Predictions on the transverse momentum spectra for charged particle production at LHC-energies from a two component model

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by  Alexander Bylinkin

Predictions on the transverse momentum spectra for charged particle production at
LHC-energies from a two component model
A. A. Bylinkin∗
Moscow Institute of Physics and Technology, MIPT, Moscow, Russia
National Research Nuclear University MEPhI, Moscow, Russia
N. S. Chernyavskaya†
Moscow Institute of Physics and Technology, MIPT, Moscow, Russia
Institute for Theoretical and Experimental Physics, ITEP, Moscow, Russia
National Research Nuclear University MEPhI, Moscow, Russia
A. A. Rostovtsev‡
Institute for Information Transmission Problems, IITP, Moscow, Russia
5
1 Transverse momentum spectra, d2σ/(dηdp2), of charged hadron production in pp-collisions are
T
0 considered in terms of a recently introduced two component model. The shapes of the particle
2 distributions vary as a function of c.m.s. energy in the collision and the measured pseudorapidity
interval. In order to extract predictions on the double-differential cross-sections d2σ/(dηdp2) of
n T
hadron production for future LHC-measurements the different sets of available experimental data
a
J have been used in this study.

1
2
I. INTRODUCTION 2]V 102 Data
] e 10
h Recently a qualitative model considering two sources G Exp
p b
- of hadroproduction has been introduced [1]. It was sug- m 10−1 Power
p gested to parametrize charged particle spectra by a sum [T
he dofisatrnibeuxtpioonnse:ntial (Boltzmann-like) and a power-law pT dpη 10−3
[ d
1 d2σ =A exp(−E /T )+ A , (1) 2/dσ 10−5
5v dηdp2T e Tkin e (1+ Tp22T·N)N ) pπT10−7
3 (cid:112) 2
2 whereETkin = p2T +M2−M withMequaltothepro- 1/ 1 10p [GeV]
(
5 duced hadron mass. Ae,A,Te,T,N are the free param- T
0 eters to be determined by fit to the data. The detailed
. argumentsforthisparticularchoicearegivenin[1]. The
1 FIG. 1. Charge particle differential cross section
0 exponential term in this model is associated with ther- (1/2πp )d2σ/dηdp [10] fitted to the function (1): the red
T T
5 malized production of hadrons by valence quarks and a (dashed)lineshowstheexponentialtermandthegreen(solid)
1 quark-gluon cloud coupled to them. While the power- one - the power law.
: law term is related to the mini-jet fragmentation of the
v
virtual partons (pomeron in pQCD) exchanged between
i
X two colliding partonic systems.
r A typical charged particle spectrum as a function of
a
transverse momentum, fitted with this function (1) is
shown in figure 1. As one can see, the exponential term
dominates the particle spectrum at low p values.
T
This model has already been shown to predict the
varies in different experiments under various conditions.
mean < p > values as a function of multiplicity in a
T In [1] it was shown that the parameters of the fit (1)
collision [3] and pseudorapidity distributions of charged
show a strong dependence on the collision energy. Un-
particles [4]. However, the major interest of many stud-
fortunately, due to the fact that different collaborations
ies in QCD is the transverse momentum spectrum itself.
measure charged particle production in their own phase
Therefore, in this article it is discussed how its shape
space and under various experimental setup, the depen-
dencesobservedin[1]weresmearedanddidnotallowto
makestrongpredictionsforfurthermeasurements. Thus,
∗ [email protected] an approach to correct the measurements in order to al-
† [email protected] low an accurate combination of different experimental
‡ [email protected] data is proposed here.
2
II. PARAMETER VARIATIONS
In [4] it was shown that two sources of hadroproduc-
tion described above, contribute to different pseudora-
pidity regions: while the power-law term of (1) prevails
in the most central region (η ∼0), the exponential term
dominates at high values of η. Since each collaboration
presents measurements on transverse momentum spec-
tra in various pseudorapidity intervals, these variations
might explain the smearing of the dependences in [1].
The idea to study parameter variations as a function of
both collision energy and pseudorapidity region has al-
ready been successfully tested in [5].
