Mathematica Package For Calculations Research Articles

AnQCD: A Mathematica package for calculations in general analytic QCD models (new version announcement)

A previously submitted Mathematica package (Ayala and Cvetič, 2015), which evaluates the QCD couplings in several analytic (holomorphic) versions of QCD, has been adjusted (anQCDv2.m) so that it now works in the newest version of Mathematica (version 11) as well. New version program summaryProgram Title: anQCDv2.m; and the supplementary packages s0r.m and Li__nu.mProgram Files doi:http://dx.doi.org/10.17632/vsnjr4bhr9.1Licensing provisions: GPLv3Programming language: Mathematica, versions 9, 10 and 11Supplementary material: file READMEanQCDv2Journal reference of previous version: Comput. Phys. Commun. 190 (2015) 182–199Does the new version supersede the previous version? For the use with Mathematica 11, yes.Reasons for the new version: The previous version (anQCD.m) does not work in Mathematica version 11.Summary of revisions: When calling the previous version (anQCD.m) with Mathematica 11, the message SystemException[“MemoryAllocationFailure”] appears, because version 11 of Mathematica apparently cannot deal efficiently with numerical dispersion integrals where the integration is over semiinfinite interval and is slowly converging. Therefore, in the new version, in the integrals J0[Nf] for the Two-delta analytic QCD model the upper integration bound is changed from ‘Infinity’ to $wmaxJ=108.5. This modification does not change the precision of evaluations in any significant manner.Nature of problem: Evaluation of spacelike physical QCD quantities D(Q2) (Q2≡−q2 nonnegative) in terms of a truncated perturbation series of the perturbative QCD coupling a(Q2)≡αs(Q2)∕π gives us a quantity which has singularities at low positive Q2, because a(Q2) has such singularities. This is incompatible with the holomorphic (analytic) properties of D(Q2) as required by the general principles of Quantum Field Theories. Further, for low |Q2| such evaluations give very unreliable results dominated by the vicinity of the mentioned (unphysical) singularities. However, if using instead a coupling A(Q2) which respects the mentioned analyticity properties in the complex Q2-plane, then such problems do not appear.Solution method: anQCDv2.m package evaluates the coupling A(Q2) and the analogs Aν(Q2) of the powers a(Q2)ν, for three versions of QCD with couplings with aforementioned analytic properties: Analytic Perturbation Theory (APT); Two-delta analytic QCD (2δanQCD); and Massive Perturbation Theory (MPT). The couplings are evaluated by numerical evaluation of the corresponding dispersion integrals.Reference:C. Ayala and G. Cvetič, anQCD: a Mathematica package for calculations in general analytic QCD models, Comput. Phys. Commun. 190 (2015) 182–199 [arXiv:1408.6868 [hep-ph]].

AnQCD: A Mathematica package for calculations in general analytic QCD models

We provide a Mathematica package that evaluates the QCD analytic couplings (in the complex domain) Aν(Q2), which are analytic analogs of the powers a(Q2)ν of the underlying perturbative QCD (pQCD) coupling a(Q2)≡αs(Q2)/π, in three analytic QCD models (anQCD): Fractional Analytic Perturbation Theory (FAPT), Two-delta analytic QCD (2δanQCD), and Massive Perturbation Theory (MPT). The analytic (holomorphic) running couplings Aν(Q2), in contrast to the corresponding pQCD expressions a(Q2)ν, reflect correctly the analytic properties of the spacelike observables D(Q2) in the complex Q2 plane as dictated by the general principles of quantum field theory. They are thus more suited for evaluations of such physical quantities, especially at low momenta |Q2|∼1GeV2. Program summaryProgram title:anQCDCatalogue identifier: AEVK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1678No. of bytes in distributed program, including test data, etc.: 46323Distribution format: tar.gzProgramming language: Mathematica versions 9.0.1 and 10.0.1.Computer: Any work-station or PC where Mathematica is running.Operating system: Linux and Mac OS X, software Mathematica 9.0.1 and Mathematica 10.0.1.Classification: 11.1, 11.5.Nature of problem:Evaluation of the values for analytic couplings Aν(Q2;Nf) in analytic QCD [the analytic analog of the power (αs(Q2;Nf)/π)ν] based on the dispersion relation; Aν represents a physical (holomorphic) function in the plane of complex squared momenta −q2≡Q2. In anQCD.m we collect the formulas for three different analytic models depending on the energy scale, Q2, number of flavors Nf, the QCD scale Λ¯Nf, and the (nonpower) index ν. The considered models are: Analytic Perturbation theory (APT), Two-delta analytic QCD (2δanQCD) and Massive Perturbation Theory (MPT).Solution method:anQCD uses Mathematica functions to perform numerical integration of spectral function for each analytic model, in order to obtain the corresponding analytic images Aν(Q2) via dispersion relation.Restrictions:It could be that for an unphysical choice of the input parameters the results are meaningless.Running time:For all operations the running time does not exceed a few seconds.

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

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 (Āν(s;nf) and Aνglob(s)) and Euclidean (Āν(Q2;nf) and Aνglob(Q2)) domains at arbitrary energy scales (s and Q2, correspondingly) for both schemes — with fixed number of active flavours nf=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, proposed in [10,69,70], and its generalization — Fractional APT, suggested in [42,46,43], needed to apply the APT imperative for renormalization-group improved hadronic observables. Program summaryProgram title: FAPTCatalogue identifier: AENJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1985No. of bytes in distributed program, including test data, etc.: 1895776Distribution format: tar.gzProgramming language: Mathematica.Computer: Any work-station or PC where Mathematica is running.Operating system: Windows XP, Mathematica (versions 5 and 7).Classification: 11.5.Nature of problem:The values of analytic images Āν(Q2) and Āν(s) of the QCD running coupling powers αsν(Q2) 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 of Āν(Q2) and Āν(s) using numerical integrations of the relevant spectral densities.Solution method: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:It could be that for an unphysical choice of the input parameters the results are without any meaning.Running time:For all operations the run time does not exceed a few seconds. Usually numerical integration is not fast, so that we advise the use of arrays of precalculated data and then to apply the routine Interpolate(as shown in supplied example of the program usage, namely in the notebook FAPT_Interp.nb).