Transverse Momentum Distributions in proton - proton Collisions at LHC Energies and Tsallis Thermodynamics

0.14 MB·

by  M. D. Azmi

Transverse Momentum Distributions in proton - proton
Collisions at LHC Energies and Tsallis Thermodynamics
4
1 M.D.Azmi1 andJ. Cleymans2
0
2 1HEPLab,DepartmentofPhysics,AligarhMuslimUniversity,Aligarh-202002,India
r 2UCT-CERNResearchCentreandDepartmentofPhysics,UniversityofCapeTown,
a
Rondebosch7701,SouthAfrica
M

1 Abstract. A detailed study of the transverse momentum distributions of charged particles
3
produced in proton - proton collisions at LHC energies is presented. This is done

] using a thermodynamically consistent form of the Tsallis distribution. All variables used
h
are thermodynamical and in particular, the temperature, T, follows from the standard
p
thermodynamicdefinition as being the derivative of the energy with respect to the (Tsallis)
-
p entropy. Themomentumdistributionofthefinalstateparticlescanbedescribedverywellby
e
theTsallisdistribution.Thevaluesoftheparametersaredeterminedfrommeasurementsbythe
h
[ ALICE,ATLASandCMScollaborationsandarediscussedindetail. Inparticular,theTsallis

parameter, q, is found with consistent values for all the transverse momentum distributions
4
v despitelargedifferencesinkinematicregionsandshowsa slightincreasewithbeamenergy,
5 reaching a value of 1.15 at 7 TeV. It is concluded that the hadronic system created in high-
3
energyp-pcollisionsatmid-rapiditycanbeseenasobeyingTsallisthermodynamics.
8
4
.
1
0 PACSnumbers:12.40.Ee,13.75.Cs,13.85.-t,05.70.-a,05.70.-a
4
1
:
v
i
X
r
a
TransverseMomentumandTsallisDistribution 2
1. Introduction
The dynamics of colliding hadrons can be determined from the distribution of the momenta
ofproduced particles. Theavailablerangeoftransversemomentahas expandedconsiderably
with the advent of the Large Hadron Collider (LHC) at CERN. Collider energies up to 7
TeV are now available in proton - proton collisions and transverse momenta of hundreds
of GeV are of a common occurrence. This helps in testing and applying the relativistic
thermodynamics and hydrodynamics in a new energy region. Thermal models are part of
the standard set of tools to analyze high-energy collisions and in this paper we investigate a
thermodynamically consistent model based on the Tsallis distribution [1, 2]. This leads to a
power law distribution which is well suited to describe the transverse momenta measured in
p - p collisions. The Tsallis distribution was first proposed about twenty-five years ago as a
generalizationoftheusualexponentialBoltzmann-Gibbsdistribution,andischaracterizedby
threeparametersq,T andV. VariousversionsoftheTsallisdistributionhavebeenconsidered
intheliterature[3,4,5,6,7,8,9]. Aformwhichissuitedfordescribingresultsinhighenergy
particlephysicsisused inthispaper.
ThevariableT used hereobeysthestandardthermodynamicrelation [10, 11],
¶ E
T = , (1)
¶ S(cid:12)
(cid:12)N,V
(cid:12)
and can, therefore, be re(cid:12)ferred as temperature. However, since the entropy used in Eq. (1) is
theTsallisentropy[1]andnotthestandardentropy,wewillcallthevariabledefinedinEq.(1)
as q-temperature. For a similar reason the volume V can be obtained from thermodynamic
relationsas,
¶ H
V = , (2)
¶ P(cid:12)
(cid:12)N,S
(cid:12)
with H =E+PV is the(cid:12)standard definition of the enthalpy and could be called q-volume as
it is obtained at fixed Tsallis entropy but we refrain from doing so and refer to it as volume
but note that it is not necessarily related to a volume deduced from other models, say HBT
