Fits to the transverse momentum distributions of charged particles produced
in p - p collisions at LHC energies based on the Tsallis distribution have been
shown to work over 14 orders of magnitude. Two versions of the Tsallis
distribution are compared and discussed, the thermodynamically consistent
version leads to a temperature of T = 74 +/- 13 MeV at the highest beam energy,
a result which is in agreement with previous analyses done with lower
transverse momentum data.Comment: 5 pages, 2 figure

Azmi, M. D.

Cleymans, J.

The European Physical Journal C

English

arXiv

M. D. Azmi

J. Cleymans

Springer - Publisher Connector

Eur. Phys. J. C (2015) 75:430
DOI 10.1140/epjc/s10052-015-3629-9
Regular Article - Theoretical Physics
The Tsallis distribution at large transverse momenta
M. D. Azmi1,a, J. Cleymans2,b
1 HEP Lab, Physics Department, Aligarh Muslim University, Aligarh 202002, India
2 Physics Department, UCT-CERN Research Centre, University of Cape Town, Rondebosch, Cape Town 7701, South Africa
Received: 16 June 2015 / Accepted: 17 August 2015 / Published online: 16 September 2015
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract The charged particle transverse momentum (pT )
spectra measured by the ATLAS and CMS collaborations for
proton–proton collisions at
√
s = 0.9 and 7 TeV have been
studied using Tsallis thermodynamics. A thermodynamically
consistent form of the Tsallis distribution is used for fitting
the transverse momentum spectra at mid-rapidity. It is found
that the fits based on the proposed distribution are working
over 14 orders of magnitude with pT values up to 200 GeV/c
and this gives a value of χ2/N DF of 0.52/24 for the CMS
data at 7 TeV. The values for dNdy |y=0 as a function of center-
of-mass energy obtained using a Tsallis distribution for both
ATLAS and CMS data sets are presented and discussed.
1 Introduction
It is well known that the Tsallis distribution gives excellent
fits to the transverse momentum distributions observed at
the Relativistic Heavy Ion Collider (RHIC) [1–3] and at the
Large Hadron Collider (LHC) [4–8] with only three parame-
ters, q, T , and dN/dy (or, alternatively, a volume V [9–11]).
The parameter q is referred to as the Tsallis parameter and
discussed elsewhere [12] in detail. The parameter T obeys
the standard thermodynamic relation,
T = ∂E
∂S
∣
∣
∣
∣
N ,V
,
however, the entropy used in the above equation is the Tsal-
lis entropy [13], not the standard Boltzmann–Gibbs (BG)
entropy. The parameter V is not necessarily related to a vol-
ume deduced from other models like HBT calculations.
It was recently shown that these fits extend to values [14]
of pT up to 200 GeV/c [15,16]. This is unexpected because in
this kinematic range hard scattering processes become impor-
a e-mail: danish.hep@gmail.com
b e-mails: jean.cleymans@gmail.com; jean.cleymans@uct.ac.za
tant [17]. A description of the high pT results has been dis-
cussed in [18] where a model using a combination of Tsallis
at low pT and QCD hard scattering at high pT was consid-
ered. The present analysis shows that the Tsallis distribution
describes measurements up to the highest pT using the same
Tsallis parameters as those obtained at low pT .
A power law based on the Tsallis distribution [13] is used
to fit the pT spectra of charged particles measured by the
ATLAS and CMS collaborations. The ATLAS collaboration
has reported the transverse momentum in an inclusive phase
space region taking into account at least two charged particles
in the kinematic range |η| < 2.5 and pT > 100 MeV [7].
The CMS collaboration has presented the differential trans-
verse momentum distribution covering a pT range up to
200 GeV/c, the largest range ever measured in a colliding
beam experiment [14]. The results for the charged particle
multiplicities are consistent with those of ATLAS measure-
ments.
Moreover, the results are compared with those obtained
in [15–17,19,20] where very good fits to transverse momen-
tum distributions were presented. We confirm the quality of
the fits but obtain different values of the parameters albeit
using a different version of the Tsallis model.
2 Tsallis distribution
Power law distributions have been widely applied in high
