long-range multiplicity and transverse momentum
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Long-Range Multiplicity and Transverse Momentum
Correlations in pp Collisions in ALICE at the LHCALICE at the LHC
(for ALICE Collaboration)
A.ASRYAN, G.FEOFILOV, A.IVANOV, A.GREBENYUK, P.NAUMENKO, V.VECHERNIN
V. Fock Institute for Physics of Saint-Petersburg State University
Reported by G.Feofilov , at the “V Workshop on Particle Correlations and Femtoscopy” , 15 October 2009
Outline
• Motivation• Long-range correlations and collectivity
effects in pp and ppbar collisions (experiment and PYTHIA)
15.10.2009 2
(experiment and PYTHIA)• ALICE • Conclusions
Space-time evolution of relativistic nuclei collisions.
I — initial state, II — QGP, III —mixed phase, IV — hadron gas, V — free particles
I.L.Rosental, Yu.A.Tarasov, UFN, v 163, № 7б 1993
15.10.2009 3
Space-time evolution of relativistic nuclei collisions.
I — initial state, II — QGP, III —mixed phase, IV — hadron gas, V — free particles
I.L.Rosental, Yu.A.Tarasov, UFN, v 163, № 7б 1993
15.10.2009 4
Can we see the initial state effects?
Motivation: Two stage scenario:
A.Capella, U.P.Sukhatme, C.--I.Tan and J.Tran Thanh Van, Phys. Lett. B81 (1979) 68; Phys. Rep. 236 (1994) 225.A.B.Kaidalov, Phys. Lett., 116B (1982) 459; A.B.Kaidalov K.A.Ter-Martirosyan, Phys. Lett., 117B (1982) 247.
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At the first stage a certain number of colour strings are formed stretched in rapidity space between the incoming partons
At the second stage these strings decay into the observed secondary hadrons.
Motivation for studies of Long-Range Correlations:“STRING FUSION”
Investigations of the charged particles long-range multiplicity correlations, measured for well separated rapidity intervals, can give us information on the number of emitting
centers and hence on the fusion of colour strings[1].
15.10.2009 6
[1] M.A.Braun, C.Pajares and V.V.Vechernin "Forward-backward multiplicity correlations, low p_t distributions in the central region and the fusion of colour strings", Internal Note/FMD, ALICE-INT-2001-16, CERN, Geneva, 2001, 13p[2] Abramovskii V.A., Kancheli O.V. “On the multiplicity distribution of secondary hadrons” Letters to JETP 31 (1980) 532
Fig.1. Quark-gluon strings and schematics for studies of Long-Range Correlations[1]
Similar picture: formation of colour flux tubes
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Similar picture:L.Mc Lerran et al.: Color Glass Condensate and Glasma
(see the report at the present Workshop)
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“Percolating color strings approach''.With growing energy and/or atomic number of colliding particles, the number of strings grows and they start to overlap, forming clusters [3-5].
E-by-E Fluctuations are due to the mixture of different type sources !
.
[3] M.A.Braun and C.Pajares, Eur. Phys. J. C16 (2000) 349.M.A.Braun,R.S.Kolevatov,C.Pajares.V.V.Vechernin, ''Correlations between multiplicities and everage transverse momentum in the percolating color strings approach'', Eur.Phys.J.C.32.535-546(2004)[4] N.Armesto, C.Saldago, U.Wiedemann, PRL94 (2005) 022002.[5] C. Pajares , arXiv:hep-ph/0501125v1 14 Jan 2005
Fluctuations !
Long-Range Correlations (LRC)CORRELATIONS BETWEEN OBSERVABLES MEASURED IN TWO
RAPIDITY INTERVALS
⟩⟨ )( yn
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“BACWARD” (B) “FORWARD”(F) Y*THEORY:• A.Capella,A.Krzywicki, Phys.Rev. D18, no11,(1978),4120-4133• A.Capella and J.tran Thanh Van, Z.Phys, C18(1983).85:• A.Krzywicki, Phys.Rev. D29,No.5,(1984)1007-1009.• N.S.Amelin, N.Armesto, M.A.Braun, E.G.Ferreiro and C.Pajares,
Phys. Rev. Lett. {\bf 73} (1994) 2813.• M.A.Braun and C.Pajares, Eur. Phys. J. {\bf C16} (2000) 349.
