# The Brick Problem for Parton Cascade Codes

The brick for Parton Cascades aims at providing a set of benchmarks for Parton Cascade Codes. In particular it allows for the calculation of elastic (collisional) energy-loss in a controlled environment in the framework of microscopic transport models (and for those models incorporating gluon splitting, radiative energy-loss can be addressed as well).

Currently the setup described here is a proposal to the TECHQM community on how to formulate such a Brick for PCMs - it has been worked out through a collaboration of the Duke and Frankfurt groups and will hopefully be adopted by the greater community in the near future.

### Basic Setup:

The basic setup of the problem is to have gluonic matter in a box in thermal equilibrium at a given temperature. Then a hard probe, i.e. a gluon, will be shot through the box at a given initial energy/momentum and its energy (or energy-loss) will be measured as a function of distance (along its trajectory) and as a function of its initial energy.

### Box Definition:

• the box should be a cube of 5 fermi length with periodic boundary conditions, i.e. a gluon leaving the box at x=+5 fm, would re-enter the box at x=0 fm with the same momentum.
• gluons will be initialized according to a thermal momentum distribution at a density of ${\displaystyle n=N/V=(16/\pi ^{2})T^{3}}$
• the box will be initialized at temperatures T=300, 400, 500 and 600 MeV

### Cross Sections:

• a Debye-screened elastic glue-glue cross section will be used - just as in expression (1) of arXiv 0711.0961 [nucl-th]
• the Debye mass is chosen to be ${\displaystyle m_{D}^{2}=(24/\pi )\alpha _{s}T^{2}}$ (i.e. calculated for Boltzmann particles) with the coupling constant fixed to ${\displaystyle \alpha _{s}=0.3}$
• a minimum c.m. energy cut-off for a parton-parton collision is set to be ${\displaystyle \Lambda _{QCD}=200}$ MeV

### Desired Calculations:

• insert an on-shell gluon into the medium with an initial momentum of 50 GeV, 100 GeV, ... up to 400 GeV in 100 GeV increments, have it propagate through the box (with periodic boundary conditions the probe may propagate arbitrary distances) and measure E vs. x with x being the distance traveled along its trajectory (alternatively one may measure E vs. time, even though this may not be as accurate for comparing to the analytic formula)
• measure the sum of all ${\displaystyle p_{\perp }^{2}}$ kicks the parton accumulates and divide it by the pathlength traveled. This provides a measure for

the transport coefficient ${\displaystyle {\hat {q}}=1/l_{x}\sum \limits _{i=1}^{N_{coll}}(\Delta p_{\perp ,i})^{2}}$

• these calculations are to be performed with the aforementioned 2-2 elastic glue-glue scattering cross section
• subsequently, take a 200 GeV gluon and propagate it through the box at temperatures T=300, 400, 500 and 600 MeV, performing the same analysis
• optionally (if the code is capable of doing so), the calculations can be repeated with 2-3 processes activated as well - this will need to be fleshed out at a later date...

### Results:

• First results for the Andong Parton Cascade code APC

(created by Steffen A. Bass on July 7th 2008, last edited by Bass 08:40, 1 July 2009 (EDT))