energy physics [1–8] to describe the transverse momentum
spectra of secondary particles produced in proton–proton col-
lisions. With the advent of the LHC, transverse momenta of
hundreds of GeV have been measured. It is of great interest to
choose a distribution which describes such high pT measure-
ments very well. A well-suited form of the Tsallis distribution
is discussed in this paper and our main criterion for choosing
such a distribution was thermodynamic consistency, which
has not always been implemented fully [21–23].
It is well established that there are numerous physical sys-
tems under which BG statistics encounters many difficulties.
123
430 Page 2 of 5 Eur. Phys. J. C (2015) 75 :430
In particular, when analyzing the pT spectra of hadrons it
is found that spectra decrease far slower than predicted by
BG statistics, and they appear to follow some power law at
high pT . The Tsallis distribution was first proposed about
27 years [13] ago as a generalization of the BG distribution.
The Tsallis form of the BG distribution is given by
f (E) ≡ expq
(
− E − μ
T
)
, (1)
where E is the energy and μ is the chemical potential and
the function expq(x) is usually defined as
expq(x) ≡
{
[1 + (q − 1)x]1/(q−1) if x > 0,
[1 + (1 − q)x]1/(1−q) if x ≤ 0, (2)
in the limit where q → 1 it reduces to the standard exponen-
tial:
lim
q→1 expq(x) → exp(x).
The expressions of the relevant thermodynamic quantities
like entropy, S, energy density, (= E/V ), pressure, P , and
particle number, N , are given below using the q-logarithm
function,
lnq(x) ≡ x
1−q − 1
1 − q ,
as integrals over the function defined in Eq. (1) [11,22,23]:
S = −gV
∫
d3 p
(2π)3
[
f q lnq f − f
]
, (3)
 = g
∫
d3 p
(2π)3
E f q , (4)
P = g
∫
d3 p
(2π)3
p2
3E
f q , (5)
N = gV
∫
d3 p
(2π)3
f q , (6)
where g is the degeneracy factor and V is the volume. The
Tsallis distribution used in this paper is thermodynamically
consistent, which means that the following relations, derived
from the differential form of the first and second laws of
thermodynamics, are satisfied [23]:
T = ∂
∂s
∣
∣
∣
∣
n
, μ = ∂
∂n
∣
∣
∣
∣
s
,
n = ∂P
∂μ
∣
∣
∣
∣
T
, s = ∂P
∂T
∣
∣
∣
∣
μ
, (7)
where s and n are the entropy and particle number densities
respectively.
The corresponding momentum distribution deduced from
Eq. (6) is given by
E
d3N
dp3
= gV E 1
(2π)3
[
1 + (q − 1) E − μ
T
]− qq−1
, (8)
which in terms of the rapidity, y, and transverse mass, mT ,
variables (E = mT cosh y) at mid-rapidity and for μ = 0
becomes [11,22]
d2N
dpT dy
∣
∣
∣
∣
y=0
= gV pT mT
(2π)2
[
1 + (q − 1)mT
T
]− qq−1
. (9)
The relationship between the above parameterization and
the one used by the RHIC and LHC collaborations [1–8] has
been discussed in [9].
Additionally, a comparison with high-energy data has
been made in [15–17,19,20], using a simplified form:
d2Nch
dpT dy
∣
∣
∣
∣
y=0
= A
[
1 + (q − 1) pT
T
]− 1q−1
(10)
where A is a normalization factor and T is a parameter used
to fit the transverse momentum distribution and not related
to the temperature in the thermodynamic sense as in Eq. (9).
Furthermore, integration of Eq. (9) over the transverse
momentum leads to [10]
dN
dy
∣
∣
∣
∣
y=0
= gV
(2π)2
∫ ∞
0
pT dpT mT
[
1 + (q − 1)mT
T
]− qq−1
= gV T
(2π)2
[
(2 − q)m2 + 2mT + 2T 2
(2 − q)(3 − 2q)
]
×
[
1 + (q − 1)m
T
]− 1q−1
, (11)
where m is the rest mass.
Using the values of parameters q, T , and V obtained from
Tsallis fit, using Eq. (9), the above equation enables us to
obtain the values of dN/dy for pions, kaons, and protons.
Their sum gives the values of dN/dy for charged particles.
3 Fit details
The charged particle transverse momentum spectra measured
by the ATLAS and CMS detectors in proton–proton colli-
sions at LHC energies are fitted using Eq. (9).
It is well established that the charged particle pT spec-
tra is composed of contribution from pions, kaons, and pro-
tons. Hence, a sum of three Tsallis distributions for pions,
π+’s, kaons, K+’s, and protons, p, has been applied to fit
the charged particle pT spectra in the following way:
123
Eur. Phys. J. C (2015) 75 :430 Page 3 of 5 430
-1
10 1 10
2
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
1
10
2
10
s VeT7=
 = 0.9 TeV s
]
-2
) 
[G
eV
T
d
p
/(
d
ch
N2
) 
(d
T
p
 1
/(
2
ev
t
1/
N
 (GeV/c)
T
p
