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We present the exact and precise (~0.1%) numerical solution of the QCD evolution equations for the parton distributions in a wide range of $Q$ and $x$ using Monte Carlo (MC) method, which relies on the so-called Markovian algorithm. We point out certain advantages of such a method with respect to the existing non-MC methods. We also formulate a challenge of constructing non-Markovian MC algorithm for the evolution equations for the initial state QCD radiation with tagging the type and $x$ of the exiting parton. This seems to be within the reach of the presently available computer CPUs and the sophistication of the MC techniques.
We present the constrained Monte Carlo (CMC) algorithm for the QCD evolution. The constraint resides in that the total longitudinal energy of the emissions in the MC and in the underlying QCD evolution is predefined (constrained). This CMC implements exactly the full DGLAP evolution of the parton distributions in the hadron with respect to the logarithm of the energy scale. The algorithm of the CMC is referred to as the non-Markovian type. The non-Markovian MC algorithm is defined as the one in which the multiplicity of emissions is chosen randomly as the first variable and not the last one, as in the Markovian MC algorithms. The former case resembles that of the fixed-order matrix element calculations. The CMC algorithm can serve as an alternative to the so-called backward evolution Markovian algorithm of Sjostrand, which is used fo...
We revisit the challenging problem of finding an efficient Monte Carlo (MC) algorithm solving the constrained evolution equations for the initial-state QCD radiation. The type of the parton (quark, gluon) and the energy fraction x of the parton exiting emission chain (entering hard process) are predefined, i.e. constrained throughout the evolution. Such a constraint is mandatory for any realistic MC for the initial state QCD parton shower. We add one important condition: the MC algorithm must not require the a priori knowledge of the full numerical exact solutions of the evolution equations, as is the case in the popular ``Markovian MC for backward evolution''. Our aim is to find at least one solution of this problem that would function in practice. Finding such a solution seems to be definitely within the reach of the currently avai...
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