Mathematica and Fortran programs for various analytic QCD couplings
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 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:
The holomorphic with IR-fixed-point 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 power-suppressed 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 :
In anQCD, the simple replacement is not correct, it leads to a strongly diverging series when increases, as argued in ; 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 . 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
The analytic analogs of powers are then obtained by combining various generalized logarithmic derivatives (with the coefficients obtained in )
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) ; 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:
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  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 )
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:= AFAPT3l[5, 1, 0, , 0.1, 0] // Timing
In:= AMPT3l[5, 1, , 0.7, 0.1] // Timing
In:= A2d3l[5, 0, 1, , 0] // Timing
In:= AFAPT3l[3, 1, 0, 0.5, 0.1, 0] // Timing
In:= AMPT3l[3, 1, 0.5, 0.7, 0.1] // Timing
In:= A2d3l[3, 0, 1, 0.5, 0] // Timing
In:= AFAPT3l[3, 0.3, 0, 0.5, 0.1, 0] // Timing
In:= AMPT3l[3, 0.3, 0.5, 0.7, 0.1] // Timing
In:= A2d3l[3, 0, 0.3, 0.5, 0] // Timing
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 .
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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