(* Mathematica script 2danQCDcouplrt0201al01181.m *)

(* Copyright: G. Cvetic (November 2018), Univ. Tecnica Federico Santa Maria
 Permission is granted for anyone to use this script for any purpose 
 on any computer. The script can be freely distributed. *)

(* The script has been checked to work in Mathematica 11.3 *)
(* ************************************************************ *)

Print["2danQCDcouplrt0201al01181: a Mathematica package for analytic running"]
Print["of the strong coupling A (the analog of pQCD coupling a=alpha_s/Pi)"]
Print["in a model with 2 deltas in the spectral function of A,"]
Print["in the c2=-4.9 Lambert renormalization scheme."]
Print["This model:"]
Print["1.) agrees with perturbative QCD at high Q2 up to (Lambda^2/Q2)^5;"]
Print["2.) reproduces correctly the tau semihadronic decay physics."]
Print["The input parameters are such that r_{tau}^{(D=0)}=0.201 ('rt0201')"]
Print["in the (leading-beta0) + (beyond-leading-beta0) resummation (rtauLBbLB)"]
Print["and alpha_s(MZ^2;MSbar)=0.1181 ('al01181')."]
Print["For additional details, see arXiv:1408.6868"]
Print["The software has been checked to work in: Mathematica 11.3"]
Print["The software can be freely distributed."];
Print["Copyright: G.Cvetic, September 2018, Univ. Tecnica Federico Santa Maria"]

(* ************************************************************ *)
(* ************************************************************ *)


(* ::Section:: *)
(*Definitions*)

