Mathematica and Fortran programs for various analytic QCD couplings
Abstract
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: gcvetic.usm.cl.
1 Why analytic QCD?
Perturbative QCD (pQCD) running coupling [, where ] has unphysical (Landau) singularities at low spacelike momenta .
For example, the oneloop pQCD running coupling
(1) 
has a Landau singularity (pole) at (). The 2loop 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 leadingtwist 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 (IRfixedpoint) behavior []. The IRfixedpoint behavior of is suggested by:

The holomorphic with IRfixedpoint behavior was proposed in various analytic QCD models, among them:
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 powersuppressed terms or .
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 :
(2) 
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
(3) 
where the construction of the analytic power analogs from was obtained.
The construction starts with logarithmic derivatives of [where ]:
(4) 
and . Using the Cauchy theorem, these quantities can be expressed in terms of the discontinuity function of anQCD coupling along its cut,
(5) 
This construction can be extended to a general noninteger
(6) 
This can be recast into an alternative form, involving () instead of
(7) 
where: and ; . This expression was obtained from Eq. (6) by the use of the following expression for the function [34]:
(8) 
The analytic analogs of powers are then obtained by combining various generalized logarithmic derivatives (with the coefficients obtained in [33])
(9) 
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
(10) 
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
(11) 
3.2 anQCD models: 2Qcd
Here, is approximated at high momenta by [], and in the unknown lowmomentum regime by two deltas:
(12) 
(13) 
The parameters and () are fixed in such a way that the resulting deviation from the underlying pQCD at high is: . The pQCDonset 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 Lambertscheme form
(14) 
where: ; , the upper (lower) sign when (), and
(15) 
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
(16) 
Since , the new coupling has no Landau singularities.
The (generalized) logarithmic derivatives are then uniquely determined
(17) 
4 Numerical implementation and results
Programs of numerical implementation in anQCD models:
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.
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 )
(18)  
Examples:
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
Out[2]=
In[3]:= AMPT3l[5, 1, , 0.7, 0.1] // Timing
Out[3]=
In[4]:= A2d3l[5, 0, 1, , 0] // Timing
Out[4]=
In[5]:= AFAPT3l[3, 1, 0, 0.5, 0.1, 0] // Timing
Out[5]=
In[6]:= AMPT3l[3, 1, 0.5, 0.7, 0.1] // Timing
Out[6]=
In[7]:= A2d3l[3, 0, 1, 0.5, 0] // Timing
Out[7]=
In[8]:= AFAPT3l[3, 0.3, 0, 0.5, 0.1, 0] // Timing
Out[8]=
In[9]:= AMPT3l[3, 0.3, 0.5, 0.7, 0.1] // Timing
Out[9]=
In[10]:= A2d3l[3, 0, 0.3, 0.5, 0] // Timing
Out[10]=
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].
Acknowledgments
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).
References
References
 [1] Cucchieri A and Mendes T 2008 Phys. Rev. Lett. 100 241601 (arXiv:0712.3517 [heplat])
