Tsallis statistics approach to the transverse momentum distributions in p-p collisions

4 MB·

by  Maciej Rybczynski

Eur.Phys.J.CmanuscriptNo.
(willbeinsertedbytheeditor)
Tsallis statistics approach to the transverse momentum distributions
in p-p collisions
MaciejRybczyn´skia,1,ZbigniewWłodarczykb,1
1InstituteofPhysics,JanKochanowskiUniversity,ul.Swietokrzyska15,PL-25406Kielce,Poland
4
1 Received:date/Accepted:date
0
2
Abstract Transversemomentumdistributionsofnegatively nonextensivestatistics[2]inwhichtheparameterqsummar-
n
a chargedpionsproducedinp-pinteractionsatbeammomenta ilydescribesallfeaturescausingadeparturefromtheusual
J 20,31,40,80and158GeV/carestudiedusingtheTsallis Boltzmann-Gibbsstatistics.Inparticularitwasshownin[5]

2 distributionasaparametrization.Resultsarecomparedwith that q−1=Var(T)/(cid:104)T(cid:105)2 and directly describes intrinsic
2
higherenergiesdataandchangesofparameterswithenergy fluctuations of temperature (however, the Tsallis distribu-

aredetermined.DifferentTsallis-likedistributionsarecom- tion also emerges from a number of other more dynamical
]
h pared. mechanisms,forexamplesee[6]formoredetailsandrefer-
p
ences).Thisapproachhasbeenshowntobeverysuccessful
- Keywords Transversemomentum·Tsallisdistribution
p indescribingmultiparticleproductionprocessesofadiffer-
e PACS 13.85.Hd·24.60.-k·25.75.Dw·89.75.-k entkind(see[6,7]forrecentreviews).Intermsoftransverse
h (cid:113)
[ momentum,transversemass,mT = m2+p2T,andrapidity

y,Eq.(1)becomes
1
v 1 Introduction
9
d2N m cosh(y)
3 Transversemomentum(p )distributionsofidentifiedhadrons =gV T ×
6 T pTdpTdy (2π)2
are the most common tools used to study the dynamics of
5 (cid:20) (cid:21) q
. high energy collisions. The p-p interactions are used as a mTcosh(y)−µ 1−q
1 × 1+(q−1) . (2)
baseline and are important to understand the particle pro- T
0
4 duction mechanism [1]. In the framework of Tsallis statis-
IthasbeenshownrepeatedlythattheTsallisdistribution
1 tics[2–4]themomentumdistributionisgivenby
givesanexcellentdescriptionof p spectrameasuredinp-
: √ T
v p collisions at RHIC ( s=62.4 and 200 GeV) and LHC
i √
X d3N gV (cid:20) E−µ(cid:21)1−qq q→1 ( s=0.9,2.76and7.0TeV)energies[3,8–11].Inpartic-
ar dp3 = (2π)3 1+(q−1) T −−→ ular changes in the transverse momentum distribution with
energy(useddataatenergies0.54,0.9,2.36and7TeV)are
(cid:18) (cid:19)
gV E−µ studied using the Tsallis distribution (2) as a parametriza-
exp − , (1)
(2π)3 T tion[12].Inthispaperweextendthisanalysistotransverse
√
momentumspectraobtainedinp-pcollisionsat s=6.27,
whereT andµ arethetemperatureandthechemicalpoten-
7.74,8.76,12.32and17.27GeVbytheNA61/SHINEcol-
tial,V is the volume and g is the degeneracy factor. In this laboration [13] 1. In addition to possibility of study colli-
form,Eq.(1)isusuallysupposedtorepresentanonextensive
generalizationoftheBoltzmann-Gibbsexponentialdistribu- 1Recently,theexperimentalresultsoninclusivespectraofnegatively
chargedpionsproducedininelasticp-pinteractionsatbeammomenta
tion,exp(−E/T),withqbeinganewparameter,inaddition
20,31,40,80and158GeV/cwerepresented[13].Themeasurements
toprevious"temperature"T.Suchanapproachisknownas
wereperformedusingthelargeacceptanceNA61/SHINEhadronspec-
trometer at the CERN Super Proton Synchrotron. Numerical results
ae-mail:[email protected] correspondingtothetwodimensionalspectraintransversemomentum
be-mail:[email protected] andrapiditycorrectedforexperimentalbiasesweregiveninRef.[14].
