(* Mathematica script AdlerFunction3lMiniMOM.m *)
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Print["AdlerFunction3lMiniMOM.m: a Mathematica script for analytic running"]
Print["It gives the coefficients d1, d2, d3 of the perturbation expansion"]
Print["for the nf=3 massless canonical Adler function d(Q^2)"]
Print["in the 3-loop Lambert-MiniMOM renormalization scheme"]
Print["as defined in paper arXiv:1608.08240"]
Print["Note that the perturbation expansion of the massless Adler function is:"]
Print["d(Q^2)_{pt} = a(Q^2) + d1 a(Q^2)^2 + d2 a(Q^2)^3 + d3 a(Q^2)^4"]
Print["where a(Q^2) = alpha_s(Q^2)/Pi in the 3-loop Lambert MiniMOM."]
Print["See Eqs. (16)-(17) of  arXiv:1608.08240"]
Print["In addition, it gives the parameters of the resummed expression"]
Print["for d(Q^2), Eq. (18) of arxiv:1608.08240:"]
Print["{tilde alpha1}=alt[2,1]; {tilde alpha2}=alt(2,2)=1-{tilde alpha1}"]
Print["kappa1= kapps(2,1); kappa2=kapps(2,2)"]
Print["be0, c1, c2MS, c3MS, c2, c3 are the RGE coefficients; the symbol explanation can be called by the command: ?symbolname"]

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


(* When using the script, the symbol explanation can be called by the command: ?symbolname *)

 (* ::Section:: *)
(*Definitions*)
nf::usage= "nf is the number of active quarks in the QCD theory, here nf=3.";
be0::usage= "be0 = beta0 RGE coefficient, here be0=9/4 (nf=3)"
c1::usage= "c1 = beta1/beta0 RGE coefficient, here c1=16/9 (nf=3)"
c2MS::usage= "c2MS = beta2(MSbar)/beta0 RGE coefficient in MSbar scheme, here c2MS=3863/864=4.47106... (nf=3)"
c3MS::usage= "c3MS = beta3(MSbar)/beta0 RGE coefficient in MSbar scheme, here c3MS=20.990247..."
c2::usage= "c2 = beta2/beta0 RGE coefficient in 3-loop Lambert-MiniMOM scheme, here c2=9.3 (nf=3) "
c3::usage= "c3 = beta3/beta0 RGE coefficient in 3-loop Lambert-MiniMOM scheme, c3=c2^2/c1 = 48.650625... (nf=3)"
d1bar::usage= "the coefficient d1 in MSbar scheme"
d2bar::usage= "the coefficient d2 in MSbar scheme"
d3bar::usage= "the coefficient d3 in MSbar scheme"
td1bar::usage= "the coefficient {tilde d}1 in MSbar scheme"
td2bar::usage= "the coefficient {tilde d}2 in MSbar scheme"
td3bar::usage= "the coefficient {tilde d}3 in MSbar scheme"
d1::usage= "the coefficient d1 in 3-loop Lambert-MiniMOM scheme [cf.arXiv:1608.08240, Eqs.(16)-(17)]"
d2::usage= "the coefficient d2 in 3-loop Lambert-MiniMOM scheme [cf.arXiv:1608.08240, Eqs.(16)-(17)]"
d3::usage= "the coefficient d3 in 3-loop Lambert-MiniMOM scheme (cf.arXiv:1608.08240)"
td1::usage= "the coefficient {tilde d}1 in 3-loop Lambert-MiniMOM scheme [cf.arXiv:1608.08240, Eqs.(16)-(17)]"
td2::usage= "the coefficient {tilde d}2 in 3-loop Lambert-MiniMOM scheme [cf.arXiv:1608.08240, Eqs.(16)-(17)]"
td3::usage= "the coefficient {tilde d}3 in 3-loop Lambert-MiniMOM scheme (cf.arXiv:1608.08240)"
alt::usage= "alt[2,1]=the coefficient {tilde alpha1}, in 3-loop Lambert-MiniMOM scheme; alt[2,1]={tilde alpha2}=1-{tilde alpha1}  [cf.arXiv:1608.08240, Eq.(18)]"
kapps::usage= "kapps[2,1]=the coefficient kappa1, in 3-loop Lambert-MiniMOM scheme; kapps[2,2]=kappa2  [cf.arXiv:1608.08240, Eq.(18)]"
(*--------------------------------------------------------------------------_*)       

nf = 3
 
be0 = 9/4
 
c1 = 16/9
 
c2 = 9.3
 
c3 = 48.65062499999999
 
c4 = 254.50358203125
 
c5 = 1331.3718635009768
 
c2MS = 3863/864
 
c3MS = 20.990247170138886
 
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)
 
td1 = 1.6398212048969851
 
td2 = -1.3731583996788679
 
td3 = 10.26091947579085
 
d1 = 1.6398212048969851
 
d2 = 1.5420792979157723
 
d3 = 19.408330460537844
 
alt[1, 1] = 1.
 
alt[2, 1] = 1.14641959100752621313863791351951317446`26.048641397486946
 
alt[2, 2] = -0.14641959100752621313863791635913234096`24.55724608854426
 
kapps[1, 1] = 
    0.4824830808572873476119276474241787660918342936356451031381`28.352182518\
111366
 
kapps[2, 1] = 0.66452627701235623409092642012074520499`27.10542036338988
 
kapps[2, 2] = 5.91621628397798760523066947474772052523`26.051187997600522
nf = 3
 
be0 = 9/4
 
c1 = 16/9
 
c2 = 9.3
 
c3 = 48.65062499999999
 
c4 = 254.50358203125
 
c5 = 1331.3718635009768
 
c2MS = 3863/864
 
c3MS = 20.990247170138886
 
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)
 
td1 = 1.6398212048969851
 
td2 = -1.3731583996788679
 
td3 = 10.26091947579085
 
d1 = 1.6398212048969851
 
d2 = 1.5420792979157723
 
d3 = 19.408330460537844
 
alt[1, 1] = 1.
 
alt[2, 1] = 1.14641959100752621313863791351951317446`26.048641397486946
 
alt[2, 2] = -0.14641959100752621313863791635913234096`24.55724608854426
 
kapps[1, 1] = 
    0.4824830808572873476119276474241787660918342936356451031381`28.352182518\
111366
 
kapps[2, 1] = 0.66452627701235623409092642012074520499`27.10542036338988
 
kapps[2, 2] = 5.91621628397798760523066947474772052523`26.051187997600522
