arXiv12042679

FAPT a Mathematica package for calculations in QCD

Fractional Analytic Perturbation Theory

Alexander P Bakuleva Vyacheslav L Khandramaiab

aBogoliubov Laboratory of Theoretical Physics JINR 141980 Dubna RussiabGomel State Technical University 246746 Gomel Belarus

Abstract

We provide here all the procedures in Mathematica which are needed for the computationof the analytic images of the strong coupling constant powers in Minkowski (Aν(snf ) andAglobν (s)) and Euclidean (Aν(Q2nf ) and Aglob

ν (Q2)) domains at arbitrary energy scales(s and Q2 correspondingly) for both schemes mdash with fixed number of active flavoursnf = 3 4 5 6 and the global one with taking into account all heavy-quark thresholdsThese singularity-free couplings are inevitable elements of Analytic Perturbation Theory(APT) in QCD [1ndash3] and its generalization mdash Fractional APT [4ndash6] needed to apply theAPT imperative for renormalization-group improved hadronic observables

PACS numbers 1238Bx 1115Bt 1110Hi

Keywords Analyticity Fractional Analytic Perturbation Theory Perturbative QCDRenormalization group evolution

Email addresses bakulevtheorjinrru (Alexander P Bakulev) vkhandramaigmailcom(Vyacheslav L Khandramai)

arX

iv1

204

2679

v3 [

hep-

ph]

24

Aug

201

2

Program Summary

Title of program FAPT

Available fromhttptheorjinrru˜bakulevfaptmatFAPTmhttptheorjinrru˜bakulevfaptmatFAPT Interpm

Computer for which the program is designed and others on which it is operable Anywork-station or PC where Mathematica is running

Operating system or monitor under which the program has been tested WindowsXP Mathematica (versions 5 and 7)

No of bytes in distributed program including test data etc47 kB (main module FAPTm) and 4 kB (interpolation module FAPT Interpm)21 kB (notebook FAPT Interpnb showing how to use the interpolation module)10 888 kB (interpolation data files AcalGlob`idat and UcalGlob`idat with ` =1 2 3 3P and 4)1

Distribution format ASCII

Nature of physical problem The values of analytic images Aν(Q2) and Aν(s) ofthe QCD running coupling powers ανs (Q2) in Euclidean and Minkowski regionscorrespondingly are determined through the spectral representation in the QCDAnalytic Perturbation Theory (APT) In the program FAPT we collect all relevantformulas and various procedures which allow for a convenient evaluation of Aν(Q2)and Aν(s) using numerical integrations of the relevant spectral densities

Method of solution FAPT uses Mathematica functions to calculate different spec-tral densities and then performs numerical integration of these spectral integrals toobtain analytic images of different objects

Restrictions on the complexity of the problem It could be that for an unphysicalchoice of the input parameters the results are out of any meaning

Typical running time For all operations the running time does not exceed a fewseconds Usually numerical integration is not fast so that we advice to use arrays ofprecalculated data and apply then the routine Interpolate (as shown in suppliedexample of the program usage namely in the notebook FAPT Interpnb)

1The notebook FAPT Interpnb and all interpolation data files are available from the same place inthe form of the zipped archive FAPT Interpzip of the size 1844 kB In order that Mathematica notebookFAPT Interpnb can use these precalculated data files one should place the directory sources withall data files in the same directory as the main file FAPT Interpnb

1

1 Introduction

QCD perturbation theory (PT) in the region of spacelike four-momentum transfer(Q2 = minusq2 gt 0 mdash hereafter we call it the Euclidean region) is based on expansionsin a series over the powers of effective charge (or running coupling constant) αs(Q

2)

which in the one-loop approximation is given by α(1)s (Q2) = (4πb0)L with b0 being the

first coefficient of the QCD beta function Eq (2)ndash(3) L = ln(Q2Λ2) and Λ = ΛQCD

is the QCD scale parameter The one-loop solution α(1)s (Q2) has a pole singularity at

L = 0 called the Landau pole The `-loop solution α(`)s (Q2) of the renormalization group

equation (2) has an `-root singularity of the type Lminus1` at L = 0 which produces thepole as well in the `-order term d` α

`s(Q

2) This prevents the application of perturbativeQCD in the low-momentum spacelike regime Q2 sim Λ2 with the effect that hadronicquantities calculated at the partonic level in terms of a power-series expansion in therunning coupling are not everywhere well defined

Such a singularity appeared first in QED [7 8] and was named ldquoghostrdquo due to thenegative residue at the corresponding propagator pole It was interpreted as an indica-tion that quantum field theory is self-contradictory However as was shown in [9 10]it is only a hint about the PT inapplicability in the region where the expansion param-eter is not small Appearance of such ldquoghostrdquo singularities from a theoretical point ofview contradicts the causality principle in quantum field theory [10 11] since it makesthe KallenndashLehmann spectral representation impossible It also complicates the deter-mination of the effective charge in the timelike region (q2 gt 0 mdash hereafter we call itthe Minkowski region) In a seminal paper by N N Bogoliubov et al of 1959 [12] theghost-free effective coupling for QED has been constructed using the dispersion relationtechnique

After the very appearance of QCD many researchers tried to determine the QCD effec-tive charge in the Minkowski region which is suitable for describing the processes of e+eminus

annihilation into hadrons as well as quarkonium and τ -lepton decays into hadrons Manysuch attempts used analytic continuation of the effective charge from the deep Euclideanregion in which perturbative QCD is known to work well into a Minkowski one whereactual experiments were performed αs(Q

2) rarr αs(s = minusQ2) In 1982 Radyushkin [13]and Krasnikov and Pivovarov [14] using the dispersion technique of [12] suggested regular(for s ge Λ2) QCD running coupling in Minkowski region the well-known πminus1 arctan(πL)

In 1995 Jones and Solovtsov using variational approach [15] constructed the effectivecouplings in Euclidean and Minkowski domains which appears to be finite for all Q2 and sand satisfy analyticity integral conditions Just in the same time Shirkov and Solovtsov [1]using the dispersion approach of [12] discovered ghost-free coupling A1(Q

2) Eq (25a)in Euclidean region and ghost-free coupling A1(s) Eq (25b) in Minkowski region whichsatisfy analyticity integral conditions

A1(Q2) = Q2

int infin0

A1(σ)

(σ +Q2)2dσ A1(s) =

1

2πi

int minuss+iεminussminusiε

A1(σ)

σdσ (1)

At the one-loop approximation the last coupling coincides with the Radyushkin one for

2

s ge Λ2 This way of making the QCDrsquos effective charge analytic in the timelike regionwas rediscovered later within an approach of fermion bubble resummation by Beneke andBraun [16] and also by Ball Beneke and Braun [17] Due to the absence of singularitiesin these couplings Shirkov and Solovtsov suggested to use this systematic approach calledAnalytic Perturbation Theory (APT) for all Q2 and s

Recently the analytic and numerical methods necessary to perform calculations intwo- and three-loop approximations were developed [18ndash24] This approach was appliedto the calculation of properties of a number of hadronic processes including the widthof inclusive τ lepton decay to hadrons [25ndash29] the scheme and renormalization-scaledependencies in the Bjorken [30 31] and GrossndashLlewellyn Smith [32] sum rules the widthof Υ meson decay to hadrons [33] etc Moreover APT was applied to the analysis ofthe processes with two scales rather than just a single scale namely the pion-photontransition form factor [34 35] and the pion electromagnetic form factor in the O(αs)order [34ndash36] To summarize we can say that APT (see reviews [37ndash39]) yields a sensibledescription of hadronic quantities in QCD though there are alternative approaches to thesingularity of effective charge in QCD mdash in particular with respect to the deep infraredregion Q2 lt Λ2 where appearance of nonzero hadronic masses may be important [40ndash42]The main advantage of the APT analysis is much more faster convergence of the APTnon-power series as compared with the standard PT power series see in [43 44]

Three-point functions used in describing the pion electromagnetic form factor orγlowastγ rarr π0 transition form factor contain logarithmic contributions at the next-to-leadingorder of the QCD PT related to the factorization scale If one set the factorization scaleproportional to the squared momentum-transfer micro2

F = Q2 then these logarithms will goto zero but additional RG factors of the type (αs(Q

2)αs(micro20))

ν with ν = γn(20) beinga fractional number will appear in the Gegenbauer coefficients of the pion distributionamplitude In both cases spectral densities used to construct analytic images of hadronicamplitudes should change This observation led Karanikas and Stefanis [45 46] to pro-pose the concept of analytization ldquoas a wholerdquo meaning that one should construct analyticimages not only of effective charge and its powers but of the whole QCD amplitude underconsideration

A QCD inspired generalization of APT to the fractional powers of effective chargecalled Fractional Analytic Perturbation Theory (FAPT) was done in [4 6] (for a recentreview see [47] for a recent generalization see [48]) followed by the application [5] tothe analysis of the factorizable contribution to the pion electromagnetic form factor Thecrucial advantage of FAPT in this case is that the perturbative results start to be lessdependent on the factorization scale This reminds the results obtained with the APTapplied to the analysis of the pion form factor in the O(α2

s) approximation where theresults also almost cease to depend on the choice of the renormalization scheme and itsscale (for a detailed review see [47] and references therein) The process of the Higgsboson decay into a bb pair of quarks was studied within a framework of FAPT in theMinkowski region at the one-loop level in [49] and at the three-loop levelmdash in [6] Resultson the resummation of non-power-series expansions of the Adler function of a scalar DSand a vector DV correlators within FAPT were presented in [50] The interplay between

3

higher orders of the perturbative QCD (pQCD) expansion and higher-twist contributionsin the analysis of recent Jefferson Lab data on the lowest moment of the spin-dependentproton structure function Γp1(Q

2) was studied in [51] using both standard QCD PTand (F)APT FAPT technique was also applied to the analysis of the structure functionF2(x) behavior at small values of x [52 53] All these successful applications of (F)APTnecessitate to have a reliable mathematical tool for calculations of spectral densities andanalytic couplings which are implemented in FAPT2

In this paper we collect all relevant formulas which are necessary for the running ofAν [L] and Aν [L] in the framework of APT and its fractional generalization FAPT Wediscuss their proper usage and provide easy-to-use Mathematica [56] procedures collectedin the package FAPT A few examples are given Here we do not consider the inclusionof analytic images of logarithms multiplied by fractional powers of couplings namely[αs(Q

2)]ν middot [ln(Q2Λ2)]

m which are needed for the full implementation of FAPT mdash we

postpone it to the next paperThe outline of the paper is as follows In the next Section we present the main formulas

of perturbative QCD which are needed for the running of the strong coupling constant upto the four-loop level Section 3 contains the basic formulas of APT and FAPT3 Finallyin Section 4 we describe the most important procedures of the package FAPT and providean example of using this package to produce some numerical estimations We hope thatfor most practical applications it should be sufficient In the Appendix we supply thecomplete collection of the developed procedures

2 Basics of the QCD running coupling

The running of the coupling constant of QCD αs(micro2) = αs[L] with L = ln[micro2Λ2] is

defined through4

dαs[L]

dL= β (αs[L]nf ) = minusαs[L]

sumkge0

bk(nf )

(αs[L]

4π

)k+1

(2)

2This task has been partially realized for both APT and its massive generalization [42] as the Maple

package QCDMAPT in [54] and as the Fortran package QCDMAPT F in [55] Both these realizations arelimited to the case of fixed number of active quarks Nf = 3 only and use approximate expressions forthe two- and higher-loop perturbative couplings compare for example Eq (33) in [54] and our Eq (7)

3Note here that FAPT includes APT as a partial case for the integer values of indices4We use notations f(Q2) and f [L] in order to specify the arguments we mean mdash squared momentum

Q2 or its logarithm L = ln(Q2Λ2) that is f [L] = f(Λ2 middot eL) and Λ2 is usually referred to nf = 3 region

4

where nf is the number of active flavours The coefficients are given by [57ndash66]

b0(nf ) = 11minus 2

3nf

b1(nf ) = 102minus 38

3nf

b2(nf ) =2857

2minus 5033

18nf +

325

54n2f

b3(nf ) =149753

6+ 3564 ζ3 minus

[1078361

162+

6508

27ζ3

]nf

+

[50065

162+

6472

81ζ3

]n2f +

1093

729n3f (3)

ζ is Riemannrsquos zeta function with values ζ2 = π26 and ζ3 asymp 1202 057 It is convenientto introduce the following notations

βf equivb0(nf )

4π a(micro2nf ) equiv βf αs(micro

2nf ) and ck(nf ) equivbk(nf )

b0(nf )k+1 (4)

Then Eq (2) in the l-loop approximation can be rewritten in the following form

da(`)[Lnf ]

dL= minus

(a(`)[Lnf ]

)2 [1 +

sumkge1

ck(nf )(a(`)[Lnf ]

)k] (5)

In the one-loop (l = 1) approximation (ck(nf ) = bk(nf ) = 0 for all k ge 1) we have asolution

a(1)[L] =1

L(6)

with the Landau pole singularity at L rarr 0 In the two-loop (l = 2) approximation(ck(nf ) = bk(nf ) = 0 for all k ge 2) the exact solution of Eq (2) is also known [67 68]

a(2)[Lnf ] =minuscminus11 (nf )

1 +Wminus1 (zW [L])with zW [L] = minuscminus11 (nf ) e

minus1minusLc1(nf ) (7)

where Wminus1[z] is the appropriate branch of Lambert functionThe three- and higher-loop solutions a(`)[Lnf ] can be expanded in powers of the

two-loop one a(2)[Lnf ] as has been suggested in [19 22ndash24 29]

a(`)[Lnf ] =sumnge1

C(`)n

(a(2)[Lnf ]

)n (8)

