2

0.4

0.6

0.8

Mass (MeV)200 400 600 800 1000 1200

q

0.7

0.8

0.9

1

1.1

1.2

%  The heavier parGcles freeze-‐ out early as compared to lighter parGcles and freeze-‐out surfaces are different for different parGcles, which shows a mass ordering of T , β and V.

%  Again , there is a similar trend of parameters in p+p and Pb+Pb peripheral collisions at 2.76 TeV: indicaGon of a similar thermodynamic system.


Possible mass ordering in T, β and V@LHC

The Number Density

The parGcle density calculated to first order in (q-‐1) normalized to parGcle density in Boltzmann gas.

The parGcle density calculated to first order in (q-‐1) normalized to parGcle density in Tsallis gas.

"  As we increase the q-‐values the contribu'on to the number density from the first order in the expansion increases.


Degree of devia'on from Boltzmannian

The Energy Density

The energy density calculated to first order in (q-‐1) normalized to energy density in Boltzmann gas.

The energy density calculated to first order in (q-‐1) normalized to energy density in Tsallis gas.

"  As we increase the q-‐values the contribu'on to the energy density from the first order in the expansion increases.



The Pressure

The pressure density calculated to first order in (q-‐1) normalized to pressure density in Boltzmann gas.

The pressure density calculated to first order in (q-‐1) normalized to pressure density in Tsallis gas.

"  As we increase the q-‐values the contribu'on to the pressure from the first order in the expansion increases.



Mass Cut-off (GeV)0 0.5 1 1.5 2 2.5

2 sC

0.1

0.15

0.2

0.25

0.3 q = 1.006 q = 1.05 q = 1.1 q = 1.15

"  The speed of sound depends on the non-‐extensive parameter q: higher is the non-‐extensivity, higher is the cs2.

"  Given q-‐value, the cs2 is independent of mass cut-‐off taken in the system above 2 GeV. Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland,

3-‐7 Nov. 2015 21

Speed of Sound for HG using Non-‐extensive Sta's'cs

c2s(T ) =

✓@P

@✏

◆

V

=s(T )

CV (T )

HT/T0 0.2 0.4 0.6 0.8 1

2 sC

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Ideal Pion Gas

Tsallis-Boltzmann Pion Gas (q=1.15)

Tsallis-Bose Pion Gas (q=1.15)

cT/T0 0.2 0.4 0.6 0.8 1

2 sC

0.1

0.15

0.2

0.25

0.3 Pion Gas

M < 0.5 GeV

M < 1 GeV

M < 1.5 GeV

M < 2.5 GeV

q = 1.05

&  Speed of sound for different values of q and different mass cut-‐offs. &  As we include more and more resonances the speed of sound decreases significantly. &  q-‐dependent cri'cality poin'ng to so^ening of EoS. &  As we increase the q value, the minimum for cs2 shi|s towards le|.

HT/T0 0.2 0.4 0.6 0.8 1

2 sC

0.1

0.15

0.2

0.25

0.3 Pion Gas

M < 0.5 GeV

M < 1 GeV

M < 1.5 GeV

M < 2.5 GeV

q = 1.005


Speed of Sound for HG using Non-‐extensive Sta's'cs

&  Taylor approximaGon in (q-‐1): study the degree of deviaGon from a thermalized Boltzmann distribuGon.

&  Inclusion of radial flow in Tsallis distribuGon gives an analyGcal way in describing the pT-‐spectra up to 3 GeV in Pb+Pb collisions at √sNN= 2.76 TeV .

&  Using different versions of Tsallis staGsGcs for pT-‐spectra fit, we found a similar trend for Tsallis parameters as a funcGon of parGcle mass in p+p and A+A peripheral collisions at RHIC and LHC energies, which shows that a similar system formaGon in both the cases.

&  Possible mass ordering in V , T and β: evidence of differen/al freeze-‐out. &  For physical hadron resonance gas, a small deviaGon in q values from 1, gives a

significant change in speed of sound. &  There is a q-‐dependent cri/cality in speed of sound. &  Above 2 GeV, the speed of sound is independent of mass cut-‐off for a physical resonance gas.


Summary


This work is done in collaboraGon with: Prof. J. Cleymans (UCT, South Africa) IIT Indore: Dr. Trambak Bhaxacharyya (Postdoc) Dr. Prakhar Garg (Postdoc) Arvind KhunGa, Dhananjay Thakur, Sushant Tripathy, Pooja Pareek and PragaG Sahoo (Ph.D. Students)

Acknowledgement


Backups

Sq(A,B) = Sq(A) + Sq(B) + (1-‐q)Sq(A) Sq(B)

The parameter |1-‐q| is a measure of the departure from addiGvity. In the limit when q = 1, S(A,B) = S(A) + S(B): entropy of an addiGve system/extensive system.

Tsallis entropy is given by

The Tsallis form of Fermi-‐Dirac and Bose-‐Einstein distribuGon is:

fT (E) ⌘ 1

expq

⇣E�µT

⌘⌥ 1

expq(x) ⌘[1 + (q � 1)x]1/(q�1) if x > 0

[1 + (1� q)x]1/(1�q) if x 0

raghunath(sahoo( indian(instute(of(technology(indore,(india(

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