Fig. 1 Charged particle multiplicities as a function of the transverse
momentum measured by the ATLAS detector for events with nch ≥ 2,
pT > 100 MeV and |η| < 2.5 at √s = 0.9 and 7 TeV in proton–proton
collisions [7] fitted with a Tsallis distribution
 (GeV/c)
T
p
-1
10 1 10
D
at
a/
F
it
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
 = 0.9 TeVs
s VeT7=
Fig. 2 Ratios of the data to the fit values measured by the ATLAS
detector at
√
s = 0.9 and 7 TeV
d2Nch
dpT dy
∣
∣
∣
∣
y=0
= 2pT V
(2π)2
3
∑
i=1
gimT,i
×
[
1 + (q − 1)mT,i
T
]− qq−1
(12)
where i = π+, K+, p. The additional factor of 2 on the
right-hand side of Eq. (12) takes into account the negatively
1 10
2
10
-16
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
1
2
10
s VeT7=
 = 0.9 TeV s
-2
) 
(G
eV
/c
)
T
d
p
/(
d
ch
N2
) 
(d
T
p
1/
(2
 (GeV/c)
T
p
Fig. 3 Charged particle differential transverse momentum yields mea-
sured within |η| < 2.4 by the CMS detector in proton–proton collisions
at
√
s = 0.9 and 7 TeV [14] fitted with a Tsallis distribution
 (GeV/c)
T
p
1 10
2
10
D
at
a/
F
it
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
 = 0.9 TeVs
s VeT7=
Fig. 4 Ratios of the data to the fit values measured by the CMS detector
at
√
s = 0.9 and 7 TeV
charged particles π−, K−, p¯. The relative weights between
particles were determined by the corresponding degeneracy
factors and are given by gπ+ = gK+ = 1 and gp = 2.
The square root of the sum of the squares of the statistical
and systematic errors (quadrature) was taken into account for
fitting the data.
The charged particle multiplicity as a function of the trans-
verse momentum measured by the ATLAS collaboration for
123
430 Page 4 of 5 Eur. Phys. J. C (2015) 75 :430
Table 1 Values of the q, T , and
R parameters and χ2/N DF
obtained using Eq. (9) to fit the
pT spectra measured by the
ATLAS [7] and CMS [14]
detectors
Experiment
√
s (TeV) q T (MeV) R (fm) χ2/N DF
ATLAS 0.9 1.129 ± 0.005 74.21 ± 3.55 4.62 ± 0.29 0.657503/36
ATLAS 7 1.150 ± 0.002 75.00 ± 3.21 5.05 ± 0.07 4.35145/41
CMS 0.9 1.129 ± 0.003 76.00 ± 0.17 4.32 ± 0.29 0.648806/17
CMS 7 1.153 ± 0.002 73.00 ± 1.42 5.04 ± 0.27 0.521746/24
Table 2 Values of dNdy
∣
∣
∣
y=0 for
pions, kaons, protons, and total
charged particles obtained using
Eq. (11) and dNchdη
∣
∣
∣
η=0
measured by ATLAS detector
Experiment
√
s (TeV) dN
π++π−
dy
∣
∣
∣
y=0
dN K
++K−
dy
∣
∣
∣
y=0
dN p+ p¯
dy
∣
∣
∣
y=0
dN
dy
∣
∣
∣
y=0
dNch
dη
∣
∣
∣
η=0
ATLAS 0.9 2.84 0.76 0.32 3.92 3.85 ± 0.18
ATLAS 7 4.27 1.37 0.72 6.36 6.25 ± 0.30
CMS 0.9 2.52 0.70 0.30 3.52
CMS 7 3.92 1.24 0.66 5.82
events with nch ≥ 2, pT > 100 MeV and |η| < 2.5 at√
s = 0.9 and 7 TeV in proton–proton collisions are shown
in Fig. 1. The parameterization given in Eq. (12) has been
used to fit the spectra. In order to show the quality of the
fits in more detail, Fig. 2 shows the ratios of the data to the
fit values at
√
s = 0.9 and 7 TeV for the ATLAS data. The
largest deviations occur at the lower energy 0.9 TeV, while
the 7 TeV data show deviations from the Tsallis fits which
are at most in the 10 % range.
The charged particle differential transverse momentum
yields in proton–proton collisions at
√
s = 0.9 and 7 TeV
have been measured within |η| < 2.4 by the CMS detector.
The CMS measurements are described by the Tsallis fit and
displayed in Fig. 3. The ratios of the data to the Tsallis fit at√
s = 0.9 and 7 TeV are shown in Fig. 4. Again, in order to
show the quality of the fits in more detail, Fig. 4 shows the
ratios of the data to the fit values at
√
s = 0.9 and 7 TeV
for the CMS data. The largest deviations occur at the highest
values of pT , where also the error bars are largest.
4 Results and discussions
A thermodynamically consistent form of the Tsallis distri-
bution, given in Eq. (9), has been used to fit the transverse
momentum spectra of the charged particle measured by both
ATLAS and CMS collaborations in proton–proton collisions
at
√
s = 0.9 and 7 TeV. It is observed from Figs. 1 and 3
that the Tsallis distribution fits well the measured pT spec-
tra, albeit that the kinematical conditions of the data of both
detectors were not identical. It is clear that the proposed form
of the Tsallis distribution fits not only the low pT but covers