Coefficients C(`)n are known and can be evaluated recursively We use in our routine for

the three-loop coupling expansion up to the 9-th power included

C(3)1 = 1 C

(3)2 = 0 C

(3)3 = c2 C

(3)4 = 0 C

(3)5 =

5

3c22 C

(3)6 =

minus1

12c1 c

22

C(3)7 =

1

20c21 c

22 +

16

5c32 C

(3)8 =

minus1

30c31 c

22 minus

23

60c1 c

32

C(3)9 =

1

42c41 c

22 +

103

420c21 c

32 +

2069

315c42 (9)

5

0 2 4 6 8 1002

03

04

05

06

Q2 [GeV2]

α(3-P)s (Q2)

α(3)s (Q2)

0 2 4 6 8 10000

002

004

006

008

010

Q2 [GeV2]

δ(Q2)

Figure 1 Left panel Comparison of the standard three-loop coupling α(3)s (Q2) (solid blue line) with the

three-loop Pade one α(3P)s (Q2) (dashed red line) Right panel Relative accuracy δ(Q2) = (α

(3P)s (Q2) minus

α(3)s (Q2))α

(3)s (Q2) of the three-loop Pade coupling as compared with the standard three-loop one

As has been shown in [24] this expansion has a finite radius of convergence which appearsto be sufficiently large for all values of nf of practical interest Note here that this methodof expressing the higher-`-loop coupling in powers of the two-loop one is equivalent to thersquot Hooft scheme where one put by hands all coefficients in β-function except b0 and b1equal to zero and effectively takes into account all higher coefficients bi by redefiningperturbative coefficients di (see for more detail in [69])

Another possibility for obtaining the ldquoexactrdquo three-loop solution is provided by theso-called Pade approximation scheme It is based on the Pade-type modification of thethree-loop beta function

β(3P) (αs) = minusα2s

4π

[b0 +

b1 αs(4π)

1minus b2 αs(4π b1)

] (10a)

da(3P)[L]

dL= minusa2(3P)[L]

[1 +

c1 a(3P)[L]

1minus c2 a(3P)[L]c1

] (10b)

The last equation can be solved exactly with the help of the same Lambert function (herethe explicit dependence on nf is not shown for shortness)

a(3P)[L] =minuscminus11

1minus c2c21 +Wminus1(z(3P)W [L]

) with z(3P)W [L] = minuscminus11 eminus1+c2c

21minusLc1 (11)

The relative accuracy of this solution as compared with numerical solution of the standardthree-loop equation (5) with l = 3 is better than 1 for Q2 ge 2 GeV2 (with Λ

(3)3 =

356 MeV) and better than 05 for Q2 ge 5 GeV2 cf Fig 1In the four-loop approximation we use the same Eq (8) with corresponding coefficients

C(4)n = C(3)

n + ∆(4)n (12a)

6

5 10 15 20 25

03

04

05

06

Q2 [GeV2]

αs(Q2)

5 10 15 20 25000

001

002

003

004

005

006

Q2 [GeV2]

δ34(Q2)

Figure 2 Left panel Comparison of the four-loop coupling α(4)s (Q2) (solid blue line) with the

three-loop one α(3)s (Q2) (dashed violet line) Right panel Relative accuracy δ34(Q2) = (α

(4)s (Q2) minus

α(3)s (Q2))α

(4)s (Q2) of the three-loop coupling as compared with the four-loop one

and

∆(4)1 = ∆

(4)2 = ∆

(4)3 = 0 ∆

(4)4 = c3 ∆

(4)5 =

minusc1 c36

∆(4)6 =

c21 c312

+ 2 c2 c3

∆(4)7 =

minusc31 c320

minus 4 c1 c2 c35

+11 c2320

∆(4)8 =

c41 c330

+9 c21 c2 c3

20+

19 c22 c33

minus 49 c1 c23

120

∆(4)9 =

c51 c342minus 41 c31 c2 c3

140minus 946 c1 c

22 c3

315+

134 c2 c23

35+

149 c21 c23

504 (12b)

In the left panel of Fig 2 we show both couplings the four-loop α(4)s (Q2) (solid blue

line) and the three-loop α(3)s (Q2) (dashed violet line) with fixed number of active flavors

nf = 4 We normalize both couplings to the same value αs(m2Z) = 0119 at the Z-boson

mass scale Numerically as can be seen in the right panel of Fig 2 the relative deviationδ34(Q

2) = (α(4)s (Q2) minus α

(3)s (Q2))α

(4)s (Q2) varies from 6 at Q2 = 1 GeV2 to 05 at

Q2 = 25 GeV2 We also compared the four-loop coupling calculated in accord withEq (8) with coupling calculated using package RunDec [70] with the same normalizationαs(m

2Z) = 0119 mdash the relative deviation appears to vary from 02 at Q2 = 1 GeV2 to

004 at Q2 = 25 GeV2

21 Global scheme

Here we consider the scheme of the so-called ldquoglobal pQCDrdquo in which the heavy-quarkthresholds are taken into account We follow here to ShirkovndashSolovtsov approach [1 18 20]with the following values of pole masses of c b and t quarks mc = 165 GeV mb =475 GeV and mt = 1725 GeV In the MS scheme of the standard pQCD one needs tomatch the running coupling values in Euclidean domain at Q2 corresponding to thesemasses M4 = mc M5 = mb and M6 = mt In order to implement these matchingconditions we need to use the original QCD coupling

α(`)s (Q2nf ) =

4 π

b0(nf )a(`)(Q2nf ) (13)

7

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

1 Introduction

QCD perturbation theory (PT) in the region of spacelike four-momentum transfer(Q2 = minusq2 gt 0 mdash hereafter we call it the Euclidean region) is based on expansionsin a series over the powers of effective charge (or running coupling constant) αs(Q

2)

which in the one-loop approximation is given by α(1)s (Q2) = (4πb0)L with b0 being the

first coefficient of the QCD beta function Eq (2)ndash(3) L = ln(Q2Λ2) and Λ = ΛQCD

is the QCD scale parameter The one-loop solution α(1)s (Q2) has a pole singularity at

L = 0 called the Landau pole The `-loop solution α(`)s (Q2) of the renormalization group

equation (2) has an `-root singularity of the type Lminus1` at L = 0 which produces thepole as well in the `-order term d` α

`s(Q

2) This prevents the application of perturbativeQCD in the low-momentum spacelike regime Q2 sim Λ2 with the effect that hadronicquantities calculated at the partonic level in terms of a power-series expansion in therunning coupling are not everywhere well defined

Such a singularity appeared first in QED [7 8] and was named ldquoghostrdquo due to thenegative residue at the corresponding propagator pole It was interpreted as an indica-tion that quantum field theory is self-contradictory However as was shown in [9 10]it is only a hint about the PT inapplicability in the region where the expansion param-eter is not small Appearance of such ldquoghostrdquo singularities from a theoretical point ofview contradicts the causality principle in quantum field theory [10 11] since it makesthe KallenndashLehmann spectral representation impossible It also complicates the deter-mination of the effective charge in the timelike region (q2 gt 0 mdash hereafter we call itthe Minkowski region) In a seminal paper by N N Bogoliubov et al of 1959 [12] theghost-free effective coupling for QED has been constructed using the dispersion relationtechnique

After the very appearance of QCD many researchers tried to determine the QCD effec-tive charge in the Minkowski region which is suitable for describing the processes of e+eminus

annihilation into hadrons as well as quarkonium and τ -lepton decays into hadrons Manysuch attempts used analytic continuation of the effective charge from the deep Euclideanregion in which perturbative QCD is known to work well into a Minkowski one whereactual experiments were performed αs(Q

2) rarr αs(s = minusQ2) In 1982 Radyushkin [13]and Krasnikov and Pivovarov [14] using the dispersion technique of [12] suggested regular(for s ge Λ2) QCD running coupling in Minkowski region the well-known πminus1 arctan(πL)

In 1995 Jones and Solovtsov using variational approach [15] constructed the effectivecouplings in Euclidean and Minkowski domains which appears to be finite for all Q2 and sand satisfy analyticity integral conditions Just in the same time Shirkov and Solovtsov [1]using the dispersion approach of [12] discovered ghost-free coupling A1(Q

2) Eq (25a)in Euclidean region and ghost-free coupling A1(s) Eq (25b) in Minkowski region whichsatisfy analyticity integral conditions

A1(Q2) = Q2

int infin0

A1(σ)

(σ +Q2)2dσ A1(s) =

1

2πi

int minuss+iεminussminusiε

A1(σ)

σdσ (1)

At the one-loop approximation the last coupling coincides with the Radyushkin one for

2

s ge Λ2 This way of making the QCDrsquos effective charge analytic in the timelike regionwas rediscovered later within an approach of fermion bubble resummation by Beneke andBraun [16] and also by Ball Beneke and Braun [17] Due to the absence of singularitiesin these couplings Shirkov and Solovtsov suggested to use this systematic approach calledAnalytic Perturbation Theory (APT) for all Q2 and s

Recently the analytic and numerical methods necessary to perform calculations intwo- and three-loop approximations were developed [18ndash24] This approach was appliedto the calculation of properties of a number of hadronic processes including the widthof inclusive τ lepton decay to hadrons [25ndash29] the scheme and renormalization-scaledependencies in the Bjorken [30 31] and GrossndashLlewellyn Smith [32] sum rules the widthof Υ meson decay to hadrons [33] etc Moreover APT was applied to the analysis ofthe processes with two scales rather than just a single scale namely the pion-photontransition form factor [34 35] and the pion electromagnetic form factor in the O(αs)order [34ndash36] To summarize we can say that APT (see reviews [37ndash39]) yields a sensibledescription of hadronic quantities in QCD though there are alternative approaches to thesingularity of effective charge in QCD mdash in particular with respect to the deep infraredregion Q2 lt Λ2 where appearance of nonzero hadronic masses may be important [40ndash42]The main advantage of the APT analysis is much more faster convergence of the APTnon-power series as compared with the standard PT power series see in [43 44]

Three-point functions used in describing the pion electromagnetic form factor orγlowastγ rarr π0 transition form factor contain logarithmic contributions at the next-to-leadingorder of the QCD PT related to the factorization scale If one set the factorization scaleproportional to the squared momentum-transfer micro2

F = Q2 then these logarithms will goto zero but additional RG factors of the type (αs(Q

2)αs(micro20))

ν with ν = γn(20) beinga fractional number will appear in the Gegenbauer coefficients of the pion distributionamplitude In both cases spectral densities used to construct analytic images of hadronicamplitudes should change This observation led Karanikas and Stefanis [45 46] to pro-pose the concept of analytization ldquoas a wholerdquo meaning that one should construct analyticimages not only of effective charge and its powers but of the whole QCD amplitude underconsideration

A QCD inspired generalization of APT to the fractional powers of effective chargecalled Fractional Analytic Perturbation Theory (FAPT) was done in [4 6] (for a recentreview see [47] for a recent generalization see [48]) followed by the application [5] tothe analysis of the factorizable contribution to the pion electromagnetic form factor Thecrucial advantage of FAPT in this case is that the perturbative results start to be lessdependent on the factorization scale This reminds the results obtained with the APTapplied to the analysis of the pion form factor in the O(α2

s) approximation where theresults also almost cease to depend on the choice of the renormalization scheme and itsscale (for a detailed review see [47] and references therein) The process of the Higgsboson decay into a bb pair of quarks was studied within a framework of FAPT in theMinkowski region at the one-loop level in [49] and at the three-loop levelmdash in [6] Resultson the resummation of non-power-series expansions of the Adler function of a scalar DSand a vector DV correlators within FAPT were presented in [50] The interplay between

3

higher orders of the perturbative QCD (pQCD) expansion and higher-twist contributionsin the analysis of recent Jefferson Lab data on the lowest moment of the spin-dependentproton structure function Γp1(Q

2) was studied in [51] using both standard QCD PTand (F)APT FAPT technique was also applied to the analysis of the structure functionF2(x) behavior at small values of x [52 53] All these successful applications of (F)APTnecessitate to have a reliable mathematical tool for calculations of spectral densities andanalytic couplings which are implemented in FAPT2

In this paper we collect all relevant formulas which are necessary for the running ofAν [L] and Aν [L] in the framework of APT and its fractional generalization FAPT Wediscuss their proper usage and provide easy-to-use Mathematica [56] procedures collectedin the package FAPT A few examples are given Here we do not consider the inclusionof analytic images of logarithms multiplied by fractional powers of couplings namely[αs(Q

2)]ν middot [ln(Q2Λ2)]

m which are needed for the full implementation of FAPT mdash we

postpone it to the next paperThe outline of the paper is as follows In the next Section we present the main formulas

of perturbative QCD which are needed for the running of the strong coupling constant upto the four-loop level Section 3 contains the basic formulas of APT and FAPT3 Finallyin Section 4 we describe the most important procedures of the package FAPT and providean example of using this package to produce some numerical estimations We hope thatfor most practical applications it should be sufficient In the Appendix we supply thecomplete collection of the developed procedures

2 Basics of the QCD running coupling

The running of the coupling constant of QCD αs(micro2) = αs[L] with L = ln[micro2Λ2] is

defined through4

dαs[L]

dL= β (αs[L]nf ) = minusαs[L]

sumkge0

bk(nf )

(αs[L]

4π

)k+1

(2)

2This task has been partially realized for both APT and its massive generalization [42] as the Maple

package QCDMAPT in [54] and as the Fortran package QCDMAPT F in [55] Both these realizations arelimited to the case of fixed number of active quarks Nf = 3 only and use approximate expressions forthe two- and higher-loop perturbative couplings compare for example Eq (33) in [54] and our Eq (7)

3Note here that FAPT includes APT as a partial case for the integer values of indices4We use notations f(Q2) and f [L] in order to specify the arguments we mean mdash squared momentum