(* Couplings, functions: *)
LL2::usage= "LL2[Nf]=Lambda_L^2, this is square the Lambert Lambda scale Lambda_L (with Nf flavors), in GeV^2, appearing in the pQCD running coupling a(Q^2) (see: a2d) underlying for the considered Nf-flavor 2danQCD";
alphaMSMZ::usage= "alphaMSMZ=alpha_s(Q^2=MZ^2,MSbar). In this case, alphaMSMZ=0.1181"
c22din::usage= "c22din=c2=beta_2/beta_0 (chosen to be c2ddin=-4.9) is the value of the c2 coefficient (for Nf=3,4,5,6) in the Lambert renormalization scheme used in the considered 2danQCD [c2=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization convention: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.]"
s0::usage= "s0[Nf]=s_0 is a dimensionless parameter for Nf flavor 2danQCD (Nf=3,4,5,6); it is equal for Nf=3,4,5,6; the pQCD-onset scale M_0^2 in Nf=3 2danQCD is then: M_0^2(Nf=3)=s0[3]*LL2[3] (in GeV^2), and similarly for other cases of Nf"
s1s0::usage= "s1s0[Nf]=s_1 is a dimensionless parameter for Nf flavor 2danQCD (Nf=3,4,5,6); the location of the first delta function of the discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the A-coupling of the Nf-flavor 2danQCD is then at sigma=M_1^2=s1s0[Nf]*LL2[Nf] (in GeV^2)"
s2s0::usage= "s2s0[Nf]=s_2 is a dimensionless parameter for Nf flavor 2danQCD (Nf=3,4,5,6); the location of the second delta function of the discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the A-coupling of the Nf-flavor 2danQCD is then at sigma=M_2^2=s2s0[Nf]*LL2[Nf] (in GeV^2)"
f1ss0::usage= "f1ss0[Nf]=f_1^2 is a dimensionless strength parameter at the first delta in rho_1(sigma) discontinuity function of A(Q^2), in the Nf-flavor 2danQCD; the first delta function in rho_1(sigma) is Pi*f1ss0[Nf]*LL2[Nf]*delta(sigma - M1^2[Nf]), where M1^2[Nf]= s1s0[Mf]*LL2[Nf] (in GeV^2)"
f2ss0::usage= "f2ss0[Nf]=f_2^2 is a dimensionless strength parameter at the second delta in rho_1(sigma) discontinuity function of A(Q^2), in the Nf-flavor 2danQCD; the second delta function in rho_1(sigma) is Pi*f2ss0[Nf]*LL2[Nf]*delta(sigma - M2^2[Nf]), where M2^2[Nf]= s2s0[Mf]*LL2[Nf] (in GeV^2)"  
a2d::usage= "a2d[Nf,u,fi]= alpha_s(Q^2=u*LL2[Nf]*Exp[I fi)/Pi =a(Q^2), where Q^2=|Q^2|*Exp[I fi] and |Q^2|=u*LL2[Nf] (see: LL2); a2d is the underlying pQCD coupling in the Lambert renormalization scheme [c2=c22din=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization conventions: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.], the pQCD coupling underlying the Nf-flavor 2danQCD coupling A (see: A1).";
apert::usage= "apert[Nf,Q2,fi]= alpha_s(Q^2=Q2*Exp[I fi)/Pi =a(Q^2), where Q^2=|Q^2|*Exp[I fi] and |Q^2|=Q2; apert is the underlying pQCD coupling in the Lambert renormalization scheme [c2=c22din=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization conventions: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.], the pQCD coupling underlying the Nf-flavor 2danQCD coupling A (see: A1); note, apert[Nf,Q2,fi]=a2d[Nf,Q2/LL2[Nf],fi].";
tr12d::usage= "tr12d[Nf,s] is the discontinuity function rho_1^{(pt)}(sigma)=Im a(Q^2=-sigma - I epsilon) of the underlying pQCD coupling a(Q^2) (see: a2d and apert) in the considered Nf-flavor 2danQCD, where s=sigma/LL2[Nf] (see: LL2)"
A1::usage= "A1[Nf,Q2,fi] = A(Q^2) = A_1(Q^2), where Q^2=Q2 Exp[I fi] (Q2=|Q^2|, in GeV^2) and A(Q^2) is the 2danQCD analog of the underlying perturbative coupling a(Q^2) = a2d[Nf,Q2/LL2[Nf],fi] (see: a2d); at high |Q^2|> 1 GeV^2 the two couplings practically coincide";
tA2::usage= "tA2[Nf,Q2,fi]={tilde A}_2(Q^2) for Q^2=Q2 Exp[I fi] (Q2=|Q^2|, in GeV^2); this is the first logarithmic derivative of A_1 (see: A1): {tilde A}_2(Q^2) = (-1)/(beta_0 1!) (d/d ln Q^2) A_1(Q^2)";
tA3::usage= "tA3[Nf,Q2,fi]={tilde A}_3(Q^2) for Q^2=Q2 Exp[I fi]; this is the second logarithmic derivative of A_1 (see: A1): {tilde A}_3(Q^2) = (-1)^2/(beta_0^2 2!) (d/d ln Q^2)^2 A_1(Q^2)";
tA4::usage= "tA4[Nf,Q2,fi]={tilde A}_4(Q^2) for Q^2=Q2 Exp[I fi]; this is the third logarithmic derivative of A_1 (see: A1): {tilde A}_4(Q^2) = (-1)^3/(beta_0^3 3!) (d/d ln Q^2)^3 A_1(Q^2)";
tA2d::usage= "tA2d[Nf,nu,Q2,fi]= {tilde A}_{nu}(Q^2), with Q^2=Q2 Exp[I fi]  (Q2=|Q^2|); {tilde A}_{nu} can have nu noninteger; {tilde A}_{nu} is the generalization (to noninteger nu) of the logarithmic derivative (-1)^{nu-1}/(beta_0^{nu-1} (nu-1)! (d/d ln Q^2)^(nu-1) A_1(Q^2)";
A2dNX::usage= "A2dNX[Nf,M,nu,Q2,fi] = A_{nu+M}(Q^2) is the 2danQCD analog of the power of the pQCD coupling a2d[Nf,Q2/LL2[Nf],fi]^(nu+M)=a(Q^2)^(nu+M) (see: a2d), for NX=N1,N2,N3,N4; here, -1 =< nu =< 1 (in general noninteger), and M is integer, M=0,1,..,X; where Q^2=Q2 Exp[I fi] and Q2=|Q^2|=u*LL2[nq] (in GeV^2). Note that for X=1 (N1), A2dN1[...] are the couplings (A_{nu}, A_{nu+1}) in the NLO sense, when in the TPS of a considered observable we only have two terms known (~a^nu and ~a^(nu+1)); for X=2 (N2), A2dN2[...] are the couplings (A_{nu}, A_{nu+1}, A_{nu+2}) in the N^2LO sense, i.e.,  when in the truncated perturbation series of a considered observable we have three terms known (~a^nu, ~a^(nu+1) and ~a^(nu+2)); etc.; Nf indicates the chosen number of active flavors of the 2danQCD (Nf=3,4,5,6), usually Nf=3."
AYNX::usage= "AYNX[Nf,Q2,fi] = A_{Y}(Q^2) is the 2danQCD analog of the power of the pQCD coupling a2d[Nf,Q2/LL2[Nf],fi]^Y=a(Q^2)^Y, for NX=N1,N2,N3,N4; here, Y is integer, Y=1,..,X+1; where Q^2=Q2*Exp[I fi] and Q2=|Q^2|= u*LL2[nq] (in GeV^2). Note that X=1 (N1) is used when we have in the truncated perturbation series of a considered observable only two terms (~a and ~a^2) known; X=2 (N2) is used when three terms are known (~a, ~a^2 and ~a^3); X=3 (N3) is used when four terms are known (~a, ~a^2, ~a^3 and ~a^4); X=4 (N4) is used when five terms are known (~a, ~a^2, ~a^3, ~a^4, and ~a^5). Nf indicates the chosen number of active flavors of the 2danQCD (Nf=3,4,5,6), usually Nf=3."
tdj::usage= "tdj={tilde d}_j (j=1,2,3) are the coefficients for the Nf=3 canonical (and massless) Adler function in the Lambert renormalization scheme used in the considered 2danQCD and in the underlying pQCD [c2=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalizations: beta_0=(11 -2 Nf/3)/4; beta_1=(102-38 Nf/3)/16; etc.]"
tdjbLB::usage= "tdjbLB={tilde d}_j - {tilde d}_j^(LB) are the 'beyond-the-leading-beta0' (bLB) part of {tilde d}_j Nf=3 coefficients of the canonical (and massless) Adler function in the Lambert renormalization scheme used in the considered 2danQCD and in the underlying pQCD (see also: tdj)."
rtauLBbLB::usage= "rtauLBbLB=r_{tau}(D=0) calculated (in the considered Lambert renormalization scheme) in the considered 2danQCD with the 'leading-beta_0 (LB) + beyond-the leading-beta_0 (bLB)' approach with Nf=3. Note that here rtauLBbLB=0.2010"
rtaudBG22::usage= "rtaudBG22=r_{tau}(D=0) calculated in the considered 2danQCD with the diagonal [2/2] Baker-Gammel approach."
rtauTPS::usage= "rtauTPS=r_{tau}(D=0) calculated in the considered 2danQCD with the truncated perturbation series approach."
aMSnf3::usage= "aMSnf3[Q2] is the pQCD coupling a(Q^2=Q2)=alpha_s(Q2)/Pi for positive Q2 (Q2 in GeV^2) in MSbar renormalization scheme, for the number of active flavors Nf=3; it corresponds to alpha_s(MZ^2;MSbar)=0.1181, and is obtained by 4-loop RGE evolution in MSbar scheme from Q2=M_Z^2 downwards, with 3-loop quark threshold conditions imposed at threshold momenta Q2=(2*mqbar)^2, where mqbar=mqbar(Q2=mqbar^2) are the MSbar quark masses of b (for Nf=5->4) and of c (for Nf=4->3); we used mbbar=4.2 GeV and mcbar=1.27 GeV."