M.A.Braun,R.S.Kolevatov,C.Pajares.V.V.Vechernin, ''Correlations between multiplicities and everage transverse momentum in the percolatin color strings approach'', Eur.Phys.J.C.32.535-546(2004)
HOW TO MEASURE THE LONG RANGE CORRELATIONS:
Observables
∑=Bn
iBtBt pp
1
For each event:1) the event mean multiplicity in BACKWARD or FORWARD rapidity windows:2) the event mean transverse momentum for BACKWARD and FORWARD rapidity windows:
FB nn ,
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∑=
=i
BtB
Bt pn
p1
FORWARD rapidity windows:
Event-by-event:We define the average value of the observable in one rapidity window at the given value of another observable in the second window(regression):
∑=
=Fn
i
iFt
FFt p
np
1
1
tFFF ptBnBnBt pornorp ⟩⟨⟩⟨⟩⟨
Usually: Correlation coefficients are defined – in the region where some linearity exists--for the absolute values of observables as:
FnnnnnB nanF
β+=⟩⟨
Here the strength of the multiplicity correlation is
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Correlation coefficients(for the normalized observables):
⟩⟨+=
⟩⟨
⟩⟨
F
FNnn
Nnn
B
nB
nn
an
nF β
Here the strength of the multiplicity correlation is measured by the coefficient
nnβ
NA49 Collaboration and Feofilov G.A., Kolevatov R.S., Kondratiev V.P.,Naumenko P.A. , Vechernin V.V., “Long-Range Correlations in PbPbCollisions at 158 AGeV”. Reported in 2004 at XVII ISHEPP. In: RelativisticNuclear Physics and Quantum Chromodynamics, Proc. XVII Internat. BaldinSeminar on High Energy Physics Problems, vol.1, JINR, Dubna, 2005, 222-231Min.bias data, 2 rapidity intervals: {-0.3,0.3} and {0.9,2.0}
05 октября 2009 13
NN PtN PtPt
Long-range correlations and collectivity effects in pp and
ppbar collisions: experiment and PYTHIA
14
Motivation: collectivity effects in pp experiment, charged particle multiplicity
correlation (ppbar collisions 0.3-1.8 TeV”, [2])
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[2] E735 Collaboration, “Charged particle multiplicity correlations in ppbar collisions at 0.3-1.8 TeV”, Physics Letters B 353 (1995) 155-160
Parameters Description
MSUB(91) Elastic Scattering
MSUB(92) Single Diffraction AB -> XB
MSUB(93) Single Diffraction AB -> AX
MSUB(94) Double Diffraction
MSUB(95) Low Pt Production
MSUB(96) Semihard QCD
PARP(85)D=33%
in v6.325“Colour correlations” - probability that an additionalinteraction in the multiple interaction formalism givestwo gluons, with colour connections to ‘nearestneighbours’ in momentum space
CollectivityEffects inPYTHIA v6.3
PYTHIA
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neighbours’ in momentum space
PARP(86)D=66%
in v6.325“Gluon-gluon string formation” - probability that an additional interaction in the multiple interaction formalism gives two gluons, either as described in PARP(85) or as a closed gluon loop. Remaining fraction is supposed to consist of quark–antiquarkpairs
Torbjörn Sjöstrand, Leif Lönnblad, StephenMrenna, Peter Skands, “PYTHIA 6.3Physics and Manual”, hep-ph/0308153, LUTP 03–38, August 2003
PYTHIAModel of multiple interactionsT. Sjöstrand, “Monte Carlo Generators”,European School of High-Energy Physics2006, Aronsborg, Sweden, 27 June 2006
Final PYTHIA parameters were tuned using<pt>Nch-Nch correlation data: (report by A.Asryan at CERN, PWG2 31.03.08)
Energy, GeVPYTHIAParameters
17 31 200 540 900 1800
MSUB(91) 0 0 1 1 1 1
MSUB(92) 1 1 1 1 1 1
Values of parameters
Elastic scattering
Single diffraction
Results are based on fittingof experimental data on Pt-Ncorrelations in p-p and p-pbar collisions from 17 to1800 GeV. MSUB(92) 1 1 1 1 1 1
