FAPT : a Mathematica package

Transcription

1 FAPT : a Mathematica package for calculations in QCD Fractional Analytic Perturbation Theory Viacheslav Khandramai ICAS, Gomel State Technical University, Belarus ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

2 Motivation Motivation Analytic Perturbation Theory (APT) [Shirkov, Solovtsov (1996,1997)] Fractional Analytic Perturbation Theory (FAPT) [Bakulev, Mikhailov, Stefanis ( )], [Bakulev, Karanikas, Stefanis (2007)] APT and FAPT: Closed theoretical scheme without singularities and additional parameters; RG-invariance, Q 2 -analyticity; Power PT set in MS-scheme {ᾱ k s (Q 2 )} a non-power APT expansion set Ā k (Q 2 ) in Euclidian domain and Ā k (s) in Minkowskian domain with both Ā k (Q 2 ) and Ā k (s) regular in the IR region. ᾱs k Ā k, Ā k dk ᾱs k d k Ā k, dk Ā k d k numbers Functions Ā k (Q 2 ) and Ā k (s) cannot be trivial calculated in higher orders. The main our goal is to simplify the calculations in framework of APT&FAPT. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

3 Introduction Introduction This talk is based on recent publication A.P. Bakulev and V.L. Khandramai Comp. Phys. Comm. 184 (2013) All relevant formulas which are necessary for the running of Ā ν[l], L = ln(q 2 /Λ 2 ) and Ā ν[l s], L s = ln(s/λ 2 ) in Minkowskian domain in framework of FAPT are collected. Outline: We provide here easy-to-use Mathematica system procedures collected in the package FAPT Our package is organized as well-known package RunDec [Chetyrkin, Kühn, Steinhauser (2000)] This task has been partially realized for APT as the Maple package QCDMAPT and as the Fortran package QCDMAPT_F [Nesterenko, Simolo (2010)] 1 Theoretical framework: from standard PT to Analytic Perturbation Theory and its generalization Fractional APT; 2 APT/FAPT Applications: Bjorken sum rule and RG-evolution; 3 Package FAPT : description of procedures and examples of usage. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

4 Theoretical Framework Standrad PT Theoretical Framework ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

5 Theoretical Framework Standrad PT Running coupling The QCD running of the coupling constant α s(µ 2 ) = (4π/b 0) a s[l] is defined through RG equation ( ) d a s[l] = as 2 c 1 as 3 c 2 as 4 c 1 as 3 µ 2..., L = ln, c d L Λ 2 k (n f ) b k(n f ) b 0(n f ). k+1 The solutions of RG equation in MS-scheme can be obtained exactly at LO and NLO a (1) s [L] = 1 L ; (LO) a s (2) c 1 1 [L; n f ] = (n f ) 1 + W 1 (z W [L]) with z W [L] = c 1 1 (n f ) e 1 L/c 1(n f ). (NLO) The higher-order solutions a s (l) [L; n f ] can be expanded in powers of the two-loop one, a s (2) [L; n f ], it has been suggested in [Kourashev, Magradze, ( )]: a s (l) [L; n f ] = ( n C n (l) a s (2) [L; n f ]). n 1 ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

6 Theoretical Framework Standrad PT Heavy quark mass thresholds For realization of so-called global scheme it needs to match the running coupling values in Euclidian domain at Q corresponding to quark masses: α (l) s [L 4(Λ 3); 3] = α (l) s [L 4(Λ 3) + λ 4; 4] ; α (l) s [L 5(Λ 3) + λ 4; 4] = α (l) s [L 5(Λ 3) + λ 5; 5] ; α (l) s [L 6(Λ 3) + λ 5; 5] = α (l) s [L 6(Λ 3) + λ 6; 6] ; where L k (Λ 3) ln ( M 2 k /Λ 2 3) at the thresholds Mk (M 4 = m c, M 5 = m b and M 6 = m t). We use all logarithms L with depending of three-flavor scale QCD Λ 3: αs glob;(l) (Q 2, Λ 3) = α s (l) [ L(Q 2 ); 3 ] θ ( Q 2 <M4 2 ) [ ] + α s (l) L(Q 2 )+λ (l) 4 (Λ3); 4 θ ( M4 2 Q 2 <M5 2 ) + α (l) s + α (l) s [ L(Q 2 )+λ (l) 5 (Λ3); 5 ] θ ( M 2 5 Q 2 <M 2 6 [ L(Q 2 )+λ (l) 6 (Λ3); 6 ] θ ( M 2 6 Q 2), and recalculate α (l) s to all other scales with the help of finite additions λ k ln ( Λ 2 3/Λ 2 k). ) ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