2
Fig. 1 (Color online) Transverse momentum distributions of nega- Fig.2 (Coloronline)Thevaluesofthenonextensivityparameterq,as
tively charged pions produced in p-p collisions as obtained by the afunctionofrapidityobtainedfromfitstothetransversemomentum
√
NA61/SHINEcollaboration[13]at s=17.27GeVinrapidityinter- distributionsatdifferentenergies.
vals0.2k<y<0.2(k+1)wherek=0,···,11fromthebottom.Data
pointsfordifferentrapiditybinswerescaledby3kforbetterreadabil-
ity.
sionsatlowincidentenergies,themeasurementsperformed
by NA61/SHINE collaboration allow us to study the low-
p partofspectra.ThevaluesofT andV areverysensitive
T
tothelow-p partofthetransversemomentumdistribution
T
and extending the analysis to lower p could bring much
T
clarificationhere.
2 Analysisoftransversemomentumdistributions
Fig.3 (Coloronline)Thevaluesofthetemperatureparameter,T,as
Transverse momentum spectra of negatively charged pions
afunctionofrapidityobtainedfromfitstothetransversemomentum
are fitted using Tsallis distribution given by Eq. (2) with distributionsatdifferentenergies.
gπ− =1andµ =0.Itisworthtobenotedthatthevariable
T andV arefunctionsofµ atfixedvaluesofq,
3 Energydependenceofparameters
The energy dependence of the various parameters is dis-
T =T +(q−1)µ, (3)
0
played in Figs. 4, 5 and 6. For comparison with higher en-
ergy data [12] which are for mid-rapidity y=0, we show
parameters as evaluated for rapidity interval 0 <y <0.2.
V =V [1+(q−1)µ/T ]q/(1−q)=V (T/T )q/(1−q) (4) All analysed parameters show a clear but weak energy de-
0 0 0 0
pendencewhichwehaveparametrizedas
andtheycanbecalculatediftheparametersT =T andV =
0
V0atµ =0areknown[12]. (cid:0)√ (cid:1)0.01326
q(s)=1.027 s (5)
TheTsallisdistributiondescribesthetransversemomen-
tum distributions of negatively charged pions in p-p colli-
sionsasobtainedbytheNA61/SHINEcollaboration[13]in
√
all rapidity intervals remarkably well as shown in Fig. 1. T(s)=0.1014(cid:0) s(cid:1)−0.03262 (6)
The values of nonextensivity parameters q needed to de-
scribethetransversemomentumdistributionsofnegatively
charged pions are shown in Fig. 2. The values of tempera- (cid:18)3V(s)(cid:19)1/3 (cid:0)√ (cid:1)0.09
tureparameterT fordifferentenergiesandrapidityintervals R(s)= =2.31 s (7)
4π
are shown in Fig. 3. The temperature parameter T shows a
clear rapidity dependence which we have parametrized as ThevalueofRisnotnecessarilyrelatedtothesizeofthe
T (cid:39)0.09cosh(y). systemasdeducedfromaHBTanalysis[15,16]butserves
3
Fig.4 (Coloronline)Energydependenceoftheparameterqappearing Fig.7 (Coloronline)dN/dyofchargedparticlesproducedinthecen-
intheTsallisdistribution.OpenpointsarefromATLAS,ALICEand tralrapidityregionasafunctionofcenter-ofmassenergyinp−pand
UA1Collaborationsdata(takenfromRef.[12]).Solidpointsarefrom p−p¯collisions.EnergydependencegivenbyEq.(9)iscomparedwith
NA61/SHINECollaborationdata[13].DataarefittedbyEq.(5). inelasticmeasurementsfromNA61/SHINE[13](p−p),NALBubble
Chamber (p−p¯), ISR (p−p), UA5 (p−p¯), PHOBOS (p−p) and
ALICE(p−p)experimentstakenfromcompilation[17].