Off[NIntegrate::inumr]

  s0[3] = 22.7
 
s0[4] = 22.7
 
s0[5] = 22.7
 
s0[6] = 22.7
 
s0 /: s0::usage = "s0[Nf]=s_0 is a dimensionless parameter for Nf flavor \
2danQCD (Nf=3,4,5,6); it is equal for Nf=3,4,5,6; the pQCD-onset scale M_0^2 \
in Nf=3 2danQCD is then: M_0^2(Nf=3)=s0[3]*LL2[3] (in GeV^2), and similarly \
for other cases of Nf"
 
s1s0[3] = 16.55814752054176
 
s1s0[4] = 16.392572873389156
 
s1s0[5] = 16.075409681616456
 
s1s0[6] = 15.280984677855036
 
s1s0 /: s1s0::usage = "s1s0[Nf]=s_1 is a dimensionless parameter for Nf \
flavor 2danQCD (Nf=3,4,5,6); the location of the first delta function of the \
discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the \
A-coupling of the Nf-flavor 2danQCD is then at sigma=M_1^2=s1s0[Nf]*LL2[Nf] \
(in GeV^2)"
 
s2s0[3] = 0.7296877632644072
 
s2s0[4] = 0.029570693495297565
 
s2s0[5] = -1.5199126812484065
 
s2s0[6] = -7.338780306188639
 
s2s0 /: s2s0::usage = "s2s0[Nf]=s_2 is a dimensionless parameter for Nf \
flavor 2danQCD (Nf=3,4,5,6); the location of the second delta function of the \
discontinuity function rho_1(sigma)=Im A(Q^2=-sigma - I epsilon) of the \
A-coupling of the Nf-flavor 2danQCD is then at sigma=M_2^2=s2s0[Nf]*LL2[Nf] \
(in GeV^2)"
 
f1ss0[3] = 0.2717974099894099
 
f1ss0[4] = 0.34990616280235204
 
f1ss0[5] = 0.49996846793571564
 
f1ss0[6] = 0.8367007259227093
 
f1ss0 /: f1ss0::usage = "f1ss0[Nf]=f_1^2 is a dimensionless strength \
parameter at the first delta in rho_1(sigma) discontinuity function of \
A(Q^2), in the Nf-flavor 2danQCD; the first delta function in rho_1(sigma) is \
Pi*f1ss0[Nf]*LL2[Nf]*delta(sigma - M1^2[Nf]), where M1^2[Nf]= \
s1s0[Mf]*LL2[Nf] (in GeV^2)"
 
f2ss0[3] = 0.5468447698518647
 
f2ss0[4] = 0.775464614745532
 
f2ss0[5] = 1.3490491556648543
 
f2ss0[6] = 4.018786134754648
 
f2ss0 /: f2ss0::usage = "f2ss0[Nf]=f_2^2 is a dimensionless strength \
parameter at the second delta in rho_1(sigma) discontinuity function of \
A(Q^2), in the Nf-flavor 2danQCD; the second delta function in rho_1(sigma) \
is Pi*f2ss0[Nf]*LL2[Nf]*delta(sigma - M2^2[Nf]), where M2^2[Nf]= \
s2s0[Mf]*LL2[Nf] (in GeV^2)"
 
M0[3] = 1.2001058355270164
 
M0[4] = 0.8518638925295077
 
M0[5] = 0.4492385565469856
 
M0[6] = 0.10876853519666786
 
LLamb[3] = 0.2518875016762538
 
LLamb[4] = 0.17879578725924464
 
LLamb[5] = 0.0942896653906982
 
LLamb[6] = 0.02282918204430111
 
LL2[3] = 0.06344731350070476
 
LL2[4] = 0.03196793354165307
 
LL2[5] = 0.00889054099948983
 
LL2[6] = 0.0005211715528118403
 
LL2 /: LL2::usage = "LL2[Nf]=Lambda_L^2, this is square the Lambert Lambda \
scale Lambda_L (with Nf flavors), in GeV^2, appearing in the pQCD running \
coupling a(Q^2) (see: a2d) underlying for the considered Nf-flavor 2danQCD"
 
be0[nq_] := (11 - 2*(nq/3))/4
 
c1[nq_] := (102 - 38*(nq/3))/(11 - 2*(nq/3))/4
 
c22d[nq_] := c22din
 
c22din = -4.9
 
c22din /: c22din::usage = "c22din=c2=beta_2/beta_0 (chosen to be c2ddin=-4.9) \
is the value of the c2 coefficient (for Nf=3,4,5,6) in the Lambert \
renormalization scheme used in the considered 2danQCD [c2=-4.9; c3=c2^2/c1. \
c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; normalization convention: \
beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; etc.]"
 
c32d[nq_] := c22d[nq]^2/c1[nq]
 
c42d[nq_] := c22d[nq]^3/c1[nq]^2
 
c52d[nq_] := c22d[nq]^4/c1[nq]^3
 
c2MS[nq_] := be2MS[nq]/be0[nq]
 
be2MS[nq_] := (2857/2 - 5033*(nq/18) + 325*(nq^2/54))/64
 
c3MS[nq_] := be3MS[nq]/be0[nq]
 
be3MS[nq_] := (1/256)*((149753/6 + 3564*Zeta[3]) - 
      (1078361/162 + (6508/27)*Zeta[3])*nq + (50065/162 + (6472/81)*Zeta[3])*
       nq^2 + (1093/729)*nq^3)
 
btP11[x_, nq_] := betaP11[x, nq]
 
betaP11[x_, nq_] := (-be0[nq])*x^2*((1 + (c1[nq] - c22d[nq]/c1[nq])*x)/
      (1 - (c22d[nq]/c1[nq])*x))
 
betaMS4l[x_, nq_] := (-be0[nq])*x^2*(1 + c1[nq]*x + c2MS[nq]*x^2 + 
      c3MS[nq]*x^3)
 
kappa2d = 2.
 