calculations.
Alotofinteresthasdevelopedrecently[12,13,14,15,16]intheTsallisdistributionand
ithasbeenusedpreviouslytodescribethetransversemomentumdataofthechargedparticles
produced in proton - protoncollisionsat RHIC and LHC energies [17, 18, 19, 20, 21, 22, 23,
24]. A comparativestudy,whichhasnotbeen donebefore, oftheTsallisfitstothetransverse
momentum distributions of the charged particles produced in proton - proton collisions
measuredbyALICE,ATLASandCMScollaborationsispresentedhere. Theparametersused
intheTsallisfitarestudiedatdifferentenergiesandunderdifferentkinematicalconditionsof
the data collection. The parameters are measured at various multiplicities. The fit results the
values of Tsallis parameter, q, in the range 1.1 to 1.15 for all the measured distributions and
found to beconsistent in all the conditionsand at all energies. The T and R parameters show
dependence on themultiplicityand on particleyields too. Theresults presented here confirm
thatthehadronicsystemcreatedinhigh-energyp-pcollisionsobeysTsallisthermodynamics.
TransverseMomentumandTsallisDistribution 3
2. TsallisDistribution
The transverse momentum distribution in heavy-ion collisions is often described by a
combination of transverse flow and a thermodynamical statistical distribution. In p - p
collisions with the Tsallis distribution such a superposition is not needed and very good fits
can beobtained.
In theframeworkofTsallisstatistics[1, 8, 12, 10, 11]integralsover
E m −q11
f = 1+(q 1) − − (3)
(cid:20) − T (cid:21)
givetheentropy,S,theparticlenumber,N, theenergy density,e , and thepressure, P.
Usingthefunction
x1 q 1
−
ln (x) − ,
q
≡ 1 q
−
often referred to as q-logarithm, it can be shown[11] that the relevant thermodynamic
quantitiesare givenby:
d3p
S = gV fqln f f , (4)
− Z (2p )3 q −
(cid:2) (cid:3)
d3p
N =gV fq, (5)
Z (2p )3
d3p
e =g E fq, (6)
Z (2p )3
d3p p2
P =g fq. (7)
Z (2p )33E
whereV isthevolumeandg isthedegeneracy factor.
Inordertousetheaboveequationsithastobeshownthattheysatisfythethermodynamic
consistency conditions. The first and second laws of thermodynamics lead to the following
twodifferentialrelations[25]:
de =T ds+m dn, (8)
dP=s dT +n dm . (9)
where, s=S/V andn=N/V are theentropyand particlenumberdensities,respectively.
Thermodynamicconsistencyrequires thatthefollowingrelationsbesatisfied:
¶e
T = , (10)
¶ s(cid:12)
(cid:12)n
¶e (cid:12)
m = (cid:12) , (11)
¶ n(cid:12)
(cid:12)s
¶ P(cid:12)
(cid:12)
n= , (12)
¶m (cid:12)
(cid:12)T
¶ P (cid:12)
(cid:12)
s= . (13)
¶ T(cid:12)
m
(cid:12)
(cid:12)
(cid:12)
TransverseMomentumandTsallisDistribution 4
Eq. (10) in particular shows that the variable T appearing in Eq. (3) indeed can be
identified as a thermodynamic temperature. As explained in the introduction we prefer to
call it q-temperatureas it is based on theTsallis form of theentropy [1]. It is straightforward
toshowthattheserelationsare indeedsatisfied [11].
It can easily be shown that the following thermodynamic consistency relation is also
satisfied:
e +P=Ts+m n. (14)
Themomentumdistributionobtainedfrom Eq.(5)is givenby:
d3N gV E m q/(q 1)
− −
= 1+(q 1) − , (15)
d3p (2p )3(cid:20) − T (cid:21)
or expressed in terms of variables used in high-energy physics, transverse momentum, p ,
T
transversemass,m = p 2+m2 and rapidity,y:
T T
p
d2N p m coshy m coshy m q/(q 1)
T T T − −
=gV 1+(q 1) − (16)
dp dy (2p )2 (cid:20) − T (cid:21)
T
At mid-rapidity, y = 0, and for zero chemical potential, as is relevant at the LHC, Eq. (16)