a wide range of pT spectra up to 200 GeV/c.
The ratios of the experimental data over the fit values,
depicted in Figs. 2 and 4, show an intriguing log-periodic
oscillation as a function of the transverse momentum not only
for high pT data sets [14] but also for low pT data sets [7].
This was first noticed in [19,20] for CMS data sets only. The
origin of such oscillations is a matter of investigation aiming
to draw some useful physics out of it.
The values of the fit parameters, q, T , and the correspond-
ing χ2/N DF obtained from the fits of the pT spectra are
given in Table 1. For the highest values of pT , reaching
200 GeV/c, a value of χ2/N DF ≈ 0.52/24 for the CMS
data is obtained.
Instead of parameter V , a parameter R ≡ [V 34π ]1/3,
referred to as radius, is listed in the table. It has been found
that the value of q increases slowly but clearly with increase
in the center-of-mass energy. The thermodynamically con-
sistent version leads to a temperature T ≈ 75 MeV at both
values of
√
s. It can be observed that the parameter R ≈ 4.5
and 5.0 fm for
√
s = 0.9 and 7 TeV, respectively, for both
ATLAS and CMS data sets. These results are in agreement
with the previous analysis done for lower pT data [9,11].
Table 2 shows the values of dNdy
∣
∣
∣
y=0 for pions, kaons,
protons, and total charged particles evaluated using Eq. (11).
The dNchdη
∣
∣
∣
η=0 values measured by the ATLAS detector is also
present. It can be found that the production of pions is less
and more kaons and protons are produced at higher energies.
The charged particle multiplicity found for ATLAS using a
Tsallis fit is close to the measured values. An estimate of the
charged particle multiplicity is presented for CMS, which
can be verified by the CMS collaboration.
5 Conclusions
We find it quite remarkable that the transverse momen-
tum distributions measured up to 200 GeV/c in pT can be
described consistently over 14 orders of magnitude by a
straightforward Tsallis distribution as presented in Eq. (12).
The values of χ2/N DF are listed in Table 1 and for the high-
est values of beam energy and pT it is found to be 0.52/24 for
123
Eur. Phys. J. C (2015) 75 :430 Page 5 of 5 430
the CMS data. The distribution used is part of a consistent
thermodynamic description satisfying the basic consistency
conditions given in Eq. (7).
In conclusion, we have shown that the proposed version of
the Tsallis distribution fits the transverse momentum spectra
with minimal values of χ2 values, and an estimate of the
particle multiplicities and particle ratio has been obtained.
OpenAccess This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
References
1. B.I. Abelev et al. (STAR collaboration), Phys. Rev. C 75, 064901
(2007)
2. A. Adare et al. (PHENIX collaboration), Phys. Rev. D 83, 052004
(2011)
3. A. Adare et al. (PHENIX collaboration), Phys. Rev. C 83, 064903
(2011)
4. K. Aamodt et al. (ALICE collaboration), Phys. Lett. B 693, 53
(2010)
5. V. Khachatryan et al. (CMS collaboration), JHEP 02, 041 (2010)
6. V. Khachatryan et al. (CMS collaboration), Phys. Rev. Lett. 105,
022002 (2010)
7. G. Aad et al. (ATLAS collaboration), New J. Phys. 13, 053033
(2011)
8. K. Aamodt et al. (ALICE collaboration), Eur. Phys. J. C 73, 2662
(2013)
9. J. Cleymans, G.I. Lykasov, A.S. Parvan, A.S. Sorin, O.V. Teryaev,
D. Worku, Phys. Lett. B 723, 351 (2013)
10. M. Rybczynski, Z. Wlodarczyk, Eur. Phys. J. C 74, 2785 (2014)
11. M.D. Azmi, J. Cleymans, J. Phys. G 41, 065001 (2014)
12. G. Wilk, Z. Wlodarczyk, Phys. Rev. Lett. 84, 2770 (2000)
13. C. Tsallis, J. Stat. Phys. 52, 479 (1988)
14. S. Chatrchyan et al. (CMS collaboration), JHEP 1108, 086 (2011)
15. C.Y. Wong, G. Wilk, Phys. Rev. D 87, 114007 (2013)
16. L.J.L. Cirto, C. Tsallis, C.Y. Wong, G. Wilk. arXiv:1409.3278 [hep-
ph]
17. C.Y. Wong, G. Wilk, L.J.L. Cirlo, C. Tsallis. arXiv:1412.0474 [hep-
ph]
18. K. Urmossy, G.G. Barnaföldi, S. Harangozó, T.S. Biró, Z. Xu.
arXiv:1501.02352 [hep-ph]
19. G. Wilk, Z. Wlodarczyk, Phys. A 413, 53 (2014)
20. G. Wilk, Z. Wlodarczyk. arXiv:1403.3508 [hep-ph]
21. J.M. Conroy, H.G. Miller, A.R. Plastino, Phys. Lett. A 374, 4581
(2010)
22. J. Cleymans, D. Worku, J. Phys. G 39, 025006 (2012)
23. J. Cleymans, D. Worku, Eur. Phys. J. A 48, 160 (2012)
123


