Markovian Monte Carlo solutions of the one-loop CCFM equations

Abstract

A systematic extension of the Monte Carlo (MC) algorithm, that solves the DGLAP equation, into the so-called the one-loop CCFM evolution is presented. Modifications are related to a z-dependent coupling constant; transverse momentum dependence is added to the x-dependence of the parton distributions. The presented Markovian algorithm for one-loop CCFM evolution is the first step in extending it to other more sophisticated schemes beyond DGLAP. In particular, implementing the complete CCFM will be the next step. The presently implemented one-loop CCFM option will be a useful tool in testing the forthcoming MC solutions. Numerical results of the new MC are confronted with other non-MC numerical solutions. The agreement within the MC statistical error of ~0.1% is found. Also, numerical results for kT-dependent structure functions are presented.

Figures (8)

• Figure 1: The upper plot shows the singlet quark distribution $xD_{q}(x,Q)$ evolved from $Q_0=1\,$GeV (black) to $Q=10$ (red), $100$ (green) and $1000$ (blue) GeV, obtained from EvolFMC (solid lines) and APCheb40 (dashed lines, hardly distinguishable), while the lower plot shows their ratio.
• Figure 2: The upper plot shows the gluon distribution $xD_{q}(x,Q)$ evolved from $Q_0=1\,$GeV (black) to $Q=10$ (red), $100$ (green) and $1000$ (blue) GeV, obtained from EvolFMC (solid lines) and APCheb40 (dashed lines, hardly distinguishable), while the lower plot shows their ratio.
• Figure 3: Gluon $k^T$ distributions integrated over $x$ for the one-loop CCFM equation of Marchesini--Webber, obtained from EvolFMC for Q = 1 (black), 10 (red), 100 (green) and 1000 (blue) GeV.
• Figure 4: Gluon $k^T$ distributions multiplied by $(k^T)^2$ integrated over $x$ for the one-loop CCFM equation of Marchesini--Webber, obtained from EvolFMC for Q = 1 (black), 10 (red), 100 (green) and 1000 (blue) GeV.
• Figure 5: The average $(k^T)^2$ of gluon as a function of $x$ for the one-loop CCFM equation of Marchesini--Webber, obtained from EvolFMC for Q = 1 (black), 10 (red), 100 (green) and 1000 (blue) GeV.