MSUB(93) 1 1 1 1 1 1
MSUB(94) 1 1 1 1 1 1
MSUB(95) 0 0 1 1 1 1
MSUB(96) 1 1 1 1 1 1
PARP(85) 90% 90% 90% 90% 90% 90%
PARP(86) 95% 95% 95% 95% 95% 95%
Low pt production
Single diffraction
Single diffraction
Double diffraction
Semihard QCD
Gluon string fusion
Gluon string formation
1800 GeV.
In PYTHIA version 6.4 andhigher these values ofparameters are set asdefault. See also R. Fields’studies in
2008 Mar 31 Page 17
http://www.phys.ufl.edu/~rfield/cdf/
0,5
0,6
0,7
0,8
0,9
PYTHIA PARP3 100%
PYTHIA PARP1 90%
PYTHIA PARP4 33%
PYTHIA PARP6 00%
EPEM Model
pp столкновения900 ГэВ
Colour correlations and Gluon-gluon string formation
in PYTHIA : influence on <p_t>Nch,p+pbar collisions, 900 GeV
Distribution of # of particle emitting sources (strings) and “collectivity”
“Collectivity” in PYTHIA
2,03,0 4,0
5,0 6,07,0
8,09,0
10,11,
12,
0
0,1
0,2
0,3
0,4
number of strings
EPEM Model
Effective Multi Pomeron Exchange Model(EPEM):N. Armesto, A. Asryan, D. Derkach, G. Feofilov,“Analysis of pt-Nch Correlations in pp andppbar Collisions”, ALICE physics week, Erice,Italy, 09 December 2005
18
<pt>Nch-Nch correlation and collectivity effects in pp and ppbar collisions at 17-1800 GeV [5]
Fig.4. Experimental data and model [5][5] N. Armesto, D. Derkach, G. A. Feofilov, “pt–Multiplicity Correlations in a Multi-pomeron Exchange Model With String Collectivity Effects”; Physics of Atomic Nuclei, 71,No.12, 2087-2095(2008).
Long-range multiplicity correlation in ppbar collisions from 0.3-1.8 TeV: experiment E735 ( Physics Letters B 353 (1995) 155-160)
and our PYTHIA 6.4 results (no fits!)
Fig.8. Correlation coefficient as a function of the rapidity gap between two rapidity intervals (“backward” and “forward” intervals of 0.2 units ).
Long-range multiplicity correlation in ppbar collisions from 0.3-1.8 TeV: experiment (E735 Physics Letters B 353 (1995) 155-160, LEFT FIGURE) and our PYTHIA 6.4 results(RIGHT)
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Fig. 9. Correlation coefficient as a function of the 2 rapidity interval width. There is no gap between the “forward” and “backward” regions.
Predictions for ALICE:pp-collisions at 10 TeV
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Fig.10. The n-n correlation strength for pp data, as a function of rapidity gap for pp collisions at √sNN = 10 TeV at ALICE for 2 sets of collectivity parameters : 0.33,0.63} (pink circles), 0.9, 0.95(black circles)
“Saturation” effect in long-range multiplicity correlation and selection of “the very central” events in
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central” events inpp-collisions at the LHC
Selection of “the very central events” using LRC dataPYTHIA-simulations: “high collectivity - left and
“ low collectivity” - rigth
Entries 78256Mean 74.9
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78256Mean 74.9
gap =3.34
=10 TeV cm W
parp(85)=0.9, parp(86)=0.95
n0 ~ 71
b= 0.761 +/- 0.006
~ 58.7 pl
<n_b>
Entries 78546Mean 102.3
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78546Mean 102.3
gap =3.34
=10 TeV cm W
parp(85)=0.33, parp(86)=0.66
n0 ~ 117
b= 0.800 +/- 0.004
~ 97.5 pl
<n_b>
N
8000
10000
Entries 78256Mean 18.46
Multiplicity
N
8000
10000
Entries 78546Mean 22.36
Multiplicity
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Entries 78256Mean 74.9
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78256Mean 74.9
gap =3.34
=10 TeV cm W
parp(85)=0.9, parp(86)=0.95
n0 ~ 71
b= 0.761 +/- 0.006
~ 58.7 pl
<n_b>
Entries 78546Mean 102.3
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78546Mean 102.3
gap =3.34
=10 TeV cm W
parp(85)=0.33, parp(86)=0.66
n0 ~ 117
b= 0.800 +/- 0.004
~ 97.5 pl
<n_b>
N
8000
10000
Entries 78256Mean 18.46
Multiplicity
N
8000
10000
Entries 78546Mean 22.36
Multiplicity
Some predictions for ALICE: Collectivity and “Saturation”
effects in long-range multiplicity correlationCollectivity effects in PYTHIA In case of large collectivity parameter set, the “plateau” observed in multiplicity correlation is expected to start earlier and to have lower value: at the level of <nB>~120 in vs.