reduces to thefollowingexpression:
d2N p m m q/(q 1)
=gV T T 1+(q 1) T − − (17)
dp dy(cid:12) (2p )2 − T
T (cid:12)y=0 h i
(cid:12)
It is worth to me(cid:12)ntion that the parameterization given in Eq. (17) is close to the
parameterization used for fitting the data taken at RHIC and LHC experiments [17, 18, 19,
20, 21, 22, 23, 24]. The parameterization used by the RHIC and LHC experiments is given
below:
d2N dN (n 1)(n 2) mT m0 −n
= p − − 1+ − , (18)
T
dp dy dy nC(nC+m (n 2))(cid:20) nC (cid:21)
T 0
−
wheren,C and m are thefit parameters usedin theparameterization.
0
At mid-rapidity, y = 0, and for zero chemical potential Eq. (18) shows the same
dependenceonthetransversemomentumasEq.(17)exceptforanadditionalfactorm which
T
ispresent inEq. (17)butnotin Eq.(18).
3. Fitdetails
The transverse momentum distributions of charged particles produced in p p collisions at
−
LHC energies were fitted using a sum of three Tsallis distributions. These consist of fits for
p +’s,K+’sand protons, p. Thefollowingexpression,at mid-rapidityand m =0, was used to
fit thedistributionsobtainedinvariousexperiments:
ddp2Ncdhy(cid:12) =2pT(2Vp )2 (cid:229) 3 gimT,i 1+(q−1)mTT,i (q−−q1) (19)
T (cid:12)y=0 i=1 h i
(cid:12)
where i = p +,K+,p. (cid:12) The relative weights between particles were determined by the
corresponding degeneracy factors and given by gp + = gK+ = 1 and gp = 2. The factor 2
TransverseMomentumandTsallisDistribution 5
on therighthand sidetakes intoaccount thecontributionsfrom antiparticles,p ,K and p¯.
− −
Insomecasesthefollowingformofthedistributionwasusedtofittheexperimentaldata
sets:
2p1p ddp2Ncdhy(cid:12) = (22pV)3 (cid:229) 3 gimT,i 1+(q−1)mTT,i (q−−q1) (20)
T T (cid:12)y=0 i=1 h i
(cid:12)
The transverse momentum s(cid:12)pectra of primary charged particles measured by the ALICE
collaboration [20] in INEL p p collisions at √s = 900 GeV ( h <0.8), normalized to the
− | |
total number of INEL events, N , for three different multiplicity selections (n ) together
evt acc
withtheTsallisfits, Eq.(17), is shownin Fig. 1.
1 102
-c)
V/ ALICE, pp, INEL
e
G 10 s = 900 GeV, |h | < 0.8
) (
hd
pT 1
d
/(h
2Ndc10-1
(1/Nevt10-2
10-3
10-4
nacc = 17 (x 200)
10-5 nacc = 7 (x 5)
nacc = 3
10-6
0.5 1 1.5 2 2.5 3 3.5 4 4.5
p (GeV/c)
T
Figure1. TransversemomentumdistributionsofchargedparticlesasmeasuredbytheALICE
collaborationin p pcollisionsat√s=0.9TeVfittedwithTsallisdistribution.
−
The charged hadron yields as measured by the CMS collaboration [21, 22] in the range
h <2.4innon-single-diffractive(NSD)eventsasafunctionof p atallthreecenter-of-mass
T
| |
energies, fitted withTsallisdistributionareshowninFig. 2.
Fig. 3 shows the charged particle multiplicities as a function of transverse momentum
measured by the ATLAS collaboration [23] for events with n 1, p > 500 MeV and
ch T
≥
h <2.5 at√s =0.9,2.36 and7 TeV fitted withTsallisdistribution.
| |
The Tsallis fits for the charged particle multiplicities as a function of transverse
momentum measured by the ATLAS collaboration [23] for events with n 2, p > 100
ch T
≥
MeV and h <2.5at √s= 0.9and 7TeV are showninFig. 4.
| |
Fig. 5 shows the Tsallis fits on the charged particle multiplicities as a function of
transverse momentum measured by the ATLAS collaboration [23] for events with n 6,
ch
≥
p >500 MeVand h <2.5at √s =0.9and 7 TeV.
T
| |
Fig. 6showstheTsallisfitsforthechargedparticlecrosssectionforthreedifferentbeam
TransverseMomentumandTsallisDistribution 6
2
-V/c)103 CMS, pp collisions, |h | < 2.4
e
G
(T102
p
d

