Quit[]

6D Recursion Relations

1. Toy Model with General (Subscript[Δ, 0])

Can evaluate ‘Definitions’ cell and go directly to ‘Solve for Anomalous Dimensions (Subscript[γ, n,l]) for General Spin-L Truncation for General (Subscript[Δ, 0]) ’

1.1   Definitions

(*for simplicity we write the scaling dimensions (Subscript[Δ, 0]) as Δ in this notebook*)
(*note that we will use the variables (x,z) instead of (z,zb) compared to the paper*)
(*conformal blocks, given as (Subscript[((G^DO)), Δ,l]) in appendix A of the paper:*)
G[Δ1_,l_,x_,z_,Δ12_,Δ34_]:=(F[0,0,x,z,Δ1,l,Δ12,Δ34]-((l+3)/(l+1))F[-1,1,x,z,Δ1,l,Δ12,Δ34]-((Δ1-4)/(Δ1-2))(((Δ1+l-Δ12)(Δ1+l+Δ12)(Δ1+l+Δ34)(Δ1+l-Δ34))/(16(Δ1+l-1)(((Δ1+l))^2)(Δ1+l+1)))F[1,1,x,z,Δ1,l,Δ12,Δ34]+(((Δ1-4)(l+3))/((Δ1-2)(l+1)))(((Δ1-l-Δ12-4) (Δ1-l+Δ12-4) (Δ1-l+Δ34-4) (Δ1-l-Δ34-4))/(16(Δ1-l-5)(((Δ1-l-4))^2)(Δ1-l-3)))F[0,2,x,z,Δ1,l,Δ12,Δ34]+2(Δ1-4)(l+3)((Δ12 Δ34)/((Δ1+l)(Δ1+l-2)(Δ1-l-4)(Δ1-l-6)))F[0,1,x,z,Δ1,l,Δ12,Δ34])
F[a_,b_,x_,z_,Δ1_,l_,Δ12_,Δ34_]:=(((((x z))^((1/2)(Δ1-l)))/(((x-z))^3))((((((-1)/2)x))^l)(x^(a+3))(z^b) f[(1/2)(Δ1+l-Δ12)+a,(1/2)(Δ1+l+Δ34)+a,Δ1+l+2a,x] f[(1/2)(Δ1-l-Δ12)-3+b,(1/2)(Δ1-l+Δ34)-3+b,Δ1-l-6+2b,z]-((((((-1)/2)z))^l)(z^(a+3))(x^b) f[(1/2)(Δ1+l-Δ12)+a,(1/2)(Δ1+l+Δ34)+a,Δ1+l+2a,z] f[(1/2)(Δ1-l-Δ12)-3+b,(1/2)(Δ1-l+Δ34)-3+b,Δ1-l-6+2b,x])))//Simplify    (*f are 2F1 hypergeometrics*)
(*to set apart the bosonic theory from the supersymmetric theory we add a 'b' to function and variable names in the following*)
 
Bb[n_,l_]/;(n<0):=0          (* Bb=anomalous dimension × (A^((0))), b for bosonic*)

(*conventions for writing the hypergeometrics*)
κ[β_,z_]:=Hypergeometric2F1[(β/2),(β/2),β,z]
κt[β_,z_]:=-((Gamma[β])/((Gamma[(β/2)])^2))Hypergeometric2F1[(β/2),(β/2),1,z]

(*definition of inner product we will use for derivation of recursion relation*)
IIb[m_,m1_,Δ_]:=Residue[((((1-x))^(m-Δ+3))/(x^(m1-Δ+5)))κt[2m,x] κ[-2m1,x],{x,0}]

(*replacement rule to avoid Gamma functions with non-integer arguments*)
GammaReplace={Gamma[b__+l+n]/;(b∉Integers)->(((2^(1-2 b-2 l-2 n)) (Sqrt[π]) (-1+2 b+2 (l+n))!)/((-(1/2)+b+l+n)!)),Gamma[b__+n]/;(b∉Integers)->(((2^(1-2 b-2 n)) (Sqrt[π]) (-1+2 b+2 n)!)/(((1/2) (-1+2 b)+n)!))};

(*(A^((0))), is derived in following section*)
Afreeb[Δ_,n_,l_]:=((2+l)Pochhammer[1+n,Δ-3] Pochhammer[3+l+n,Δ-3](2Δ-3+l+2 n) (2Δ-2+l+2 n) (Δ-3+n)! (Δ-1+ (l+n))! Gamma[2Δ-5+n] Gamma[2Δ-3+l+n])/((1/2) ((Gamma[-2+Δ])^2) ((Gamma[Δ])^2) (2Δ-6+2 n)! (2Δ-2+2 (l+n))!)      (*for l...even, l...odd: Afreeb=0*)