The Tsallis distribution at large transverse momenta

The charged particle transverse momentum ($$p_T$$) spectra measured by the ATLAS and CMS collaborations for proton–proton collisions at $$\sqrt{s} = 0.9$$ and 7 TeV have been studied using Tsallis thermodynamics. A thermodynamically consistent form of the Tsallis distribution is used for fitting the transverse momentum spectra at mid-rapidity. It is found that the fits based on the proposed distribution are working over 14 orders of magnitude with $$p_T$$ values up to 200 GeV/c and this gives a value of $$\chi ^2/NDF$$ of 0.52/24 for the CMS data at 7 TeV. The values for $$\frac{\mathrm{d}N}{\mathrm{d}y}|_{y=0}$$ as a function of center-of-mass energy obtained using a Tsallis distribution for both ATLAS and CMS data sets are presented and discussed

EDP Sciences OAI-PMH repository (1.2.0)

Crossref

Eur. Phys. J. C (2015) 75:430DOI 10.1140/epjc/s10052-015-3629-9Regular Article - Theoretical PhysicsThe Tsallis distribution at large transverse momentaM. D. Azmi1,a, J. Cleymans2,b1 HEP Lab, Physics Department, Aligarh Muslim University, Aligarh 202002, India2 Physics Department, UCT-CERN Research Centre, University of Cape Town, Rondebosch, Cape Town 7701, South AfricaReceived: 16 June 2015 / Accepted: 17 August 2015 / Published online: 16 September 2015© The Author(s) 2015. This article is published with open access at Springerlink.comAbstract The charged particle transverse momentum (pT )spectra measured by the ATLAS and CMS collaborations forproton–proton collisions at√s = 0.9 and 7 TeV have beenstudied using Tsallis thermodynamics. A thermodynamicallyconsistent form of the Tsallis distribution is used for fittingthe transverse momentum spectra at mid-rapidity. It is foundthat the fits based on the proposed distribution are workingover 14 orders of magnitude with pT values up to 200 GeV/cand this gives a value of χ2/N DF of 0.52/24 for the CMSdata at 7 TeV. The values for dNdy |y=0 as a function of center-of-mass energy obtained using a Tsallis distribution for bothATLAS and CMS data sets are presented and discussed.1 IntroductionIt is well known that the Tsallis distribution gives excellentfits to the transverse momentum distributions observed atthe Relativistic Heavy Ion Collider (RHIC) [1–3] and at theLarge Hadron Collider (LHC) [4–8] with only three parame-ters, q, T , and dN/dy (or, alternatively, a volume V [9–11]).The parameter q is referred to as the Tsallis parameter anddiscussed elsewhere [12] in detail. The parameter T obeysthe standard thermodynamic relation,T = ∂E∂S∣∣∣∣N ,V,however, the entropy used in the above equation is the Tsal-lis entropy [13], not the standard Boltzmann–Gibbs (BG)entropy. The parameter V is not necessarily related to a vol-ume deduced from other models like HBT calculations.It was recently shown that these fits extend to values [14]of pT up to 200 GeV/c [15,16]. This is unexpected because inthis kinematic range hard scattering processes become impor-a e-mail: danish.hep@gmail.comb e-mails: jean.cleymans@gmail.com; jean.cleymans@uct.ac.zatant [17]. A description of the high pT results has been dis-cussed in [18] where a model using a combination of Tsallisat low pT and QCD hard scattering at high pT was consid-ered. The present analysis shows that the Tsallis distributiondescribes measurements up to the highest pT using the sameTsallis parameters as those obtained at low pT .A power law based on the Tsallis distribution [13] is usedto fit the pT spectra of charged particles measured by theATLAS and CMS collaborations. The ATLAS collaborationhas reported the transverse momentum in an inclusive phasespace region taking into account at least two charged particlesin the kinematic range |η| < 2.5 and pT > 100 MeV [7].The CMS collaboration has presented the differential trans-verse momentum distribution covering a pT range up to200 GeV/c, the largest range ever measured in a collidingbeam experiment [14]. The results for the charged particlemultiplicities are consistent with those of ATLAS measure-ments.Moreover, the results are compared with those obtainedin [15–17,19,20] where very good fits to transverse momen-tum distributions were presented. We