7 Theoretical Framework Standrad PT Why we need APT and FAPT? For standard QCD PT in Euclidian domain the expressions for hadronic quantities F ( Q 2 = µ 2, ā s ) are based on expansions in a series over the powers of running coupling F [L] = 1 + f 1 ā s[l] + f 2 ā 2 s [L] + f 3 ā 3 s [L] + f 4 ā 4 s [L] Hadronic quantities calculated in terms of a power-series are not everywhere well defined because of unphysical coupling singularities: ā s (1) [L] = 1/L, ā s (2) [L] 1/ L + c 1lnc 1,... In standard QCD PT we have not only power series but also: RG-improvement to account for higher-orders { } γ(a) Z[L] = exp β(a) da 1-loop [ā s[l]] γ 0/(2β 0 ) ; Factorization (ā s[l]) n L m ; Two-loop case (ā s) ν ln(ā s). ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

8 Theoretical Framework Standrad PT Basics of APT The analytic images of the strong coupling powers: ā n Euclidian Ā (l) n [L; n f ]= 0 are defined through the spectral density Leading order (integer powers): ρ ν (l) (σ; n f ) dσ, ā n Minkowskian Ā (l) ρ (l) n (σ; n f ) σ + Q 2 n [L s; n f ]= dσ s σ ρ (l) n [L; n f ] = 1 π Im ( ᾱ (l) s [L iπ; n f ]) n. ρ (1) 1 (σ) = 4 1 Im b 0 L σ iπ = 4π 1 b 0 L 2 σ + π. 2 A (1) 1 [Shirkov, Solovtsov (1996, 1997)] and A (1) 1 [Jones, Solovtsov (1995); Jones, Solovtsov, Solovtsova (1995); Milton, Solovtsov (1996)]: Ā (1) 4π 1 [L] = b 0 ( 1 L 1 Ā (1) 1 [Ls] = 4 b 0 arccos ), L = ln ( Q 2 /Λ 2) ; e L 1 ( ) L s, L s = ln ( s/λ 2). L 2 s + π 2 ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

9 Theoretical Framework Standrad PT APT Running Coupling All APT couplings have the following properties: Universal finite IR values: Ā(0) = Ā(0) = 4π/b 0; Week dependence on the number of loops; Correspondence with perturbative ᾱ s(q 2 ) at Q 2 1 GeV l 1 [0]~1.4 2,3,4 1 [Q 2 ] [ ] Q 2, GeV ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

10 Theoretical Framework FAPT FAPT: one-loop Euclidian Ā ν [L] Euclidian coupling: Ā (1) ν [L] = 4π b 0 F (z, ν) is Lerch transcendent function. [ 1 L F ] (e L, 1 ν), L = ln ν Γ(ν) ( Q 2 Λ 2 ). Properties: Ā 0[L] = 1; Ā m[l] = L m for m N; Ā m[l] = ( 1) m Ā m[ L] for m 2, m N; Ā m[± ] = 0 for m 2, m N Ν [L] Ν=2 Ν=2.25 Ν=2.5 Ν=3 Ν= L ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

11 Theoretical Framework FAPT FAPT: one-loop Minkowskian Ā ν [L] Minkowskian coupling: Ā (1) ν [L] = 4 sin [ (ν 1)arccos ( L/ π 2 + L 2)] ( s ), L b 0 (ν 1) (π 2 + L 2 ) (ν 1)/2 s = ln. Λ 2 Here we need the elementary functions only. Properties: Ā 0[L] = 1; Ā 1[L] = L; Ā 2[L] = L 2 π2, Ā 3 [L] = 3 L ( L 2 π 2),... ; Ā m[l] = ( 1) m Ā m[ L] for m 2, m N; Ā m[± ] = 0 for m 2, m N Ν [L] Ν=2.25 Ν=2 Ν=2.5 Ν=2.75 Ν= L ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

