anQCD: Fortran programs for couplings at complex momenta in various analytic QCD models

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• 作者：Cé ; sar Ayala^a ; ^{c.ayala86@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Gorazd Cvetič^b ; ^{gorazd.cvetic@usm.cl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Analytic (holomorphic) QCD coupling ; Fractional Analytic Perturbation Theory ; Two-delta analytic QCD model ; Massive Perturbation Theory ; Perturbative QCD ; Renormalization group evolution
• 刊名：Computer Physics Communications
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：199
• 期：Complete
• 页码：114-117
• 全文大小：373 K

文摘

We provide three Fortran programs which evaluate the QCD analytic (holomorphic) couplings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si13.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=e4a820aa6567326215a4510077b300dd" title="Click to view the MathML source">A_ν(Q²)class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\nu }\left(Q^2\right)$ for complex or real squared momenta class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si14.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=fca067473fd13b0281b55ea14d2b9dc3" title="Click to view the MathML source">Q²class="mathContainer hidden">class="mathCode">$Q^2$. These couplings are holomorphic analogs of the powers class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si15.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=5b214025cf0b59668f29e8a9df11b29d" title="Click to view the MathML source">a(Q²)^νclass="mathContainer hidden">class="mathCode">$a{\left(Q^2\right)}^{\nu }$ of the underlying perturbative QCD (pQCD) coupling class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si16.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=db198305e7da5efecb6d784911b27f14" title="Click to view the MathML source">a(Q²)≡α_s(Q²)/πclass="mathContainer hidden">class="mathCode">$a\left(Q^2\right)\equiv {\alpha }_s\left(Q^2\right)/\pi$, in three analytic QCD models (anQCD): Fractional Analytic Perturbation Theory (FAPT), Two-delta analytic QCD (2class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si17.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=45f3da5ae59968517c624c577eeb9668" title="Click to view the MathML source">δclass="mathContainer hidden">class="mathCode">$\delta$anQCD), and Massive Perturbation Theory (MPT). The index class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si18.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=51f031137635085f3656f0bb83a9e75d" title="Click to view the MathML source">νclass="mathContainer hidden">class="mathCode">$\nu$ can be noninteger. The provided programs do basically the same job as the Mathematica package anQCD.m published by us previously (Ayala and Cvetič, 2015), but are now written in Fortran.

Program summary

Program title: AanQCDext

Catalogue identifier: AEYK_v1_0

Program summary URL:class="interref" data-locatorType="url" data-locatorKey="http://cpc.cs.qub.ac.uk/summaries/AEYK_v1_0.html">http://cpc.cs.qub.ac.uk/summaries/AEYK_v1_0.html

Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland

Licensing provisions: Standard CPC licence, class="interref" data-locatorType="url" data-locatorKey="http://cpc.cs.qub.ac.uk/licence/licence.html">http://cpc.cs.qub.ac.uk/licence/licence.html

No. of lines in distributed program, including test data, etc.: 12105

No. of bytes in distributed program, including test data, etc.: 98822

Distribution format: tar.gz

Programming language: Fortran.

Computer: Any work-station or PC where Fortran 95/2003/2008 (gfortran) is running.

Operating system: Operating system Linux (Ubuntu and Scientific Linux), Windows (in all cases using gfortran).

Classification: 11.1, 11.5.

