raghunath(sahoo( indian(instute(of(technology(indore,(india(
TRANSCRIPT
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 4
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
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 5
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).
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 6
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 7
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
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 8
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
⇤+ ....
)
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 9
Taylor expansion in (q-‐1)
Using Taylor expansion, transverse momentum distribution is given by,
where (q-‐1) ≅ x and (E-‐μ)/T = Φ
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 10
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 11
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⇠)
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 12
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 13
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
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 14
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 15
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 16
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 17
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 18
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 19
Degree of devia'on from Boltzmannian
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 20
Degree of devia'on from Boltzmannian
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
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 22
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.
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 23
Summary
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 24
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
Raghunath Sahoo, WPCF-‐2015, Warsaw, Poland, 3-‐7 Nov. 2015 25
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