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Fig.11. Collectivity and “Saturation” effect in long-range multiplicity correlation: PYTHIA predictions for ALICE, pp collisions at √sNN = 10 TeV for 2 sets of collectivity parameters : 0.33,0.63} (lower raw), 0.9, 0.95(upper raw) .Figures are for 2 values of rapidity gap: 0 (left column) and 3.34 (right column). Backward and forward rapidity intervals are of 0.2 units of rapidity.
level of <nB>~120 in vs. <nB>~170 (see left column for 0 gap). Similar is the result for the gap=3.4 units(right column)
“Saturation” effects in long-range multiplicity correlation vs. “collectivity in PYTHIA” (PARP(85) and PARP(86)):
Multiplicity “plateau”
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“Saturation” effects in long-range <p_t>-multiplicity correlationvs. “collectivity in PYTHIA” (PARP(85) and PARP(86)::
<p_t>-”plateau”
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“Saturation” effects in long-range <p_t>-multiplicity correlation for strange
particles Λ,<multipllclty> -“plateau” vs. “collectivity in PYTHIA” (PARP(85) and PARP(86)):
_
Λ_
Λ
_
Λ
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For “plateau” events:“High” collectivity:
<nB_lamdas>/<Nch_plateau>= 1.8/60.3=0.03 (!)
“Low” collectivity:<nB_lambdas>/<Nch_plateau>=
2.3/98=0,023
ΛΛ
Features of “the plateau” (PYTHIA) events:
increase of collectivity effects in PYTHIA(of Colour correlations and Gluon-gluon string formation)
leads to:
1) Increased formation of the number of paticleemitting sources
2) Damping of the mean multipliicty of “plateau” events
3) Increased <p_t> values for “plateau” events3) 1.5 increase of yields Λ/<Nch> for “plateau”
events
à The very central рр-collisions!29
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The 1st measurements of <n>n, <p_t>n and <p_t>p_tcorrelations in рр collisions in ALICE at the LHC:
1) 1) to start with SCAN IN to start with SCAN IN η∈ [-1.0, +1.0], δ = 0. 22)2) TO SCAN INTO SCAN IN [0, 2π],3) TO SCAN IN3) TO SCAN IN p_tp_t, <1 , <1 GevGev/c, 1/c, 1--2GeV/c, 2GeV/c, и и >2 >2 GeVGeV/c/c
-- 1.01.0 ��0 1.0
∈
3) TO SCAN IN3) TO SCAN IN p_tp_t, <1 , <1 GevGev/c, 1/c, 1--2GeV/c, 2GeV/c, и и >2 >2 GeVGeV/c/c
Future : Future : “net“net--charge” LRCcharge” LRCFuture : Future : LRC for Strange and LRC for Strange and multistrangemultistrange hyperonshyperonsFuture : Future : TO SCAN INTO SCAN IN: η∈ [-2.0, +2.0]
DetectorsDetectors: : ITS, TPC, FMDITS, TPC, FMD
Future : Future : TO SCAN INTO SCAN IN: η∈ [-3.7,+4.7]
The following set of pseudoarpidity intervals (“windows”) were selected within the TPC/ALICE domain (see Fig.1):1. Correlation in full rapidity interval [-0.8 .. 0.8]
Pseudorapidity intervals for the 1st pp-collisions study (in the TPC domain)
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0-0.8 -0.4 0.4 0.8
2. Backward-forward correlation using data from symmetrical –variable width- windows of 0.2,0.4,0.6,0.8 pseudorapidity units
3. Correlation for symmetrical windows of 0.2,0.4,0.6, units, - scanning of the gap width
Fig/1. The complete set of 11 pseudoarpidity intervals proposed for LRC study within the TPC/ALICE domain
LRC Library
LRC Library includes 5 classes: TFit, TLRC, TNN, TPtN, TPtPt, Long Range Correlation library algorythms provide: Vizualization of regression curves, calculation of correlation coffecients, statistical errors estimates of N-N, Pt-N and Pt-Ptcorrelation coefficients.