12 APT/FAPT Applications APT/FAPT Applications ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

13 APT/FAPT Applications Non-power APT expansions The polarized Bjorken Sum Rule is given by a sum of two series in powers of α s and OPE higher twists corrections µ p n 2i : Γ p n 1 (Q 2 ) = g A 6 CBj + µ p n 2i Q, 2i 2 i=2 C Bj(Q 2 ) 1 Bj(Q 2 ), g A = ± Perturbative power-correction: [Baikov, Chetyrkin, Kühn (2010)] PT Bj (Q 2 ) = ᾱ s(q 2 ) ᾱ 2 s(q 2 ) ᾱ 3 s(q 2 ) ᾱ 4 s(q 2 ) +... Instead of universal power-in-α s expansion in APT one should use non-power functional expansions: APT Bj (Q 2 ) = 0.318Ā 1(Q 2 ) Ā 2(Q 2 ) Ā 3(Q 2 ) Ā 4(Q 2 ) +... ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

14 APT/FAPT Applications Loop stabilization Loop stabilization The Γ p n 1 (Q 2 ) (with HT=0) in both the standard PT and APT approaches with the combined set of the Jefferson Lab and SLAC data. p-n APT PT NLO PT N 2 LO PT N 3 LO Q 2 (GeV 2 ) The behavior of the PT curves is changed order by order. The APT curves in all three orders practically coincide with each other. The deviation of the APT curves from the data shows for necessity of the higher-twist contribution. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

15 APT/FAPT Applications Convergence Convergence The relative contributions of separate terms in PT and APT expansions for Bj(Q 2 ), N i (Q 2 ) = δ i (Q 2 )/ Bj(Q 2 ), as a functions of Q 2 from [Khandramai et. al (2012)] 342 V.L. Khandramai etv.l. al. / Khandramai Physics Letters al. B 706 / Physics (2012) Letters B 706 (2012) dependence Fig. 2. Theof Qthe 2 -dependence relative contributions of the relative the contributions four-loop level at the in four-loop the Fig. level 3. inthe theq 2 -dependence Fig. 3. Theof Q the 2 -dependence relative contributions of the relative of the contributions perturbative ofexpan- 2GeV terms 2,soin Eq. sion (5) terms in the inapt Eq. approach. (5) in Third APT and approach. fourth Third order and contributions fourth order the pertu ur-loop PT approach. PT order Four-loop overshoots PT the order three-loop overshoots onethe atthree-loop Q 2 2GeV one 2,so at Q 2 sion Asymptotic structure manifestation: in the region rove it does the accuracy not improve of the theptaccuracy prediction of the compared PT prediction to the compared three-loopto the three-loop amount Q < less1 than GeV 5% total, theso dominant amount to less than 5% total, so the NLO APT approximation the NLO APT is approximation sufficient for description of the low scription energyof JLab thedata low at energy the current JLab data level at of theexperimental current levelaccuracy. of is su one. experime contribution to the PT correction comes from the αs-term; 4 its relative contribution increases with decreasing Q. hesame deviation time, of theapt deviation curve from of APT thecurve data from clearly theshows data clearly shows of forthe necessity HT contribution of the HTwhich contribution is also quite whichstable is also[4]. quite stable [4]. Good loop convergence in APT: the higher order contributions are stable at all Q 2 tion This may situation be considered may be as considered a hint of the as transition a hint of the of transition of the PTasymptotic series values; the regime asymptotic the(while 3 rd regime and APT series 4(while th terms remains APT series contribute con- remains con GeV for 2 Q. 2 We 0.7 explore GeV 2. this Wepossibility explore this in possibility more in more less than 5% and 1% respectively. Qvergent) detail. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