confirm the quality ofthe fits but obtain different values of the parameters albeitusing a different version of the Tsallis model.2 Tsallis distributionPower law distributions have been widely applied in highenergy physics [1–8] to describe the transverse momentumspectra of secondary particles produced in proton–proton col-lisions. With the advent of the LHC, transverse momenta ofhundreds of GeV have been measured. It is of great interest tochoose a distribution which describes such high pT measure-ments very well. A well-suited form of the Tsallis distributionis discussed in this paper and our main criterion for choosingsuch a distribution was thermodynamic consistency, whichhas not always been implemented fully [21–23].It is well established that there are numerous physical sys-tems under which BG statistics encounters many difficulties.123430 Page 2 of 5 Eur. Phys. J. C (2015) 75 :430In particular, when analyzing the pT spectra of hadrons itis found that spectra decrease far slower than predicted byBG statistics, and they appear to follow some power law athigh pT . The Tsallis distribution was first proposed about27 years [13] ago as a generalization of the BG distribution.The Tsallis form of the BG distribution is given byf (E) ≡ expq(− E − μT), (1)where E is the energy and μ is the chemical potential andthe function expq(x) is usually defined asexpq(x) ≡{[1 + (q − 1)x]1/(q−1) if x > 0,[1 + (1 − q)x]1/(1−q) if x ≤ 0, (2)in the limit where q → 1 it reduces to the standard exponen-tial:limq→1 expq(x) → exp(x).The expressions of the relevant thermodynamic quantitieslike entropy, S, energy density, (= E/V ), pressure, P , andparticle number, N , are given below using the q-logarithmfunction,lnq(x) ≡ x1−q − 11 − q ,as integrals over the function defined in Eq. (1) [11,22,23]:S = −gV∫d3 p(2π)3[f q lnq f − f], (3) = g∫d3 p(2π)3E f q , (4)P = g∫d3 p(2π)3p23Ef q , (5)N = gV∫d3 p(2π)3f q , (6)where g is the degeneracy factor and V is the volume. TheTsallis distribution used in this paper is thermodynamicallyconsistent, which means that the following relations, derivedfrom the differential form of the first and second laws ofthermodynamics, are satisfied [23]:T = ∂∂s∣∣∣∣n, μ = ∂∂n∣∣∣∣s,n = ∂P∂μ∣∣∣∣T, s = ∂P∂T∣∣∣∣μ, (7)where s and n are the entropy and particle number densitiesrespectively.The corresponding momentum distribution deduced fromEq. (6) is given byEd3Ndp3= gV E 1(2π)3[1 + (q − 1) E − μT]− qq−1, (8)which in terms of the rapidity, y, and transverse mass, mT ,variables (E = mT cosh y) at mid-rapidity and for μ = 0becomes [11,22]d2NdpT dy∣∣∣∣y=0= gV pT mT(2π)2[1 + (q − 1)mTT]− qq−1. (9)The relationship between the above parameterization andthe one used by the RHIC and LHC collaborations [1–8] hasbeen discussed in [9].Additionally, a comparison with high-energy data hasbeen made in [15–17,19,20], using a simplified form:d2NchdpT dy∣∣∣∣y=0= A[1 + (q − 1) pTT]− 1q−1(10)where A is a normalization factor and T is a parameter usedto fit the transverse momentum distribution and not relatedto the temperature in the thermodynamic sense as in Eq. (9).Furthermore, integration of Eq. (9) over the transversemomentum leads to [10]dNdy∣∣∣∣y=0= gV(2π)2∫ ∞0pT dpT mT[1 + (q − 1)mTT]− qq−1= gV T(2π)2[(2 − q)m2 + 2mT + 2T 2(2 − q)(3 − 2q)]×[1 + (q − 1)mT]− 1q−1, (11)where m is the rest mass.Using the values of parameters q, T , and V obtained fromTsallis fit, using Eq. (9), the above equation enables us toobtain the values of dN/dy for pions, kaons, and protons.Their sum gives the values of dN/dy for charged particles.3 Fit detailsThe charged particle transverse momentum spectra measuredby the ATLAS and CMS detectors in proton–proton colli-sions at LHC energies are fitted using Eq. (9).It is well established that the charged particle pT spec-tra is composed of contribution from pions, kaons, and pro-tons. Hence, a sum of three Tsallis distributions for pions,π+’s, kaons, K+’s, and protons, p, has been applied to fitthe charged particle pT spectra in the following way:123Eur. Phys. J. C (2015) 75 :430 Page 3 of 5 430-110 1 10210-1010-910-810-710-610-510-410-310-210-110110210s VeT7= = 0.9 TeV s]-2) [GeVTdp/(dchN2) (dTp 1/(2evt1/N (GeV/c)TpFig. 