-- JuanCastillo - 24 Feb 2008

Event types

After the cross section (CERN-HERA group parametrization) [1]

  • Total cross section in mb: ${\sigma}_{total} = 48 + 0.533(ln p)^2 +(-4.51)(ln p) $
  • Elastic cross section in mb: ${\sigma}_{elastic} = 11.9 + 26.9 p^{-1.21} + 0.169(ln p)^2 +(-1.85)(ln p)$
  • Inelastic cross section in mb: ${\sigma}_{inelastic} = {\sigma}_{total}-{\sigma}_{elastic}$

( p= laboratory momentum of incident hadron in Gev/c)

Contributions to cross section:
  • Total: ${\sigma}_{total}$ = 40 mb.
  • inelastic: ${\sigma}_{inelastic}$ =30 mb.
    • diffractive: 10% of ${\sigma}_{inelastic}$Two types [2]
      • Single diffractive events (SD)
      • Double diffractive (DD).
    • non-diffractive: 90% of ${\sigma}_{inelastic}$.

Characterization:
  • inelastic diffractive events: Small number of particles produced. Small energy transfer. [1] No significant hadronic particle activity over a large region of rapidity or pseudorapidity (rapidity gap)[6] Types:
    • Single diffractive events (SD). Only one beam particle is excited. Highly assimetric.
    • Double diffractive (DD). Both beam particle are excited.
  • inelastic non-diffractive events: Large number of particles produced. Large energy transfer.[1]

It is defined as non-single-diffractive event (NSD) any inelastic hadron-hadron interaction that cannot be regarded as a single diffractive (SD) event: in the framework of the PYTHIA hadronic interactions, the “non single-diffractive” sample includes the 2 → 2 partonic processes and the double diffractive (DD) hadron-hadron interactions. [4]
From an experimental point, minimum biased events (MB) are usually identified with NSD events at low $E_t$.

Average multiplicity

Definition: Total number of particles produced in a collision [1]. Since usually only charged particles are detected, we speak of charged multiplicity.

Parametrizations

After Thome [1]
$<N_{ch}> = 0.88 + 0.44lns + 0.188(lns)^2$
Or, in general [5]:
$<N_{ch}> = a_0 + a_1 (lns) + a_2 (lns)^2$

Charged rapidity density as a function of center-of-mass energy for AA and pp collisions
Fig 1: Charged rapidity density as a function of center-of-mass energy for AA and pp collisions. The general parametrization is displayed as dotted, dashed and solid cuvers(from [3])

The multiplicity of charged particles in pp collision is lower than corresponding multiplicity for e+e- collision for the same center-of-mass energy due to leading particle effect.

Charged multiplicity for different hadron-hadron collisions
Fig 2: Averaged charged multiplicity for different hadronic systems. a) muon-proton b) electron-positron annihilation c) comparison of electron-positron annihilation with proton-proton d) comparison after a transformation (from [5]).

Models

Landau

Explosion.

Bjorken

Assumptions [7]:
  • Particle production for the same rapidity is having the same shape, but different height depending on the center-of-mass energy (true for nucleon-nucleon and for nucleus-nucleus collision)
  • Landau Hydrodynamical model evolution.
  • Entropy is conserved during the expansion (assumption of Landau)
  • Final multiplicity proportional to entropy density.

The assumption of hydrodynamic is still valid for nucleon-nucleon collision if we begin with an entropy density (instead of an energy density) and is independent of the equation of state.

Multiplicity distribution

Shape uncertain. It depends on the processes occurred in the energy range considered (therefore, on cross section estimations).

Models

Clan Structure Scenario

Multiplicity distribution as a sum of contribution of individual negative binomial (NB) "Pascal" multiplicity distributions [10]

Charged multiplicity after the clan model

Fig 3: Charged multiplicity after the clan model, as a result of the contribution of 2 different NB(dashed and dotted)(from [10])

KNO scaling.

  • Idea of scaling: homogeneous functions. ${\psi({\lambda}x)} = {\lambda}^p {\psi(x)}$ (grade p)
  • Based on the validity of Feymann scaling for many-particle inclusive cross sections: cross section as a function of Feymann scaling variable $x_F$ independent of incident energy [1],[9].

Suppose the reaction $a+b{\longrightarrow}c+X$
Coordinates of 4-momentum will be identified with associated particles (a,b,c,X)
The Feymann scaling variable for a detected particle is defined $x_F = {\frac{c*_z}{c*_z(max)}}$, (center-of-mass system). It can be positive or negative. $c_z$ is maximum when X is at rest. In the limit of very high energies, and positive values, $x_F$ can be identified with the forward-light cone variable $x_+ = {\frac{c_0+c_z}{b_0+b_z}}$

General formulation: ${P(n,s)= {\frac{1}{\lambda(s)}} {\psi}({\frac{n+c(s)}{\lambda(s)}}) }$
If the scaling parameter ${\lambda(s)}$ grows, the average multiplicity grows.

KNO scaling Energy Dependence

Fig 4: KNO scaling Energy Dependence

Simulation using event generators

FORTRAN Monte Carlo codes. Amongst others[1]:

  • PYTHIA. Based on Lund string fragmentation . With a lot of user-tunable parameters corresponding to several "physical" assertments.
  • HERWIG. Based on cluster hadronization. Almost free of parameters.

