Advanced Collision Theory

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-- JuanCastillo - 25 Mar 2008

Beta Function(s): (QCD). It is describing the change with an arbitrary parameter ${\mu^2}$ of the renormalization constant ${\alpha_s}$ and of the charge e. See also RGE

Comover model: "Co-movers". Parton interaction model.
In a AB collision, being s the coordinates in A rest frame, b the impact parameter, it is possible to define survival probabilities S(b,s) for nuclear absortion (abs) and comover interaction (co).
This is giving a differential cross section for J/psi event samples:
$I_{AB}^{J/\psi} = A \int { d^2s {\sigma}_{AB}(b){S_{abs}}(b,s){S_{co}}(b,s) }  $
  • ${\sigma}_{AB}(b)$ is depending on ${\sigma}_{pp} $ and profile functions from the Wood-Saxon nuclear densites
  • $S_{abs}(b,s)$ depends on nuclear Wood-Saxon profile functions.
  • $S_{co}(b,s)$ depends on comover densities calculated using the DPM formalism.

DGLAP: Dokshitser-Gribov-Lipatov-Altarelli-Parisi equation.
Fundamental equation in the theory of perturbative QCD. Equivalent to the equation describing the running of the strong coupling constant ${\alpha_s}$ with the renormalization scale ${\mu^2}$ ( See also RGE ).
A non-rigorous derivation has the form:
${ {\mu^2}{\frac{{\partial}q(x,{\mu^2})}{{\partial}{\mu^2}}} = {\frac{\alpha_s({\mu^2})}{2{\pi}}}{\int_{x}^1}{\frac{dy}{y}}{P_{qq}(x/y)}q(y,{\mu^2}) +{\theta}({\alpha_s^2})  }$
Where $q(y,{\mu^2})$ is the quark distribution function including Bremstrahlung corrections.
It is obtained from making the structure function ($F_1$) independient of the arbitrary scaling factor ${\mu^2}$, that is, from ${{\partial}F_1 }/ {{\partial} {\mu^2}} = 0$

Dual Parton Model:

Glauber model: In heavy ion collisions (HIC) it describes a number of interaction processes over a wide range of energies from near the Coulomb barrier to higher energies. Assumptions (from this UTC seminar):
  • Nucleus-nucleus interaction treated in terms of interaction between nucleons.
  • Nucleons are distributed according to a density function (e.g. Woods-Saxon)
  • Nucleons travel in straight lines and are not deflected as they pass through the other nucleus.
  • Nucleons interact according to the inelastic cross section ${\sigma}$NN measured in pp collisions, even after interacting.

Hadron-String Dynamics model: (AKA HSD model) It pretends to be an unique model for nuclear dynamics, creation of dense and hot hadronic matter and in-medium modification of hadron properties. Here you find a web about HSD

QCD Lagrangian: It describes the strong interaction between quarks and gluons. Determined by Gauge invariance under the group SU(N) and renormalizability, it reads:
${L_{QCD}}=  - {\frac{1}{4}}F^{A}_{{\mu}{\nu}}F^{A}_{{\mu}{\nu}} + {\Sigma} {\bar{q}} (i{\gamma^{\mu}}D_{\mu} -m_i)q_i$
  • The sum is over spinor quark fields of different flavour, labelled $q_i$ each one of mass $m_i$.
  • Dirac matrices ${\gamma^{\mu}}$ appear due to fermionic nature of quarks.
  • The covariant derivative $D_{\mu}$ is obtained from the gluon field $A^{A}_{\mu}$, in this way:
$(D_{\mu})_{ab} = {\partial}{\delta}_{ab} + ig(t^{A}A^{A}_{\mu})_{ab}$ (a,b =1,...,N are color index)
  • The matrices ${t^A}$ are the generating matrices of SU(N) in the fundamental representation ((A =1,...,${N^2+1}$) is the color index in the adjoint representation)
  • The field-strength tensor for the gluon field is ${F^{A}_{{\mu}{\nu}}}$ being ${f^{ABC}}$ the structure constants of SU(3)
  • $g$ is the QCD coupling constant and it's the only free parameter (together with the quark masses)
From ${L_{QCD}}$ using standard QCD one can compute several observables, like cross sections or decay rates, in powers of ${\alpha_s(E)}$

QMD model: Quantum Molecular Dynamics model. Developed after the VUU model, for dealing with fluctuations and correlations, not treated when using a one-body distribution funtion (like in VUU model). A brief theoretical discussion you will find here

RGE: Renormalization Group Equation. For S= $ {\frac{\sigma}{\sigma_{pt}}}$ it has the form:
$ ({ {\mu { \frac{\delta}{{\delta}{\mu}} {\vert_{e_{\mu}}}}} +       {\beta(e_{\mu}){\frac{\delta}{{\delta}e_{\mu}}}} }) S( ({\vert}q^2{\vert}/{\mu^2}),e_{\mu}) =0$
  • $ {\mu^2} $ is our renormalization scale.
  • $ {e_{\mu}}$ is the "renormalization scale dependent" coupling constant, and
  • $ {\beta(e_{\mu})} $ = ${{\mu}{\frac{de_{\mu}}{d{\mu}}}}$ is the beta function (Callan-Symanzyk equation)
In terms of $ {\mu^2} $ and ${\alpha}$, it has the form:
$ ({ {\mu^2 { \frac{\delta}{{\delta}{\mu^2}} {\vert_{\alpha_{\mu}}}}} +       {\beta(\alpha_{\mu}){\frac{\delta}{{\delta}{\alpha_{\mu}}}}} }) S( ({\vert}q^2{\vert}/{\mu^2}),{\alpha_{\mu}}) =0$
Where $ {\beta({\alpha_{\mu}})} $ = ${{\mu^2}{\frac{d{\alpha_{\mu}}}{d{\mu^2}}}}{\vert_{e_0}}$ is the beta function, describing the running of the coupling ${\alpha_s}$ with the renormalization scale ${\mu^2}$

VUU model: Vlasov-Uehling-Uhlenbeck model. One of the first models for RHIC. It solves the transport equation for the one-body distribution function, and it's successfully describing "one-body" observables like collective flow, stopping and particle spectra. More info here. See also QMD model.

Woods-Saxon potential: Potential with the depth of the well proportional to the density of nucleons.
This is given for the spherical case by: $ {\rho(r)} = {\frac{\rho_0}{1+exp({\frac{r-R}{a}})}}$
  • r = distance nucleon-center of the nucleus
  • R = radius of the nucleus
  • a = surface thickness
  • $ {\rho_0} $ = constant
That is is corresponding to this potential (spherical Woods-Saxon potential):
$ V = - {\frac{V_0}{1+exp({\frac{r-R}{a}})}}$
From here

Topic revision: r2 - 2008-04-08, JuanCastillo
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