16 APT/FAPT Applications The RG evolution of the higher-twist µ p n 4 (Q 2 ) The RG evolution of the higher-twist µ p n 4 (Q 2 ) See [Pasechnik et al. (PRD,2010)] The expression for the evolution of the higher twist µ p n 4 in the Bjorken sum rule is known: [ ] αs(q µ p n 4,PT (Q2 ) = µ p n 2 ν ) 4,PT (Q2 0 ) α s(q0 2), ν = γ0, γ 0 = 16 8πβ 0 3 C F, C F = 4 3. For the RG-evolution of the µ p n 4 in the Fractional APT we need change the fractional powers of the QCD running coupling constant by the corresponding analytic images: µ p n 4,APT (Q2 ) = µ p n 4,APT (Q2 0 ) A(1) ν (Q 2 ) A ν (1) (Q0 2). Note, at the same time the evolution of the higher twist µ p n 6, µ p n 8,... is still unknown. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

17 APT/FAPT Applications The RG evolution of the higher-twist µ p n 4 (Q 2 ) The RG evolution of the higher-twist µ p n 4 (Q 2 ) The Q 2 -dependence of evolution factors in the standard PT and Fractional APT Μ 4 p n Q 2 Μ 4 p n Q APT PT Q 2, GeV The influence of non-physical singularities complicate the analysis of the higher-twist evolution in the low-energy region of the standard PT. The evolution from 1 GeV 2 to Λ 2 QCD in the APT increases the absolute value of µ p n 4 (normalized at Q 2 0 = 1 GeV 2 ) by about 20 %. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

18 APT/FAPT Applications The RG evolution of the higher-twist µ p n 4 (Q 2 ) The RG evolution of the higher-twist µ p n 4 (Q 2 ) Using the above-mentioned total APT expressions for Bjorken sum rule fitted to experimental data we extract the coefficients of the higher twist µ 4 OPE corrections without and with their RG-evolution. Method Q 2 min GeV 2 µ 4/M 2 µ 6/M 4 µ 8/M (3) - - NNLO APT (4) 0.008(2) - no evolution (4) 0.010(3) (3) (3) - - NNLO APT (4) (4) - with evolution (4) (6) (8) Note, we do not study account RG-evolution for the PT calculations since the only effect of that would be the enhancement of the Landau singularities by extra divergences at Λ QCD, whereas at higher Q 2 the evolution is negligible. Extracted values of µ p n 4 in APT become more stable with respect to Q min variations. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

19 Higher twist analysis p-n APT/FAPT Applications Q 2 (GeV 2 ) Higher twist analysis PT NLO PT N 2 LO PT N 3 LO APT Q 2 (GeV 2 ) APT 0.08 Figure 5: The one-parametric µ 4-fits of the BSR JLab data in various (NLO, N 2 LO, N 3 LO) orders of the PT and the all-order APT spondingly, the valu expansions. In the PT case, the four-loop result does not improve out to be considerab the data description compared to the three-loop one. In the APT The three-parametric case, (µ the 4 + NLO µ 6 approximation + µ 8)-fits of is sufficient the Bjorken due to higher-loop SR datastability in various in orders the PTof approach the PT and the all-orderof APT. the APT expansion (see also Fig. 3). 4 /M NLO N 2 LO N 3 LO Figure 7: Value of the h Figure 6: The three-parametric µ 4,6,8-fits of the BSR JLab data in JLab data using the PT The APT application various leads (NLO, ton 2 accurate LO, N 3 LO) orders dataof description the PT and the all-order down APT to Q 2 with 0.1 errorgev bands. 2 Ver expansions. tainty ranges in Λ-para always at the two-loop APT level. and N 3 LO approximati other, so the four-loop r variations compared to 3.2. Sensitivity of the higher twists to Λ QCD variations ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

20 FAPT package Package FAPT ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

21 FAPT package FAPT package review FAPT package review Title of program: FAPT Available from: bakulev/fapt.mat/fapt.m bakulev/fapt.mat/fapt_interp.m Computer: Any work-station or PC where Mathematica is running. Operating system or monitor under which the program has been tested: Mathematica (versions 5,7,8). FAPT package contains the calculations of the required objects: 1 ᾱ s (l) [L, n f ], ᾱ s (l);glob 2 ρ (l) [L σ, n f, ν], ρ (l);glob [L σ, ν, Λ nf =3] 3 Ā (l) ν [L, n f ], A (l);glob ν [L, ν, Λ nf =3] 4 Ā (l) ν [L, n f ], A (l);glob ν [L, ν, Λ nf =3] ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

