Mathematica and Fortran programs for various analytic QCD couplings

Mathematica and Fortran programs for various analytic QCD couplings

César Ayala and Gorazd Cvetič Department of Physics, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

We outline here the motivation for the existence of analytic QCD models, i.e., QCD frameworks in which the running coupling has no Landau singularities. The analytic (holomorphic) coupling is the analog of the underlying pQCD coupling , and any such defines an analytic QCD model. We present the general construction procedure for the couplings which are analytic analogs of the powers . Three analytic QCD models are presented. Applications of our program (in Mathematica) for calculation of in such models are presented. Programs in both Mathematica and Fortran can be downloaded from the web page:

111Preprint USM-TH-330. Based on the presentation given by G.C. at the 16th International workshop on Advanced Computing and Analysis Techniques in physics research (ACAT 2014), Prague, Czech Republic, September 1-5, 2014. To appear in the proceedings by the IOP Conference Series publishing.

1 Why analytic QCD?

Perturbative QCD (pQCD) running coupling [, where ] has unphysical (Landau) singularities at low spacelike momenta .

For example, the one-loop pQCD running coupling


has a Landau singularity (pole) at (). The 2-loop pQCD coupling has a Landau pole at and Landau cut at .

It is expected that the true QCD coupling has no such singularities. Why?

General principles of QFT dictate that any spacelike observable (correlators of currents, structure functions, etc.) is an analytic (holomorphic) function of in the entire complex plane with the exception of the timelike axis: , where GeV is a threshold mass (). If can be evaluated as a leading-twist term, then it is a function of the running coupling where : . Then the argument is expected to have the same analyticity properties as , which is not the case with the pQCD coupling in the usual renormalization schemes (, ’t Hooft, etc.).

A QCD coupling with holomorphic behavior for , represents an analytic QCD model (anQCD).

Such holomorphic behavior comes usually together with (IR-fixed-point) behavior []. The IR-fixed-point behavior of is suggested by:

  • lattice calculations [1, 2, 3]; calculations based on Dyson-Schwinger equations (DSE) [4, 5]; Gribov-Zwanziger approach [6, 7];

  • The holomorphic with IR-fixed-point behavior was proposed in various analytic QCD models, among them:

    1. Analytic Perturbation Theory (APT) of Shirkov, Solovtsov et al. [8, 9, 10, 11, 12];

    2. its extension Fractional APT (FAPT) [13, 14, 15];

    3. analytic models with very close to at high : with or , [16, 17, 18, 19];

    4. Massive Perturbation Theory (MPT), [20, 21, 22, 23].

Perturbative QCD (pQCD) can give analytic coupling in specific schemes with IR fixed point; the condition of reproduction of the correct value of the (strangeless and massless) semihadronic lepton decay ratio strongly restricts such schemes [24, 25, 26].

If the analytic coupling is not perturbative, differs from the pQCD couplings at GeV by nonperturbative (NP) terms, typically by some power-suppressed terms or .

An analytic QCD model which gives was constructed in [27, 28, 29].

2 The formalism of constructing in general anQCD

Having [the analytic analog of ] specified, we want to evaluate the physical QCD quantities in terms of such .

Usually is known as a (truncated) power series in terms of the pQCD coupling :


In anQCD, the simple replacement is not correct, it leads to a strongly diverging series when increases, as argued in [30]; a different formalism was needed, and was developed for general anQCD, first for the case of integer [31, 32], and then for the case of general [33]. It results in the replacements


where the construction of the analytic power analogs from was obtained.

The construction starts with logarithmic derivatives of [where ]:


and . Using the Cauchy theorem, these quantities can be expressed in terms of the discontinuity function of anQCD coupling along its cut,


This construction can be extended to a general noninteger


This can be recast into an alternative form, involving () instead of


where: and ; . This expression was obtained from Eq. (6) by the use of the following expression for the function [34]:


The analytic analogs of powers are then obtained by combining various generalized logarithmic derivatives (with the coefficients obtained in [33])


3 The considered anQCD models

We constructed Mathematica and Fortran programs for three anQCD models: 1.) Fractional Analytic Perturbation Theory (FAPT) [13, 14, 15]; 2.) 2 analytic QCD (2anQCD) [19]; 3.) Massive Perturbation Theory (MPT) [20, 21, 22, 23]. These three models are described below.

