FAPT : A Mathematica package for QCD calculations

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1 FAPT : A Mathematica package for QCD calculations Viacheslav Khandramai Gomel State Technical University Belarus Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 1 / 40

In memory of Alexander Bakulev In memory of Alexander Bakulev This talk based on recent publication A.P. Bakulev and V.L. Khandramai Comp. Phys. Comm. 184, Iss. 1 (2013) 183-193.

2 In memory of Alexander Bakulev In memory of Alexander Bakulev This talk based on recent publication A.P. Bakulev and V.L. Khandramai Comp. Phys. Comm. 184, Iss. 1 (2013) Plan of talk: 1 Theoretical framework: from standard PT to Analytic Perturbative Theory and its generalization Fractional APT; 2 : DIS SR Analysis; Renorm-group Q 2 -evolution; Work on FAPT package in time of AB visit to Gomel (October, 2011) Adler D-function; 3 Package FAPT : description of procedures and examples of usage. Viacheslav Khandramai (Gomel State Technical FAPT University : A Mathematica Belarus) package for QCD calculations 2 / 40

3 Motivaton Motivation Analytic Perturbative Theory, APT, [Shirkov, Solovtsov (1996,1997)] Fractional Analytic Perturbative Theory, (F)APT, [Bakulev, Mikhailov, Stefanis ( )], [Bakulev, Karanikas, Stefanis (2007)]: Analytic PT: Closed theoretical scheme without Landau singularities and additional parameters; RG-invariance, Q 2 -analyticity; Power PT set {ᾱ k s (Q 2 )} a non-power APT expansion set {A k (Q 2 ), A k (s)} with all A k (Q 2 ), A k (s) regular in the IR region. dk α k s d k A k Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 3 / 40

4 Introduction Introduction The main goal is to simplify calculations in the framework of APT&(F)APT. For this purpose we collect 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 the framework of APT and (F)APT. Note, We provide here easy-to-use Mathematica system procedures collected in the package FAPT organized as package RunDec [Chetyrkin, Kühn, Steinhauser (2000)] This task has been partially realized for both APT and its massive generalization [Nesterenko, Papavassiliou (2005)] as the Maple package QCDMAPT and as the Fortran package QCDMAPT_F [Nesterenko, Simolo (2010)]. iacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 4 / 40

5 Theoretical Framework Standrad PT Theoretical Framework Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 5 / 40

6 Theoretical Framework Standrad PT Running coupling Running coupling α s(µ 2 ) = (4π/b 0) a s[l] with L = ln(µ 2 /Λ 2 ) obtained from RG equation d a s[l] d L = a 2 s c 1 a 3 s c 2 a 4 s c 1 a 3 s..., c k (n f ) b k(n f ) b 0(n f ) k+1, Exact solutions of RGE known only at LO and NLO a (1) [L] = 1 L (LO) a (2) [L; n f ] = c 1 1 (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-loop solutions a (l) [L; n f ] can be expanded in powers of the two-loop one, a (2) [L; n f ], as has been suggested in [Kourashev, Magradze, ( )]: a (l) [L; n f ] = n 1 C (l) n ( a(2) [L; n f ] )n. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 6 / 40

7 Theoretical Framework Standrad PT Heavy quark mass thresholds s (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 glob;(l) + 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) ) Α S 4 Q Α S global; 4 Q n f n f 4 n f Q 2, GeV Figure: Graphical comparison: Fixed-n f α s (4) [Q 2, n f ] Global α glob;(4) s [Q 2 ]. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 7 / 40

8 Theoretical Framework Standrad PT Problems Coupling singularities LO solution generates Landau pole singularity: a s[l] = 1/L NLO solution generates square-root singularity: a s[l] 1/ L + c 1 lnc 1 PT power-series expansion of D(Q 2, µ 2 = Q 2 ) D in the running coupling: D[L] = 1 + d 1a s[l] + d 2a 2 s [L] + d 3a 3 s [L] + d 4a 4 s [L] +..., are not everywhere well defined RG evolution: B(Q 2 ) = [ Z(Q 2 )/Z(µ 2 ) ] B(µ 2 ) reduces in 1-loop approximation to Z a ν [L] ν fractional ν=ν0 γ 0 /(2b 0 ), Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 8 / 40

9 Theoretical Framework Standrad PT Basics of APT The analytic images of the strong coupling powers: Ā n (l) [L; n f ]= 0 define through spectral density ρ ν (l) (σ; n f ) dσ, Ā (l) ρ (l) n (σ; n f ) σ + Q 2 n [L s; n f ]= dσ s σ ρ n (l) [L; n f ] = 1 ( ) n π Im α s (l) sin[n ϕ (l) [L; n f ]] [L iπ; n f ] = π (β f R (l) [L; n f ]). n One-loop: 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)] и 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 Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 9 / 40

