The Fragmentation of Heavy Quarks

The probability to produce a hadron h from a heavy quark Q can be split in a short- and a long-range part [26]  

The short-distance, perturbative part Dq(x, fip) models the evolution of a quark produced off-shell at the scale /x/7 via gluon emissions to a quark on its mass shell. This is what is usually implemented in the parton shower algorithms of the Monte Carlo simulation programs. A parton shower develops through successive splitting until the perturbative approach becomes unreliable ( Aqcd)- The parton shower represents an approximative perturbative treatment of QCD dynamics based on the DGLAP evolution equations. It improves the fixed order pQCD calculation by taking into account soft and collinear enhanced terms to all orders. 

The most widely used formula for modeling the fragmentation of heavy quarks is the Peterson fragmentation function. The probability that the hadron receives a momentum fraction z from the quark is given by [27] 

In case of cluster fragmentation models color-singlet clusters of partons form after the perturbative phase of jet development and then decay into the observed hadrons. The clusters originate from gluon splitting in quark pairs and subsequent recombination with neighboring quarks and antiquarks. Afterward, the clusters are assumed to decay isotropically in their rest frame into pairs of hadrons, where the branching ratios are determined by the density of states. 

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