Q2 or its logarithm L = ln(Q2Λ2) that is f [L] = f(Λ2 middot eL) and Λ2 is usually referred to nf = 3 region

4

where nf is the number of active flavours The coefficients are given by [57ndash66]

b0(nf ) = 11minus 2

3nf

b1(nf ) = 102minus 38

3nf

b2(nf ) =2857

2minus 5033

18nf +

325

54n2f

b3(nf ) =149753

6+ 3564 ζ3 minus

[1078361

162+

6508

27ζ3

]nf

+

[50065

162+

6472

81ζ3

]n2f +

1093

729n3f (3)

ζ is Riemannrsquos zeta function with values ζ2 = π26 and ζ3 asymp 1202 057 It is convenientto introduce the following notations

βf equivb0(nf )

4π a(micro2nf ) equiv βf αs(micro

2nf ) and ck(nf ) equivbk(nf )

b0(nf )k+1 (4)

Then Eq (2) in the l-loop approximation can be rewritten in the following form

da(`)[Lnf ]

dL= minus

(a(`)[Lnf ]

)2 [1 +

sumkge1

ck(nf )(a(`)[Lnf ]

)k] (5)

In the one-loop (l = 1) approximation (ck(nf ) = bk(nf ) = 0 for all k ge 1) we have asolution

a(1)[L] =1

L(6)

with the Landau pole singularity at L rarr 0 In the two-loop (l = 2) approximation(ck(nf ) = bk(nf ) = 0 for all k ge 2) the exact solution of Eq (2) is also known [67 68]

a(2)[Lnf ] =minuscminus11 (nf )

1 +Wminus1 (zW [L])with zW [L] = minuscminus11 (nf ) e

minus1minusLc1(nf ) (7)

where Wminus1[z] is the appropriate branch of Lambert functionThe three- and higher-loop solutions a(`)[Lnf ] can be expanded in powers of the

two-loop one a(2)[Lnf ] as has been suggested in [19 22ndash24 29]

a(`)[Lnf ] =sumnge1

C(`)n

(a(2)[Lnf ]

)n (8)

Coefficients C(`)n are known and can be evaluated recursively We use in our routine for

the three-loop coupling expansion up to the 9-th power included

C(3)1 = 1 C

(3)2 = 0 C

(3)3 = c2 C

(3)4 = 0 C

(3)5 =

5

3c22 C

(3)6 =

minus1

12c1 c

22

C(3)7 =

1

20c21 c

22 +

16

5c32 C

(3)8 =

minus1

30c31 c

22 minus

23

60c1 c

32

C(3)9 =

1

42c41 c

22 +

103

420c21 c

32 +

2069

315c42 (9)

5

0 2 4 6 8 1002

03

04

05

06

Q2 [GeV2]

α(3-P)s (Q2)

α(3)s (Q2)

0 2 4 6 8 10000

002

004

006

008

010

Q2 [GeV2]

δ(Q2)

Figure 1 Left panel Comparison of the standard three-loop coupling α(3)s (Q2) (solid blue line) with the

three-loop Pade one α(3P)s (Q2) (dashed red line) Right panel Relative accuracy δ(Q2) = (α

(3P)s (Q2) minus

α(3)s (Q2))α

(3)s (Q2) of the three-loop Pade coupling as compared with the standard three-loop one

As has been shown in [24] this expansion has a finite radius of convergence which appearsto be sufficiently large for all values of nf of practical interest Note here that this methodof expressing the higher-`-loop coupling in powers of the two-loop one is equivalent to thersquot Hooft scheme where one put by hands all coefficients in β-function except b0 and b1equal to zero and effectively takes into account all higher coefficients bi by redefiningperturbative coefficients di (see for more detail in [69])

Another possibility for obtaining the ldquoexactrdquo three-loop solution is provided by theso-called Pade approximation scheme It is based on the Pade-type modification of thethree-loop beta function

β(3P) (αs) = minusα2s

4π

[b0 +

b1 αs(4π)

1minus b2 αs(4π b1)

] (10a)

da(3P)[L]

dL= minusa2(3P)[L]

[1 +

c1 a(3P)[L]

1minus c2 a(3P)[L]c1

] (10b)

The last equation can be solved exactly with the help of the same Lambert function (herethe explicit dependence on nf is not shown for shortness)

a(3P)[L] =minuscminus11

1minus c2c21 +Wminus1(z(3P)W [L]

) with z(3P)W [L] = minuscminus11 eminus1+c2c

21minusLc1 (11)

The relative accuracy of this solution as compared with numerical solution of the standardthree-loop equation (5) with l = 3 is better than 1 for Q2 ge 2 GeV2 (with Λ

(3)3 =

356 MeV) and better than 05 for Q2 ge 5 GeV2 cf Fig 1In the four-loop approximation we use the same Eq (8) with corresponding coefficients

C(4)n = C(3)

n + ∆(4)n (12a)

6

5 10 15 20 25

03

04

05

06

Q2 [GeV2]

αs(Q2)

5 10 15 20 25000

001

002

003

004

005

006

Q2 [GeV2]

δ34(Q2)

Figure 2 Left panel Comparison of the four-loop coupling α(4)s (Q2) (solid blue line) with the

three-loop one α(3)s (Q2) (dashed violet line) Right panel Relative accuracy δ34(Q2) = (α

(4)s (Q2) minus

α(3)s (Q2))α

(4)s (Q2) of the three-loop coupling as compared with the four-loop one

and

∆(4)1 = ∆

(4)2 = ∆

(4)3 = 0 ∆

(4)4 = c3 ∆

(4)5 =

minusc1 c36

∆(4)6 =

c21 c312

+ 2 c2 c3

∆(4)7 =

minusc31 c320

minus 4 c1 c2 c35

+11 c2320

∆(4)8 =

c41 c330

+9 c21 c2 c3

20+

19 c22 c33

minus 49 c1 c23

120

∆(4)9 =

c51 c342minus 41 c31 c2 c3

140minus 946 c1 c

22 c3

315+

134 c2 c23

35+

149 c21 c23

504 (12b)

In the left panel of Fig 2 we show both couplings the four-loop α(4)s (Q2) (solid blue

line) and the three-loop α(3)s (Q2) (dashed violet line) with fixed number of active flavors

nf = 4 We normalize both couplings to the same value αs(m2Z) = 0119 at the Z-boson

mass scale Numerically as can be seen in the right panel of Fig 2 the relative deviationδ34(Q

2) = (α(4)s (Q2) minus α

(3)s (Q2))α

(4)s (Q2) varies from 6 at Q2 = 1 GeV2 to 05 at

Q2 = 25 GeV2 We also compared the four-loop coupling calculated in accord withEq (8) with coupling calculated using package RunDec [70] with the same normalizationαs(m

2Z) = 0119 mdash the relative deviation appears to vary from 02 at Q2 = 1 GeV2 to

004 at Q2 = 25 GeV2

21 Global scheme

Here we consider the scheme of the so-called ldquoglobal pQCDrdquo in which the heavy-quarkthresholds are taken into account We follow here to ShirkovndashSolovtsov approach [1 18 20]with the following values of pole masses of c b and t quarks mc = 165 GeV mb =475 GeV and mt = 1725 GeV In the MS scheme of the standard pQCD one needs tomatch the running coupling values in Euclidean domain at Q2 corresponding to thesemasses M4 = mc M5 = mb and M6 = mt In order to implement these matchingconditions we need to use the original QCD coupling

α(`)s (Q2nf ) =

4 π

b0(nf )a(`)(Q2nf ) (13)

7

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

s ge Λ2 This way of making the QCDrsquos effective charge analytic in the timelike regionwas rediscovered later within an approach of fermion bubble resummation by Beneke andBraun [16] and also by Ball Beneke and Braun [17] Due to the absence of singularitiesin these couplings Shirkov and Solovtsov suggested to use this systematic approach calledAnalytic Perturbation Theory (APT) for all Q2 and s

Recently the analytic and numerical methods necessary to perform calculations intwo- and three-loop approximations were developed [18ndash24] This approach was appliedto the calculation of properties of a number of hadronic processes including the widthof inclusive τ lepton decay to hadrons [25ndash29] the scheme and renormalization-scaledependencies in the Bjorken [30 31] and GrossndashLlewellyn Smith [32] sum rules the widthof Υ meson decay to hadrons [33] etc Moreover APT was applied to the analysis ofthe processes with two scales rather than just a single scale namely the pion-photontransition form factor [34 35] and the pion electromagnetic form factor in the O(αs)order [34ndash36] To summarize we can say that APT (see reviews [37ndash39]) yields a sensibledescription of hadronic quantities in QCD though there are alternative approaches to thesingularity of effective charge in QCD mdash in particular with respect to the deep infraredregion Q2 lt Λ2 where appearance of nonzero hadronic masses may be important [40ndash42]The main advantage of the APT analysis is much more faster convergence of the APTnon-power series as compared with the standard PT power series see in [43 44]

Three-point functions used in describing the pion electromagnetic form factor orγlowastγ rarr π0 transition form factor contain logarithmic contributions at the next-to-leadingorder of the QCD PT related to the factorization scale If one set the factorization scaleproportional to the squared momentum-transfer micro2

F = Q2 then these logarithms will goto zero but additional RG factors of the type (αs(Q

2)αs(micro20))

ν with ν = γn(20) beinga fractional number will appear in the Gegenbauer coefficients of the pion distributionamplitude In both cases spectral densities used to construct analytic images of hadronicamplitudes should change This observation led Karanikas and Stefanis [45 46] to pro-pose the concept of analytization ldquoas a wholerdquo meaning that one should construct analyticimages not only of effective charge and its powers but of the whole QCD amplitude underconsideration

A QCD inspired generalization of APT to the fractional powers of effective chargecalled Fractional Analytic Perturbation Theory (FAPT) was done in [4 6] (for a recentreview see [47] for a recent generalization see [48]) followed by the application [5] tothe analysis of the factorizable contribution to the pion electromagnetic form factor Thecrucial advantage of FAPT in this case is that the perturbative results start to be lessdependent on the factorization scale This reminds the results obtained with the APTapplied to the analysis of the pion form factor in the O(α2

s) approximation where theresults also almost cease to depend on the choice of the renormalization scheme and itsscale (for a detailed review see [47] and references therein) The process of the Higgsboson decay into a bb pair of quarks was studied within a framework of FAPT in theMinkowski region at the one-loop level in [49] and at the three-loop levelmdash in [6] Resultson the resummation of non-power-series expansions of the Adler function of a scalar DSand a vector DV correlators within FAPT were presented in [50] The interplay between

3

higher orders of the perturbative QCD (pQCD) expansion and higher-twist contributionsin the analysis of recent Jefferson Lab data on the lowest moment of the spin-dependentproton structure function Γp1(Q

2) was studied in [51] using both standard QCD PTand (F)APT FAPT technique was also applied to the analysis of the structure functionF2(x) behavior at small values of x [52 53] All these successful applications of (F)APTnecessitate to have a reliable mathematical tool for calculations of spectral densities andanalytic couplings which are implemented in FAPT2

In this paper we collect all relevant formulas which are necessary for the running ofAν [L] and Aν [L] in the framework of APT and its fractional generalization FAPT Wediscuss their proper usage and provide easy-to-use Mathematica [56] procedures collectedin the package FAPT A few examples are given Here we do not consider the inclusionof analytic images of logarithms multiplied by fractional powers of couplings namely[αs(Q

2)]ν middot [ln(Q2Λ2)]

m which are needed for the full implementation of FAPT mdash we

postpone it to the next paperThe outline of the paper is as follows In the next Section we present the main formulas

of perturbative QCD which are needed for the running of the strong coupling constant upto the four-loop level Section 3 contains the basic formulas of APT and FAPT3 Finallyin Section 4 we describe the most important procedures of the package FAPT and providean example of using this package to produce some numerical estimations We hope thatfor most practical applications it should be sufficient In the Appendix we supply thecomplete collection of the developed procedures

2 Basics of the QCD running coupling

The running of the coupling constant of QCD αs(micro2) = αs[L] with L = ln[micro2Λ2] is

defined through4

dαs[L]

dL= β (αs[L]nf ) = minusαs[L]

sumkge0

bk(nf )

(αs[L]

4π

)k+1

(2)

2This task has been partially realized for both APT and its massive generalization [42] as the Maple

package QCDMAPT in [54] and as the Fortran package QCDMAPT F in [55] Both these realizations arelimited to the case of fixed number of active quarks Nf = 3 only and use approximate expressions forthe two- and higher-loop perturbative couplings compare for example Eq (33) in [54] and our Eq (7)

3Note here that FAPT includes APT as a partial case for the integer values of indices4We use notations f(Q2) and f [L] in order to specify the arguments we mean mdash squared momentum

Q2 or its logarithm L = ln(Q2Λ2) that is f [L] = f(Λ2 middot eL) and Λ2 is usually referred to nf = 3 region

4

where nf is the number of active flavours The coefficients are given by [57ndash66]

b0(nf ) = 11minus 2

3nf

b1(nf ) = 102minus 38

3nf

b2(nf ) =2857

2minus 5033

18nf +

325

54n2f

b3(nf ) =149753

6+ 3564 ζ3 minus

[1078361

162+

6508

27ζ3

]nf

+

[50065

162+

6472

81ζ3

]n2f +

1093

729n3f (3)

ζ is Riemannrsquos zeta function with values ζ2 = π26 and ζ3 asymp 1202 057 It is convenientto introduce the following notations

βf equivb0(nf )

4π a(micro2nf ) equiv βf αs(micro

2nf ) and ck(nf ) equivbk(nf )

b0(nf )k+1 (4)

Then Eq (2) in the l-loop approximation can be rewritten in the following form

da(`)[Lnf ]

dL= minus

(a(`)[Lnf ]