22 FAPT package Numerical parameters Numerical parameters The pole masses of heavy quarks and Z-boson, collected in the set NumDefFAPT (all mass variables and parameters are measured in GeVs): MQ4 : M c = 1.65 GeV, MQ5 : M b = 4.75 GeV ; MQ6 : M t = GeV, MZboson : M Z = GeV. *The package RunDec is using the set NumDef with slightly different values of these parameters (M c = 1.6 GeV, M b = 4.7 GeV, M t = 175 GeV, M Z = GeV). Collection in the set setbetafapt the following rules of substitutions b i b i (n f ) b0 : b n f, b1, b2, b3. *Here we follow the same substitution strategy as in RunDec, but our b i differ from bi RunDec by factors 4 i+1 : b i = 4 i+1 bi RunDec. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

23 FAPT package FAPT package: αs α s calculations \[CapitalLambda]l[Λ 3, n f ] returns l-loop QCD scales with n f matching conditions and three-flavors QCD scale Λ 3: active quarks by using \[CapitalLambda]l[Λ, k] = Λl[Λ 3, n f = k] = Λ (l) k (Λ3), (l = 1 4 ; k = 2, 4 6), \[Alpha]Barl[Q 2, n f, Λ] returns l-loop running QCD couplings at scale Q 2 with fixed n f and QCD scale Λ: \[Alpha]Barl[Q 2, n f, Λ] = αbarl[q 2, n f, Λ] = ᾱ (l) s [ln(q 2 /Λ 2 ); n f ], (l = 1 4) \[Alpha]Globl[Q 2, Λ 3] returns global l-loop QCD coupling at scale Q 2 and QCD scale Λ 3, corresponds to n f = 3: \[Alpha]Globl[Q 2, Λ 3] = αglobl[q 2, Λ 3] = α glob;(l) s (Q 2, Λ 3), (l = 1 4) ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

24 FAPT package Example 1 (for Standard PT) Example 1 (for Standard PT) We assume that the three-loop QCD scale is fixed at the value Λ 3 = GeV. We want to evaluate the corresponding values of αs glob;(l) (Q 2, Λ) at the scale Q = 4 GeV. I n [ 1 ] : = S e t D i r e c t o r y [ N o t e b o o k D i r e c t o r y [ ] ] ; << FAPT.m << RunDEC.m I n [ 2 ] : = L33 =0.387; I n [ 3 ] : = Q1=4; I n [ 4 ] : = A=\[ Alpha ] Glob3 [Mb^2, L33 ] Out [4]= TEST w i t h RunDEC WE USE OBTAINED RESULT AS INITIAL VALUE FOR AsRunDec I n [ 5 ] : = Q2=10; I n [ 6 ] : = { \ [ Alpha ] Glob3 [ Q2^2, L33 ], AsRunDec [ A, Q1, Q2, 3 ] } Out [6]= { , } >>> ERROR LESS THEN 0.5% DUE TO SMALL DIFFERENCE IN POLE QUARK MASSES ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

25 FAPT package ρν calculations ρ ν calculations RhoBarl[L, n f, ν] returns l-loop spectral density ρ (l) ν of fractional-power ν at L = ln(q 2 /Λ 2 ) and at fixed number of active quark flavors n f : RhoBarl[L, k, ν] = ρ (l) ν [L; n f = k], (l = 1 4) RhoGlobl[L, ν, Λ 3] returns the global l-loop spectral density ρ ν (l);glob [L; Λ 3] of fractional-power ν at L = ln(q 2 /Λ 2 3) and with Λ 3 corresponds to n f = 3: RhoGlobl[L, ν, Λ 3] = ρ (l);glob ν [L; Λ 3], (l = 1 4) I n [ 7 ] : = Show [ P l o t [ RhoGlob3 [ L, 1, ], {L, 14, 10}, P l o t S t y l e > { Thick, Red } ], P l o t [ RhoBar3 [ L, 3, 1 ], {L, 14, 10}, P l o t S t y l e > { Thick, Dashed, Blue } ] ] Out [7]= s e e i n F i g. r i g h t Ρ 1 glob; 1 [L] Ρ 1.5 glob; 1 [L] ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31 L

