Parametrization of gK from arXiv:1412.3820 (Updated Dec 14, 2014).
References:
	 	    [1]   Strong Coupling   S. Bethke, Nucl.Phys.Proc.Suppl. 234 (2013) 229-234 
	 	    [2]  “BLNY”                   Phys. Rev. D67, 073016 (2003), hep-ph/0212159
	               [3]  “KN”                       Phys. Lett. B633, 710 (2006), arXiv:hep-ph/0506225
	             [4]   New From:  	arXiv:1412.3820

Clear["Global *"];

Basic Parameters, etc...

Lambda = 0.339;   (* ==== From Bethke [1] ==== *)
nf = 3;
CF = 4/3;
beta0 := 11-2*nf/3;
beta1 := 102-38*nf/3;

stdloops = 2;
alpiN[mu_,loops_] := 2/( beta0 * Log[mu/Lambda]) +
                                        If[ loops > 1, (-beta1*Log[2*Log[mu/Lambda]] / (beta0^3 * Log[mu/Lambda]^2 )), 0];
alpi[mu_] := alpiN[mu,stdloops]
C1 := 2*Exp[-EulerGamma]
muQ[Q_] := Q
bstar[b_,bmax_] := If[ bmax == Infinity, b, b/Sqrt[1+b^2/bmax^2] ]
mub[b_,bm_] := C1 /bstar[b,bm]

Pi alpi[1.78] (* ==== tau lepton mass value of (Subscript[α, s]) ==== *)

Anomalous dimensions, CS kernel, etc ...

Integral of (Subscript[γ, K])(μ)/μ from (Subscript[Q, i]) to (Subscript[Q, f]).  This is needed for calculations of (OverTilde[K]) and L; See Eq.(23) and Eq.(60) of paper:

Intgam[Qi_,Qf_]:= (2 CF (beta1(1 + Log[2]) + (beta1 + 2 (beta0^2) Log[Qf/Lambda])Log[Log[Qf/Lambda]])/((beta0^3) Log[Qf/Lambda])) - 
(2 CF ( beta1 (1 + Log[2]) + (beta1 +2 (beta0^2) Log[Qi/Lambda])Log[Log[Qi/Lambda]])/((beta0^3) Log[Qi/Lambda]))

(* ==== Evolving from (Subscript[μ, b*]) to Q. See, e.g., Eq.(23) from arXiv:1412.3820. ===== *)

IntgamQ[Q_,b_,bm_]:= Intgam[mub[b,bm],Q]

(* ==== LO (OverTilde[K])((Subscript[b, T]),μ), Eq.(9), from arXiv:1412.3820. ===== *)

KtildeLO[b_,mu_] := - alpi[mu] CF ( Log[b^2 mu^2/4] + 2 EulerGamma)

(* ==== LO (OverTilde[K])((SubStar[b]),(Subscript[μ, b*])). ===== *)

KtildeLOb[b_,bm_]:=KtildeLO[bstar[b,bm],mub[b,bm]]

(* ==== The result is zero with the choice of (Subscript[C, 1]) above. ===== *)

KtildeLOb[b,bm]

The parametrization (See Eqs. (79, 81) from arXiv:1412.3820):
	- bm0: Reference value for bmax. 1.5 GeV-1in paper.
	- bm:   bmax
	- gm:   Maximum value of gK at bT = ∞, with bmax = bm0. Called g0 in paper.

(* ==== The value of (Subscript[g, m]) as a function of (Subscript[b, max]) . Eq.(81) from arXiv:1412.3820 ===== *)

gmax[gm_,bm0_,bm_] := gm+ Intgam[ 2*Exp[-EulerGamma]/bm0,2*Exp[-EulerGamma]/bm]

(* ==== Negative of the parametrization for (Subscript[g, K]): -Eq.(79) from arXiv:1412.3820 ==== *)

minusgK[b_,bm_,gm_] := - gmax[gm,1.5,bm] (1 - Exp[-CF alpi[mub[b,bm]]b^2 /(gmax[gm,1.5,bm] bm^2)] )

(* ==== Negative of derivative of (Subscript[g, K]) wrt log((Subscript[b, T]^2)): needed for Eq.(31) from arXiv:1412.3820 ==== *)

minusdgK[b_,bm_,gm_] := (u/2)D[minusgK[u,bm,gm],u] /. u -> b

(* ==== Calculation of (OverTilde[K])(b,(Subscript[μ, Q])): see Eq.(25) from arXiv:1412.3820 ==== *)

Ktilde[Q_,b_,bm_,gm_] := -IntgamQ[Q,b,bm] +minusgK[b,bm,gm]

Plots of  -(Subscript[g, K]) and negative of the integral of γK ; gives terms in Eq.(25):

gKplots = Plot[{minusgK[b,1.5,0.5] ,minusgK[b,1.3,0.5]},{b,0,8},PlotStyle-> {{Thick,Red},{Thick,Dashed,Red}},PlotRange-> {-1.5,.5}];

Kregplots = Plot[{-IntgamQ[2,b,1.5] ,-IntgamQ[2,b,1.3]} ,{b,0,8},PlotStyle-> {{Thick,Blue},{Thick,Dashed,Blue}},PlotRange-> {-1.5,.5}];

Totalplots = Plot[{-IntgamQ[2,b,1.5] + minusgK[b,1.5,.5],-IntgamQ[2,b,1.3]+ minusgK[b,1.3,0.5]} ,{b,0,8},PlotStyle-> {{Thick,Green},{Thick,Dashed,Green}},PlotRange-> {-1.5,.5}];