)2 [1 +

sumkge1

ck(nf )(a(`)[Lnf ]

)k] (5)

In the one-loop (l = 1) approximation (ck(nf ) = bk(nf ) = 0 for all k ge 1) we have asolution

a(1)[L] =1

L(6)

with the Landau pole singularity at L rarr 0 In the two-loop (l = 2) approximation(ck(nf ) = bk(nf ) = 0 for all k ge 2) the exact solution of Eq (2) is also known [67 68]

a(2)[Lnf ] =minuscminus11 (nf )

1 +Wminus1 (zW [L])with zW [L] = minuscminus11 (nf ) e

minus1minusLc1(nf ) (7)

where Wminus1[z] is the appropriate branch of Lambert functionThe three- and higher-loop solutions a(`)[Lnf ] can be expanded in powers of the

two-loop one a(2)[Lnf ] as has been suggested in [19 22ndash24 29]

a(`)[Lnf ] =sumnge1

C(`)n

(a(2)[Lnf ]

)n (8)

Coefficients C(`)n are known and can be evaluated recursively We use in our routine for

the three-loop coupling expansion up to the 9-th power included

C(3)1 = 1 C

(3)2 = 0 C

(3)3 = c2 C

(3)4 = 0 C

(3)5 =

5

3c22 C

(3)6 =

minus1

12c1 c

22

C(3)7 =

1

20c21 c

22 +

16

5c32 C

(3)8 =

minus1

30c31 c

22 minus

23

60c1 c

32

C(3)9 =

1

42c41 c

22 +

103

420c21 c

32 +

2069

315c42 (9)

5

0 2 4 6 8 1002

03

04

05

06

Q2 [GeV2]

α(3-P)s (Q2)

α(3)s (Q2)

0 2 4 6 8 10000

002

004

006

008

010

Q2 [GeV2]

δ(Q2)

Figure 1 Left panel Comparison of the standard three-loop coupling α(3)s (Q2) (solid blue line) with the

three-loop Pade one α(3P)s (Q2) (dashed red line) Right panel Relative accuracy δ(Q2) = (α

(3P)s (Q2) minus

α(3)s (Q2))α

(3)s (Q2) of the three-loop Pade coupling as compared with the standard three-loop one

As has been shown in [24] this expansion has a finite radius of convergence which appearsto be sufficiently large for all values of nf of practical interest Note here that this methodof expressing the higher-`-loop coupling in powers of the two-loop one is equivalent to thersquot Hooft scheme where one put by hands all coefficients in β-function except b0 and b1equal to zero and effectively takes into account all higher coefficients bi by redefiningperturbative coefficients di (see for more detail in [69])

Another possibility for obtaining the ldquoexactrdquo three-loop solution is provided by theso-called Pade approximation scheme It is based on the Pade-type modification of thethree-loop beta function

β(3P) (αs) = minusα2s

4π

[b0 +

b1 αs(4π)

1minus b2 αs(4π b1)

] (10a)

da(3P)[L]

dL= minusa2(3P)[L]

[1 +

c1 a(3P)[L]

1minus c2 a(3P)[L]c1

] (10b)

The last equation can be solved exactly with the help of the same Lambert function (herethe explicit dependence on nf is not shown for shortness)

a(3P)[L] =minuscminus11

1minus c2c21 +Wminus1(z(3P)W [L]

) with z(3P)W [L] = minuscminus11 eminus1+c2c

21minusLc1 (11)

The relative accuracy of this solution as compared with numerical solution of the standardthree-loop equation (5) with l = 3 is better than 1 for Q2 ge 2 GeV2 (with Λ

(3)3 =

356 MeV) and better than 05 for Q2 ge 5 GeV2 cf Fig 1In the four-loop approximation we use the same Eq (8) with corresponding coefficients

C(4)n = C(3)

n + ∆(4)n (12a)

6

5 10 15 20 25

03

04

05

06

Q2 [GeV2]

αs(Q2)

5 10 15 20 25000

001

002

003

004

005

006

Q2 [GeV2]

δ34(Q2)

Figure 2 Left panel Comparison of the four-loop coupling α(4)s (Q2) (solid blue line) with the

three-loop one α(3)s (Q2) (dashed violet line) Right panel Relative accuracy δ34(Q2) = (α

(4)s (Q2) minus

α(3)s (Q2))α

(4)s (Q2) of the three-loop coupling as compared with the four-loop one

and

∆(4)1 = ∆

(4)2 = ∆

(4)3 = 0 ∆

(4)4 = c3 ∆

(4)5 =

minusc1 c36

∆(4)6 =

c21 c312

+ 2 c2 c3

∆(4)7 =

minusc31 c320

minus 4 c1 c2 c35

+11 c2320

∆(4)8 =

c41 c330

+9 c21 c2 c3

20+

19 c22 c33

minus 49 c1 c23

120

∆(4)9 =

c51 c342minus 41 c31 c2 c3

140minus 946 c1 c

22 c3

315+

134 c2 c23

35+

149 c21 c23

504 (12b)

In the left panel of Fig 2 we show both couplings the four-loop α(4)s (Q2) (solid blue

line) and the three-loop α(3)s (Q2) (dashed violet line) with fixed number of active flavors

nf = 4 We normalize both couplings to the same value αs(m2Z) = 0119 at the Z-boson

mass scale Numerically as can be seen in the right panel of Fig 2 the relative deviationδ34(Q

2) = (α(4)s (Q2) minus α

(3)s (Q2))α

(4)s (Q2) varies from 6 at Q2 = 1 GeV2 to 05 at

Q2 = 25 GeV2 We also compared the four-loop coupling calculated in accord withEq (8) with coupling calculated using package RunDec [70] with the same normalizationαs(m

2Z) = 0119 mdash the relative deviation appears to vary from 02 at Q2 = 1 GeV2 to

004 at Q2 = 25 GeV2

21 Global scheme

Here we consider the scheme of the so-called ldquoglobal pQCDrdquo in which the heavy-quarkthresholds are taken into account We follow here to ShirkovndashSolovtsov approach [1 18 20]with the following values of pole masses of c b and t quarks mc = 165 GeV mb =475 GeV and mt = 1725 GeV In the MS scheme of the standard pQCD one needs tomatch the running coupling values in Euclidean domain at Q2 corresponding to thesemasses M4 = mc M5 = mb and M6 = mt In order to implement these matchingconditions we need to use the original QCD coupling

α(`)s (Q2nf ) =

4 π

b0(nf )a(`)(Q2nf ) (13)

7

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

higher orders of the perturbative QCD (pQCD) expansion and higher-twist contributionsin the analysis of recent Jefferson Lab data on the lowest moment of the spin-dependentproton structure function Γp1(Q

2) was studied in [51] using both standard QCD PTand (F)APT FAPT technique was also applied to the analysis of the structure functionF2(x) behavior at small values of x [52 53] All these successful applications of (F)APTnecessitate to have a reliable mathematical tool for calculations of spectral densities andanalytic couplings which are implemented in FAPT2

In this paper we collect all relevant formulas which are necessary for the running ofAν [L] and Aν [L] in the framework of APT and its fractional generalization FAPT Wediscuss their proper usage and provide easy-to-use Mathematica [56] procedures collectedin the package FAPT A few examples are given Here we do not consider the inclusionof analytic images of logarithms multiplied by fractional powers of couplings namely[αs(Q

2)]ν middot [ln(Q2Λ2)]

m which are needed for the full implementation of FAPT mdash we

postpone it to the next paperThe outline of the paper is as follows In the next Section we present the main formulas

of perturbative QCD which are needed for the running of the strong coupling constant upto the four-loop level Section 3 contains the basic formulas of APT and FAPT3 Finallyin Section 4 we describe the most important procedures of the package FAPT and providean example of using this package to produce some numerical estimations We hope thatfor most practical applications it should be sufficient In the Appendix we supply thecomplete collection of the developed procedures

2 Basics of the QCD running coupling

The running of the coupling constant of QCD αs(micro2) = αs[L] with L = ln[micro2Λ2] is

defined through4

dαs[L]

dL= β (αs[L]nf ) = minusαs[L]

sumkge0

bk(nf )

(αs[L]

4π

)k+1

(2)

2This task has been partially realized for both APT and its massive generalization [42] as the Maple

package QCDMAPT in [54] and as the Fortran package QCDMAPT F in [55] Both these realizations arelimited to the case of fixed number of active quarks Nf = 3 only and use approximate expressions forthe two- and higher-loop perturbative couplings compare for example Eq (33) in [54] and our Eq (7)

3Note here that FAPT includes APT as a partial case for the integer values of indices4We use notations f(Q2) and f [L] in order to specify the arguments we mean mdash squared momentum

Q2 or its logarithm L = ln(Q2Λ2) that is f [L] = f(Λ2 middot eL) and Λ2 is usually referred to nf = 3 region

4

where nf is the number of active flavours The coefficients are given by [57ndash66]

b0(nf ) = 11minus 2

3nf

b1(nf ) = 102minus 38

3nf

b2(nf ) =2857

2minus 5033

18nf +

325

54n2f

b3(nf ) =149753

6+ 3564 ζ3 minus

[1078361

162+

6508

27ζ3

]nf

+

[50065

162+

6472

81ζ3

]n2f +

1093

729n3f (3)

ζ is Riemannrsquos zeta function with values ζ2 = π26 and ζ3 asymp 1202 057 It is convenientto introduce the following notations

βf equivb0(nf )

4π a(micro2nf ) equiv βf αs(micro

2nf ) and ck(nf ) equivbk(nf )

b0(nf )k+1 (4)

Then Eq (2) in the l-loop approximation can be rewritten in the following form

da(`)[Lnf ]

dL= minus

(a(`)[Lnf ]

)2 [1 +

sumkge1

ck(nf )(a(`)[Lnf ]

)k] (5)

In the one-loop (l = 1) approximation (ck(nf ) = bk(nf ) = 0 for all k ge 1) we have asolution

a(1)[L] =1

L(6)

with the Landau pole singularity at L rarr 0 In the two-loop (l = 2) approximation(ck(nf ) = bk(nf ) = 0 for all k ge 2) the exact solution of Eq (2) is also known [67 68]

a(2)[Lnf ] =minuscminus11 (nf )

1 +Wminus1 (zW [L])with zW [L] = minuscminus11 (nf ) e

minus1minusLc1(nf ) (7)

where Wminus1[z] is the appropriate branch of Lambert functionThe three- and higher-loop solutions a(`)[Lnf ] can be expanded in powers of the

two-loop one a(2)[Lnf ] as has been suggested in [19 22ndash24 29]

a(`)[Lnf ] =sumnge1

C(`)n

(a(2)[Lnf ]

)n (8)

Coefficients C(`)n are known and can be evaluated recursively We use in our routine for

the three-loop coupling expansion up to the 9-th power included

C(3)1 = 1 C

(3)2 = 0 C

(3)3 = c2 C

(3)4 = 0 C

(3)5 =

5

3c22 C

(3)6 =

minus1

12c1 c

22

C(3)7 =

1

20c21 c

22 +

16

5c32 C

(3)8 =

minus1

30c31 c

22 minus

23

60c1 c

32

C(3)9 =

1

42c41 c

22 +

103

420c21 c

32 +

2069

315c42 (9)

5

0 2 4 6 8 1002

03

04

05

06

Q2 [GeV2]

α(3-P)s (Q2)

α(3)s (Q2)

0 2 4 6 8 10000

002

004

006

008

010

Q2 [GeV2]

δ(Q2)

Figure 1 Left panel Comparison of the standard three-loop coupling α(3)s (Q2) (solid blue line) with the

three-loop Pade one α(3P)s (Q2) (dashed red line) Right panel Relative accuracy δ(Q2) = (α

(3P)s (Q2) minus

α(3)s (Q2))α

(3)s (Q2) of the three-loop Pade coupling as compared with the standard three-loop one

As has been shown in [24] this expansion has a finite radius of convergence which appearsto be sufficiently large for all values of nf of practical interest Note here that this methodof expressing the higher-`-loop coupling in powers of the two-loop one is equivalent to thersquot Hooft scheme where one put by hands all coefficients in β-function except b0 and b1equal to zero and effectively takes into account all higher coefficients bi by redefiningperturbative coefficients di (see for more detail in [69])

Another possibility for obtaining the ldquoexactrdquo three-loop solution is provided by theso-called Pade approximation scheme It is based on the Pade-type modification of thethree-loop beta function

β(3P) (αs) = minusα2s

4π

[b0 +

b1 αs(4π)

1minus b2 αs(4π b1)

] (10a)

da(3P)[L]

dL= minusa2(3P)[L]

[1 +

c1 a(3P)[L]

1minus c2 a(3P)[L]c1

] (10b)

The last equation can be solved exactly with the help of the same Lambert function (herethe explicit dependence on nf is not shown for shortness)

a(3P)[L] =minuscminus11

1minus c2c21 +Wminus1(z(3P)W [L]

) with z(3P)W [L] = minuscminus11 eminus1+c2c

21minusLc1 (11)

The relative accuracy of this solution as compared with numerical solution of the standardthree-loop equation (5) with l = 3 is better than 1 for Q2 ge 2 GeV2 (with Λ

(3)3 =

356 MeV) and better than 05 for Q2 ge 5 GeV2 cf Fig 1In the four-loop approximation we use the same Eq (8) with corresponding coefficients

C(4)n = C(3)

n + ∆(4)n (12a)

6

5 10 15 20 25

03

04

05

06

Q2 [GeV2]

αs(Q2)

5 10 15 20 25000

001

002

003

004

005

006

Q2 [GeV2]

δ34(Q2)

Figure 2 Left panel Comparison of the four-loop coupling α(4)s (Q2) (solid blue line) with the

three-loop one α(3)s (Q2) (dashed violet line) Right panel Relative accuracy δ34(Q2) = (α