10 Theoretical Framework Standrad PT IR-behavior In the IR-region Universal finite IR values: Ā(0) = Ā(0) = 4π/b 0 1.4; Loop stabilization at two-loop level. This yields practical weak loop dependence of Ā(Q 2 ), Ā(s), and higher expansion functions: l 1 [0]~1.4 2,3,4 1 [Q 2 ] [ ] Q 2, GeV Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 10 / 40

11 Theoretical Framework (F)APT Why we need (F)APT? In standard QCD PT we have not only power series F [L] = m f m a m s [L], but also: RG-improvement to account for higher-orders { } as [L] γ(a) Z[L] = exp β(a) da 1-loop [a s[l]] γ 0/(2β 0 ) Factorization (a s[l]) n L m Two-loop case (a s) ν ln(a s) New functions: (a s) ν ( done in FAPT package) (a s) ν ln(a s), (a s) ν L m, (in preparation) Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 11 / 40

12 Theoretical Framework (F)APT (F)APT: one-loop Euclidian Ā ν [L] Euclidean coupling (L = ln(q 2 /Λ 2 )): Ā ν[l] = 4π b 0 ( 1 L F ) (e L, 1 ν) ν Γ(ν) Here F (z, ν) is reduced Lerch transcendent function (analytic function in ν). 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; iacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 12 / 40

13 Theoretical Framework (F)APT (F)APT: one-loop Euclidian Ā ν [L] Ν [L] Ν=2 Ν=2.25 Ν=2.5 Ν= Ν= L Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 13 / 40

14 Theoretical Framework (F)APT (F)APT: one-loop Minkowskian Ā ν [L] Minkowskian coupling (L = ln(s/λ 2 )): Here we need only elementary functions. Properties: Ā 0[L] = 1; Ā 1[L] = L; Ā ν[l] = 4 b 0 sin [ (ν 1)arccos ( L/ π 2 + L 2)] (ν 1) (π 2 + L 2 ) (ν 1)/2 Ā 2[L] = L 2 π2 3, Ā 3 [L] = 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 iacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 14 / 40

15 Theoretical Framework (F)APT (F)APT: one-loop Minkowskian Ā ν [L] Ν [L] Ν=2 Ν= Ν=2.25 Ν= Ν= L Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 15 / 40

16 Theoretical Framework (F)APT Non-power APT expansions Instead of universal power-in-α s expansion in APT one should use non-power functional expansions. In Euclidian space Adler D-function D PT (Q 2 ) = d 0 + d 1 α s(q 2 ) + d 2 α 2 s(q 2 ) + d 3 α 3 s(q 2 ) + d 4 α 4 s(q 2 ) D APT (Q 2 ) = d 0 + d 1 Ā 1(Q 2 ) + d 2 Ā 2(Q 2 ) + d 3 Ā 3(Q 2 ) + d 4 Ā 4(Q 2 ) In Minkowskian space R-ratio R PT (s) = r 0 + r 1 α s(s) + r 2 α 2 s(s) + r 3 α 3 s(s) + r 4 α 4 s(s) R APT (s) = d 0 + d 1 Ā 1(s) + d 2 Ā 2(s) + d 3 Ā 3(s) + d 4 Ā 4(s) Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 16 / 40

17 Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 17 / 40

18 Loop stabilization Loop stabilization Perturbative power-correction of the polarized Bjorken Sum Rule (see [Khandramai et. al (PLB, 2012)]) Γ p n 1 (Q 2 ) = g A 6 CBj, CBj(Q2 ) 1 PT Bj (Q 2 ), g A = ± p-n APT PT NLO PT N 2 LO PT N 3 LO Q 2 (GeV 2 ) Loop stabilization of IR behavior at two-loop level Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 18 / 40

19 Scale-dependence Scale-dependence [Baikov, Chetyrkin, Kühn (2010)] C Bj(Q 2, x µ = µ 2 /Q 2 ) = α s ( ln x µ)α 2 s Weak scale dependence of observables p-n ( ln x µ ln 2 x µ)α 3 s ( ln x µ ln 2 x µ ln 3 x µ)α 4 s x =2.0 N 2 LO PT x =0.5 N 2 LO PT x =2.0 N 3 LO PT x =0.5 N 3 LO PT x =2.0 N 3 LO APT x =0.5 N 3 LO APT Q 2 (GeV 2 ) Figure: The µ-scale ambiguities for the perturbative part of the BSR versus Q 2 from [Khandramai et al. (2012)] Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 19 / 40