alphaMSMZ = 0.1181
 
alphaMSMZ /: alphaMSMZ::usage = "alphaMSMZ=alpha_s(Q^2=MZ^2,MSbar). In this \
case, alphaMSMZ=0.1185, or 0.1181, or 0.1189"
 
apert[nq_, Q2_, fi_] := a2d[nq, Q2/LL2[nq], fi]
 
a2d[nq_, u_, fi_] := If[fi >= 0, (-(1/c1[nq]))*
      (1/(1 - c22d[nq]/c1[nq]^2 + ProductLog[-1, zu2d[nq, u, fi]])), 
     (-(1/c1[nq]))*(1/(1 - c22d[nq]/c1[nq]^2 + ProductLog[1, 
         zu2d[nq, u, fi]]))]
 
a2d /: a2d::usage = "a2d[Nf,u,fi]= alpha_s(Q^2=u*LL2[Nf]*Exp[I fi)/Pi \
=a(Q^2), where Q^2=|Q^2|*Exp[I fi] and |Q^2|=u*LL2[Nf] (see: LL2); a2d is the \
underlying pQCD coupling in the Lambert renormalization scheme \
[c2=c22din=-4.9; c3=c2^2/c1. c4=c2^3/c1^2, etc., and cj=beta_j/beta_0; \
normalization conventions: beta_0=(11 -2*Nf/3)/4; beta_1=(102-38*Nf/3)/16; \
etc.], the pQCD coupling underlying the Nf-flavor 2danQCD coupling A (see: \
A1)."
 
zu2d[nq_, u_, fi_] := ((-(1/c1[nq]/E))*Exp[(-I)*(be0[nq]/c1[nq])*fi])/
     u^(be0[nq]/c1[nq])
 
aMSnf3[Qf2_] := First[y[Log[Qf2/Q4^2]] /. sol1a]
 
Q4 = 2.54
 
sol1a = {{y -> InterpolatingFunction[{{-2.8926644601027207, 20.}}, 
        {5, 7, 1, {656}, {4}, 0, 0, 0, 0, Automatic, {}, {}, False}, 
        {{-2.8926644601027207, -2.892664460102677, -2.892664460102634, 
         -2.892664460102591, -2.892664460102548, -2.892664460102496, 
         -2.892664460102444, -2.892664460102392, -2.89266446010234, 
         -2.8926644601022775, -2.8926644601022153, -2.8926644601021527, 
         -2.89266446010209, -2.892664460102015, -2.8926644601019396, 
         -2.8926644601018645, -2.892664460101789, -2.8926644601016984, 
         -2.892664460101608, -2.8926644601015172, -2.8926644601014266, 
         -2.8926644601013174, -2.8926644601012086, -2.8926644601010993, 
         -2.89266446010099, -2.8926644601008586, -2.8926644601007276, 
         -2.892664460100596, -2.8926644601004647, -2.8926644601003066, 
         -2.8926644601001485, -2.8926644600999905, -2.8926644600998324, 
         -2.892664460099642, -2.8926644600994513, -2.8926644600992613, 
         -2.8926644600990707, -2.8926644600988416, -2.8926644600986124, 
         -2.8926644600983833, -2.8926644600981537, -2.892664460097878, 
         -2.892664460097602, -2.892664460097326, -2.89266446009705, 
         -2.8926644600967175, -2.8926644600963853, -2.892664460096053, 
         -2.892664460095721, -2.892664460095321, -2.8926644600949207, 
         -2.8926644600945206, -2.8926644600941205, -2.892664460093639, 
         -2.8926644600931573, -2.8926644600926754, -2.892664460092194, 
         -2.892664460091614, -2.892664460091034, -2.892664460090454, 
         -2.892664460089874, -2.8926644600891755, -2.8926644600884774, 
         -2.892664460087779, -2.892664460087081, -2.89266446008624, 
         -2.892664460085399, -2.8926644600845584, -2.8926644600837172, 
         -2.892664460082705, -2.8926644600816926, -2.89266446008068, 
         -2.8926644600796676, -2.8926644600784486, -2.8926644600772295, 
         -2.89266446007601, -2.892664460074791, -2.8926644600733233, 
         -2.892664460071855, -2.892664460070387, -2.8926644600689193, 
         -2.8926644600671514, -2.892664460065384, -2.892664460063616, 
         -2.8926644600618485, -2.8926644600597196, -2.892664460057591, 
         -2.8926644600554625, -2.8926644600533336, -2.8926644600507703, 
         -2.892664460048207, -2.8926644600456437, -2.8926644600430804, 
         -2.8926644600399936, -2.8926644600369067, -2.89266446003382, 
         -2.892664460030733, -2.8926644600270155, -2.892664460023298, 
         -2.8926644600195806, -2.892664460015863, -2.892664460011386, 
         -2.8926644600069094, -2.8926644600024325, -2.8926644599979556, 
         -2.892664459992564, -2.8926644599871727, -2.892664459981781, 
         -2.8926644599763893, -2.892664459969896, -2.8926644599634024, 
         -2.8926644599569094, -2.892664459950416, -2.8926644599425955, 
         -2.8926644599347746, -2.892664459926954, -2.892664459919134, 
         -2.8926644599097147, -2.8926644599002955, -2.892664459890876, 
         -2.892664459881457, -2.892664459870112, -2.892664459858767, 
         -2.8926644598474223, -2.892664459836077, -2.892664459822412, 
         -2.8926644598087474, -2.8926644597950824, -2.8926644597814173, 
         -2.8926644597649576, -2.892664459748498, -2.892664459732038, 
         -2.8926644597155784, -2.8926644596957516, -2.8926644596759243, 
         -2.8926644596560975, -2.8926644596362707, -2.8926644596123867, 
         -2.892664459588503, -2.892664459564619, -2.8926644595407356, 
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A1[nq_, Q2_, fi_] := (1/Pi)*NIntegrate[tr12d[nq, Exp[w]]/
        (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w]), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] + 
     f1ss0[nq]/((Q2/LL2[nq])*Exp[I*fi] + s1s0[nq]) + 
     f2ss0[nq]/((Q2/LL2[nq])*Exp[I*fi] + s2s0[nq]) + J0[nq] - 
     (Q2/LL2[nq])*Exp[I*fi]*J1[nq] + (Q2/LL2[nq])^2*Exp[I*2*fi]*J2[nq]
 