(4)s (Q2) minus

α(3)s (Q2))α

(4)s (Q2) of the three-loop coupling as compared with the four-loop one

and

∆(4)1 = ∆

(4)2 = ∆

(4)3 = 0 ∆

(4)4 = c3 ∆

(4)5 =

minusc1 c36

∆(4)6 =

c21 c312

+ 2 c2 c3

∆(4)7 =

minusc31 c320

minus 4 c1 c2 c35

+11 c2320

∆(4)8 =

c41 c330

+9 c21 c2 c3

20+

19 c22 c33

minus 49 c1 c23

120

∆(4)9 =

c51 c342minus 41 c31 c2 c3

140minus 946 c1 c

22 c3

315+

134 c2 c23

35+

149 c21 c23

504 (12b)

In the left panel of Fig 2 we show both couplings the four-loop α(4)s (Q2) (solid blue

line) and the three-loop α(3)s (Q2) (dashed violet line) with fixed number of active flavors

nf = 4 We normalize both couplings to the same value αs(m2Z) = 0119 at the Z-boson

mass scale Numerically as can be seen in the right panel of Fig 2 the relative deviationδ34(Q

2) = (α(4)s (Q2) minus α

(3)s (Q2))α

(4)s (Q2) varies from 6 at Q2 = 1 GeV2 to 05 at

Q2 = 25 GeV2 We also compared the four-loop coupling calculated in accord withEq (8) with coupling calculated using package RunDec [70] with the same normalizationαs(m

2Z) = 0119 mdash the relative deviation appears to vary from 02 at Q2 = 1 GeV2 to

004 at Q2 = 25 GeV2

21 Global scheme

Here we consider the scheme of the so-called ldquoglobal pQCDrdquo in which the heavy-quarkthresholds are taken into account We follow here to ShirkovndashSolovtsov approach [1 18 20]with the following values of pole masses of c b and t quarks mc = 165 GeV mb =475 GeV and mt = 1725 GeV In the MS scheme of the standard pQCD one needs tomatch the running coupling values in Euclidean domain at Q2 corresponding to thesemasses M4 = mc M5 = mb and M6 = mt In order to implement these matchingconditions we need to use the original QCD coupling

α(`)s (Q2nf ) =

4 π

b0(nf )a(`)(Q2nf ) (13)

7

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

where nf is the number of active flavours The coefficients are given by [57ndash66]

b0(nf ) = 11minus 2

3nf

b1(nf ) = 102minus 38

3nf

b2(nf ) =2857

2minus 5033

18nf +

325

54n2f

b3(nf ) =149753

6+ 3564 ζ3 minus

[1078361

162+

6508

27ζ3

]nf

+

[50065

162+

6472

81ζ3

]n2f +

1093

729n3f (3)

ζ is Riemannrsquos zeta function with values ζ2 = π26 and ζ3 asymp 1202 057 It is convenientto introduce the following notations

βf equivb0(nf )

4π a(micro2nf ) equiv βf αs(micro

2nf ) and ck(nf ) equivbk(nf )

b0(nf )k+1 (4)

Then Eq (2) in the l-loop approximation can be rewritten in the following form

da(`)[Lnf ]

dL= minus

(a(`)[Lnf ]

)2 [1 +

sumkge1

ck(nf )(a(`)[Lnf ]

)k] (5)

In the one-loop (l = 1) approximation (ck(nf ) = bk(nf ) = 0 for all k ge 1) we have asolution

a(1)[L] =1

L(6)

with the Landau pole singularity at L rarr 0 In the two-loop (l = 2) approximation(ck(nf ) = bk(nf ) = 0 for all k ge 2) the exact solution of Eq (2) is also known [67 68]

a(2)[Lnf ] =minuscminus11 (nf )

1 +Wminus1 (zW [L])with zW [L] = minuscminus11 (nf ) e

minus1minusLc1(nf ) (7)

where Wminus1[z] is the appropriate branch of Lambert functionThe three- and higher-loop solutions a(`)[Lnf ] can be expanded in powers of the

two-loop one a(2)[Lnf ] as has been suggested in [19 22ndash24 29]

a(`)[Lnf ] =sumnge1

C(`)n

(a(2)[Lnf ]

)n (8)

Coefficients C(`)n are known and can be evaluated recursively We use in our routine for

the three-loop coupling expansion up to the 9-th power included

C(3)1 = 1 C

(3)2 = 0 C

(3)3 = c2 C

(3)4 = 0 C

(3)5 =

5

3c22 C

(3)6 =

minus1

12c1 c

22

C(3)7 =

1

20c21 c

22 +

16

5c32 C

(3)8 =

minus1

30c31 c

22 minus

23

60c1 c

32

C(3)9 =

1

42c41 c

22 +

103

420c21 c

32 +

2069

315c42 (9)

5

0 2 4 6 8 1002

03

04

05

06

Q2 [GeV2]

α(3-P)s (Q2)

α(3)s (Q2)

0 2 4 6 8 10000

002

004

006

008

010

Q2 [GeV2]

δ(Q2)

Figure 1 Left panel Comparison of the standard three-loop coupling α(3)s (Q2) (solid blue line) with the

three-loop Pade one α(3P)s (Q2) (dashed red line) Right panel Relative accuracy δ(Q2) = (α

(3P)s (Q2) minus

α(3)s (Q2))α

(3)s (Q2) of the three-loop Pade coupling as compared with the standard three-loop one

As has been shown in [24] this expansion has a finite radius of convergence which appearsto be sufficiently large for all values of nf of practical interest Note here that this methodof expressing the higher-`-loop coupling in powers of the two-loop one is equivalent to thersquot Hooft scheme where one put by hands all coefficients in β-function except b0 and b1equal to zero and effectively takes into account all higher coefficients bi by redefiningperturbative coefficients di (see for more detail in [69])

Another possibility for obtaining the ldquoexactrdquo three-loop solution is provided by theso-called Pade approximation scheme It is based on the Pade-type modification of thethree-loop beta function

β(3P) (αs) = minusα2s

4π

[b0 +

b1 αs(4π)

1minus b2 αs(4π b1)

] (10a)

da(3P)[L]

dL= minusa2(3P)[L]

[1 +

c1 a(3P)[L]

1minus c2 a(3P)[L]c1

] (10b)

The last equation can be solved exactly with the help of the same Lambert function (herethe explicit dependence on nf is not shown for shortness)

a(3P)[L] =minuscminus11

1minus c2c21 +Wminus1(z(3P)W [L]

) with z(3P)W [L] = minuscminus11 eminus1+c2c

21minusLc1 (11)

The relative accuracy of this solution as compared with numerical solution of the standardthree-loop equation (5) with l = 3 is better than 1 for Q2 ge 2 GeV2 (with Λ

(3)3 =

356 MeV) and better than 05 for Q2 ge 5 GeV2 cf Fig 1In the four-loop approximation we use the same Eq (8) with corresponding coefficients

C(4)n = C(3)

n + ∆(4)n (12a)

6

5 10 15 20 25

03

04

05

06

Q2 [GeV2]

αs(Q2)

5 10 15 20 25000

001

002

003

004

005

006

Q2 [GeV2]

δ34(Q2)

Figure 2 Left panel Comparison of the four-loop coupling α(4)s (Q2) (solid blue line) with the

three-loop one α(3)s (Q2) (dashed violet line) Right panel Relative accuracy δ34(Q2) = (α

(4)s (Q2) minus

α(3)s (Q2))α

(4)s (Q2) of the three-loop coupling as compared with the four-loop one

and

∆(4)1 = ∆

(4)2 = ∆

(4)3 = 0 ∆

(4)4 = c3 ∆

(4)5 =

minusc1 c36

∆(4)6 =

c21 c312

+ 2 c2 c3

∆(4)7 =

minusc31 c320

minus 4 c1 c2 c35

+11 c2320

∆(4)8 =

c41 c330

+9 c21 c2 c3

20+

19 c22 c33

minus 49 c1 c23

120

∆(4)9 =

c51 c342minus 41 c31 c2 c3

140minus 946 c1 c

22 c3

315+

134 c2 c23

35+

149 c21 c23

504 (12b)

In the left panel of Fig 2 we show both couplings the four-loop α(4)s (Q2) (solid blue

line) and the three-loop α(3)s (Q2) (dashed violet line) with fixed number of active flavors

nf = 4 We normalize both couplings to the same value αs(m2Z) = 0119 at the Z-boson

mass scale Numerically as can be seen in the right panel of Fig 2 the relative deviationδ34(Q

2) = (α(4)s (Q2) minus α

(3)s (Q2))α

(4)s (Q2) varies from 6 at Q2 = 1 GeV2 to 05 at

Q2 = 25 GeV2 We also compared the four-loop coupling calculated in accord withEq (8) with coupling calculated using package RunDec [70] with the same normalizationαs(m

2Z) = 0119 mdash the relative deviation appears to vary from 02 at Q2 = 1 GeV2 to

004 at Q2 = 25 GeV2

21 Global scheme

Here we consider the scheme of the so-called ldquoglobal pQCDrdquo in which the heavy-quarkthresholds are taken into account We follow here to ShirkovndashSolovtsov approach [1 18 20]with the following values of pole masses of c b and t quarks mc = 165 GeV mb =475 GeV and mt = 1725 GeV In the MS scheme of the standard pQCD one needs tomatch the running coupling values in Euclidean domain at Q2 corresponding to thesemasses M4 = mc M5 = mb and M6 = mt In order to implement these matchingconditions we need to use the original QCD coupling

α(`)s (Q2nf ) =

4 π

b0(nf )a(`)(Q2nf ) (13)

7

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

0 2 4 6 8 1002

03

04

05

06

Q2 [GeV2]

α(3-P)s (Q2)

α(3)s (Q2)

0 2 4 6 8 10000

002

004

006

008

010

Q2 [GeV2]

δ(Q2)

Figure 1 Left panel Comparison of the standard three-loop coupling α(3)s (Q2) (solid blue line) with the

three-loop Pade one α(3P)s (Q2) (dashed red line) Right panel Relative accuracy δ(Q2) = (α

(3P)s (Q2) minus

α(3)s (Q2))α

(3)s (Q2) of the three-loop Pade coupling as compared with the standard three-loop one

As has been shown in [24] this expansion has a finite radius of convergence which appearsto be sufficiently large for all values of nf of practical interest Note here that this methodof expressing the higher-`-loop coupling in powers of the two-loop one is equivalent to thersquot Hooft scheme where one put by hands all coefficients in β-function except b0 and b1equal to zero and effectively takes into account all higher coefficients bi by redefiningperturbative coefficients di (see for more detail in [69])

Another possibility for obtaining the ldquoexactrdquo three-loop solution is provided by theso-called Pade approximation scheme It is based on the Pade-type modification of thethree-loop beta function

β(3P) (αs) = minusα2s

4π

[b0 +

b1 αs(4π)

1minus b2 αs(4π b1)

] (10a)

da(3P)[L]

dL= minusa2(3P)[L]

[1 +

c1 a(3P)[L]

1minus c2 a(3P)[L]c1

] (10b)

The last equation can be solved exactly with the help of the same Lambert function (herethe explicit dependence on nf is not shown for shortness)

a(3P)[L] =minuscminus11

1minus c2c21 +Wminus1(z(3P)W [L]

) with z(3P)W [L] = minuscminus11 eminus1+c2c

21minusLc1 (11)

The relative accuracy of this solution as compared with numerical solution of the standardthree-loop equation (5) with l = 3 is better than 1 for Q2 ge 2 GeV2 (with Λ

(3)3 =

356 MeV) and better than 05 for Q2 ge 5 GeV2 cf Fig 1In the four-loop approximation we use the same Eq (8) with corresponding coefficients

C(4)n = C(3)

n + ∆(4)n (12a)

6

5 10 15 20 25

03

04

05

06

Q2 [GeV2]

αs(Q2)

5 10 15 20 25000

001

002

003

004

005

006

Q2 [GeV2]

δ34(Q2)

Figure 2 Left panel Comparison of the four-loop coupling α(4)s (Q2) (solid blue line) with the

three-loop one α(3)s (Q2) (dashed violet line) Right panel Relative accuracy δ34(Q2) = (α

(4)s (Q2) minus

α(3)s (Q2))α

(4)s (Q2) of the three-loop coupling as compared with the four-loop one

and

∆(4)1 = ∆

(4)2 = ∆

(4)3 = 0 ∆

(4)4 = c3 ∆

(4)5 =

minusc1 c36

∆(4)6 =

c21 c312

+ 2 c2 c3

∆(4)7 =

minusc31 c320

minus 4 c1 c2 c35

+11 c2320

∆(4)8 =

c41 c330

+9 c21 c2 c3

20+

19 c22 c33

minus 49 c1 c23

120

∆(4)9 =

c51 c342minus 41 c31 c2 c3

140minus 946 c1 c

22 c3

315+

134 c2 c23

35+

149 c21 c23

504 (12b)

In the left panel of Fig 2 we show both couplings the four-loop α(4)s (Q2) (solid blue

line) and the three-loop α(3)s (Q2) (dashed violet line) with fixed number of active flavors

nf = 4 We normalize both couplings to the same value αs(m2Z) = 0119 at the Z-boson

mass scale Numerically as can be seen in the right panel of Fig 2 the relative deviationδ34(Q

2) = (α(4)s (Q2) minus α

(3)s (Q2))α

(4)s (Q2) varies from 6 at Q2 = 1 GeV2 to 05 at

Q2 = 25 GeV2 We also compared the four-loop coupling calculated in accord withEq (8) with coupling calculated using package RunDec [70] with the same normalizationαs(m