Tofurtherstudythevariationsofthespectrashapeas
afunctionofpseudorapidityweusethedatapublishedby
FIG. 3. Variation of the N parameter of (1) obtained from
the UA1 experiment [2] which are presented by charged
the fits to the experimental data [6, 8, 10] (full points) as a
particle spectra in five pseudorapidity bins, covering the √
functionofc.m.s. energy sinacollision. Solidlineshowsa
total rapidity interval |η|<3.0. Figure 2 shows how the
fitofthisvariation. Inaddition,openpointsshowtheparam-
parameter N varies with pseudorapidity together with a
eters for the data measured in another pseudorapidity inter-
power-law fit of this variation. Note, that the parameter
val [11, 12] and pointed line shows the predictions calculated
showsagrowthwithpseudorapidity[4],thatisexplained according to (2).
by higher thermalization of the spectra, as found in [4].
Figure 3 shows the parameter N variation as a func-
tion of c.m.s. energy in a collision. One can notice that
it can be characterized by the falling N-value. It is re-
latedtothefactthattheprobabilitytoproduceahigh-p
√ T
mini-jet should grow with s. Notably, this behaviour
correlateswiththefactthatN decreaseswhentherapid-
ityintervalbetweenthesecondaryhadronandtheinitial
proton increases.
One can extract the following regularity from the fit
shown in figure 31:
N =2+5.25·s−0.093 (3)
Remarkably, in the s → ∞ limit N → 2, which corre-
sponds to d2σ/dp2 ∝1/p4 in the proposed parametriza-
T T
FIG. 2. Variation of the N parameter of (1) obtained from tion (1). Such behaviour can be expected just from the
the fits to the experimental data [2] as a function of pseudo- dimensional counting in pQCD. However, in a real colli-
rapidity. Line shows a power-law fit of this variation. sion one should take the initial conditions and the kine-
maticrestrictionsintoaccount,resultinginahighervalue
√
This variation can be parametrized in the following of N for lower s. Obviously, such initial conditions
way: should become negligible in the s → ∞ limit, that is
confirmed by the observed behavior (3).
N =N0·(1+0.06·|η|1.52) (2)
In addition, figure 3 shows UA1 [11] and CMS [12]
where N0 denotes the parameter value at η ∼ 0 and data measured under different experimental conditions.
In these measurements the pseudorapidity interval was
η might be taken as a mean < η > of the measured
much wider (|η| < 2.5) than in [6, 8, 10]. Therefore,
pseudorapidity interval.
onecancomparetheparametervaluesobtainedfromthe
Since the variations of the parameter as function of
fit of these data (open points in figure 3) to the values
pseudorapidity have been found, it is desirable to ex-
extrapolated from (2) with N0 calculated according to
clude its influence when studying the dependences of N
(3) and |η| = 1.25 (pointed lines) and see a rather good
on the c.m.s. energy in a collision. This is possible, if
agreement. Notably, this behaviour correlates with the
onecombineonlythosedatathathavebeenmeasuredin
more or less the same pseudorapidity intervals. Hence,
wesuggesttolookfirstofallatISR[6],PHENIX[8]and
ALICE [10] data that were measured in the most central
√
(|η|<0.8) pseudorapidity region. 1 Inthe(3)andbelow sisgiveninGeV.
3
factthatN decreaseswhentherapidityintervalbetween This dependence is shown in the figure 5. One can see,
the secondary hadron and the same side beam proton that larger N corresponds to the smaller p . That
Tmax
increases. should correspond to the x → 1 limit of PDFs, where
Let us now check this correlation explicitly and calcu- the fall-off of the perturbative cross section is modified
latetherapidityintervalinthemovingprotonrestframe by the fall-off of the PDFs.
according to a simple formula:
√
η(cid:48) =|η|−log( s/2mp), (4) N 9
|η| < 0.8, 31 GeV < s < 7 TeV, pp
wherem isthemassoftheincomingproton. Theresults 8
p |η| < 2.4, 200 GeV < s < 7 TeV, pp
of such a procedure are shown in the figure 4. Surpris- 0.6 < η < 3,s = 630 GeV, pp
7 |η| < 0.8, 100 GeV < s = 13 TeV, pp, MC
ingly, all the points came to a single line in this inter-
pretation. To understand the origin of this universality 6
one might use MC-generators: hard processes at large
5 s=31GeV s=64GeV
p are known to be described by MC generators pretty s=100GeV
T s=200GeV
well, thus it is expected to get the value of N-parameter 4 s=200GeV s=200Gse=V630GeV
s=500GeV s=900GeV
ftroomthethreeafiltsdaotfa,thbeutMwCi-tghenaerhaitgehderspaecccturraacryathanerdcilnosae 3 s=900GeV ss==990000GGeesVV=7TeVs=4sT=se2=V766T0eGVesV=s7=T1e3VTeV
s=7TeV
wider collision energy range. To check this universality, 2
0 1 2 3 4 5 6 7 8 9 10
we have produced the Monte Carlo samples for proton- log(p )
proton collisions at different energies for inelastic(INEL) Tmax
events with the PYTHIA 8.2 generator [7]. Indeed, the
FIG.5. ThedependenceoftheNparameteronthemaximal
values of the parameter N extracted from the fits to the
kinematicallyallowedtransversemomentumofthesecondary
MC-generatedspectraarenicelyplacedatthesameline.
hadron. The data points from different experiments [2, 6, 8,
Thus, a universal parameter describing the shape of the
10–12] are shown together with the generated data(MC).
transverse momentum spectra in pp-collisions has been
found.
