FAPT: A Mathematica package for calculations in QCD Fractional Analytic Perturbation Theory

Abstract

We provide here all the procedures in Mathematica which are needed for the computation of the analytic images of the strong coupling constant powers in Minkowski (Ā ν (s; n f ) and A glob ν (s)) and Euclidean (Ā ν (Q 2 ; n f ) and A glob ν (Q 2 )) domains at arbitrary energy scales (s and Q 2 , correspondingly) for both schemes -with fixed number of active flavours n f = 3, 4, 5, 6 and the global one with taking into account all heavy-quark thresholds. These singularity-free couplings are inevitable elements of Analytic Perturbation Theory (APT) in QCD [1-3], and its generalization -Fractional APT [4][5][6], needed to apply the APT imperative for renormalization-group improved hadronic observables.Computer for which the program is designed and others on which it is operable: Any work-station or PC where Mathematica is running.Operating system or monitor under which the program has been tested: Windows XP, Mathematica (versions 5 and 7).No. of bytes in distributed program including test data etc.: 47 kB (main module FAPT.m) and 4 kB (interpolation module FAPT Interp.m); 21 kB (notebook FAPT Interp.nb showing how to use the interpolation module); 10 888 kB (interpolation data files: AcalGlob i.dat and UcalGlob i.dat with = 1, 2, 3, 3P, and 4) 1 Distribution format: ASCIINature of physical problem: The values of analytic imagesĀ ν (Q 2 ) andĀ ν (s) of the QCD running coupling powers α ν s (Q 2 ) in Euclidean and Minkowski regions, correspondingly, are determined through the spectral representation in the QCD Analytic Perturbation Theory (APT). In the program FAPT we collect all relevant formulas and various procedures which allow for a convenient evaluation ofĀ ν (Q 2 ) andĀ ν (s) using numerical integrations of the relevant spectral densities. Method of solution:FAPT uses Mathematica functions to calculate different spectral densities and then performs numerical integration of these spectral integrals to obtain analytic images of different objects.Restrictions on the complexity of the problem: It could be that for an unphysical choice of the input parameters the results are out of any meaning.Typical running time: For all operations the running time does not exceed a few seconds. Usually numerical integration is not fast, so that we advice to use arrays of precalculated data and apply then the routine Interpolate (as shown in supplied example of the program usage, namely in the notebook FAPT Interp.nb).

How to perform a QCD analysis of DIS in analytic perturbation theory

“…The authors of [11] fixed the Λ value directly from the comparison with the data in the leading order (where Q (n f = 3) for simplicity (recalling that an appropriate value should be taken from the analysis of the experimental data but this goes beyond the scope of this work). The couplings in pQCD and in FAPT were calculated with the Mathematica package developed by Bakulev and Khandramai in [36] where the heavy flavour thresholds were taken into account.…”

“…For our purpose we use here only the two-loop couplings like a ν (2)s , A (2) ν . Extensions up to four-loops can be found in [36]. Now we implement this formalism with the help of numerical calculation with the main module Mathematica package FAPT.m of [36] (confirmed by a recent program in [37]).…”

“…Now we implement this formalism with the help of numerical calculation with the main module Mathematica package FAPT.m of [36] (confirmed by a recent program in [37]). According to this and using the corresponding notation from [36] in the RHS of Eqs. (3.8-3.10), we have…”

“…The FAPT evolution relation (3.13) for the moments is the main result of the Section. Further, we shall use code (3.8) from [36] to obtain A . This analytic version of the moment evolution does not face any problems at low energies due to the boundedness of couplings and rapid convergence of the FAPT series.…”

The polarized nonsinglet $\Delta q_3$ and nonsinglet fragmentation function $D^{\pi^{+}}_{u_v}$ in the analytic approach to QCD

“…The mathematical tool for numerical calculations of A ν for any ν up to four-loop order is given in Refs. [11,12]. It should be stressed that values of the QCD scale parameter Λ are different in the PT and APT approaches.…”