20 Convergence Convergence Better loop convergence: the 3 rd and 4 th terms contribute less than 5% and 1% respectively. Again the 2-loop (N 2 LO) level is sufficient. 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 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- sion terms Eq. (5) in the APT approach. Third Bj APT(Q 2 approach. ), the pertu PTFigure: approach. Four-loop The relative PT order overshoots contributions the three-loop of one separate at Q 2 2GeV terms 2,soin PT sion expansion terms in Eq. (5) for in the ur-loop PT order overshoots the three-loop one at Q 2 2GeV 2,so and fourth Third order and contributions fourth order rove it does not improve the accuracy of the PT prediction compared to the three-loop amount less than 5% total, so the NLO APT approximation is su Nthe accuracy of the PT prediction compared to the three-loop amount to less than 5% total, so the NLO APT approximation is sufficient for description of the low scription energyof JLab thedata low at energy the current JLab data level at of theexperimental current levelaccuracy. of experime i (Q 2 ) = δ i (Q 2 )/ Bj (Q 2 ), as a function of Q 2 from [Khandramai et. al (2012)] one. 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]. tion This may situation be considered may be as considered a hint of the as transition a hint of the of transition of Viacheslav the PTasymptotic serieskhandramai the regime asymptotic (Gomel (whileregime State APT series Technical (while remains FAPT : APT University series con- A Mathematica remains Belarus) con- package for QCD calculations 20 / 40

21 DIS Sum Rules DIS Sum Rules See [Pasechnik et al. (PRD,2010)] The total expression for the perturbative part of Γ p,n 1 (Q2 ) including the higher twist contributions reads Γ p,n 1 (Q2 ) = 1 12 [(±a ) a8 E NS (Q 2 ) + 4 ] 3 a0 E S(Q 2 ) + i=2 µ p,n 2i Q 2i 2, where E S and E NS are the singlet and nonsinglet Wilson coefficients (for n f = 3): ( E NS (Q 2 ) = 1 αs π αs ) 2 ( αs ) O(α 4 s ), π π ( E S (Q 2 ) = 1 αs π αs ) 2 O(α 3 s ). π In Γ p n 1 the singlet and octet contributions are canceled out, giving rise to more fundamental Bjorken SR: Γ p n 1 (Q 2 ) = g A 6 E NS(Q 2 ) + i=2 µ p n 2i (Q 2 ) Q 2i 2. The triplet and octet axial charges a 3 g A = ± and a 8 = ± Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 21 / 40

22 The RG evolution of the axial singlet charge a 0 (Q 2 ) The RG evolution of the axial singlet charge a 0 (Q 2 ) { a0 PT (Q 2 ) = a0 PT (Q0 2 ) 1 + γ2 a APT 0 (Q 2 ) = a APT 0 (Q 2 0 ) } [α s(q 2 ) α s(q (4π) )] β 0 { 1 + γ2 (4π) 2 β 0 [A 1(Q 2 ) A 1(Q 2 0 )], }, γ 2 = 16n f. The evolution from 1 GeV 2 to Λ QCD in the APT increases the absolute value of a 0 by about 10 % a 0 Q 2 a 0 Q PT 1.2 Q 2, GeV 2 APT Figure: Evolution of a 0 (Q 2 ) normalized at Q0 2 = 1 GeV2 in APT and PT. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 22 / 40

23 The RG evolution of the axial singlet charge a 0 (Q 2 ) The RG evolution of the axial singlet charge a 0 (Q 2 ) Note, the Q 2 -evolution of µ p 4 (Q2 ) leads to close fit results within error bars. Therefore considered only of a 0(Q 2 ) Table: Combined fit results of the proton Γ p 1 (Q2 ) data (elastic contribution excluded). APT fit results a 0 and µ APT 4,6,8 (at the scale Q2 0 = 1 GeV2 ) are given without and with taking into account the RG Q 2 evolution of a 0 (Q 2 ). Method Qmin 2 GeV2 a 0 µ 4 /M 2 µ 6 /M 4 µ 8 /M (4) (4) 0 0 NNLO APT (3) (4) (8) 0 no evolution (3) (4) (9) (5) (4) (4) 0 0 NNLO APT (3) (4) (8) 0 with evolution (4) (4) (9) (5) The fit results become more stable with respect to Q min variations Obtained values are very close to the corresponding COMPASS [Alexakhin et al. (2007)] and HERMES [Airapetian et al. (2007)] results 0.35 ± Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 23 / 40