A1 /: A1::usage = "A1[Nf,Q2,fi] = A(Q^2) = A_1(Q^2), where Q^2=Q2 Exp[I fi] \
(Q2=|Q^2|, in GeV^2) and A(Q^2) is the 2danQCD analog of the underlying \
perturbative coupling a(Q^2) = a2d[Nf,Q2/LL2[Nf],fi] (see: a2d); at high \
|Q^2|> 1 GeV^2 the two couplings practically coincide"
 
tr12d[nq_, s_] := If[s != 0, If[s > 0, Im[a2d[nq, s, -Pi]], 
      Im[(1/c1[nq])*(1/(1 - c22d[nq]/c1[nq]^2 + ProductLog[-1, 
           zu2d[nq, -s, 0]]))]], 0]
 
tr12d /: tr12d::usage = "tr12d[Nf,s] is the discontinuity function \
rho_1^{(pt)}(sigma)=Im a(Q^2=-sigma - I epsilon) of the underlying pQCD \
coupling a(Q^2) (see: a2d) in the considered Nf-flavor 2danQCD, where \
s=sigma/LL2[Nf] (see: LL2)"
 
lns0[3] = 3.122364924487357
 
lns0[4] = 3.122364924487357
 
lns0[5] = 3.122364924487357
 
lns0[6] = 3.122364924487357
 
lnsm = 92.10340371976183
 
J0[3] = 0.00466999455549764309108360437157522531`16.
 
J0[4] = 0.00507753887074683231808795530064138381`16.
 
J0[5] = 0.00557750058581599984275926991902285832`16.
 
J0[6] = 0.00622202997429364348515952541839318206`16.
 
J1[3] = 4.84346028881497324758771242290045`16.*^-45
 
J1[4] = 5.29827956855580615903950025463651`16.*^-45
 
J1[5] = 5.87569345738507775148605975808154`16.*^-45
 
J1[6] = 6.66537630861582027402462127582015`16.*^-45
 
J2[3] = 2.446809702456800300219863101055`16.*^-85
 
J2[4] = 2.676734109529465657890833417244`16.*^-85
 
J2[5] = 2.968727412713396780222296111734`16.*^-85
 
J2[6] = 3.368285035422341381188279148526`16.*^-85
 
tA2[nq_, Q2_, fi_] := (1/be0[nq]/Pi)*((Q2/LL2[nq])*Exp[I*fi])*
     (NIntegrate[Exp[-w]*(tr12d[nq, Exp[w]]/
         (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^2), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] + 
      (f1ss0[nq]*Pi)/((Q2/LL2[nq])*Exp[I*fi] + s1s0[nq])^2 + 
      (f2ss0[nq]*Pi)/((Q2/LL2[nq])*Exp[I*fi] + s2s0[nq])^2 + J1[nq] - 
      2*(Q2/LL2[nq])*Exp[I*fi]*J2[nq] + 3*(Q2/LL2[nq])^2*Exp[I*2*fi]*J3[nq])
 
tA2 /: tA2::usage = "tA2[Nf,Q2,fi]={tilde A}_2(Q^2) for Q^2=Q2 Exp[I fi] \
(Q2=|Q^2|, in GeV^2); this is the first logarithmic derivative of A_1 (see: \
A1): {tilde A}_2(Q^2) = (-1)/(beta_0 1!) (d/d ln Q^2) A_1(Q^2)"
 
J3[3] = 1.63689500729422241956655214488`16.*^-125
 
J3[4] = 1.79074905870213668885243599517`16.*^-125
 
J3[5] = 1.98615801578900018756979502896`16.*^-125
 
J3[6] = 2.25360240588900587649899107709`16.*^-125
 
tA3[nq_, Q2_, fi_] := (1/2/be0[nq]^2/Pi)*((Q2/LL2[nq])*Exp[I*fi])*
     (NIntegrate[Exp[-w]*(tr12d[nq, Exp[w]]/
         (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^2), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] - 
      2*NIntegrate[Exp[-w]*(tr12d[nq, Exp[w]]/
          (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^3), {w, lns0[nq], lnsm}, 
        MaxRecursion -> 30, WorkingPrecision -> 16] + 
      (f1ss0[nq]*Pi)*((-s1s0[nq] + (Q2/LL2[nq])*Exp[I*fi])/
        (s1s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^3) + (f2ss0[nq]*Pi)*
       ((-s2s0[nq] + (Q2/LL2[nq])*Exp[I*fi])/
        (s2s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^3))
 
tA3 /: tA3::usage = "tA3[Nf,Q2,fi]={tilde A}_3(Q^2) for Q^2=Q2 Exp[I fi]; \
this is the second logarithmic derivative of A_1 (see: A1): {tilde A}_3(Q^2) \
= (-1)^2/(beta_0^2 2!) (d/d ln Q^2)^2 A_1(Q^2)"
 
tA4[nq_, Q2_, fi_] := (1/6/be0[nq]^3/Pi)*((Q2/LL2[nq])*Exp[I*fi])*
     (NIntegrate[Exp[-w]*(1 - 4*Exp[-w]*(Q2/LL2[nq])*Exp[I*fi] + 
         Exp[-2*w]*(Q2/LL2[nq])^2*Exp[2*I*fi])*(tr12d[nq, Exp[w]]/
         (1 + (Q2/LL2[nq])*Exp[I*fi]*Exp[-w])^4), {w, lns0[nq], lnsm}, 
       MaxRecursion -> 30, WorkingPrecision -> 16] + 
      (f1ss0[nq]*Pi)*((s1s0[nq]^2 - 4*s1s0[nq]*(Q2/LL2[nq])*Exp[I*fi] + 
         (Q2/LL2[nq])^2*Exp[2*I*fi])/(s1s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^4) + 
      (f2ss0[nq]*Pi)*((s2s0[nq]^2 - 4*s2s0[nq]*(Q2/LL2[nq])*Exp[I*fi] + 
         (Q2/LL2[nq])^2*Exp[2*I*fi])/(s2s0[nq] + (Q2/LL2[nq])*Exp[I*fi])^4))
 