1 Charged particle multiplicities as a function of the transversemomentum measured by the ATLAS detector for events with nch ≥ 2,pT > 100 MeV and |η| < 2.5 at √s = 0.9 and 7 TeV in proton–protoncollisions [7] fitted with a Tsallis distribution (GeV/c)Tp-110 1 10Data/Fit0.60.70.80.911.11.21.3 = 0.9 TeVss VeT7=Fig. 2 Ratios of the data to the fit values measured by the ATLASdetector at√s = 0.9 and 7 TeVd2NchdpT dy∣∣∣∣y=0= 2pT V(2π)23∑i=1gimT,i×[1 + (q − 1)mT,iT]− qq−1(12)where i = π+, K+, p. The additional factor of 2 on theright-hand side of Eq. (12) takes into account the negatively1 10210-1610-1410-1210-1010-810-610-410-2101210s VeT7= = 0.9 TeV s-2) (GeV/c)Tdp/(dchN2) (dTp1/(2 (GeV/c)TpFig. 3 Charged particle differential transverse momentum yields mea-sured within |η| < 2.4 by the CMS detector in proton–proton collisionsat√s = 0.9 and 7 TeV [14] fitted with a Tsallis distribution (GeV/c)Tp1 10210Data/Fit0.60.70.80.911.11.21.3 = 0.9 TeVss VeT7=Fig. 4 Ratios of the data to the fit values measured by the CMS detectorat√s = 0.9 and 7 TeVcharged particles π−, K−, p¯. The relative weights betweenparticles were determined by the corresponding degeneracyfactors and are given by gπ+ = gK+ = 1 and gp = 2.The square root of the sum of the squares of the statisticaland systematic errors (quadrature) was taken into account forfitting the data.The charged particle multiplicity as a function of the trans-verse momentum measured by the ATLAS collaboration for123430 Page 4 of 5 Eur. Phys. J. C (2015) 75 :430Table 1 Values of the q, T , andR parameters and χ2/N DFobtained using Eq. (9) to fit thepT spectra measured by theATLAS [7] and CMS [14]detectorsExperiment√s (TeV) q T (MeV) R (fm) χ2/N DFATLAS 0.9 1.129 ± 0.005 74.21 ± 3.55 4.62 ± 0.29 0.657503/36ATLAS 7 1.150 ± 0.002 75.00 ± 3.21 5.05 ± 0.07 4.35145/41CMS 0.9 1.129 ± 0.003 76.00 ± 0.17 4.32 ± 0.29 0.648806/17CMS 7 1.153 ± 0.002 73.00 ± 1.42 5.04 ± 0.27 0.521746/24Table 2 Values of dNdy∣∣∣y=0 forpions, kaons, protons, and totalcharged particles obtained usingEq. (11) and dNchdη∣∣∣η=0measured by ATLAS detectorExperiment√s (TeV) dNπ++π−dy∣∣∣y=0dN K++K−dy∣∣∣y=0dN p+ p¯dy∣∣∣y=0dNdy∣∣∣y=0dNchdη∣∣∣η=0ATLAS 0.9 2.84 0.76 0.32 3.92 3.85 ± 0.18ATLAS 7 4.27 1.37 0.72 6.36 6.25 ± 0.30CMS 0.9 2.52 0.70 0.30 3.52CMS 7 3.92 1.24 0.66 5.82events with nch ≥ 2, pT > 100 MeV and |η| < 2.5 at√s = 0.9 and 7 TeV in proton–proton collisions are shownin Fig. 1. The parameterization given in Eq. (12) has beenused to fit the spectra. In order to show the quality of thefits in more detail, Fig. 2 shows the ratios of the data to thefit values at√s = 0.9 and 7 TeV for the ATLAS data. Thelargest deviations occur at the lower energy 0.9 TeV, whilethe 7 TeV data show deviations from the Tsallis fits whichare at most in the 10 % range.The charged particle differential transverse momentumyields in proton–proton collisions at√s = 0.9 and 7 TeVhave been measured within |η| < 2.4 by the CMS detector.The CMS measurements are described by the Tsallis fit anddisplayed in Fig. 3. The ratios of the data to the Tsallis fit at√s = 0.9 and 7 TeV are shown in Fig. 4. Again, in order toshow the quality of the fits in more detail, Fig. 4 shows theratios of the data to the fit values at√s = 0.9 and 7 TeVfor the CMS data. The largest deviations occur at the highestvalues of pT , where also the error bars are largest.4 Results and discussionsA thermodynamically consistent form of the Tsallis distri-bution, given in Eq. (9), has been used to fit the transversemomentum spectra of the charged particle measured by bothATLAS and CMS collaborations in proton–proton collisionsat√s = 0.9 and 7 TeV. It is observed from Figs. 1 and 3that the Tsallis distribution fits well the measured pT spec-tra, albeit that the kinematical conditions of the data of bothdetectors were not identical. It is clear that the proposed formof the Tsallis distribution fits not only the low pT but coversa wide range of pT spectra up to 200 GeV/c.The ratios of the experimental data over the fit values,depicted in Figs. 2 and 4, show an intriguing log-periodicoscillation as a function of the transverse momentum not onlyfor high pT data sets [14] but also for low pT data sets [7].This was first