arXiv subjects
Publications and source records attributed to O. Gituliar.
We present analytical expressions for the 31 five-particle phase-space master integrals in massless QCD as an $ε$-series with coefficients being multiple zeta values of weight up to 12. In addition, we provide a computer code for the Monte-Carlo integration in higher dimensions, based on the RAMBO algorithm, that has been used to numerically cross-check the obtained results in 4, 6, and 8 dimensions.
We present analytical expressions for the 31 five-particle phase-space master integrals in massless QCD as an $ε$-series with coefficients being multiple zeta values of weight up to 12. In addition, we provide computer code for the Monte-Carlo integration in higher dimensions, based on the RAMBO algorithm, that has been used to numerically cross-check the obtained results in 4, 6, and 8 dimensions.
In this talk we discuss some aspects of the analytical calculation of energy correlations in electron-positron annihilation at a next-to-leading order in QCD. Our primary focus is on the most difficult task: the calculation of master integrals for real-emission contributions, which are functions of two dimensionless variables and the dimensional regulator. We use a method of differential equations and their so-called epsilon-form which is constructed with the help of the Fuchsia program based on Lee's algorithm.
This document collects the proceedings of the "Parton Radiation and Fragmentation from LHC to FCC-ee" workshop (http://indico.cern.ch/e/ee\_jets16) held at CERN in Nov. 2016. The writeup reviews the latest theoretical and experimental developments on parton radiation and parton-hadron fragmentation studies --including analyses of LEP, B-factories, and LHC data-- with a focus on the future perspectives reacheable in $e^+e^-$ measurements at the Future Circular Collider (FCC-ee), with multi-ab$^{-1}$ integrated luminosities yielding 10$^{12}$ and 10$^{8}$ jets from Z and W bosons decays as well as 10$^5$ gluon jets from Higgs boson decays. The main topics discussed are: (i) parton radiation and parton-to-hadron fragmentation functions (splitting functions at NNLO, small-$z$ NNLL resummations, global FF fits including Monte Carlo (MC) and neural-network analyses of the latest Belle/BaBar high-precision data, parton shower MC generators), (ii) jet properties (quark-gluon discrimination, $e^+e^-$ event shapes and multi-jet rates at NNLO+N$^{n}$LL, jet broadening and angularities, jet substructure at small-radius, jet charge determination, $e^+e^-$ jet reconstruction algorithms), (iii) heavy-quark jets (dead cone effect, charm-bottom separation, gluon-to-$b\bar{b}$ splitting), and (iv) non-perturbative QCD phenomena (colour reconnection, baryon and strangeness production, Bose-Einstein and Fermi-Dirac final-state correlations, colour string dynamics: spin effects, helix hadronization).
We present $\text{Fuchsia}$ $-$ an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $\partial_x\,\mathbf{f}(x,ε) = \mathbb{A}(x,ε)\,\mathbf{f}(x,ε)$ finds a basis transformation $\mathbb{T}(x,ε)$, i.e., $\mathbf{f}(x,ε) = \mathbb{T}(x,ε)\,\mathbf{g}(x,ε)$, such that the system turns into the epsilon form: $\partial_x\, \mathbf{g}(x,ε) = ε\,\mathbb{S}(x)\,\mathbf{g}(x,ε)$, where $\mathbb{S}(x)$ is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator $ε$. That makes the construction of the transformation $\mathbb{T}(x,ε)$ crucial for obtaining solutions of the initial equations. In principle, $\text{Fuchsia}$ can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals.
We report on the recent progress in reducing differential equations for Feynman master integrals to canonical form with the help of a method proposed by Roman Lee. For the first time, we present Fuchsia --- our open-source implementation of the Lee algorithm written in Python using mathematical routines of a free computer algebra system SageMath. We demonstrate Fuchsia by reducing differential equations for NLO contributions to splitting functions in QCD, which contain both loops and legs integrals.
We report on the status an ab initio computation of the time-like splitting functions at next-to-next-to-leading order in QCD. Time-like splitting functions govern the collinear kinematics of inclusive hadron production in $e^+e^-$ annihilation and the evolution of the parton fragmentation distributions. Current knowledge about them at three loops has been inferred by means of crossing symmetry from their related space-like counterparts, the deep-inelastic structure functions and parton densities. In this approach certain parts of the off-diagonal quark-gluon splitting function are left undetermined, which calls for an independent calculation from first principles. We outline the method for calculating master integrals from differential equations which are required to attack the problem.
A method for calculating phase-space master integrals for the decay process $1 \to n$ massless partons in QCD using integration-by-parts and differential equations techniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G)HPLs as a series in the dimensional regulator $ε$ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.
We report on re-calculation of the next-to-leading order DGLAP evolution kernels performed in a scheme suited for Monte Carlo simulations of parton cascades (parton showers).
In this note we outline the Monte Carlo project {\tt KrkMC}. The goal of this project is to construct a QCD Parton Shower accurate to NLO level in both coefficient function and splitting function (shower) parts. We discuss in detail one of its aspects --- the evolution kernels. The kernels had to be recalculated in a new regularisation scheme, called NPV. In this scheme all the singularities in the plus component of the integration momenta are regularised by means of principal value prescription. This is in contrast to the standard approach, in which only the spurious axial singularities are regularised by principal value. As a result, the triple poles in the dimensional regularisation parameter $ε$ are replaced by a combination of $ε$-poles and logarithms of geometrical cut-off $δ$. The resulting exclusive parton densities are more suitable for stochastic applications in four dimensions. Simultaneously, at the inclusive level, the standard and new prescriptions give the same results provided appropriate real and virtual contributions are added.
We report on the progress in calculating NLO DGLAP splitting functions for $x<1$ using the New Principal Value prescription, which is a modification of the standard Principal Value approach proposed by Curci, Furmanski and Petronzio in 1980. The new prescription reproduces the standard results on the inclusive (integrated) level, but simplifies individual contributions and restricts the cancellations between real and virtual diagrams which makes it useful for Monte Carlo simulations.
We propose a modified use of the Principal Value prescription for regularizing the infrared singularities in the light-cone axial gauge by applying it to all singularities in the light-cone plus component of integration momentum. The modification is motivated by and applied to the re-calculation of the QCD NLO splitting functions for the purpose of Monte Carlo implementations. The final results agree with the standard PV prescription whereas contributions from separate graphs get simplified.
Construction of a QCD cascade at the NLO level requires recalculation of the splitting functions in a different manner [1]. We describe the calculation of some of the virtual contributions to the non-singlet splitting function. In order to be compatible with the earlier calculated real contributions [2], the principal value prescription for regularizing the infrared singularities must be used in a new way. We illustrate this new scheme on simple examples. For the calculations we wrote a Mathematica package called Axiloop. We describe its current status.
The project of constructing a complete NLO-level Parton Shower Monte Carlo for the QCD processes developed in IFJ PAN in Krakow is reviewed. Four issues are discussed: (1) the extension of the standard inclusive collinear factorization into a new, fully exclusive scheme; (2) reconstruction of the LO Parton Shower in the new scheme; (3) inclusion of the exclusive NLO corrections into the hard process and (4) inclusion of the exclusive NLO corrections into the evolution (ladder) part.