15.10.2009 33
Examples of the Long-Range correlations analysis AliROOT for ALICE at the LHC
10TeV
Fn0 10 20 30 40 50 60 70
Fn
>B
<P
t
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Entries = 1185195
0.000672±a = 0.480082
0.000101±b = 0.008974
/(n2) = 6.0880882χ = 2χ> = 6.409446 F<n
> = 0.528295Fn>B<<Pt
>> = 7.381883 F<<n>> = 0.209237
Fn>B<<<Pt
= 0.432154Fn>Bd<Pt
PtN
PDC’09 P-P data (run LHC9a4) PythiaPt -Nch correlation for [-0.8..0],[0..0.8] rapidity windows (September 2009)b ~ 0.009 GeV/c(PRELIMINARY)
Fn0 10 20 30 40 50 60 70
Fn
>B
<P
t
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Entries = 1185195
0.000672±a = 0.480082
0.000101±b = 0.008974
/(n2) = 6.0880882χ = 2χ> = 6.409446 F<n
> = 0.528295Fn>B<<Pt
>> = 7.381883 F<<n>> = 0.209237
Fn>B<<<Pt
= 0.432154Fn>Bd<Pt
PtN
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PDC’09 P-P data (run LHC9a4) PythiaPt -Nch correlation for [-0.2..0],[0..0.2] rapidity windows (September 2009)b ~ 0.016 GeV/c(PRELIMINARY)
Summary1) It is proposed to start study of the <n>n, <p_t>n and
<p_t> p_t Long-Range Correlations (LRC) in рр collisionsin ALICE at the LHC.
2) This will provide a new information on the number ofparticle emitting sources, formed in the collisions and toselect a class of unique events that might be
15.10.2009 36
select a class of unique events that might becharacterized as “the very central collisions” , possessingthe maximal energy density
3) “Saturation effects” in <n>n or <p_t>n correlations in ppcollisions, - in case if they are confirmed experimentally, -are important for the future analysis of possible formationand hadronisation processes of the QGP.
AcknowledgemntsAuthors are grateful to T. Sjöstrand
for permanent interest to these studies andfruitful discussions.
THANK YOU FOR YOUR ATTENTION!