3.1 anQCD models: FAPT

Application of the Cauchy theorem to the function gives


In FAPT, the integration over the Landau part of the cut in the above integral is eliminated; since , the Landau cut is . This leads to the FAPT coupling


3.2 anQCD models: 2Qcd

Here, is approximated at high momenta by [], and in the unknown low-momentum regime by two deltas:


The parameters and () are fixed in such a way that the resulting deviation from the underlying pQCD at high is: . The pQCD-onset scale is determined so that the model reproduces the measured (strangeless and massless) tau lepton semihadronic decay ratio .

The underlying pQCD coupling is chosen in 2anQCD, for calculational convenience, in the Lambert-scheme form


where: ; , the upper (lower) sign when (), and


3.3 anQCD models: MPT

Nonperturbative physics suggests that the gluon acquires at low momenta an effective (dynamical) mass GeV, and that the coupling then has the form


Since , the new coupling has no Landau singularities.

The (generalized) logarithmic derivatives are then uniquely determined


4 Numerical implementation and results

Programs of numerical implementation in anQCD models:

  • for integer power analogs in APT and in “massive QCD” [35, 36]: Nesterenko and Simolo, 2010 (in Maple) [37], and 2011 (in Fortran) [38];

  • for general power analogs in FAPT: Bakulev and Khandramai, 2013 (in Mathematica) [39];

  • for general power analogs in 2anQCD, MPT and FAPT: the presented work in Mathematica [40] and Fortran (programs in both languages can be downloaded from the web page:

The basic relations for the numerical implementation of are: in FAPT Eq. (11); in 2anQCD Eqs. (13) and (9); in MPT Eqs. (17) and (9).

In Mathematica, is implemented as . In Mathematica 9.0.1 it is unstable for . Therefore, we provide a subroutine Li__nu.m (which is called by the main Mathematica program anQCD.m) and gives a stable version under the name . This problem does not exist in Mathematica 10.0.1.

In Fortran, program Vegas [41] is used for integrations. However, in Fortran, function is not implemented for general (complex) , and is evaluated as an integral Eq. (8). Therefore, the evaluation of ’s in 2anQCD is somewhat more time consuming in Fortran than in Mathematica. Further, more care has to be taken in Fortran to deal correctly with singularities of the integrands.

Figure 1: in three anQCD models with and , as a function of for ; the underlying pQCD coupling is included for comparison: (a) 2anQCD coupling and pQCD coupling, in the Lambert scheme with (and for ); (b) FAPT and MPT in 4-loop scheme and with ; MPT is with .
Figure 2: The same as in Fig. 1, but with (). is calculated from using the relation (9) with , and truncation at in 2anQCD, and at in MPT; and in FAPT using Eq. (11). Figs. 1 and 2 are taken from [40].

5 Main procedures in Mathematica for three analytic QCD models

1.) gives -loop analytic FAPT coupling with real power index , with fixed number of active quark flavors , in the Euclidean domain [ and ]

2.) gives “-loop” 2anQCD coupling , with power index ( and real; ), with number of active quark flavors , in the Euclidean domain. It is used in the truncation approach [where in (9): and , and we truncate at ]

3.) gives -loop analytic MPT coupling , with real power index () and with number of active quark flavors , in the Euclidean domain ( and )



Input scale of the underlying pQCD for FAPT and MPT is . The times are for a typical laptop, using Mathematica 9.0.1; the first entry in the results is the time of calculation, in .

In[1]:= anQCD.m;

In[2]:= AFAPT3l[5, 1, 0, , 0.1, 0] // Timing


In[3]:= AMPT3l[5, 1, , 0.7, 0.1] // Timing


In[4]:= A2d3l[5, 0, 1, , 0] // Timing


In[5]:= AFAPT3l[3, 1, 0, 0.5, 0.1, 0] // Timing


In[6]:= AMPT3l[3, 1, 0.5, 0.7, 0.1] // Timing


In[7]:= A2d3l[3, 0, 1, 0.5, 0] // Timing


In[8]:= AFAPT3l[3, 0.3, 0, 0.5, 0.1, 0] // Timing


In[9]:= AMPT3l[3, 0.3, 0.5, 0.7, 0.1] // Timing


In[10]:= A2d3l[3, 0, 0.3, 0.5, 0] // Timing


6 Conclusions

We constructed programs, in Mathematica and Fortran, which evaluate couplings in three models of analytic QCD (FAPT, 2anQCD, and MPT). These couplings are holomorphic functions (free of Landau singularities) in the complex plane with the exception of the negative semiaxis, and are analogs of powers of the underlying perturbative QCD. We checked that our results in FAPT model agree with those of Mathematica program [39].


This work was supported by FONDECYT (Chile) Grant No. 1130599 and DGIP (UTFSM) internal project USM No. 11.13.12 (C.A and G.C).



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