Exact solutions of the QCD evolution equations using Monte Carlo method

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

QED challenges at FCC-ee precision measurements

The expected experimental precision of the rates and asymmetries in the Future Circular Collider with electron positron beams (FCC-ee) in the centre of the mass energy range 88-365GeV considered for construction in CERN, will be better by a factor 5-200. This will be thanks to very high luminosity, factor up to $10^5$ higher than in the past LEP experiments. This poses the extraordinary challenge of improving the precision of the Standard Model predictions by a comparable factor. In particular the perturbative calculations of the trivial QED effects, which have to be removed from the experimental data, are considered to be a major challenge for almost all quantities to be measured at FCC-ee. The task of this paper is to summarize on the "state of the art" in this class of the calculations left from the LEP era and to examine what is to be done to match the precision of the FCC-ee experiments -- what kind of technical advancements are necessary. The above analysis will be done for most important observables of the FCC-ee like the total cross sections near $Z$ and $WW$ threshold, charge asymmetries, the invisible width of $Z$ boson, the spin asymmetry from $\tau$ lepton decay and the luminosity measurement.Comment: Corrected author's name in ref. [106

Non-Markovian Monte Carlo Algorithm for the Constrained Markovian Evolution in QCD

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 available computer CPUs and the sophistication of the modern MC techniques. We describe in this work the first example of an efficient solution of this kind. Its numerical implementation is still restricted to the pure gluon-strahlung. As expected, it is not in the class of the so-called Markovian MCs. For this reason we refer to it as belonging to a class of non-Markovian MCs. We show that numerical results of our new MC algorithm agree very well (to 0.2%) with the results of the other MC program of our own (unconstrained Markovian) and another non-MC program QCDnum16. This provides a proof of the existence of the new class of MC techniques, to be exploited in the precision perturbative QCD calculations for the Large Hadron Collider

Constrained non-Markovian Monte Carlo modelling of the evolution equation in QCD

A new class of the constrained Monte Carlo (CMC) algorithms for the QCD evolution equation was recently discovered. The constraint is imposed on the type and the total longitudinal energy of the parton exiting QCD evolution and entering a hard process. The efficiency of the new CMCs is found to be reasonable.Comment: Contribution to HERA-LHC worksho

How to Generate Four-Fermion Phase Space

We present a scheme for integrating the matrix element of an arbitrary e^+e^-\to f_1f_2\bar f_3\bar f_4 process over the complete four-fermion phase space, or its any part, by means of the Monte Carlo technique. The presented algorithm has been successfully implemented in the KORALW Monte Carlo code.Comment: 16 page

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

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 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

Exclusive Monte Carlo modelling of NLO DGLAP evolution

The next-to-leading order (NLO) evolution of the parton distribution functions (PDFs) in QCD is a common tool in the lepton-hadron and hadron-hadron collider data analysis. The standard NLO DGLAP evolution is formulated for inclusive (integrated) PDFs and done using inclusive NLO kernels. We report here on the ongoing project, called KRKMC, in which NLO DGLAP evolution is performed for the exclusive multiparton (fully unintegrated) distributions (ePDFs) with the help of the exclusive kernels. These kernels are calculated within the two-parton phase space for the non-singlet evolution, using Curci-Furmanski-Petronzio factorization scheme. The multiparton distribution, with multiple use of the exclusive NLO kernels, is implemented in the Monte Carlo program simulating multi-gluon emission from single quark emitter. High statistics tests ($\sim 10^{10}$ events) show that the new scheme works perfectly well in practice and, at the inclusive (integrated) level, is equivalent with the traditional inclusive NLO DGLAP evolution. Once completed, this new technique is aimed as a building block for the new more precise NLO parton shower Monte Carlo, for W/Z production at LHC and for ep scattering, as well as a starting point for other perturbative QCD based Monte Carlo projects.Comment: Contribution RADCOR 2009 Int. Symposiu

NLO evolution kernels: Monte Carlo versus MSbar

We investigate the differences between the NLO evolution kernels in the Curci-Furmanski-Petronzio (CFP) and Monte Carlo (MC) factorization schemes for the non-singlet case. We show the origin of these differences and present them explicitly. We examine the influence of the choice of the factorization scale in the MC scheme (given by the upper phase space limit) on the evolution kernels in this scheme.Comment: Contribution to Cracow Epiphany Conference 201