hd 10
/h
Nc
2d 1
) T
p
10-1
p2
1/(
10-2
10-3
10-4 s = 7 TeV (x 100)
s = 2.36 TeV (x 10)
10-5
s = 0.9 TeV
10-6
0 1 2 3 4 5 6
p (GeV/c)
T
Figure2. TransversemomentumdistributionsofchargedparticlesasmeasuredbytheCMS
collaborationin p pcollisionsat√s=0.9,2.36and7TeVfittedwithTsallisdistribution.
−
energies, 0.9,2.76 and7 TeV measured recentlyby theALICE collaboration[24].
In all cases thefits are verygood.
4. Results
ThetransversemomentumspectrumoftheprimarychargedparticlesmeasuredbytheALICE,
CMSandATLAScollaborationsandcrosssectionmeasuredbytheALICEcollaboration[20,
21, 22, 23, 24] in p p collisions at LHC energies were fitted using the Tsallis distribution
−
givenin Eq.(17).
Though,thekinematicalconditionsofthedatatakingofallthethreecollaborationswere
different. The Tsallis distribution fits well to the measured p spectrum as well as the cross
T
section.
The values of the Tsallis parameter, q, the q-temperature, T, and the radius, R, defined
as R (3V/4p )1/3 obtained from the fits of the p spectrum are shown in Figs. 7, 8 and 9,
T
≡
respectively. The values of the q parameter shows a clear increase with beam energy. The
collaborations show consistent compatible results. It is to be noted that the parameter q can
bedeterminedwithafairly highprecision.
The results for the q-temperature, T, shown in Fig. 8 are compatible and show no clear
dependenceonthebeamenergy. Theonlypointsthatareclearlyhigherarethoseobtainedby
theATLAScollaborationfordatawithahighnumberofcharged particles(n 6).
ch
≥
TheresultsfortheparameterRagainshowaclearincreasewithbeamenergy,whichwas
notseen in apreviousanalysis[16]. Againtheresultsare compatible.
TransverseMomentumandTsallisDistribution 7
2 104
-V/c)103 nch ‡ 1, pT > 500 MeV, |h | < 2.5
e
G102
(pT10
d
1
hd
/Nch10-1
2) d10-2
T
p10-3
p2
1/(10-4
Nev10-5
1/10-6
10-7
10-8 s = 7 TeV (x 100)
10-9
s = 2.36 TeV (x 10)
10-10
s = 0.9 TeV
10-11
1 10 102
p (GeV/c)
T
Figure3.TransversemomentumdistributionsofchargedparticlesasmeasuredbytheATLAS
collaborationatallthreecenter-of-massenergiestogetherwithTsallisfit.
2 103
-V/c)102 nch ‡ 2, pT > 100 MeV, |h | < 2.5
e
G 10
(pT 1
d