24 The RG evolution of the higher-twist µ p n 4 (Q 2 ) [ ] αs(q µ p n 4,PT (Q2 ) = µ p n 2 ν ) 4,PT (Q2 0 ) α s(q0 2), The RG evolution of the higher-twist µ p n 4 (Q 2 ) µ p n 4,APT (Q2 ) = µ p n 4,APT (Q2 0 ) A(1) ν (Q 2 ) A ν (1) (Q0 2), ν = γ0, γ 0 = 16 8πβ 0 3 C F, C F = 4 3. The evolution from 1 GeV 2 to Λ QCD in the APT increases the absolute value of µ p n 4 by about 20 % Μ 4 p n Q 2 Μ 4 p n Q APT PT Q 2, GeV Figure: Evolution of µ p n 4 (Q 2 ) normalized at Q 2 0 = 1 GeV2 in APT and PT. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 24 / 40

25 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 ) Table: Combined fit results of the Γ p n 1 data. APT fit results µ APT 4,6,8 (at the scale Q2 0 = 1 GeV2 ) are given without and with taking into account the RG Q 2 -evolution of µ p n 4. Method Q 2 min GeV 2 µ 4/M 2 µ 6/M 4 µ 8/M (3) 0 0 NNLO APT (4) 0.008(2) 0 no evolution (4) 0.010(3) (3) (3) 0 0 NNLO APT (4) (4) 0 with evolution (4) (6) (8) Account of this evolution, which is most important at low Q 2, improves the stability of the extracted parameters whose Q 2 dependence diminishes Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 25 / 40

26 The M 4 and M 8 moments evolution The M 4 and M 8 moments evolution [Bakulev, Ayala (In preparation)] Figure: The M 4 (solid curves) and M 8 (dashed curves) moments evolution normilized at the scale Q 2 0 = 4 GeV2 in the APT (blue curves) and PT (red curves). Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 26 / 40

27 Adler D-function analysis Adler D-function analysis Π µν(q 2 ) = i e iqx T {J µ(x)j ν(0)} 0 d 4 x, Π µν(q 2 ) = ( q µq ν g µνq 2) Π(Q 2 ). D(Q 2 ) = Q 2 dπ( Q 2 ) dq 2 = Q 2 The OPE-representation for the D-function D OPE(Q 2 ) = D PT(Q 2 ) + D NP(Q 2 ) 0 R(s) ds, R(s) = Im Π(s)/π. (s + Q 2 ) α s α 2 s α 3 s α 4 s + A Q 4 + A simple model for the function R V (s) (see [Peris, Perrottet, de Rafael (1998), Dorokhov (2004)]) ( ) RV had (s) = 2π m 2 gv 2 V δ(s mv 2 ) α(0) s θ(s s 0), π ( ) DV had (Q 2 ) = 2π gv 2 Q 2 m 2 V (Q 2 + m 2 V ) α(0) s π Q 2 Q 2 + s 0, which reproduces well the experimental" curve D exp V (Q2 ) with the parameters: m V = 770 MeV, g 2 V 2.1, α s (0) 0.4, and s GeV 2. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 27 / 40

28 Adler D-function analysis Adler D-function analysis Table: Fit results of the Adler D-function data based on hadron model. Method Order Qmin 2, GeV2 A, GeV 4 χ 2 d.o.f LO PT NLO N 2 LO N 3 LO LO APT NLO N 2 LO N 3 LO Standard PT provides: the results strongly changes from order to order; APT gives stable values of non-perturbative O ( 1/Q 4) -correction and allow to describe data up to Q min = 0.2 GeV 2. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 28 / 40

29 Adler D-function analysis Adler D-function analysis 2.0 D Q 2 N 3 LO APT PT 0.5 Model APT A Q 4 PT A Q 4 Q 2, GeV Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 29 / 40

30 Adler D-function analysis Package FAPT Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 30 / 40

31 FAPT package FAPT package review Title of program: FAPT Available from: bakulev/fapt.mat/fapt.m bakulev/fapt.mat/fapt_interp.m Computer for which the program is designed and others on which it is operable: Any work-station or PC where Mathematica is running. Operating system or monitor under which the program has been tested: Windows XP, Mathematica (versions 5,7,8). FAPT package contains: 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] Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 31 / 40

32 FAPT package 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 by factors 4 i+1 : b i = 4 i+1 b RunDec b RunDec i i. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 32 / 40