Show[{gKplots,Kregplots}] (** Compare Thick((Subscript[b, max]) = 1.5 (GeV^(-1))) with Dashed ((Subscript[b, max]) = 1.3 (GeV^(-1))) **)
Show[{gKplots,Kregplots,Totalplots}] (** Check that adding Thick curves and Thin curves gives nearly same curve. **)

(OverTilde[K])(b,(Subscript[μ, Q])) with other parametrizations of gK:

No  (Subscript[g, K ]),but (SubStar[b]) prescription to regulate large (Subscript[b, T])

nogK1Q2[b_] := -IntgamQ[2,b,0.5]
nogK2Q2[b_] := -IntgamQ[2,b,1.5]
nogK1Q10[b_] := -IntgamQ[2,b,0.5]
nogK2Q10[b_] := -IntgamQ[2,b,1.5]
nogK1Q100[b_] := -IntgamQ[2,b,0.5]
nogK2Q100[b_] := -IntgamQ[2,b,1.5]

BLNY [2] and KN [3] (Subscript[g, K ])

BLNYgKQ2[b_] := -IntgamQ[2,b,0.5]- (.68/2) b^2
KNgKQ2[b_]:= -IntgamQ[2,b,1.5]- (.184/2) b^2
BLNYgKQ10[b_] := -IntgamQ[2,b,0.5]- (.68/2) b^2
KNgKQ10[b_]:= -IntgamQ[2,b,1.5]- (.184/2) b^2
BLNYgKQ100[b_] := -IntgamQ[2,b,0.5]- (.68/2) b^2
KNgKQ100[b_]:= -IntgamQ[2,b,1.5]- (.184/2) b^2

Kplots1=Plot[{Ktilde[2,b,0.5,0.3],Ktilde[2,b,1.5, 0.3],nogK1Q2[b] ,nogK2Q2[b],BLNYgKQ2[b],KNgKQ2[b]},{b,0,8},PlotRange->{-2,.6},PlotStyle-> {{Thick,Red,DotDashed},{Thick,Blue},{Dashed,Red},{Dotted,Blue},{Thick,Dashed,Black},{Thick,Dotted,Black}},AxesLabel->{"(Subscript[b, T]) ((GeV^(-1)))","(OverTilde[K])((Subscript[b, T]); Q = 2 GeV)"}]

Export["Kplots1.pdf",Kplots1]

Kplots2=Plot[{Ktilde[2,b,0.5,0.3],Ktilde[2,b,1.5, 0.3],Ktilde[10,b,0.5,0.3],Ktilde[10,b,1.5, 0.3],Ktilde[100,b,0.5,0.3],Ktilde[100,b,1.5, 0.3]},{b,0,8},PlotRange->{-2,.6},PlotStyle-> {{Thick,Red,DotDashed},{Thick,Blue},{Dashed,Red},{Dotted,Blue},{Thick,Dashed,Black},{Thick,Dotted,Black}},AxesLabel->{"(Subscript[b, T]) ((GeV^(-1)))","(OverTilde[K])((Subscript[b, T]); Q = 2, 10, 100 GeV)"}]

Export["Kplots2.pdf",Kplots2]

Calculations of master function, A(b):

(* ==== Perturbatively calculable part of A(b); second term of fifth line of Eq.(31) arXiv:1412.3820 ===== *)

Astar[b_,bm_]:= CF alpi[mub[b,bm]](bstar[b,bm]^2/b^2)

(* ==== Complete calculation of A(b) using above parameterization of (Subscript[g, k]). See also Eq.(79) arXiv:1412.3820 ===== *)

A[b_,bm_,gm_]:= -minusdgK[b,bm,gm] + Astar[b,bm]

(* ==== Complete calculation of A(b) using the quadratic models of (Subscript[g, k]) from [1,2]. See also Eq.(50) arXiv:1412.3820 ===== *)

Aquad[b_,bm_,g2_] := g2 b^2 / 2 + Astar[b,bm]

minusdK[Q_,b_,bm_,gm_] := -(u/2)D[Ktilde[mu,u,bm,gm],u] /. u -> b

(* ==== Checking A(b) ==== *)

minusdK[1,.5,1.5,.3]
A[.5,1.5,.3]

Using BLNY and KN parametrizations of (Subscript[g, K])

BLNYA[b_]:= Aquad[b,.5,.68]
KNA[b_] := Aquad[b,1.5,.184]

Aplots1 =Plot[{A[b,0.5,0.3],A[b,1.5,0.3],Astar[b,0.5],Astar[b,1.5],BLNYA[b],KNA[b]},{b,0,5},PlotRange->{0,1},PlotStyle-> {{Thick,Red,DotDashed},{Thick,Blue},{Dashed,Red},{Dotted,Blue},{Dashed,Black,Thick},{Dotted,Black,Thick}},AxesLabel->{"(Subscript[b, T]) ((GeV^(-1)))","A((Subscript[b, T]))"}]
Export["Aplots1.pdf",Aplots1]

Aplots2 =Plot[{A[b,.9,0.3],A[b,1.0,0.3],A[b,1.25,0.3],A[b,1.5,0.3],A[b,1.6,0.3]},{b,0,5},PlotRange->{0,1},PlotStyle-> {{Thick,DotDashed,Red},{Thick,Blue},{Thick,Dashed,Green},{Thick,Dotted,Purple},{Black}},AxesLabel->{"(Subscript[b, T]) ((GeV^(-1)))","A((Subscript[b, T]))"}]
Export["Aplots2.pdf",Aplots2]