Remarkably,similarlytoN,T andT alsoshowdepen-
e √
dencesasafunctionofboththecollisionsenergy sand
N =5.04+0.27η(cid:48) (5) the measured pseudorapidity interval η. The variations
of the T and T parameters were studied in [5]. In [5]
e
the possible theoretical explanation of the thermalized
particle production was presented and the following pro-
N 7 portionalities were established:
6.5 |η| < 0.8, 31 GeV < s < 7 TeV, pp T =409·(√s)0.06·exp(0.06|η|) MeV (7)
|η| < 2.4, 200 GeV < s < 7 TeV, pp
6
0.6 < η < 3., s = 630 GeV, pp
√
5.5 |η| < 0.8, 100 GeV < s = 13 TeV, pp, MC s=31GeV Te =98·( s)0.06·exp(0.06|η|) MeV (8)
5
s=64GeV The parametrizations for T and T differes only by a
4.5 e
4 s=900Gs=e6V30GeVs=200GeV s=500GeVs=200GeV constant factor. However, both T and Te parameters
3.5 s=7TeVs=2760GeV s=900GeV ss==290000GGsee=VV100GeV reerflgieecstatnhde twhheernmaclloizsaetriotno twhheicvhaliesnscteroqnugaerrksa.tThihgehreerfoerne-,
3 s=7Tse=V6TesV=4TeVs=7TeV the (7,8) parametrizations which are functions of centre
2.5 s=13TeV
of mass energy and rapidity interval can be rewritten
2
−9 −8 −7 −6 −5 −4 −3 −2 −1 in the form with only one universal parameter. This
η’ universal parameter is the rapidity distance η(cid:48)(cid:48) from the
farther incoming proton.
FIG.4. Thedependenceofthe Nparameteronthepseudo-
√
rapidity of the secondary hadron in the moving proton rest η(cid:48)(cid:48) =|η|+log( s/2m ) (9)
frame. Thedatapointsfromdifferentexperiments[2,6,8,10– p
12] are shown together with the generated data(MC). Using (7,8) we get the universal dependence2:
T =409·exp(0.06η(cid:48)(cid:48))·(2m )0.06 MeV, (10)
The further check of the observed phenomena can be p
made if one plot N as a function of the logarithm of
T =98·exp(0.06η(cid:48)(cid:48))·(2m )0.06 MeV. (11)
the maximal kinematically allowed transverse momen- p
tum pTmax of the secondary hadron at the specific pseu- The dependences are shown on figure 6.
dorapidity η.
√
s
pTmax = 2 sin[2·tan−1(exp(−|η|))] (6) 2 Inthe(10,11,22)mp isgiveninunitsof1GeV/c2.
4
√
η =0.692+0.293·ln s, (15)
exp
V]
Ge √
[Te σexp =0.896+0.136·ln s, (16)
T,
|η| < 0.8, 31 GeV < s < 7 TeV, T
|η| < 2.4, 200 GeV < s < 7 TeV, T Apower =0.13·s0.175, (17)
|η| < 0.8, 31 GeV < s < 7 TeV, T
e
|η| < 2.4, 200 GeV < s < 7 TeV, T
e
0.409 Exp(0.06η’’)(2⋅0.939)0.06
0.098 Exp(0.06η’’)(2⋅0.939)0.06 Aexp =0.76·s0.106, (18)
Now, with the knowledge of the variations of T, N
and T parameters and the exponential and power-law
10−1 e
2 3 4 5 6 7 8 9 10 contributions one can calculate the normalisation pa-
η’’
rameters A and A in (1) in the following way.