tofixthenormalizationofthedistribution(2).Inparticular,
wehave
(cid:12)
dN(cid:12) gVT (cid:104) m(cid:105) 1
(cid:12) = 1+(q−1) 1−q×
dy(cid:12)(cid:12) (2π)2 T
y=0
(2−q)m2+2mT+2T2
× . (8)
(2−q)(3−2q)
For evaluated above energy dependence of parameters
Fig.5 (Coloronline)Energydependenceofthetemperatureparameter q(s),T(s)andR(s)givenbyEqs.(5-7)wehave
T appearingintheTsallisdistribution.OpenpointsarefromATLAS,
ALICE and UA1 Collaborations data (taken from Ref. [12]). Solid
(cid:12)
pboyiEntqs.a(r6e).fromNA61/SHINECollaborationdata[13].Dataarefitted ddNy(cid:12)(cid:12)(cid:12) (cid:39)0.1+0.56(cid:0)√s(cid:1)0.24. (9)
(cid:12)
y=0
EnergydependenceofdN/dyinthecentralrapidityre-
gionincomparisonwithinelasticmeasurementsisshownin
Fig.7.
We can treat the size of the system, R, more seriously.
TheradiusgivenbyEq.(7)iscalculatedforµ=0.Forother
valuesofchemicalpotential,thesizeissmaller(cf.Eqs.(3)
and (4)). Comparing R(s) with experimental data deduced
fromHBTanalysiswecanseethatR (cid:39)R/κ,whereκ=
HBT
3.5.InFig.8wedisplayedR(s)/κ incomparisonwithdata
obtainedfromHBTanalysis[18].
Followingthisobservationweassume
Fig.6 (Coloronline)EnergydependenceoftheradiusRappearingin
thevolumefactor,V=4/3πR3.OpenpointsarefromATLAS,ALICE Vµ=0=Vµ·κ3 (10)
andUA1Collaborationsdata(takenfromRef.[12]).Solidpointsare
fromNA61/SHINECollaborationdata[13].DataarefittedbyEq.(7). andfromEqs.(3)and(4)wehave
T (cid:16) (cid:17)
µ = µ=0 κ3(q−1)/q−1 (11)
q−1
4
unperturbative theory or model, and the high-p region is
T
governedbyhardphysicsrepresentedbyperturbativeQCD.
In(15),thenonextensiveformulaworksinthewholerange
of p and it is not derived from some particular theory. It
T
is only a generalization of the regular statistical mechan-
ics and just offers the kind of universal unifying principle,
namelytheexistenceofsomekindofequilibriumaffecting
all scales of p , which is described by two parameters, T
T
and q. The temperature T characterize its mean properties
and the parameter q, known as the nonextensivity parame-
ter,expressesactionofthepotentiallynon-triviallongrange
effectsbelievedtobecausedbyfluctuations[5](butalsoby
some correlations or long memory effects [2]). It is worth
Fig.8 (Coloronline)EnergydependenceoftheradiusRµ=Rµ=0/3.5 tobenotedthattheinvariantmomentumdistributioninthe
(solidpoints)incomparisonwithHBTmeasurementsofsourceradii
obtainedinhadron-hadronreactions[18](openpoints). form(cf.Eq.(1))
and using parametrizations (5) and (6) we have energy de- d3N gV (cid:20) E(cid:21)1−qq
E = 1+(q−1) , (16)
pendenceofchemicalpotentialintheform dp3 (2π)3 T
(cid:0)√ (cid:1)−0.022 result in Eq. (2) without pre-factor mTcosh(y) in the right
µ(s)(cid:39)0.39 s . (12)
hand side of the equation. For the non-relativistic energies
(E=p2/(2m)),Eq.(16)correspondstoTsallisdistribution
4 Differentparametrizations
d3N gV (cid:20) p2 (cid:21)1−qq
Almost fifty years ago Hagedorn develop a statistical de- E = 1+(q−1) , (17)
scriptionofmomentumspectraobservedinmultiparticlepro- dp3 (2π)3 2mT
ductionprocesses[19].Hagedorn’sapproachpredictsanex-
originatedfrommultiplicativenoise[21,22]2.