hd10-1
/h
Nc10-2
2d
) T10-3
p
10-4
p2
1/(10-5
Nev10-6
1/
10-7
10-8
10-9
s = 7 TeV (x 10)
10-10
10-11
s = 0.9 TeV
10-12
10-1 1 10 102
p (GeV/c)
T
Figure4.TransversemomentumdistributionsofchargedparticlesasmeasuredbytheATLAS
collaborationinthemostinclusivephase-spacespaceregionat√s=0.9and7TeVtogether
withTsallisfit.
TransverseMomentumandTsallisDistribution 8
2 103
-V/c)102 nch ‡ 6, pT > 500 MeV, |h | < 2.5
e
G 10
(T
p 1
d

hd10-1
/h
Nc10-2
2d
) 1T0-3
p
10-4
p2
1/(10-5
Nev10-6
1/
10-7
10-8
10-9
s = 7 TeV (x 10)
10-10
10-11 s = 0.9 TeV
10-12
1 10 102
p (GeV/c)
T
Figure 5. Transverse momentum distributions of charged particles with the smallest
contribution from diffractive events as measured by the ATLAS collaboration at √s = 0.9
and7TeVtogetherwithTsallisfit.
2)c104 ALICE, pp, INEL, |h | < 0.8
-2V103
e
G102
b
m 10
) (pT 1
d
h d10-1
)/(ch10-2
2sd10-3
) (pT10-4

p1/(210-5
10-6
10-7
s = 7 TeV
10-8
s = 2.76 TeV
10-9
s = 0.9 TeV
10-10
10-1 1 10 102
p (GeV/c)
T
Figure6. CrosssectionofchargedparticlesasmeasuredbytheALICEcollaboration[24]at
differentenergies.
TransverseMomentumandTsallisDistribution 9
1.3
1.25
11..22
q 1.15
1.11
1.05
0.9 0.9 0.9 0.9 2.36 7 0.9 2.36 7 0.9 2.76 7
0.81
ALICE CMS ATLAS AALLIICCEE
Figure 7. The Tsallis parameter q obtained from fits to the p spectrum. Results from the
T
ALICE collaboration are indicated with square points, CMS by triangles and ATLAS by
circles.Thebeamenergyisgivenonthex-axis.ThefirstthreepointsfromALICEcorrespond
todifferentvaluesofacceptedchargedparticles,nacc=3,7and17respectively.
120
110
100
90
)
V
e
M
(
T 80
70
60
0.9 0.9 0.9 0.9 2.36 7 0.9 2.36 7 0.9 2.76 7
50
ALICE CMS ATLAS ALICE
Figure 8. The q-temperature, T, obtained from fits to the p spectrum. Results from the
T
ALICE collaboration are indicated with square points, from CMS by triangles, those from
ATLAS by circles. The first three points were obtained at a beam energy of 900 GeV and
correspondtodifferentvaluesofacceptedchargedparticles,nacc=3,7and17respectively.
TransverseMomentumandTsallisDistribution 10
6
5
4
)
m
R (f nch _> 1 nch _> 2 nch _> 6
3
2
0.9 2.36 7 0.9 7 0.9 7 0.9 2.36 7 0.9 2.76 7
1
ATLAS CMS ALICE
Figure 9. The radius, R, obtained from fits to the p spectrum. The results from ATLAS
T
areindicatedbycircles, CMSbytrianglesandALICEbysquarepoints. TheATLASpoints
correspondtodifferentacceptancesofchargedparticles. Thebeamenergiesaregivenonthe
x-axis.
5. Conclusions
A thermodynamically consistent form of the Tsallis distribution, Eq. (17), gives an excellent
descriptionofthetransversemomentumspectraandcrosssectionoftheprimarychargedpar-
ticlesmeasured in p p collisionsat √s=0.9, 2.36,7 TeVand 2.76 TeV, respectively.
−
The values of the Tsallis parameter, q, are found between 1.11 to 1.15 and show a clear
increasewithbeam energy.
As observed from the fit of p spectra measured by the ALICE, ATLAS and CMS col-
T
laborations [20, 23, 21, 22, 24], the q-temperature, T, is consistent with being constant as a
functionofbeam energy.
Thevaluesofradius,R, showsasmallincreasewithbeam energy.
Overall, the values obtained for the Tsallis parameter, q, are remarkably consistent, a
featurewhich doesnot becomeapparent when usingtheparameterization ofEq.(18).
It is concluded that the hadronic system created in high-energy p - p collisions at central ra-
piditycan beseen asobeyingTsallisthermodynamics.