tA4 /: tA4::usage = "tA4[Nf,Q2,fi]={tilde A}_4(Q^2) for Q^2=Q2 Exp[I fi]; \
this is the third logarithmic derivative of A_1 (see: A1): {tilde A}_4(Q^2) = \
(-1)^3/(beta_0^3 3!) (d/d ln Q^2)^3 A_1(Q^2)"
 
tA[nq_, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi]
 
tA2d[nq_, nupp_, Q2_, fi_] := If[Abs[nupp] <= 2., 
     tAMA1l[nq, nupp, Q2/LL2[nq], fi] + 
      (-(Pi*be0[nq]^(nupp - 1)*Gamma[nupp])^(-1))*
       ((Pi*(f1ss0[nq]/s1s0[nq]))*PolyLog[-nupp + 1, (-s1s0[nq])*Exp[(-I)*fi]*
           (LL2[nq]/Q2)] + (Pi*(f2ss0[nq]/s2s0[nq]))*PolyLog[-nupp + 1, 
          (-s2s0[nq])*Exp[(-I)*fi]*(LL2[nq]/Q2)] - 
        NIntegrate[(tr1MA1l[nq, ssig]/ssig)*PolyLog[-nupp + 1, 
           (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], {ssig, 0, s0[nq]}] + 
        NIntegrate[((tr12d[nq, ssig] - tr1MA1l[nq, ssig])/ssig)*
          PolyLog[-nupp + 1, (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], 
         {ssig, s0[nq], 10*s0[nq]}] + NIntegrate[
         (tr12d[nq, Exp[w]] - tr1MA1l[nq, Exp[w]])*PolyLog[-nupp + 1, 
           (-Exp[w])*Exp[(-I)*fi]*(LL2[nq]/Q2)], {w, Log[10*s0[nq]], 10.^9}, 
         MaxRecursion -> 30, WorkingPrecision -> 16]), 
     tAMA1l[nq, nupp, Q2/LL2[nq], fi] + 
      (-(Pi*be0[nq]^(nupp - 1)*Gamma[nupp])^(-1))*
       ((Pi*(f1ss0[nq]/s1s0[nq]))*PolyLog[-nupp + 1, (-s1s0[nq])*Exp[(-I)*fi]*
           (LL2[nq]/Q2)] + (Pi*(f2ss0[nq]/s2s0[nq]))*PolyLog[-nupp + 1, 
          (-s2s0[nq])*Exp[(-I)*fi]*(LL2[nq]/Q2)] - 
        NIntegrate[(tr1MA1l[nq, ssig]/ssig)*PolyLog[-nupp + 1, 
           (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], {ssig, 0, s0[nq]}] + 
        NIntegrate[((tr12d[nq, ssig] - tr1MA1l[nq, ssig])/ssig)*
          PolyLog[-nupp + 1, (-ssig)*Exp[(-I)*fi]*(LL2[nq]/Q2)], 
         {ssig, s0[nq], 10*s0[nq]}] + NIntegrate[
         (tr12d[nq, Exp[w]] - tr1MA1l[nq, Exp[w]])*PolyLog[-nupp + 1, 
           (-Exp[w])*Exp[(-I)*fi]*(LL2[nq]/Q2)], {w, Log[10*s0[nq]], 10.^6}, 
         MaxRecursion -> 30, WorkingPrecision -> 16])]
 
tA2d /: tA2d::usage = "tA2d[Nf,nu,Q2,fi]= {tilde A}_{nu}(Q^2), with Q^2=Q2 \
Exp[I fi]  (Q2=|Q^2|); {tilde A}_{nu} can have nu noninteger; {tilde A}_{nu} \
is the generalization (to noninteger nu) of the logarithmic derivative \
(-1)^{nu-1}/(beta_0^{nu-1} (nu-1)! (d/d ln Q^2)^(nu-1) A_1(Q^2)"
 
tAMA1l[nq_, nu_, u_, fi_] := If[Abs[u*Exp[I*fi] - 1] > Delt[nu], 
     (1/be0[nq]^nu)*(1/(Log[u] + I*fi)^nu - 
       PolyLog[-nu + 1, (1/u)*Exp[(-I)*fi]]/Gamma[nu]), 
     (1/be0[nq]^nu)*Limt1l[u, nu, fi]]
 
Delt[nu_] := 10.^(-2) + (HeavisideTheta[nu - 3]*(nu - 3)*2)/10.^2
 
Limt1l[u_, nu_, fi_] := -(Zeta[1 - nu]/Gamma[nu]) + 
     ((u*Exp[I*fi] - 1)*Zeta[-nu])/Gamma[nu] - 
     ((u*Exp[I*fi] - 1)^2*(Zeta[-1 - nu] + Zeta[-nu]))/(2*Gamma[nu])
 
tr1MA1l[nq_, s_] := (Pi/be0[nq])*(1/(Log[s]^2 + Pi^2))
 
AN4[nq_, M_, nupp_, Q2_, fi_] := A2dN4[nq, M, nupp, Q2, fi]
 