ponentialdecayofmomentumdistribution
ExponentialfunctionEq.(13)describeddataonlyinthe
d3N (cid:16) p (cid:17) limitedrangeoftransversemomentum,0.15<p <0.6[13].
T T
E (cid:39)Cexp − (13)
dp3 T AsshowninFig.1,theTsallisdistributiongivenbyEq.(2)
describesall p rangeremarkablywell.
T
for transverse momenta, whereas in experiments one ob-
AllTsallis-likedistributionsleadtoapowerlawtail
servesnon-exponentialbehaviorforlargetransversemomenta.
Subsequently,Hagedornproposedthe"QCDinspired"em-
piricalformuladescribingthedataoftheinvariantmomen- d2N
∝p−n (18)
tumdistributionofhadronsasafunctionof pT overawide pTdpTdy T
range[20]:
ofthedistributionforsufficientlylargetransversemomenta.
d3N (cid:18) p (cid:19)−n (cid:40)exp(−np /p ) for p →0 Thedifferencebetweenthemcanbeseeninlow pT region,
T T 0 T
E =C 1+ → where
dp3 p0 (pT/p0)−n for pT →∞
(14)
(cid:40)
d2N α−βp +γp2 forEqs.(13),(14),(15)
withC,p0andnbeingfitparameters.Thisbecomespureex- ∝ T T
ponentialforsmall pT andpurepowerlawforlarge pT.For pTdpTdy α−γp2T forEqs.(1),(16),(17)
n=q/(q−1) and p =T/(q−1), the Hagedorn formula (19)
0
(14)coincideswithTsallisdistribution[2],
d3N (cid:104) p (cid:105) q 2TheLangevinequationdp/dt+γ(t)p=ξ(t)wherebothγ(t)and
E =C 1−(1−q) T 1−q. (15) ξ(t)denotestochasticprocesses(traditionalmultiplicativenoiseand
dp3 T
additivenoise,respectively)leadstoapower-lawtailofthedistribu-
tion for sufficiently large momenta. As shown in [21] in the case of
Thebasicconceptualdifferencebetween(14)and(15)is
Cov(γ,ξ)=0andE(ξ)=0(i.e.,for,respectively,nocorrelationbe-
intheunderlyingphysicalpicture.In(14)thelow-p region
T tweennoisesandnodrifttermduetotheadditivenoise)thesolutionis
iscontrolledbysoftphysicsrepresentedbysomeunknown givenbythenon-normalizedTsallisdistributionforthevariablep2.
5
For distributions with the same mean transverse momen-
tum, (cid:104)p (cid:105), the parameter T evaluated from Eq. (13) is
T exp
connected with parameter T evaluated from Eq. (2) by the
relation
T (cid:39)a+b·T, (20)
exp
where, numerically, a=0.31−0.654q+0.354q2 and b=
27.35−55q+29.07q2.Moreover,itisremarkabletonotice
thatparametrization(1)proposedbyCleymans[3,4]isfor
momentum distribution, d3N/dp3 while the other Tsallis-
likeparametrizations(14)-(17)areforinvariantdistribution
Ed3N/dp3.
Acknowledgements ThisresearchwassupportedbytheNationalSci-
enceCenter(NCN)undercontracts:2011/03/B/ST2/02617
and2012/04/M/ST2/00816.
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