 [2] Bogolubsky I L, Ilgenfritz E M, MüllerPreussker M and Sternbeck A 2009 Phys. Lett. B 676 69 (arXiv:0901.0736 [heplat])
 [3] Furui S 2009 PoS LAT 2009 227 (arXiv:0908.2768 [heplat])
 [4] Lerche C and von Smekal L 2002 Phys. Rev. D 65 125006 (hepph/0202194)
 [5] Aguilar A C and Papavassiliou J 2008 Eur. Phys. J. A 35 189 (arXiv:0708.4320 [hepph])
 [6] Zwanziger D 2004 Phys. Rev. D 69 016002 (hepph/0303028)
 [7] Dudal D, Gracey J A, Sorella S P, Vandersickel N and Verschelde H 2008 Phys. Rev. D 78 065047 (arXiv:0806.4348 [hepth])
 [8] Shirkov D V and Solovtsov I L 1996 JINR Rapid Commun. 2[76] 5–10 (arXiv:hepph/9604363)
 [9] Shirkov D V and Solovtsov I L 1997 Phys. Rev. Lett. 79 1209–1212 (arXiv:hepph/9704333)
 [10] Milton K A and Solovtsov I L 1997 Phys. Rev. D 55 5295–98 (arXiv:hepph/9611438)
 [11] Shirkov D V 2001 Eur. Phys. J. C 22 331 (hepph/0107282)
 [12] Karanikas A I and Stefanis N G 2001 Phys. Lett. B 504 225 (hepph/0101031)
 [13] Bakulev A P, Mikhailov S V and Stefanis N G 2005 Phys. Rev. D 72 074014 (arXiv:hepph/0506311)
 [14] Bakulev A P, Mikhailov S V and Stefanis N G 2007 Phys. Rev. D 75 056005 (arXiv:hepph/0607040)
 [15] Bakulev A P, Mikhailov S V and Stefanis N G 2010 JHEP 1006 085 (arXiv:1004.4125 [hepph])
 [16] Webber B R 1998 JHEP 9810 012 (hepph/9805484)
 [17] Alekseev A I 2006 Few Body Syst. 40 57 (arXiv:hepph/0503242)
 [18] Contreras C, Cvetič G, Espinosa O and Martínez H E 2010 Phys. Rev. D 82 074005 (arXiv:1006.5050)
 [19] Ayala C, Contreras C and Cvetič G 2012 Phys. Rev. D 85 114043 (arXiv:1203.6897 [hepph])
 [20] Simonov Yu A 1995 Phys. Atom. Nucl. 58 107 (hepph/9311247)
 [21] Simonov Yu A 2010 arXiv:1011.5386 [hepph]
 [22] Badalian A M and Kuzmenko D S 2001 Phys. Rev. D 65 016004 (hepph/0104097)
 [23] Shirkov D V 2013 Phys. Part. Nucl. Lett. 10 186 (arXiv:1208.2103 [hepth])
 [24] Cvetič G, Kögerler R and Valenzuela C 2010 J. Phys. G 37 075001 (arXiv:0912.2466 [hepph])
 [25] Cvetič G, Kögerler R and Valenzuela C 2010 Phys. Rev. D 82 114004 (arXiv:1006.4199 [hepph])
 [26] Contreras C, Cvetič G, Kögerler R, Kröger P and Orellana O 2014 arXiv:1405.5815 [hepph].
 [27] Nesterenko A V 2001 Phys. Rev. D 64 116009 (arXiv:hepph/0102124)
 [28] Nesterenko A V 2003 Int. J. Mod. Phys. A 18 5475 (arXiv:hepph/0308288)
 [29] Aguilar A C, Nesterenko A V and Papavassiliou J 2005 J. Phys. G 31 997 (hepph/0504195)
 [30] Cvetič G 2014 Phys. Rev. D 89 036003 (arXiv:1309.1696 [hepph]).
 [31] Cvetič G and Valenzuela C 2006 J. Phys. G 32 L27 (arXiv:hepph/0601050)
 [32] Cvetič G and Valenzuela C 2006 Phys. Rev. D 74 114030 (arXiv:hepph/0608256)
 [33] Cvetič G and Kotikov A V 2012 J. Phys. G 39 065005 (arXiv:1106.4275 [hepph])
 [34] Kotikov A V, Krivokhizhin V G and Shaikhatdenov B G 2012 Phys. Atom. Nucl. 75 507
 [35] Nesterenko A V and Papavassiliou J 2005 Phys. Rev. D 71 016009 (hepph/0410406)
 [36] Nesterenko A V 2009 Nucl. Phys. Proc. Suppl. 186 207 (arXiv:0808.2043 [hepph])
 [37] Nesterenko A V and Simolo C 2010 Comput. Phys. Commun. 181 1769 (arXiv:1001.0901 [hepph])
 [38] Nesterenko A V and Simolo C 2011 Comput. Phys. Commun. 182 2303 (arXiv:1107.1045 [hepph])
 [39] Bakulev A P and Khandramai V L 2013 Comput. Phys. Commun. 184 183 (arXiv:1204.2679 [hepph])
 [40] Ayala C and Cvetič G 2014 arXiv:1408.6868 [hepph].
 [41] Lepage G P 1980 CLNS80/447.