2Z) = 0119 mdash the relative deviation appears to vary from 02 at Q2 = 1 GeV2 to

004 at Q2 = 25 GeV2

21 Global scheme

Here we consider the scheme of the so-called ldquoglobal pQCDrdquo in which the heavy-quarkthresholds are taken into account We follow here to ShirkovndashSolovtsov approach [1 18 20]with the following values of pole masses of c b and t quarks mc = 165 GeV mb =475 GeV and mt = 1725 GeV In the MS scheme of the standard pQCD one needs tomatch the running coupling values in Euclidean domain at Q2 corresponding to thesemasses M4 = mc M5 = mb and M6 = mt In order to implement these matchingconditions we need to use the original QCD coupling

α(`)s (Q2nf ) =

4 π

b0(nf )a(`)(Q2nf ) (13)

7

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

5 10 15 20 25

03

04

05

06

Q2 [GeV2]

αs(Q2)

5 10 15 20 25000

001

002

003

004

005

006

Q2 [GeV2]

δ34(Q2)

Figure 2 Left panel Comparison of the four-loop coupling α(4)s (Q2) (solid blue line) with the

three-loop one α(3)s (Q2) (dashed violet line) Right panel Relative accuracy δ34(Q2) = (α

(4)s (Q2) minus

α(3)s (Q2))α

(4)s (Q2) of the three-loop coupling as compared with the four-loop one

and

∆(4)1 = ∆

(4)2 = ∆

(4)3 = 0 ∆

(4)4 = c3 ∆

(4)5 =

minusc1 c36

∆(4)6 =

c21 c312

+ 2 c2 c3

∆(4)7 =

minusc31 c320

minus 4 c1 c2 c35

+11 c2320

∆(4)8 =

c41 c330

+9 c21 c2 c3

20+

19 c22 c33

minus 49 c1 c23

120

∆(4)9 =

c51 c342minus 41 c31 c2 c3

140minus 946 c1 c

22 c3

315+

134 c2 c23

35+

149 c21 c23

504 (12b)

In the left panel of Fig 2 we show both couplings the four-loop α(4)s (Q2) (solid blue

line) and the three-loop α(3)s (Q2) (dashed violet line) with fixed number of active flavors

nf = 4 We normalize both couplings to the same value αs(m2Z) = 0119 at the Z-boson

mass scale Numerically as can be seen in the right panel of Fig 2 the relative deviationδ34(Q

2) = (α(4)s (Q2) minus α

(3)s (Q2))α

(4)s (Q2) varies from 6 at Q2 = 1 GeV2 to 05 at

Q2 = 25 GeV2 We also compared the four-loop coupling calculated in accord withEq (8) with coupling calculated using package RunDec [70] with the same normalizationαs(m

2Z) = 0119 mdash the relative deviation appears to vary from 02 at Q2 = 1 GeV2 to

004 at Q2 = 25 GeV2

21 Global scheme

Here we consider the scheme of the so-called ldquoglobal pQCDrdquo in which the heavy-quarkthresholds are taken into account We follow here to ShirkovndashSolovtsov approach [1 18 20]with the following values of pole masses of c b and t quarks mc = 165 GeV mb =475 GeV and mt = 1725 GeV In the MS scheme of the standard pQCD one needs tomatch the running coupling values in Euclidean domain at Q2 corresponding to thesemasses M4 = mc M5 = mb and M6 = mt In order to implement these matchingconditions we need to use the original QCD coupling

α(`)s (Q2nf ) =

4 π

b0(nf )a(`)(Q2nf ) (13)

7

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

where the indicator (`) signals about the loop order of the approximation we use5

In what follows we use all logarithms L with respect to three-flavor scale Λ23

L(Q2) = ln(Q2Λ2

3

) (14)

Recalculation to all other scales is realized with the help of finite additions

ln(Q2Λ2

k

)= L(Q2) + λk with λk equiv ln

(Λ2

3Λ2k

) (15)

and Λk mdash the corresponding to the specified value nf = k scale of QCD We also definethe corresponding logarithmic values at the thresholds Mk (k = 4divide 6)

Lk(Λ3) equiv ln(M2

kΛ23

) (16)

All QCD scales Λf f = 4 5 6 we treat as functions of the single parameter namely thethree-flavor scale Λ3

Λf rarr Λf (Λ3) with Λ3 gt Λ4(Λ3) gt Λ5(Λ3) gt Λ6(Λ3) (17)

which should be defined from matching conditions for the running coupling at the heavy-quark thresholds

For an illustration we consider here the two-loop approximation with the runningcoupling α

(2)s [Lnf ]

α(2)s [Lnf ] =

minus4 π

b0(nf )c1(nf ) [1 +Wminus1(zW [Lnf ])](18)

with zW [Lnf ] = (1c1(nf )) exp [minus1 + iπ minus Lc1(nf )] Then matching conditions are

α(2)s [L4(Λ3) 3] = α(2)

s [L4(Λ3) + λ4 4] (19a)

α(2)s [L5(Λ3) + λ4 4] = α(2)

s [L5(Λ3) + λ5 5] (19b)

α(2)s [L6(Λ3) + λ5 5] = α(2)

s [L6(Λ3) + λ6 6] (19c)

These relations define constants λk with k = 4divide 6 as functions of variable Λ3 namely

λk rarr λ(2)k (Λ3) (20)

and as a consequence the continuous global effective QCD coupling

αglob(2)s (Q2Λ3) = α(2)

s

[L(Q2) 3

]θ(Q2ltM2

4

)+ α(2)

s

[L(Q2)+λ

(2)4 (Λ3) 4

]θ(M2

4 leQ2 ltM25

)+ α(2)

s

[L(Q2)+λ

(2)5 (Λ3) 5

]θ(M2

5 leQ2 ltM26

)+ α(2)

s

[L(Q2)+λ

(2)6 (Λ3) 6

]θ(M2

6 leQ2) (21)

5Note here that the dependence a(`)(Q2nf ) on nf is the consequence of Eq (5) where for l gt 1 onehas nf -dependent coefficients ck(nf )

8

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

Here is the list of partial values of Λ(2)f (Λ3) λ

(2)f (Λ3) and Lf (Λ3) with f = 4 5 6 for

Λ3 = 400 MeV

Λ(2)4 = 333 MeV Λ

(2)5 = 233 MeV Λ

(2)6 = 98 MeV (22a)

λ(2)4 = 0367 λ

(2)5 = 108 λ

(2)6 = 282 (22b)

L4 = 2197 L5 = 4750 L6 = 12162 (22c)

In our m-file we use the following realizations The QCD scales are encoded asΛ1[Λ nf ] Λ2[Λ nf ] and Λ3[Λ nf ] (in Mathematica capital Greek symbol Λ can be writ-ten as [CapitalLambda])

[CapitalLambda]`[Λ k] = Λ`[Λ nf = k] = Λ(`)k (Λ) (` = 1divide 4 3P k = 4divide 6) (23a)

the threshold logarithms mdash as λ`4[Λ] λ`5[Λ] and λ`6[Λ] (in Mathematica Greek symbolλ can be written as [Lambda])

[Lambda]`k[Λ] = λ`k[Λ] = ln(Λ2Λ`[Λ k]2

) (` = 1divide 4 3P k = 4divide 6) (23b)

the running QCD couplings with fixed nf mdash as αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] andαBar3[Q2 nf Λ] (in Mathematica Greek symbol α can be written as [Alpha])

[Alpha]Bar`[Q2 nf Λ] = αBar`[Q2 nf Λ] = α(`)s [ln(Q2Λ2)nf ] (` = 1divide 4 3P) (23c)

and the global running QCD couplings mdash as αGlob1[Q2Λ] αGlob2[Q2Λ] andαGlob3[Q2Λ]

[Alpha]Glob`[Q2Λ] = αGlob`[Q2Λ] = αglob(`)s (Q2Λ) (` = 1divide 4 3P) (23d)

To be more specific we consider here an example We assume that the two-loop αs

is given at the Z-boson scale as α(2)s [ln(m2

ZΛ2) 5] = 0119 We want to evaluate the

corresponding values of the QCD scales Λ3 Λ4 and Λ5 and the coupling αglob(`)s (Q2Λ)

at the scale Q = M5 We show a possible Mathematica realization of this task

In[1]= ltltFAPTm

Comment NumDefFAPT is a set of Mathematica rules in our package which assigns typicalvalues to the physical parameters used in our procedures

In[2]= MZ = MZbosonNumDefFAPT Mb=MQ5NumDefFAPT

Out[2]= 9119 475

Comment evaluation of L23= Λ(2)3 from α

(2)s [ln(m2

ZΛ2) 5] based on the explicit solution

Eq (18) Eq (21)

In[3]= L23=lxFindRoot[[Alpha]Glob2[MZ^2lx]==0119 lx0103]

Out[3]= 0387282

9

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

Comment evaluation of L24= Λ(2)4 and L25= Λ

(2)5 from L23= Λ

(2)3 based on Eq (23a)

In[4]= L24=[CapitalLambda]2[L234] L25=[CapitalLambda]2[L235]

Out[4]= 0321298 0224033

Comment evaluation of αglob(2)s (M2

b ) from L23= Λ(2)3

In[5]= [Alpha]Glob2[Mb^2L23]

Out[5]= 0218894

3 Basics of FAPT

In the end of the previous section we used for the running QCD couplings with fixednf the Bar notations mdash αBar1[Q2 nf Λ] αBar2[Q2 nf Λ] and αBar3[Q2 nf Λ] We didit on purpose to have a direct connection to our previous papers on the subject [4ndash6 47]where we used the normalized coupling a(micro2) = βf αs(micro

2) cf Eq (4) To be in line withthese definitions we also introduce analogous expressions for the fixed-Nf quantities withstandard normalization ie

Aν(Q2) =Aν(Q2)

βνf Aν(s) =

Aν(s)

βνf (24)

which correspond to the analytic couplings Aν and Aν in the ShirkovndashSolovtsov terminol-ogy [1]

The basic objects in the (F)APT approach are spectral densities ρ(`)ν (σnf ) which enter

the KallenndashLehmann spectral representation for the analytic couplings

A(`)ν [Lnf ] =

int infin0

ρ(`)ν (σnf )

σ +Q2dσ =

int infinminusinfin

ρ(`)ν [Lσnf ]

1 + exp(Lminus Lσ)dLσ (25a)

A(`)ν [Lsnf ] =

int infins

ρ(`)ν (σnf )

σdσ =

int infinLs

ρ(`)ν [Lσnf ] dLσ (25b)

It is convenient to use the following representation for spectral functions

ρ(`)ν [Lnf ] =1

πIm(α(`)s [Lminus iπnf ]

)ν=

sin[ν ϕ(`)[Lnf ]]

π (βf R(`)[Lnf ])ν (26)

which is based on the module-phase representation of a complex number

α(`)s [Lminus iπnf ] =

a(`) [Lminus iπnf ]

βf (nf )=

eiϕ(`)[Lnf ]

βf (nf )R(`)[Lnf ] (27)

In the one-loop approximation the corresponding functions have the most simple form

ϕ(1)[L] = arccos

(Lradic

L2 + π2

) R(1)[L] =

radicL2 + π2 (28)

10

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

and do not depend on nf whereas at the two-loop order they have a more complicatedform

R(2)[Lnf ] = c1(nf )∣∣∣1 +W1 (zW [Lminus iπnf ])

∣∣∣ (29a)

ϕ(2)[Lnf ] = arccos

[Re

(minusR(2)[Lnf ]

1 +W1 (zW [Lminus iπnf ])

)](29b)

with W1[z] being the appropriate branch of Lambert function In the three-loop approx-imation we use either Eq (8) and then obtain

R(3)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(3)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(30a)

ϕ(3)[L] = arccos

[R(3)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(3)k

R(3)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](30b)

or Eq (11) mdash and then obtain

R(3P)[L] = c1

∣∣∣∣1minus c2c21

+W1

(z(3P)W [Lminus iπ]

) ∣∣∣∣ (31a)

ϕ(3P)[L] = arccos

Re

minusR(3P)[L]

1minus (c2c21) +W1

(z(3P)W [Lminus iπ]

) (31b)

In the four-loop approximation we use Eq (8) and then obtain

R(4)[L] =

∣∣∣∣∣ei ϕ(2)[L]

R(2)[L]+sumkge3

C(4)k

ei k ϕ(2)[L]

Rk(2)[L]

∣∣∣∣∣minus1

(32a)

ϕ(4)[L] = arccos

[R(4)[L] cos

(ϕ(2)[L]

)R(2)[L]

+sumkge3

C(4)k

R(4)[L] cos(k ϕ(2)[L]

)Rk

(2)[L]

](32b)

Here we do not show explicitly the nf dependence of the corresponding quantities mdash it goes

inside through R(2)[L] = R(2)[Lnf ] ϕ(2)[L] = ϕ(2)[Lnf ] C(3)k = C

(3)k [nf ] C

(4)k = C

(4)k [nf ]

ck = ck(nf ) with k = 1divide 3 and z(3P)W [L] = z

(3P)W [Lnf ] In the left panel of Fig 3 we show

both spectral densities in comparison On the right panel of this figure we show therelative deviation of the Pade spectral density from the standard one one can see that itvaries from +1 at L asymp minus7 reduces to minus2 at L asymp 0 and then reaches the maximumof +2 at L asymp 35

In accordance with Eq (21) the global spectral densities are constructed through thenf -fixed ones in the following manner

ρ(`)globν [LσΛ3] = ρ(`)ν [Lσ 3] θ (Lσ lt L4(Λ3)) + ρ(`)ν

[Lσ + λ

(`)6 (Λ3) 6

]θ (L6(Λ3) le Lσ)

+ ρ(`)ν

[Lσ + λ

(`)4 (Λ3) 4

]θ (L4(Λ3) le Lσ lt L5(Λ3))