Difference between HERWIG and PYTHIA

Fig 5: Difference between HERWIG (solid) and PYTHIA (dashed) : the peak at low multiplicity for PYTHIA is due to the contribution of diffractive events. Default parameters have been used for both generators (from [3])

Xu Statistical Approach

Assuptions [8]:

  • NSD events.
  • 3 different system corresponding to rapidity regions: Central (C) Projectile(P) and Target(T).
  • two energy-momentum sources 1 and 2 (target and projectile)
  • Probability for the system i of obtaining $E_1$ from "source 1": $f_1(E_1) = A_1exp(-BE_1)$
  • Isotropic decay of hadronic clusters: number of charged hadrons proportional to energy of the system i (i=C,P,T)

After this model, $ <n_i>P(n_i)= 4{\frac{n_i}{<n_i>}}exp[-2{\frac{n_i}{<n_i>}}]$
$P(n_i)$ is the probability that the system i produces n charged particles. In the central rapidity region, where the interaction C-C is dominant, the charged multiplicity distribution should have that form.

Charged multiplicity distribution for different pseudorapidity windows
Fig 6: Charged multiplicity distribution for different pseudorapidity windows (from [8]) Only contributions from the C system have been included

QCD multi-modal distribution

Multiplicity distribution fluctuations

Xu Statistical approach solution.

Isotropic decay of clusters implies that the probability density to observe at a given rapidity y an hadron produced by the cluster j located in Y is:

${\rho(y,Y)}= {\frac{1}{2cosh^2(y-Y)}}$

Integrating on a rapidity window (W) for a "cluster position" interval we find the probability of finding an hadron in W. Supposing that each cluster generates one positive and one negative hadron, the probability of finding a charged hadron in a rapidity window W is:

$ <n_W>P(n_W)= 4{\frac{n_W}{<n_W>}}exp[-2{\frac{n_{W}}{<n_{W}>}}]$

Where the discrepancy for high rapidities ( ${\eta_W}$<5) is due to neglecting P* and T* system contributions. Dependences on "normalized charged-mulitiplicity moments" can also be traced.

Information we can obtain (from [3])

  • contribution of soft and hard events.
  • "intermitent behaviuour" as a signature of phase transition to QGP.

Bibliography (Commented)

[1] Introduction to High-Energy Heavy-Ion Collisions. Cheuk-Yin Wong - Science - 1994 p.27-28. Thome fit.
[2] Proton-Antiproton Collider Physics. G. Altarelli - Science - 1989. p.87. KNO scaling implies the shape of P(N) should assimptoticlaly become energy independent. p.90. There were indincations from the early UA1 data that KNO scaling held reasonable well in the central region. This appeared to be confirmed by UA5 data. p.103. Table "multiplicity"
[3] ALICE PPR I. p. 1620.Phase space covered by TPC: ${\eta}<0.9$ p.1736.PYTHIA tunning
[4] BOTTOM PRODUCTION. arXiv:hep-ph/0003142v2
[5] GENERAL CHARACTERISTICS OF HADRON–HADRON COLLISIONS. W. Kittel. PACS numbers: 12.38.Qk, 13.66.Bc, 13.85.Hd, 13.87.Fh p.17-18.
[6] Hard Single Diffraction in ¯pp Collisions at ps = 630 and 1800 GeV. arXiv:hep-ex/9912061v1
[7] High relativistic nucleus-nucleus collision: The central rapidity region. J.D.Bjorken. Phys.Rev.D (1983)
[8] Statistical approach to non-diffractive hadron-hadron collisions: Multiplicity distributions and correlations in different rapidity intervals. Cai Xu et al. Phys.Rev.D (1986)
[9] Is There Koba-Nielsen-Olesen Scaling at Fermilab Tevatron Collider Energies (1600-2000 GeV)? Ina Sarcevic. PRL. 27 July 1987.
[10] Scenarios for multiplicity distributions in pp collisions in the TeV energy region. Roberto Ugoccioni and Alberto Giovannini. Journal of Physics: Conference Series 5 (2005) 199–208.

Topic attachments
I Attachment Action Size Date Who Comment
Herwig_pythia.pngpng Herwig_pythia.png manage 33.0 K 2008-02-24 - 14:50 JuanCastillo Difference between HERWIG and PYTHIA
KNO_scaling_E_dep.pngpng KNO_scaling_E_dep.png manage 10.3 K 2008-02-27 - 20:23 JuanCastillo KNO scaling Energy Dependence
charged_ALICE.pngpng charged_ALICE.png manage 44.5 K 2008-02-25 - 09:04 JuanCastillo Charged rapidity density as a function of center-of-mass energy for AA and pp collisions
charged_clan.pngpng charged_clan.png manage 14.3 K 2008-02-27 - 20:05 JuanCastillo Charged multiplicity after the clan model
charged_general.pngpng charged_general.png manage 54.0 K 2008-02-24 - 15:41 JuanCastillo Charged multiplicity for different hadron-hadron collisions
charged_statistical.pngpng charged_statistical.png manage 43.1 K 2008-02-25 - 18:22 JuanCastillo Charged multiplicity distribution for different pseudorapidity windows
Topic revision: r14 - 2008-09-18, JuanCastillo
 
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