Solving constrained Markovian evolution in QCD with the help of the non-Markovian Monte Carlo

Abstract

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 Sjöstrand, which is used for modelling the initial-state parton shower in modern QCD MC event generators. We test practical feasibility and efficiency of our CMC implementation in a series of numerical exercises, comparing its results with those from other MC and non-MC programs, in a wide range of Q and x, down to the 0.1% precision level. In particular, satisfactory numerical agreement is found with the results of the Markovian MC program of our own and the other non-MC program. The efficiency of the new constrained MC is found to be quite good.

Collinear and TMD quark and gluon densities from parton branching solution of QCD evolution equations

“…In this sense, it is close in spirit to the works in refs. [43][44][45][46][47][48][49][50][51], in refs. [52][53][54][55][56][57][58], and in refs.…”

Constrained MC for QCD evolution with rapidity ordering and minimum kT

“…In addition, our new CMCs will not only generalize the solutions of ref. [8], but will be described in such a way that any future extension to other types of evolution kernels will be rather easy. In the following sub-chapters we shall present the details of our generalized solutions in detail.…”

“…[8] the first CMC algorithm of the class I was found and tested for DGLAP kernel, that is for our case (A). This algorithm is based on the observation that for the product of steeply rising functions, proportional to 1/y i , the δ-function constraint is effectively resolved by a single (let's say) y k , while all other y i , can be considered as unconstrained.…”

“…(2) for which a CMC was constructed already in refs. [8] and [9]. The new kernels are based on the following generic form where…”

“…Two distinct versions of this relatively new technique were proposed and tested in refs. [8,9,10] for the DGLAP [11] evolution in the leading-logarithmic (LL) approximation.…”