+ ρ(`)ν

[Lσ + λ

(`)5 (Λ3) 5

]θ (L5(Λ3) le Lσ lt L6(Λ3)) (33)

11

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

-10 -5 0 5 10000

002

004

006

008

010

L

ρ(3)1 [L]

-10 -5 0 5 10

-002

-001

000

001

002

L

δ(3)1 [L]

Figure 3 Left panel Comparison of the standard three-loop spectral density ρ(3)1 [L] (solid blue line)

with the three-loop Pade one ρ(3P)1 [L] (dashed red line) Right panel Relative accuracy δ

(3)1 [L] =

(ρ(3P)1 [L]minus ρ(3)1 [L])ρ

(3)1 [L] of the three-loop Pade spectral density as compared with the standard three-

loop one

with Lσ equiv ln(σΛ23) and the corresponding global analytic couplings are

A(`)globν [LΛ3] =

int infinminusinfin

ρ(`)globν [LσΛ3]

1 + exp(Lminus Lσ)dLσ (34a)

A(`)globν [LΛ3] =

int infinL

ρ(`)globν [LσΛ3] dLσ (34b)

4 FAPT Procedures

In our package FAPTm we use the following realizations for the spectral densitiesRhoBar`[L nf ν] returns `-loop spectral density ρ

(`)ν (` = 1 2 3 3P 4) of fractional-power

ν at L = ln(Q2Λ2) and at fixed number of active quark flavors nf

RhoBar`[L k ν] = ρ(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (35a)

whereas RhoGlob`[L νΛ3] returns the global `-loop spectral density ρ(`)globν [L Λ3] (` =

1 2 3 3P 4) of fractional-power ν at L = ln(Q2Λ23) cf and with Λ3 being the QCD

nf = 3-scale

RhoGlob`[L νΛ3] = ρ(`)globν [L Λ3] (` = 1divide 4 3P) (35b)

Analogously AcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of

fractional-power ν coupling A(`)ν [Lnf ] in Euclidean domain

AcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36a)

and UcalBar`[L nf ν] returns `-loop (` = 1 2 3 3P 4) analytic image of fractional-power

ν coupling A(`)ν [L nf ] in Minkowski domain

UcalBar`[L k ν] = A(`)ν [Lnf = k] (` = 1divide 4 3P k = 3divide 6) (36b)

12

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

In global case AcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν

coupling A(`)globν [LΛ3] in Euclidean domain

AcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37a)

and UcalGlob`[L νΛ3] returns `-loop analytic image of fractional-power ν coupling

A(`)globν [LΛ3] in Minkowski domain

UcalGlob`[L νΛ3] = A(`)globν [LΛ3] (` = 1divide 4 3P) (37b)

We consider here an example of using this quantities in case of Mathematica 7 Weassume that the two-loop QCD scale Λ3 is fixed at the value Λ3 = 0387 GeV defined atthe end of section 21

In[1]= ltltFAPTm

In[2]= L23=0387

We determine the value of the two-loop QCD scale L23APT = Λ(2)APT

3 in APT corre-sponding to the same value 0119 as before but now for the global analytic coupling

In[3]= MZ = MZboson NumDefFAPT

Out[3]= 9119

In[4]= L23APT=lxFindRoot[AcalGlob2[Log[MZ^2lx^2]1lx]

== 0119lx035045]

Out[4]= 0379788

Now we evaluate the value of A(2)globν [LL23APT] for L = minus50 minus30 minus10 10 30 50

with indication of the needed time

In[5]= L0=-5 AcalGlob2[L01L23APT]Timing

Out[5]= 0734 -5 0929485

In[6]= L0=-3 AcalGlob2[L01L23APT]Timing

Out[6]= 0421 -30786904

In[7]= L0=-1 AcalGlob2[L01L23APT]Timing

Out[7]= 0422 -1060986

In[8]= L0=1 AcalGlob2[L01L23APT]Timing

Out[8]= 0437 10434041

In[9]= L0=3 AcalGlob2[L01L23APT]Timing

Out[9]= 0469 30301442

13

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

-2 0 2 4 6 8 1000

02

04

06

08

10

L

A(2)globν [L]

-2 0 2 4 6 8 1000

02

04

06

08

10

L

(2)globν

[L]

Figure 4 Left panel Graphics produced in Out[10] for A(2)globν [LL23APT] as a function of L Right

panel Graphics produced in Out[11] for A(2)globν [LL23APT] as a function of L

In[10]= L0=5 AcalGlob2[L01L23APT]Timing

Out[10]= 0531 50219137

Now we create a two-dimensional plot of A(2)globν [LL23APT] and A

(2)globν [LL23APT] for

L isin [minus3 11] with indication of the needed time

In[11]= Plot[AcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[11]= 19843 Graphics (see in the left panel of Fig4)

In[12]= Plot[UcalGlob2[L1L23APT]L-311MaxRecursion-gt1]Timing

Out[12]= 14656 Graphics (see in the right panel of Fig4)

5 Interpolation

The calculation of the spectral integrals is a computational task requiring a long timeespecially if one is using the result in another numerical integration procedure Thereforeit seems reasonable to pre-compute analytic images of couplings for a fixed set of argumentvalues consisting of N points for each argument For example we will consider in whatfollows the case of A(1)glob

ν [L νΛ(1)3 ] We will be interested in the following ranges of

arguments L isin [minus5 5] Λ(1)3 isin [02 05] and ν =isin [05 15] Then

Lmin = minus5 Lmax = 5 DL = (Lmaxminus Lmin)(N minus 1) νmin = 05 νmax = 15 Dν = (νmaxminus νmin)(N minus 1) Λmin = 02 Λmax = 05 DΛ = (Λmaxminus Λmin)(N minus 1)

(38)

The table of calculated values is generated by Mathematica using the following command

DATA = Flatten[Table[Li νjΛk AcalGlob1[Li νjΛk] i N jN kN] 2]

where Li = Lmin + (iminus 1)DL νj = νmin + (j minus 1)Dν and Λk = Λmin + (k minus 1)DΛ Thenwe save all calculated results in the file ldquoAcalGlob1idatrdquo

14

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

In[2] XY = N[DATA] outFile = OpenWrite[AcalGlob1idat]

Write[outFile XY] Close[outFile]

Out[2] OutputStream[AcalGlob1idat 15] Null AcalGlob1idat

After that we can read them and use interpolation to reproduce function A(1)globν [L νΛ

(1)3 ]

in the considered ranges of arguments values

In[3] DATA = Read[AcalGlob1idat]

AcalGlob1Interp = Interpolation[DATA]

in order to select the appropriate value of N Now we can analyze the accuracy of inter-polation In Fig 5 we show the dependencies of interpolation errors on the number ofthe used points N One can see that using the interpolation at N = 6 for A(`)glob

ν andA

(`)globν provided accuracy not worse 0005

In the previous case we investigated the dependence of the accuracy of interpolationon the number of points at fixed L ν and Λ3 Let us now consider how the accuracy of

4 6 8 10 12 14 16 18 20-001

000

001

002

003

004

005

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

Number of points4 6 8 10 12 14 16 18 20

-001

000

001

002

003

004

005

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

Number of points

Figure 5 Relative errors of the interpolation procedure for A(`)globν (left panel) and A

(`)globν (right

panel) calculated at various loop orders with fixed L = 35 ν = 11 and Λ3 = 036 GeV

0 1 2 3 4 5

00000

00025

00050

00075

00100

1glob

2glob

3glob

3Pglob

Rel

ativ

e er

ror

L0 1 2 3 4 5

00000

00025

00050

00075

00100

A1glob

A2glob

A3glob

A3Pglob

Rel

ativ

e er

ror

L

Figure 6 Relative error of the interpolation procedure for Aglobν=11 (left panel) and Aglobν=11 (right panel)calculated at various loop orders with Λ3 = 036 GeV for N = 11 number of points

15

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

the interpolation depends on the L These results are shown in Fig 6 From the lastfigure one can see that the maximum error of interpolation corresponds to the regionL = 0divide 5 The error in A(1)glob

ν=06 is less than in A(1)globν=11 In any case using N = 11 points

for interpolation of pre-computed data for each parameter L ν and Λ3 provides an errorless than 001

To obtain the results much faster one can use module FAPT Interpm which consists ofprocedures AcalGlob`i[L νΛ3] and UcalGlob`i[L νΛ3] They are based on interpolationusing the basis of the precalculated data in the ranges L = [minus5 13] ν1-loop = [05 40] andΛ1-loopnf=3 = [0150 0300] ν2-loop = [05 50] and Λ2-loop

nf=3 = [0300 0450] ν3-loop = [05 60]

and Λ3-loopnf=3 = [0300 0450] ν4-loop = [05 70] and Λ4-loop

nf=3 = [0300 0450] For examplein the four-loop case module FAPT Interpm contains procedures

AcalGlob4i = Interpolation[Read[sourcesAcalGlob4idat]]

UcalGlob4i = Interpolation[Read[sourcesUcalGlob4idat]]

which should be used with the same arguments L ν and Λ3 as the original proceduresAcalGlob`[L νΛ3] and UcalGlob`[L νΛ3] They provide much faster results of calcula-tions with high enough accuracy

In[1]= Timing[AcalGlob4i[1 11 036]]

Out[1]= 0 039298

In[2]= Timing[AcalGlob4[1 11 036]]

Out[2]= 0405 0392964

In[3]= Timing[UcalGlob4i[1 11 036]]

Out[3]= 0 0375421

In[4]= Timing[UcalGlob4[1 11 036]]

Out[4]= 0359 0375372

Acknowledgments

We would like to thank Andrei Kataev Sergey Mikhailov Irina Potapova DmitryShirkov and Nico Stefanis for stimulating discussions and useful remarks This workwas supported in part by the Russian Foundation for Fundamental Research (Grant No11-01-00182) and the BRFBRndashJINR Cooperation Program under contract No F10D-002

Appendix A Numerical parameters

Here we shortly describe numerical parameters used in the package

16

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

First in FAPTm we use the pole masses of heavy quarks and Z-boson collected in theset NumDefFAPT

MQ4 Mc = 165 GeV MQ5 Mb = 475 GeV MQ6 Mt = 1725 GeV MZboson MZ = 9119 GeV

(A1)

Note here that all mass variables and parameters are measured in GeVs That meansfor example that in all procedures of our package the following value MQ4 = 165 is usedThe package RunDec of [70] is using the set NumDef with slightly different values of theseparameters (Mc = 16 GeV Mb = 47 GeV Mt = 175 GeV MZ = 9118 GeV)

Second we collect in the set setbetaFAPT the following rules of substitutions bi rarrbi(nf ) cf Eq (3)

b0 b0 rarr 11minus 2

3nf b1 b1 rarr 102minus 38

3nf

b2 b2 rarr2857

2minus 5033

18nf +

325

54n2f (A2)

b3 b3 rarr149753

6minus 1078361

162nf +

50065

162n2f +

1093

729n3f

+

[3564minus 6508

27nf +

6472

81n2f

]ζ[3]

Here we follow the same substitution strategy as in [70] but our bi differ from theirs bCKSi

by factors 4i+1 bi = 4i+1 bCKSi In parallel the set setbetaFAPT4Pi defines substitutions

bi rarr bi(nf )(4π) which are more appropriate to determine coefficients ci(nf )

Appendix B Description of the main procedures

Here we shortly describe the main procedures of our package which can be useful forpractical calculations

bull RhoBar`[LNfNu]

general it computes the `-loop spectral density ρ(`)[Lσ nf ν]

input the logarithmic argument L=Lσ = ln[σΛ2] the number of active flavorsNf=nf and the power index Nu=ν

output ρ(`)

example In order to compute the value of the four-loop spectral densityρ(4)[395 4 162] = 00247209 one has to use the commandRhoBar4[395 4 162]

bull RhoGlob`[LNuLam]

general it computes the `-loop global spectral density ρ(`)glob[Lσ νΛnf=3]

17

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output ρ(`)glob

example In order to compute the value of the four-loop spectral densityρ(4)glob[395 162 0350] = 00221662 one has to use the commandRhoGlob4[395 162 035]

18

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

bull AcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Euclidean

domain

input the logarithmic argument L=ln[Q2Λ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA(3)

162[395 4] = 011352 one has to use the commandAcalBar3[395 4 162]

bull UcalBar`[LNfNu]

general it computes the `-loop nf -fixed analytic coupling A(`)ν [L nf ] in Minkowski

domain

input the logarithmic argument L=ln[sΛ2] the number of active flavors Nf=nf and the power index Nu=ν

output A(`)ν

example In order to compute the value of the three-loop spectral densityA

(3)162[395 4] = 01011 one has to use the command

UcalBar3[395 4 162]

bull AcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Euclidean domain

input the logarithmic argument L=Lσ = ln[σΛ2nf=3] the power index Nu=ν and

the QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

example In order to compute the value of the two-loop analytic couplingA(2)glob

162 [395 0350] = 0103858 one has to use the commandAcalGlob2[395 162 035]

bull UcalGlob`[LNuLam]

general it computes the `-loop global analytic coupling A(`)globν [L νΛnf=3] in

Minkowski domain

input the logarithmic argument L=ln[sΛ2nf=3] the power index Nu=ν and the

QCD scale parameter Lam=Λnf=3 (in GeV)

output A(`)globν

19

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

example In order to compute the value of the two-loop analytic couplingA

(2)glob162 [395 0350] = 00932096 one has to use the command

UcalGlob2[395 162 035]

All Λnf=3 are in GeV all squared momentum transfer Q2 (Euclidean) central-of-massenergy squared s (Minkowski) and spectral-integration variables σ are in GeV2 Thenumber of loops ` is everywhere specified in the name of a procedure

