Advanced Collision Theory
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JuanCastillo - 25 Mar 2008
Beta Function(s): (QCD). It is describing the change with an arbitrary parameter
of the renormalization constant
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:
where:
- is depending on and profile functions from the Wood-Saxon nuclear densites
- depends on nuclear Wood-Saxon profile functions.
- 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
with the renormalization scale
( See also
RGE ).
A non-rigorous derivation has the form:
Where
is the quark distribution function including Bremstrahlung corrections.
It is obtained from making the structure function (
) independient of the arbitrary scaling factor
, that is, from
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 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:
- The sum is over spinor quark fields of different flavour, labelled each one of mass .
- Dirac matrices appear due to fermionic nature of quarks.
- The covariant derivative is obtained from the gluon field , in this way:
(a,b =1,...,N are color index)
- The matrices are the generating matrices of SU(N) in the fundamental representation ((A =1,...,) is the color index in the adjoint representation)
- The field-strength tensor for the gluon field is being the structure constants of SU(3)
- is the QCD coupling constant and it's the only free parameter (together with the quark masses)
From
using standard QCD one can compute several observables, like cross sections or decay rates, in powers of
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=
it has the form:
Where:
- is our renormalization scale.
- is the "renormalization scale dependent" coupling constant, and
- = is the beta function (Callan-Symanzyk equation)
In terms of
and
, it has the form:
Where
=
is the
beta function, describing the running of the coupling
with the renormalization scale
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:
Where:
- r = distance nucleon-center of the nucleus
- R = radius of the nucleus
- a = surface thickness
- = constant
That is is corresponding to this potential (spherical Woods-Saxon potential):
From
here