e
FIG. 6. The dependence of the T and T parameter on the
e
pseudorapidity of the secondary hadron in the moving oppo-
site side proton rest frame. The data points from different dσ =(cid:18)dσ(cid:19) +(cid:18)dσ(cid:19) =(cid:90) ∞ d2σ dp2 (19)
experiments [2, 6, 8, 10–12] are shown. dη dη dη dηdP2 T
exp pow 0 T
III. PREDICTION FOR FURTHER (cid:18)dσ(cid:19) (cid:90) ∞ A ANT2
= dp2 = (20)
MEASUREMENTS dη pow 0 (1+ p2T )N T N −1
T2·N
Though MC was shown to provide a nice description
of the high-pt part of the spectra, the nature of the soft (cid:18)dσ(cid:19) (cid:90) ∞
particle production still remains ambiguous and varies dη = Aeexp(−ETkin/Te)dp2T =2AeTe(m+Te)
for different MC-generators. Therefore, the next step in exp 0
(21)
understanding the underlying dynamics of high energy
Thus, we get the set of equations allowing us to make
hadronic processes was done in a recent analysis [5].
a prediction for a double differential cross section, using
In [4] it was shown, that the introduced approach is
the formula (1):
able to give predictions on the pseudorapidity distribu-
tions in high energy collisions for non-single diffractive
events (NSD). Using the parameterizations from [4] in 
|η|=(|η| +|η| )/2,
adfudensdccirttiiiboonnesowtnhiltyehoshf(2aa)p-ce(es8n)otrfoencohefarcmgaeandsspperanorvetriidgcyelesaansdpfoearcmtmruael,aabsutehrineadgt TN==450.094·m+eaxx0p.(207.0η6(cid:48)mη(cid:48)i(cid:48)n)·(2mp)0.06,
(22)
pseudorapidity region. Let us now summarize all the T =98·exp(0.06η(cid:48)(cid:48))·(2m )0.06,
equations to obtain the final result3. AAe==(N2T−e1(m)1(+dTσe/)d(dησ)/dη,)expp,
NT2 pow
(cid:18)dσ(cid:19) (cid:18) η2 (cid:19) where η(cid:48) and η(cid:48)(cid:48)can be calculated using (4) and (9)
dη =Apower·exp −2σ2 , (12) correspondingly.
power power Now, one can calculate double differential cross sec-
tionsd2σ/(dηdp2)ofchargedparticleproductioninhigh
T
(cid:18)dσ(cid:19) (cid:18) (η−η )2(cid:19) energy collisions at different energies for NSD events.
dη =Aexp·exp − 2σ2exp + These predictions are shown in figure 7 for |η|<0.8 and
exp exp (13) |η| < 2.4 pseudorapidity intervals together with the ex-
A ·exp(cid:18)−(η+ηexp)2(cid:19), perimental data measured by ALICE [9] and CMS [12].
exp 2σ2 Agoodagreementofthepredictionwiththedatacanbe
exp
observed. Thus,theseresults(22)giveusapowerfultool
for predicting the spectral shapes in NSD events.
√
σ =0.217+0.235·ln s, (14)
power
IV. CONCLUSION
3 Inthe(12)andbelowAandAeismeasuredinmb/GeV2,Apow In conclusion, transverse momentum spectra in pp-
andAexp inmb collisions have been considered using a two component
5
2] 10 model. Variations of the parameters obtained from the
V LHC 14 TeV, NSD, | η | < 0.8
Ge LHC 7 TeV, NSD, |η | < 2.4 fit have been studie√d as a function of pseudorapidity η
) [η10−1 CLHMCS 9d0a0ta G 7e TVe, VN,S NDS, D| η, || η < | 0<. 82.4 aranmdect.emr.dse.scerniberingygasshainpethoeftchoellsispieocnt.raAinupnpiv-ceorsllaisliopnas-
d
pT ALICE data 900 GeV, NSD, | η | < 0.8 was found to be a preudorapidity of a secondary hadron
d
N/10−3 in moving proton rest frame. Finally, the observed
2d dependences, together with previous investigations al-
(
)T lowed to make predictions on double differential spectra
p
2 π10−5 dth2eσ/a(vdapil2Tadbηle)eaxtpehriigmheernteanledragtiaes., successfully tested on
1 /
(
10−7
2 4 6 8 10
p [GeV]
T ACKNOWLEDGMENTS
FIG. 7. Predictions the yield of charged particles
The authors thank Prof. Mikhail Ryskin, Prof. Torb-
(1/2πp )d2N/(dηdp )inhighenergycollisionsinNSDevents
T T
jorn Sjostrand and Dr. Peter Skands for fruitful discus-
together with data points from ALICE [9] and CMS [12] ex-
periments. sions and help provided during the preparation of this
paper.
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