33 FAPT package α s calculations The QCD scales Λl[Λ, n f ]: \[CapitalLambda]l[Λ, k] = Λl[Λ, n f = k] = Λ (l) k (Λ), (l = 1 4, 3P ; k = 4 6), The threshold logarithms as λl4[λ], λl5[λ], and λl6[λ]: \[Lambda]lk[Λ] = λlk[λ] = ln ( Λ 2 /Λl[Λ, k] 2), (l = 1 4, 3P ; k = 4 6), The running QCD couplings with fixed n f as αbarl[q 2, n f, Λ]: \[Alpha]Barl[Q 2, n f, Λ] = αbarl[q 2, n f, Λ] = α (l) s [ln(q 2 /Λ 2 ); n f ], (l = 1 4, 3P), The global running QCD couplings αglobl[q 2, Λ], : \[Alpha]Globl[Q 2, Λ] = αglobl[q 2, Λ] = α glob;(l) s (Q 2, Λ), (l = 1 4, 3P), iacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 33 / 40

34 FAPT package Example 1 We assume that the two-loop QCD scale Λ 3 is fixed at the value Λ 3 = GeV. We want to evaluate the corresponding values of the coupling αs glob;(l) (Q 2, Λ) at the scale Q = M 5. Possible Mathematica realization of this task 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 I n [ 2 ] : = L23 =0.387; I n [ 3 ] : = Mb=MQ5/. NumDefFAPT Out [3]= I n [ 4 ] : = \ [ Alpha ] Glob2 [Mb^2, L23 ] Out [4]= Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 34 / 40

35 FAPT package ρ ν calculations RhoBarl[L, n f, ν] returns l-loop spectral density ρ (l) ν (l = 1, 2, 3, 3P, 4) 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, 3P ; k = 3 6) RhoGlobl[L, ν, Λ 3] returns the global l-loop spectral density ρ (l);glob ν [L; Λ 3] (l = 1, 2, 3, 3P, 4) of fractional-power ν at L = ln(q 2 /Λ 2 3), cf. and with Λ 3 being the QCD n f = 3-scale: RhoGlobl[L, ν, Λ 3] = ρ (l);glob ν [L; Λ 3], (l = 1 4, 3P) iacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 35 / 40

36 FAPT package Ā ν and Ā ν calculations AcalBarl[L, n f, ν] returns l-loop (l = 1, 2, 3, 3P, 4) analytic image of fractional-power ν coupling Ā ν (l) [L; n f ] in Euclidean domain, AcalBarl[L, k, ν] = Ā (l) ν [L; n f = k], (l = 1 4, 3P ; k = 3 6) 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, 3P) UcalBarl[L, n f, ν] returns l-loop (l = 1, 2, 3, 3P, 4) analytic image of fractional-power ν coupling Ā ν (l) [L, n f ] in Minkowski domain UcalBarl[L, k, ν] = Ā (l) ν [L; n f = k], (l = 1 4, 3P ; k = 3 6) UcalGlobl[L, ν, Λ 3] returns l-loop analytic image of fractional-power ν coupling [L, Λ 3] in Minkowski domain A (l);glob ν UcalGlobl[L, ν, Λ 3] = A (l);glob ν [L, Λ 3], (l = 1 4, 3P) iacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 36 / 40

37 FAPT package Example 2 Creation of a two-dimensional 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: I n [ 5 ] : = P l o t [ AcalGlob2 [ L, 1, L23APT ], { L, 3,11}]// Timing Out [5]= { , 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 [ 6 ] : = P l o t [ UcalGlob2 [ L, 1, L23APT ], { L, 3,11}]// Timing Out [6]= { , 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 Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 37 / 40

38 FAPT package Interpolation To obtain the results much faster one can use module 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. 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 Figure: Relative error of the interpolation procedure for A glob ν=1.1 (left panel) and Aglob ν=1.1 (right panel), calculated at various loop orders with Λ 3 = 0.36 GeV for N = 11 number of points. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 38 / 40

39 Summary Summary APT provides natural way for coupling and related quantities with Universal (loop & scheme independent) IR limit; Weak loop dependence; Practical scheme independence. (F)APT provides effective tool to apply APT approach for renormgroup improved perturbative amplitudes. This approaches are used in many applications, for example: 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)]. I collect in FAPT package all the procedures in APT and (F)APT which are needed to compute analytic images of the standard QCD coupling powers up to 4-loops of renormalization group running and to use it for both schemes: with fixed number of active flavours n f, A ν (Q 2 ; n f ), A ν (s; n f ), and the global one with taking into account all heavy-quark thresholds, A glob ν (Q 2 ), A glob ν (s) based on the system Mathematica. Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 39 / 40

40 Summary Thanks for your attention! Viacheslav Khandramai (Gomel State Technical FAPT : University A Mathematica Belarus) package for QCD calculations 40 / 40

FAPT : a Mathematica package

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.