Nature of problem:   Calculation of values of the running analytic couplings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si19.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=2dc8be28bfaa4459c548e29af8aca2d0" title="Click to view the MathML source">A_ν(Q²;N_f)class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\nu }(Q^2;N_f)$ for general complex squared momenta class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si20.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=d135870c7f4ef01c03c938b8966f8ca1" title="Click to view the MathML source">Q²≡−q²class="mathContainer hidden">class="mathCode">$Q^2\equiv -q^2$, in three analytic QCD models, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si19.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=2dc8be28bfaa4459c548e29af8aca2d0" title="Click to view the MathML source">A_ν(Q²;N_f)class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\nu }(Q^2;N_f)$ is the analytic (holomorphic) analog of the power class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si22.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=39dedf0afef1af62a9128a7e2eb77c37" title="Click to view the MathML source">(α_s(Q²;N_f)/π)^νclass="mathContainer hidden">class="mathCode">${({\alpha }_s(Q^2;N_f)/\pi )}^{\nu }$. Here, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si19.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=2dc8be28bfaa4459c548e29af8aca2d0" title="Click to view the MathML source">A_ν(Q²;N_f)class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\nu }(Q^2;N_f)$ is a holomorphic function in the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si14.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=fca067473fd13b0281b55ea14d2b9dc3" title="Click to view the MathML source">Q²class="mathContainer hidden">class="mathCode">$Q^2$ complex plane, with the exception of the negative semiaxis class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si25.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=d2ac023f55fa9830396e659f4f0efd06">class="imgLazyJSB inlineImage" height="19" width="91" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0010465515003719-si25.gif">class="mathContainer hidden">class="mathCode">$(-\infty ,-M_{\mathrm{thr}}^2]$, reflecting the analyticity properties of the spacelike renormalization invariant quantities class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si26.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=3f92da7c453ca4b414f0ac96fa6d9edb" title="Click to view the MathML source">D(Q²)class="mathContainer hidden">class="mathCode">$D\left(Q^2\right)$ in QCD. In contrast, the perturbative QCD power class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si22.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=39dedf0afef1af62a9128a7e2eb77c37" title="Click to view the MathML source">(α_s(Q²;N_f)/π)^νclass="mathContainer hidden">class="mathCode">${({\alpha }_s(Q^2;N_f)/\pi )}^{\nu }$ has singularities even outside the negative semiaxis (Landau ghosts). The three considered models are: Analytic Perturbation theory (APT); Two-delta analytic QCD (2class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si17.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=45f3da5ae59968517c624c577eeb9668" title="Click to view the MathML source">δclass="mathContainer hidden">class="mathCode">$\delta$anQCD); Massive Perturbation Theory (MPT). We refer to Ref. [1] for more details and literature.

Solution method:   The Fortran programs for FAPT and 2class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si17.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=45f3da5ae59968517c624c577eeb9668" title="Click to view the MathML source">δclass="mathContainer hidden">class="mathCode">$\delta$anQCD models contain routines and functions needed to perform two-dimensional numerical integrations involving the spectral function, in order to evaluate class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si13.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=e4a820aa6567326215a4510077b300dd" title="Click to view the MathML source">A_ν(Q²)class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\nu }\left(Q^2\right)$ couplings. In MPT model, one-dimensional numerical integration involving class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si31.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=151d453b716c1e08041c6ce51d56b017" title="Click to view the MathML source">A₁(Q²)class="mathContainer hidden">class="mathCode">${\mathcal{A}}_1\left(Q^2\right)$ is sufficient to evaluate any class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si13.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=e4a820aa6567326215a4510077b300dd" title="Click to view the MathML source">A_ν(Q²)class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\nu }\left(Q^2\right)$ coupling.

Restrictions:   For unphysical choices of the input parameters the results are meaningless. When class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si14.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=fca067473fd13b0281b55ea14d2b9dc3" title="Click to view the MathML source">Q²class="mathContainer hidden">class="mathCode">$Q^2$ is close to the cut region of the couplings (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si14.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=fca067473fd13b0281b55ea14d2b9dc3" title="Click to view the MathML source">Q²class="mathContainer hidden">class="mathCode">$Q^2$ real negative), the calculations can take more time and can have less precision.

Running time:   For evaluation of a set of about 10 related couplings, the times vary in the range class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si35.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=82a053e32ab16f64fbdc674118fe4e79" title="Click to view the MathML source">t∼10¹–10²class="mathContainer hidden">class="mathCode">$t\sim {10}^1\text{–}{10}^2$ s. MPT requires less time, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0010465515003719&_mathId=si36.gif&_user=111111111&_pii=S0010465515003719&_rdoc=1&_issn=00104655&md5=31a5699b270b7c98a24b14524226fef1" title="Click to view the MathML source">t∼1–10¹class="mathContainer hidden">class="mathCode">$t\sim 1\text{–}{10}^1$ s.

References:

class="listitem" id="list_l000005">

class="label">[1]

C. Ayala and G. Cvetic, anQCD: a Mathematica package for calculations in general analytic QCD models, Comput. Phys. Commun. 190 (2015) 182. arXiv:1408.6868 [hep-ph].