A2dN4[nq_, 4, nupp_, Q2_, fi_] := tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 3, nupp_, Q2_, fi_] := tA2d[nq, 3 + nupp, Q2, fi] + 
     tk1f[nq, 3 + nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 2, nupp_, Q2_, fi_] := tA2d[nq, 2 + nupp, Q2, fi] + 
     tk1f[nq, 2 + nupp]*tA2d[nq, 3 + nupp, Q2, fi] + 
     tk2f[nq, 2 + nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi] + 
     tk1f[nq, 1 + nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk2f[nq, 1 + nupp]*tA2d[nq, 3 + nupp, Q2, fi] + 
     tk3f[nq, 1 + nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
A2dN4[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi] + 
     tk2f[nq, nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk3f[nq, nupp]*tA2d[nq, 3 + nupp, Q2, fi] + 
     tk4f[nq, nupp]*tA2d[nq, 4 + nupp, Q2, fi]
 
tk1f[nq_, s_] := -k1f[nq, s]
 
k1f[nq_, s_] := c1[nq]*s*(-1 + HarmonicNumber[s])
 
tk2f[nq_, s_] := k1f[nq, s]*k1f[nq, s + 1] - k2f[nq, s]
 
k2f[nq_, s_] := ((s - 1)*(s/2))*c22d[nq] + 
     c1[nq]^2*((s*((s + 1)/2))*((PolyGamma[s + 1] - PolyGamma[1])^2 + 
         (PolyGamma[1, s + 1] - PolyGamma[1, 1]) + 1) - 
       s^2*(PolyGamma[s + 1] - PolyGamma[1]))
 
tk3f[nq_, s_] := (-k1f[nq, s])*k1f[nq, s + 1]*k1f[nq, s + 2] + 
     k1f[nq, s]*k2f[nq, s + 1] + k1f[nq, s + 2]*k2f[nq, s] - k3f[nq, s]
 
k3f[nq_, s_] := (s*(s - 1)*((s + 4)/12))*c32d[nq] + 
     (s*(s + 1)*((s + 2)/2))*c1[nq]*c22d[nq]*((s/(s + 2))*ES[1, s + 1] + 
       2/(s + 1) + 1/(s + 2) - 11/6) + (s*(s + 1)*((s + 2)/6))*c1[nq]^3*
      (Z3[s + 2] - 3*Z2[s + 1] + 3*ES[1, s] - 1)
 
ES[m_, n_] := ((-1)^(m - 1)/(m - 1)!)*(PolyGamma[m - 1, n + 1] - 
      PolyGamma[m - 1, 1])
 
Z3[n_] := ES[1, n]^3 - 3*ES[1, n]*ES[2, n] + 2*ES[3, n]
 
Z2[n_] := ES[1, n]^2 - ES[2, n]
 
tk4f[nq_, s_] := k1f[nq, s]*k1f[nq, s + 1]*k1f[nq, s + 2]*k1f[nq, s + 3] - 
     k1f[nq, s]*k1f[nq, s + 1]*k2f[nq, s + 2] - k1f[nq, s]*k1f[nq, s + 3]*
      k2f[nq, s + 1] - k1f[nq, s + 2]*k1f[nq, s + 3]*k2f[nq, s] + 
     k2f[nq, s]*k2f[nq, s + 2] + k1f[nq, s]*k3f[nq, s + 1] + 
     k1f[nq, s + 3]*k3f[nq, s] - k4f[nq, s]
 
k4f[nq_, s_] := ((s*(s + 1)*(s + 2)*(s + 3))/6)*
     (c42d[nq]*(1/12 - 2/((s + 1)*(s + 2)*(s + 3))) + 
      c22d[nq]^2*(5/12 - (3*s^2 + 6*s + 1)/((s + 1)*(s + 2)*(s + 3))) + 
      (c1[nq]*(c32d[nq]/2))*((1 - 6/((s + 2)*(s + 3)))*Z1[s + 1] + 
        4/(s + 1) - 1/(s + 2) - 1/(s + 3) - 13/6) + (c1[nq]^2*(c22d[nq]/2))*
       ((3*((s + 1)/(s + 3)))*Z2[s + 2] + (12/(s + 2) + 6/(s + 3) - 11)*
         Z1[s + 1] - 6/(s + 1) - 5/(s + 2) - 11/(s + 3) + 38/3) + 
      (c1[nq]^4/4)*(Z4[s + 3] - 4*Z3[s + 2] + 6*Z2[s + 1] - 4*Z1[s] + 1))
 
Z1[n_] := ES[1, n]
 
Z4[n_] := ES[1, n]^4 - 6*ES[1, n]^2*ES[2, n] + 3*ES[2, n]^2 + 
     8*ES[1, n]*ES[3, n] - 6*ES[4, n]
 
AN3[nq_, M_, nupp_, Q2_, fi_] := A2dN3[nq, M, nupp, Q2, fi]
 
A2dN3[nq_, 3, nupp_, Q2_, fi_] := tA2d[nq, 3 + nupp, Q2, fi]
 
A2dN3[nq_, 2, nupp_, Q2_, fi_] := tA2d[nq, 2 + nupp, Q2, fi] + 
     tk1f[nq, 2 + nupp]*tA2d[nq, 3 + nupp, Q2, fi]
 
A2dN3[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi] + 
     tk1f[nq, 1 + nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk2f[nq, 1 + nupp]*tA2d[nq, 3 + nupp, Q2, fi]
 
A2dN3[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi] + 
     tk2f[nq, nupp]*tA2d[nq, 2 + nupp, Q2, fi] + 
     tk3f[nq, nupp]*tA2d[nq, 3 + nupp, Q2, fi]
 
A5N4[nq_, Q2_, fi_] := tA2d[nq, 5, Q2, fi]
 
A4N4[nq_, Q2_, fi_] := tA4[nq, Q2, fi] + tk1f[nq, 4]*tA2d[nq, 5, Q2, fi]
 
A3N4[nq_, Q2_, fi_] := tA3[nq, Q2, fi] + tk1f[nq, 3]*tA4[nq, Q2, fi] + 
     tk2f[nq, 3]*tA2d[nq, 5, Q2, fi]
 