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BACK-UP SLIDES
15.10.2009 38
Since 2002: Long-Range Correlations as a method to study string fusion
pheneomenon in ALICE
• P.A.Bolokhov, M.A.Braun, G.A.Feofilov, V.P.Kondratiev, V.V.Vechernin, Internal Note/PHY.ALICE-INT-2002-20(2002)16p;
• P.A.Bolokhov, M.A.Braun, G.A.Feofilov,V.P.Kondratiev and V.V.Vechernin,\\``Forward-Backward Correlations in Relativistic Heavy Ion Collisions''.In: Relativistic Nuclear Physics and Quantum Collisions''.In: Relativistic Nuclear Physics and Quantum Chromodynamics.\\Proceedings of the XVI International Baldin Seminar on High Energy PhysicsProblems (2002),vol.1, JINR, Dubna, 2004, pp. 263-271
• P.A.Bolokhov, M.A.Braun, G.A.Feofilov,V.P.Kondratiev and V.V.Vechernin,\\ ``Experimental Studies of Colour String Fusion at ALICE'‘ , In: Relativistic Nuclear Physics and Quantum Chromodynamics.\\Proceedings of the XVI International Baldin Seminar on High Energy Physics Problems (2002), vol.1.,JINR, Dubna, 2004, pp. 272-278
• ALICE collaboration “ALICE: Physics Performance Report, Volume II”, J. Phys. G: Nucl. Part. Phys. 32 (2006) 1295-2040 (Section: 6.5.15 - Long-range correlations, p.1749)
Feasibility of the measurement of the longFeasibility of the measurement of the long--range range correlations in terms of statistics and systematic errors correlations in terms of statistics and systematic errors during the first pp run at 0.9/10 TeV in 2008 in ALICE during the first pp run at 0.9/10 TeV in 2008 in ALICE
1). Short 900 GeV min bias pp run is interesting for this analysis to start and to compare to the existing data on multiplicity correlation
N-N long-range correlation : Statistics about 1 - 2 x 10^5 eventsshould be sufficient for the comparison to the existing data at 900 GeV(E735 Collaboration, “Charged particle multiplicity correlations in ppbar collisions at 0.3-1.8 TeV”, Physics Letters B 353 (1995) 155-160 ) ,where about 150000 events data were accumulated
10/15/2009
40
data were accumulated
2). 10 TeV min bias run has to provide 2-3 x 10^5 of events for p_t-n correlation coefficient determination (This corresponds to [-1..0], [0..1]
rapidity windows). In a case of smaller windows of 0.2 units of rapidity the minimal required
statistics will be 1 - 2 x 10^6 .
N-N long-range correlation saturation: study will require statistics of about 10^7 events.
0.2
0.25
0.3
0.35
Nchb Max Cut = 20
Nchb Max Cut = 40
Nchb Max Cut = 60
Distribution of # of particle emitting sources (strings)
and Nch cuts for pp- collisions at 10 TeV (PYTHIA)
15.10.2009 41-0.05
0
0.05
0.1
0.15
0.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Nchb Max Cut = 60
Nchb Max Cut = 80
Nchb Max Cut = 100
Nchb Max Cut = 120
# of strings (PYTHIA)
Результаты PYTHIA v6.325
p-pbar при 200 ГэВ p-pbar при 540 ГэВ
p-antip столкновения p-antip столкновенияp-antip столкновения200 ГэВ|η| ≤ 2.5SppS (UA1)
p-antip столкновения540 ГэВ|η| ≤ 2.5ISR (ABCCDHW)
05 октября 2009 42
Selection of “the very central
events” using LRC data
Entries 78256Mean 74.9
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78256Mean 74.9
gap =3.34
=10 TeV cm W
parp(85)=0.9, parp(86)=0.95
n0 ~ 71
b= 0.761 +/- 0.006
~ 58.7 pl
<n_b>
Entries 78546Mean 102.3
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78546Mean 102.3
gap =3.34
=10 TeV cm W
parp(85)=0.33, parp(86)=0.66
n0 ~ 117
b= 0.800 +/- 0.004
~ 97.5 pl
<n_b>
F n0 20 40 60 80 100 120 140 160
N
0
2000
4000
6000
8000
10000
Entries 78256Mean 18.46
Multiplicity
F n0 20 40 60 80 100 120 140 160
N
0
2000
4000
6000
8000
10000
Entries 78546Mean 22.36
Multiplicity PYTHIA-
simulations
15.10.2009 43
Entries 78256Mean 74.9
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78256Mean 74.9
gap =3.34
=10 TeV cm W
parp(85)=0.9, parp(86)=0.95
n0 ~ 71
b= 0.761 +/- 0.006
~ 58.7 pl
<n_b>
Entries 78546Mean 102.3
F n0 20 40 60 80 100 120 140 160
>
B <
n
0
20
40
60
80
100
120
140
Entries 78546Mean 102.3
gap =3.34
=10 TeV cm W
parp(85)=0.33, parp(86)=0.66
n0 ~ 117
b= 0.800 +/- 0.004
~ 97.5 pl
<n_b>
F n0 20 40 60 80 100 120 140 160
N
0
2000
4000
6000
8000
10000
Entries 78256Mean 18.46
Multiplicity
F n0 20 40 60 80 100 120 140 160
N
0
2000
4000
6000
8000
10000
Entries 78546Mean 22.36
Multiplicity
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