noticed in [19,20] for CMS data sets only. Theorigin of such oscillations is a matter of investigation aimingto draw some useful physics out of it.The values of the fit parameters, q, T , and the correspond-ing χ2/N DF obtained from the fits of the pT spectra aregiven in Table 1. For the highest values of pT , reaching200 GeV/c, a value of χ2/N DF ≈ 0.52/24 for the CMSdata is obtained.Instead of parameter V , a parameter R ≡ [V 34π ]1/3,referred to as radius, is listed in the table. It has been foundthat the value of q increases slowly but clearly with increasein the center-of-mass energy. The thermodynamically con-sistent version leads to a temperature T ≈ 75 MeV at bothvalues of√s. It can be observed that the parameter R ≈ 4.5and 5.0 fm for√s = 0.9 and 7 TeV, respectively, for bothATLAS and CMS data sets. These results are in agreementwith the previous analysis done for lower pT data [9,11].Table 2 shows the values of dNdy∣∣∣y=0 for pions, kaons,protons, and total charged particles evaluated using Eq. (11).The dNchdη∣∣∣η=0 values measured by the ATLAS detector is alsopresent. It can be found that the production of pions is lessand more kaons and protons are produced at higher energies.The charged particle multiplicity found for ATLAS using aTsallis fit is close to the measured values. An estimate of thecharged particle multiplicity is presented for CMS, whichcan be verified by the CMS collaboration.5 ConclusionsWe find it quite remarkable that the transverse momen-tum distributions measured up to 200 GeV/c in pT can bedescribed consistently over 14 orders of magnitude by astraightforward Tsallis distribution as presented in Eq. (12).The values of χ2/N DF are listed in Table 1 and for the high-est values of beam energy and pT it is found to be 0.52/24 for123Eur. Phys. J. C (2015) 75 :430 Page 5 of 5 430the CMS data. The distribution used is part of a consistentthermodynamic description satisfying the basic consistencyconditions given in Eq. (7).In conclusion, we have shown that the proposed version ofthe Tsallis distribution fits the transverse momentum spectrawith minimal values of χ2 values, and an estimate of theparticle multiplicities and particle ratio has been obtained.OpenAccess This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.References1. B.I. Abelev et al. (STAR collaboration), Phys. Rev. C 75, 064901(2007)2. A. Adare et al. (PHENIX collaboration), Phys. Rev. D 83, 052004(2011)3. A. Adare et al. (PHENIX collaboration), Phys. Rev. C 83, 064903(2011)4. K. Aamodt et al. (ALICE collaboration), Phys. Lett. B 693, 53(2010)5. V. Khachatryan et al. (CMS collaboration), JHEP 02, 041 (2010)6. V. Khachatryan et al. (CMS collaboration), Phys. Rev. Lett. 105,022002 (2010)7. G. Aad et al. (ATLAS collaboration), New J. Phys. 13, 053033(2011)8. K. Aamodt et al. (ALICE collaboration), Eur. Phys. J. C 73, 2662(2013)9. J. Cleymans, G.I. Lykasov, A.S. Parvan, A.S. Sorin, O.V. Teryaev,D. Worku, Phys. Lett. B 723, 351 (2013)10. M. Rybczynski, Z. Wlodarczyk, Eur. Phys. J. C 74, 2785 (2014)11. M.D. Azmi, J. Cleymans, J. Phys. G 41, 065001 (2014)12. G. Wilk, Z. Wlodarczyk, Phys. Rev. Lett. 84, 2770 (2000)13. C. Tsallis, J. Stat. Phys. 52, 479 (1988)14. S. Chatrchyan et al. (CMS collaboration), JHEP 1108, 086 (2011)15. C.Y. Wong, G. Wilk, Phys. Rev. D 87, 114007 (2013)16. L.J.L. Cirto, C. Tsallis, C.Y. Wong, G. Wilk. arXiv:1409.3278 [hep-ph]17. C.Y. Wong, G. Wilk, L.J.L. Cirlo, C. Tsallis. arXiv:1412.0474 [hep-ph]18. K. Urmossy, G.G. Barnaföldi, S. Harangozó, T.S. Biró, Z. Xu.arXiv:1501.02352 [hep-ph]19. G. Wilk, Z. Wlodarczyk, Phys. A 413, 53 (2014)20. G. Wilk, Z. Wlodarczyk. arXiv:1403.3508 [hep-ph]21. J.M. Conroy, H.G. Miller, A.R. Plastino, Phys. Lett. A 374, 4581(2010)22. J. Cleymans, D. Worku, J. Phys. G 39, 025006 (2012)23. J. Cleymans, D. Worku, Eur. Phys. J. A 48, 160 (2012)123

file:///data/remote/core/dit/data/Springer-OA/pdf/ff9/aHR0cDovL2xpbmsuc3ByaW5nZXIuY29tLzEwLjExNDAvZXBqYy9zMTAwNTItMDE1LTM2MjktOS5wZGY=.pdf

The Tsallis Distribution at Large Transverse Momenta

Abstract

Similar works

Full text

Available Versions

Springer - Publisher Connector

EDP Sciences OAI-PMH repository (1.2.0)

Crossref

Springer - Publisher Connector