References

[1] D V Shirkov I L Solovtsov Analytic QCD running coupling with finite IR be-haviour and universal αs(0) value JINR Rapid Commun 2[76] (1996) 5ndash10 arXivhep-ph9604363

Analytic model for the QCD running coupling with universal αs(0) value Phys RevLett 79 (1997) 1209ndash1212 arXivhep-ph9704333

[2] K A Milton I L Solovtsov Analytic perturbation theory in QCD and Schwingerrsquosconnection between the beta function and the spectral density Phys Rev D55 (1997)5295ndash5298 arXivhep-ph9611438

[3] I L Solovtsov D V Shirkov Analytic approach to perturbative QCD and renor-malization scheme dependence Phys Lett B442 (1998) 344ndash348 arXivhep-ph

9711251

[4] A P Bakulev S V Mikhailov N G Stefanis QCD analytic perturbation theoryFrom integer powers to any power of the running coupling Phys Rev D72 (2005)074014 Erratum ibid D72 (2005) 119908(E) arXivhep-ph0506311

[5] A P Bakulev A I Karanikas N G Stefanis Analyticity properties of three-pointfunctions in QCD beyond leading order Phys Rev D72 (2005) 074015 arXiv

hep-ph0504275

[6] A P Bakulev S V Mikhailov N G Stefanis Fractional analytic perturbation theoryin Minkowski space and application to Higgs boson decay into a bb pair Phys RevD75 (2007) 056005 Erratum ibid D77 (2008) 079901(E) arXivhep-ph0607040

[7] L D Landau A Abrikosov L Halatnikov On the quantum theory of fields NuovoCim Suppl 3 (1956) 80ndash104

[8] D J Gross The discovery of asymptotic freedom and the emergence of QCD ProcNat Acad Sci 102 (2005) 9099ndash9108

[9] N N Bogolyubov D V Shirkov Introduction to the theory of quantized fieldsIntersci Monogr Phys Astron 3 (1959) 1ndash720

[10] N N Bogolyubov D V Shirkov Introduction to the Theory of Quantum FieldsWiley New York 1959 1980

20

[11] N N Bogolyubov A A Logunov I T Todorov Introduction to Axiomatic QuantumField Theory Benjamin Cummings Reading Massachusetts 1975

[12] N N Bogolyubov A A Logunov D V Shirkov The method of dispersion relationsand perturbation theory Soviet Physics JETP 10 (1960) 574

[13] A V Radyushkin Optimized lambda-parametrization for the QCD running couplingconstant in space-like and time-like regions JINR Rapid Commun 78 (1996) 96ndash99[JINR Preprint E2-82-159 26 Febr 1982] arXivhep-ph9907228

[14] N V Krasnikov A A Pivovarov The influence of the analytical continuation effectson the value of the QCD scale parameter Λ extracted from the data on charmoniumand upsilon hadron decays Phys Lett B116 (1982) 168ndash170

[15] H F Jones I L Solovtsov QCD running coupling constant in the timelike regionPhys Lett B349 (1995) 519ndash524 arXivhep-ph9501344

[16] M Beneke V M Braun Naive non-Abelianization and resummation of fermionbubble chains Phys Lett B348 (1995) 513ndash520 arXivhep-ph9411229

[17] P Ball M Beneke V M Braun Resummation of (β0αs)n corrections in QCD

Techniques and applications to the tau hadronic width and the heavy quark pole massNucl Phys B452 (1995) 563ndash625 arXivhep-ph9502300

[18] B A Magradze Analytic approach to perturbative QCD Int J Mod Phys A15(2000) 2715ndash2734 arXivhep-ph9911456

[19] D S Kourashev The QCD observables expansion over the scheme-independent two-loop coupling constant powers the scheme dependence reduction hep-ph9912410(1999) arXivhep-ph9912410

[20] B A Magradze QCD coupling up to third order in standard and analytic per-turbation theories Dubna preprint E2-2000-222 2000 [hep-ph0010070] arXiv

hep-ph0010070

[21] D S Kourashev B A Magradze Explicit expressions for Euclidean and Min-kowskian QCD observables in analytic perturbation theory Preprint RMI-2001-182001 [hep-ph0104142] (2001) arXivhep-ph0104142

[22] B A Magradze Practical techniques of analytic perturbation theory of QCD PreprintRMI-2003-55 2003 [hep-ph0305020] (2003) arXivhep-ph0305020

[23] D S Kourashev B A Magradze Explicit expressions for timelike and spacelike ob-servables of quantum chromodynamics in analytic perturbation theory Theor MathPhys 135 (2003) 531ndash540

[24] B A Magradze A novel series solution to the renormalization group equation inQCD Few Body Syst 40 (2006) 71ndash99 arXivhep-ph0512374

21

[25] K A Milton I L Solovtsov O P Solovtsova V I Yasnov Renormalization schemeand higher loop stability in hadronic tau decay within analytic perturbation theoryEur Phys J C14 (2000) 495ndash501 arXivhep-ph0003030

[26] K A Milton I L Solovtsov O P Solovtsova Remark on the perturbative componentof inclusive tau decay Phys Rev D65 (2002) 076009 arXivhep-ph0111197

[27] G Cvetic C Valenzuela Various versions of analytic QCD and skeleton-motivatedevaluation of observables Phys Rev D74 (2006) 114030 arXivhep-ph0608256

[28] G Cvetic R Kogerler C Valenzuela Analytic QCD coupling with no power termsin UV regime J Phys G37 (2010) 075001 arXiv09122466

[29] B A Magradze Testing the Concept of Quark-Hadron Duality with the ALEPH τDecay Data Few Body Syst 48 (2010) 143ndash169 arXiv10052674

[30] K A Milton I L Solovtsov O P Solovtsova The Bjorken sum rule in the analyticapproach to perturbative QCD Phys Lett B439 (1998) 421ndash427 arXivhep-ph

9809510

[31] R S Pasechnik D V Shirkov O V Teryaev Bjorken Sum Rule and pQCD frontieron the move Phys Rev D78 (2008) 071902 arXiv08080066

[32] K A Milton I L Solovtsov O P Solovtsova The GrossndashLlewellyn Smith sumrule in the analytic approach to perturbative QCD Phys Rev D60 (1999) 016001arXivhep-ph9809513

[33] D V Shirkov A V Zayakin Analytic perturbation theory for practitioners andUpsilon decay Phys Atom Nucl 70 (2007) 775ndash783 arXivhep-ph0512325

[34] N G Stefanis W Schroers H-C Kim Pion form factors with improved infraredfactorization Phys Lett B449 (1999) 299 arXivhep-ph9807298

[35] N G Stefanis W Schroers H-C Kim Analytic coupling and Sudakov effects inexclusive processes Pion and γlowastγ rarr π0 form factors Eur Phys J C18 (2000)137ndash156 arXivhep-ph0005218

[36] A P Bakulev K Passek-Kumericki W Schroers N G Stefanis Pion form factorin QCD From nonlocal condensates to NLO analytic perturbation theory Phys RevD70 (2004) 033014 arXivhep-ph0405062

[37] I L Solovtsov D V Shirkov The analytic approach in quantum chromodynamicsTheor Math Phys 120 (1999) 1220ndash1244 arXivhep-ph9909305

[38] D V Shirkov Analytic perturbation theory for QCD observables Theor Math Phys127 (2001) 409ndash423 arXivhep-ph0012283

22

[39] D V Shirkov I L Solovtsov Ten years of the analytic perturbation theory in QCDTheor Math Phys 150 (2007) 132ndash152 arXivhep-ph0611229

[40] L V Dung H D Phuoc O V Tarasov The influence of quark masses on theinfrared behavior of αs(Q

2) in QCD Sov J Nucl Phys 50 (1989) 1072ndash1079

[41] Y A Simonov Perturbative expansions in QCD and analytic properties of αs PhysAtom Nucl 65 (2002) 135ndash152 arXivhep-ph0109081

[42] A V Nesterenko J Papavassiliou The massive analytic invariant charge in QCDPhys Rev D71 (2005) 016009 arXivhep-ph0410406

[43] A P Bakulev D V Shirkov Inevitability and Importance of Non-PerturbativeElements in Quantum Field Theory in B Dragovich Z Rakic (Eds) Proceedingsof the 6th MATHEMATICAL PHYSICS MEETING Summer School and Conferenceon Modern Mathematical Physics September 14ndash23 2010 Belgrade Serbia Instituteof Physics Belgrade (Serbia) 2011 pp 27ndash53 arXivprotectvrulewidth0pt

protecthrefhttparxivorgabs11022380arXiv11022380

[44] D V Shirkov A Few Lessons from pQCD Analysis at Low Energies arXiv12023220[hep-ph] (2012) arXiv12023220

[45] A I Karanikas N G Stefanis Analyticity and power corrections in hard-scatteringhadronic functions Phys Lett B504 (2001) 225ndash234 Erratum ibid B636 (2006)330 arXivhep-ph0101031

[46] N G Stefanis Perturbative logarithms and power corrections in QCD hadronicfunctions A unifying approach Lect Notes Phys 616 (2003) 153ndash166 arXiv

hep-ph0203103

[47] A P Bakulev Global Fractional Analytic Perturbation Theory in QCD with SelectedApplications Phys Part Nucl 40 (2009) 715ndash756 arXiv08050829[hep-ph]

[48] G Cvetic A V Kotikov Analogs of noninteger powers in general analytic QCD JPhys G39 (2012) 065005 arXiv11064275

[49] D J Broadhurst A L Kataev C J Maxwell Renormalons and multiloop estimatesin scalar correlators Higgs decay and quark-mass sum rule Nucl Phys B592 (2001)247ndash293 arXivhep-ph0007152

[50] A P Bakulev S V Mikhailov N G Stefanis Higher-order QCD perturbation theoryin different schemes From FOPT to CIPT to FAPT JHEP 1006 (2010) 085 (1ndash38)arXiv10044125

[51] R S Pasechnik D V Shirkov O V Teryaev O P Solovtsova V L Khandra-mai Nucleon spin structure and pQCD frontier on the move Phys Rev D81 (2010)016010 arXiv09113297

23

[52] G Cvetic A Y Illarionov B A Kniehl A V Kotikov Small-x behavior of thestructure function F2 and its slope part ln(F2)part ln(1x) for rsquofrozenrsquo and analytic strong-coupling constants Phys Lett B679 (2009) 350ndash354 arXiv09061925

[53] A V Kotikov V G Krivokhizhin B G Shaikhatdenov Analytic and rsquofrozenrsquo QCDcoupling constants up to NNLO from DIS data Phys Atom Nucl 75 (2012) 507ndash524arXiv10080545

[54] A V Nesterenko C Simolo QCDMAPT program package for Analytic approach toQCD Comput Phys Commun 181 (2010) 1769ndash1775 arXiv10010901

[55] A Nesterenko C Simolo QCDMAPT F Fortran version of QCDMAPT packageComput Phys Commun 182 (2011) 2303ndash2304 arXiv11071045

[56] S Wolfram Mathematica mdash a system for doing mathematics by computer Addison-Wesley New York 1988

[57] D J Gross F Wilczek Ultraviolet behavior of nonabelian gauge theories Phys RevLett 30 (1973) 1343ndash1346

[58] D J Gross F Wilczek Asymptotically free gauge theories 1 Phys Rev D8 (1973)3633ndash3652

[59] H D Politzer Reliable perturbative results for strong interactions Phys Rev Lett30 (1973) 1346ndash1349

[60] D R T Jones Two-Loop Diagrams in YangndashMills Theory Nucl Phys B75 (1974)531 doi1010160550-3213(74)90093-5

[61] W E Caswell Asymptotic Behavior of Non-Abelian Gauge Theories to Two-LoopOrder Phys Rev Lett 33 (1974) 244 doi101103PhysRevLett33244

[62] E Egorian O V Tarasov Two-loop renormalization of the QCD in an arbitrarygauge Theor Math Phys 41 (1979) 863ndash 869

[63] O V Tarasov A A Vladimirov A Y Zharkov The Gell-MannndashLow Functionof QCD in the Three-Loop Approximation Phys Lett B93 (1980) 429ndash432 doi

1010160370-2693(80)90358-5

[64] S A Larin J A M Vermaseren The three-loop QCD beta function and anoma-lous dimensions Phys Lett B303 (1993) 334ndash336 arXivhep-ph9302208 doi

1010160370-2693(93)91441-O

[65] T van Ritbergen J A M Vermaseren S A Larin The four-loop beta functionin quantum chromodynamics Phys Lett B400 (1997) 379ndash384 arXivhep-ph

9701390

24

[66] M Czakon The Four-loop QCD beta-function and anomalous dimensions NuclPhys B710 (2005) 485ndash498 arXivhep-ph0411261

[67] B A Magradze The gluon propagator in analytic perturbation theory in F LBezrukov V A Matveev V A Rubakov A N Tavkhelidze S V Troitsky (Eds)Proceedings of the 10th International Seminar Quarksrsquo98 Suzdal Russia 18ndash24 May1998 INR RAS Moscow 1999 pp 158ndash171 arXivhep-ph9808247

[68] E Gardi G Grunberg M Karliner Can the QCD running coupling have a causalanalyticity structure JHEP 07 (1998) 007 arXivhep-ph9806462

[69] A V Garkusha A L Kataev The absence of QCD β-function factorization propertyof the generalized Crewther relation in the rsquot Hooft MS-based scheme Phys LettB705 (2011) 400ndash404 arXiv11085909

[70] K G Chetyrkin J H Kuhn M Steinhauser RunDec A Mathematica package forrunning and decoupling of the strong coupling and quark masses Comput Phys Com-mun 133 (2000) 43ndash65 arXivhep-ph0004189 doi101016S0010-4655(00)

00155-7

25