A2N4[nq_, Q2_, fi_] := tA2[nq, Q2, fi] + tk1f[nq, 2]*tA3[nq, Q2, fi] + 
     tk2f[nq, 2]*tA4[nq, Q2, fi] + tk3f[nq, 2]*tA2d[nq, 5, Q2, fi]
 
A1N4[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
A4N3[nq_, Q2_, fi_] := tA4[nq, Q2, fi]
 
A3N3[nq_, Q2_, fi_] := tA3[nq, Q2, fi] + tk1f[nq, 3]*tA4[nq, Q2, fi]
 
A2N3[nq_, Q2_, fi_] := tA2[nq, Q2, fi] + tk1f[nq, 2]*tA3[nq, Q2, fi] + 
     tk2f[nq, 2]*tA4[nq, Q2, fi]
 
A1N3[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
AN2[nq_, M_, nupp_, Q2_, fi_] := A2dN2[nq, M, nupp, Q2, fi]
 
A2dN2[nq_, 2, nupp_, Q2_, fi_] := tA2d[nq, 2 + nupp, Q2, fi]
 
A2dN2[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi] + 
     tk1f[nq, 1 + nupp]*tA2d[nq, 2 + nupp, Q2, fi]
 
A2dN2[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi] + 
     tk2f[nq, nupp]*tA2d[nq, 2 + nupp, Q2, fi]
 
A3N2[nq_, Q2_, fi_] := tA3[nq, Q2, fi]
 
A2N2[nq_, Q2_, fi_] := tA2[nq, Q2, fi] + tk1f[nq, 2]*tA3[nq, Q2, fi]
 
A1N2[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
AN1[nq_, M_, nupp_, Q2_, fi_] := A2dN1[nq, M, nupp, Q2, fi]
 
A2dN1[nq_, 1, nupp_, Q2_, fi_] := tA2d[nq, 1 + nupp, Q2, fi]
 
A2dN1[nq_, 0, nupp_, Q2_, fi_] := tA2d[nq, nupp, Q2, fi] + 
     tk1f[nq, nupp]*tA2d[nq, 1 + nupp, Q2, fi]
 
A2N1[nq_, Q2_, fi_] := tA2[nq, Q2, fi]
 
A1N1[nq_, Q2_, fi_] := A1[nq, Q2, fi]
 
rtaudBG22 = 0.18502118322073985
 
rtaudBG22 /: rtaudBG22::usage = "rtaudBG22=r_{tau}(D=0) calculated in the \
considered 2danQCD with the diagonal [2/2] Baker-Gammel approach."
 
rtauTPS = 0.17492221481086054
 
rtauTPS /: rtauTPS::usage = "rtauTPS=r_{tau}(D=0) calculated in the \
considered 2danQCD with the truncated perturbation series approach."
 
rtauLBbLB = 0.20098024836442593
 
rtauLBbLB /: rtauLBbLB::usage = "rtauLBbLB=r_{tau}(D=0) calculated (in the \
considered Lambert renormalization scheme) in the considered 2danQCD with the \
'leading-beta_0 (LB) + beyond-the leading-beta_0 (bLB)' approach with Nf=3. \
Note that here rtauLBbLB=0.2010, or 0.2030, or 0.1990"
 
d1bar = 1/12 + (9*(11/2 - 4*Zeta[3]))/4
 
d2bar = -12521/288 + (81*(302/9 - (76*Zeta[3])/3))/16 + 13*Zeta[3] + 
     (3*(401 - 424*Zeta[3] + 200*Zeta[5]))/16
 
d3bar = 144939499/20736 - (5693495*Zeta[3])/864 + (5445*Zeta[3]^2)/8 + 
     9*(1045381/15552 - (40655*Zeta[3])/864 + (5*Zeta[3]^2)/6 - 
       (260*Zeta[5])/27) + 27*(-6131/5832 + (203*Zeta[3])/324 + 
       (5*Zeta[5])/18) + (65945*Zeta[5])/288 - (7315*Zeta[7])/48 + 
     3*(-13044007/10368 + (12205*Zeta[3])/12 - 55*Zeta[3]^2 + 
       (29675*Zeta[5])/432 + (665*Zeta[7])/72)
 
td1bar = 1/12 + (9*(11/2 - 4*Zeta[3]))/4
 
td2bar = -12521/288 - (16*(1/12 + (9*(11/2 - 4*Zeta[3]))/4))/9 + 
     (81*(302/9 - (76*Zeta[3])/3))/16 + 13*Zeta[3] + 
     (3*(401 - 424*Zeta[3] + 200*Zeta[5]))/16
 
td3bar = 144939499/20736 + (8891*(1/12 + (9*(11/2 - 4*Zeta[3]))/4))/2592 - 
     (5693495*Zeta[3])/864 + (5445*Zeta[3]^2)/8 + 
     9*(1045381/15552 - (40655*Zeta[3])/864 + (5*Zeta[3]^2)/6 - 
       (260*Zeta[5])/27) + 27*(-6131/5832 + (203*Zeta[3])/324 + 
       (5*Zeta[5])/18) + (65945*Zeta[5])/288 - 
     (40*(-12521/288 + (81*(302/9 - (76*Zeta[3])/3))/16 + 13*Zeta[3] + 
        (3*(401 - 424*Zeta[3] + 200*Zeta[5]))/16))/9 - (7315*Zeta[7])/48 + 
     3*(-13044007/10368 + (12205*Zeta[3])/12 - 55*Zeta[3]^2 + 
       (29675*Zeta[5])/432 + (665*Zeta[7])/72)
 
d1 = 1.6398212048969851
 
d2 = 15.742079297915776
 
d3 = 83.55174899975434
 
td1 = 1.6398212048969851
 
td2 = 12.826841600321135
 
td3 = 34.57868801343341
 
td1bLB = 0.08333333333333326
 
td2bLB = -2.8843605694609113
 
td3bLB = 9.746656167968794