26 FAPT package Aν and Aν calculations Ā ν and Ā ν calculations AcalBarl[L, n f, ν] returns l-loop analytic image of fractional-power ν coupling Ā (l) ν [L; n f ] in Euclidian domain, AcalBarl[L, k, ν] = Ā (l) ν [L; n f = k], (l = 1 4) AcalGlobl[L, ν, Λ 3] returns l-loop analytic image of fractional-power ν coupling [L, Λ 3] in Euclidean domain A (l);glob ν AcalGlobl[L, ν, Λ 3] = A (l);glob ν [L, Λ 3], (l = 1 4) UcalBarl[L, n f, ν] returns l-loop analytic image of fractional-power ν coupling Ā (l) ν [L, n f ] in Minkowskian domain UcalBarl[L, k, ν] = Ā (l) ν [L; n f = k], (l = 1 4) UcalGlobl[L, ν, Λ 3] returns l-loop analytic image of fractional-power ν coupling [L, Λ 3] in Minkowskian domain A (l);glob ν UcalGlobl[L, ν, Λ 3] = A (l);glob ν [L, Λ 3], (l = 1 4) ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

27 FAPT package Example 2 (for FAPT) Example 2 (for FAPT) Construction of a two plot of A (2);glob ν [L, L23APT] and A (2);glob ν [L, L23APT] for L [ 3, 11] with indication of the needed time (in seconds): I n [ 8 ] : = P l o t [ AcalGlob2 [ L, 1, L23APT ], { L, 3,11}]// Timing Out [8]= { , G r a p h i c s ( s e e i n the l e f t p a n e l o f F i g. below )} I n [ 9 ] : = P l o t [ UcalGlob2 [ L, 1, L23APT ], { L, 3,11}]// Timing Out [9]= { , G r a p h i c s ( s e e i n the r i g h t p a n e l o f F i g. below )} A (2);glob ν [L] (2);glob ν [L] L L ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

28 FAPT package Interpolation Interpolation To obtain the fast results one can use the package FAPT_Interp which consists of procedures AcalGlobli[L, ν, Λ 3] and UcalGlobli[L, ν, Λ 3], which are based on interpolation using the basis of the precalculated data. Relative error of the interpolation procedure for A glob ν=1.1 and Aglob ν=1.1. 0,0100 0,0100 Relative error, % 0,0075 0,0050 0, glob 2 glob 3 glob 3P glob Relative error, % 0,0075 0,0050 0,0025 A 1 glob A 2 glob A 3 glob A 3P glob 0,0000 0, L L The relative error for A (1);glob ν is less than for A (1);glob ν. In any case, using the same pre-computed data provides an error less than 0.01 %. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

29 Summary Summary APT provides natural way for coupling constant and related quantities with Universal (loop independent) IR limit; Weak dependence on the number loops. Fractional APT provides effective tool to apply Analytic approach for RG improved perturbative amplitudes. This approaches are used in several applications: Higgs boson decay [Bakulev, Mikhailov, Stefanis (2007)]; calculation of binding energies and masses of quarkonia [Ayala, Cvetič (2013)]; analysis of the structure function F 2(x) behavior at small values of x [Kotikov, Krivokhizhin, Shaikhatdenov (2012)]; resummation approach [Bakulev, Potapova (2011)]. We collected in FAPT package all the procedures in APT and FAPT which are needed to compute the analytic images of the ᾱ s powers up to N 3 LO based on the system Mathematica. ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

30 In memory of Alexander Bakulev In memory of Alexander Bakulev Professor Alexander Bakulev Leading Researcher, Bogolyubov Lab, JINR, Dubna About 70 papers (average citations per paper = 20), including popular papers on Fractional APT: A. Bakulev, S. Mikhailov, N. Stefanis, Phys.Rev. D72 (2005) [CI=60] A. Bakulev, S. Mikhailov, N. Stefanis, Phys.Rev. D75 (2007) [CI=47] A. Bakulev, Phys.Part.Nucl. 40 (2009) [CI=30] ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31

31 In memory of Alexander Bakulev Thanks for your attention! And I thank especially organizers of ACAT2013 for happy possibility to take part in conference! ICAS (Gomel State Tech. University) V. Khandramai May 18, / 31