1.2   Free Disconnected Contribution (Subscript[((A^((0)))), n,l])

Fb[Δ_,x_,z_]:=(1+(1/(((x z))^Δ))+(1/((((1-x)(1-z)))^Δ)))    (*free disconnected part of 4-pt correlator, its CPW expansion gives the leading contribution to the OPE coefficients*)

(*The free OPE coefficients are calculated from the CPW expansion in CoeffListFreeMNb[Δ,max], where the (A^((0))) are given in terms of m and n and we take the sum up to n,m=max+1*)
(*m and n are related to the scaling dimension Δ and the spin l as: Δ=2n+l+2(Subscript[Δ, 0]) and l=m-n *)

CoeffListFreeMNb[Δ1_,maximum_]:=CoeffListFreeMNb[Δ1,maximum]=Module[{Δ=Δ1,Max=maximum},     
max=Max;

G1b[Δ,x,z]=Sum[Cb[m,n]((2^(m-n)) (1+m-n) G[m+n+2Δ,m-n,x,z,0,0]/.f->Hypergeometric2F1),{n,0,max+1},{m,n,max+1}];

Part1[Δ]=Series[(1/(((x z))^Δ))G1b[Δ,x,z],{x,0,max},{z,0,max}]//Normal//Expand;
Part2[Δ]=Series[Fb[Δ,x,z]-(1/(((x z))^Δ)),{x,0,max},{z,0,max}]//Normal//Expand;
EqGb[Δ]=Part1[Δ]-Part2[Δ];

CoeffList0b=SolveAlways[(EqGb[Δ])==0,{x,z}]//Flatten;
PosCb=Position[Table[CoeffList0b[[i]][[2]],{i,1,Length[CoeffList0b]}]/.(a_+b_ Cb[c_,d_])->A/.b_ Cb[c_,d_]->A,A];
Delete[CoeffList0b,PosCb] (*only output the coefficients which are solved for*)
]

CoeffListFreeMNb[4,6]

CoeffListFreeMNb[6,6]

CoeffListFreeNLb[Δ_,Max_]:=CoeffListFreeMNb[Δ,Max]/.Cb[m__,n__]->Cb[n,m-n]//Simplify;  (*rewrite everything in terms of spin l=m-n and n*)

CoeffListFreeNLb[4,6]

compare to general formula

Table[{C[n,l],Afreeb[4,n,l]},{n,0,4},{l,0,6,2}]

CoeffListFreeNLb[6,6]

compare to general formula

Table[{C[n,l],Afreeb[6,n,l]},{n,0,4},{l,0,6,2}]

Results:

Afreeb[Δ_,n_,l_]:=((2+l)Pochhammer[1+n,Δ-3] Pochhammer[3+l+n,Δ-3](2Δ-3+l+2 n) (2Δ-2+l+2 n) (Δ-3+n)! (Δ-1+ (l+n))! Gamma[2Δ-5+n] Gamma[2Δ-3+l+n])/((1/2) ((Gamma[-2+Δ])^2) ((Gamma[Δ])^2) (2Δ-6+2 n)! (2Δ-2+2 (l+n))!)

1.3   Crossing Equation

(*LHSb[i]/RHSb[i] denote the left-hand/right-hand side of the crossing equation*)

LHSb[1]=Coefficient[Coefficient[D[(((2))^l)(l+1)(1/(((x z))^Δ))G[2n+l+(2 Δ),l,x,z,0,0]/.f[a__,b__,c__,z]->kt[c,1-z] Log[1-z]/.f[a__,b__,c__,x]->k[c,x],n],Log[x z]],Log[1-z]];
RHSb[1]=Coefficient[Coefficient[D[(((2))^l)(l+1)(1/((((1-x)(1-z)))^Δ))G[2n+l+(2 Δ),l,1-x,1-z,0,0]/.f[a__,b__,c__,1-x]->kt[c,x] Log[x]/.f[a__,b__,c__,1-z]->k[c,1-z],n],Log[(-1+x)(-1+z)]],Log[x]];

LHSb[2]= (List@@Collect[(LHSb[1]/.k[a__,b__]:>k[a//Simplify,b]/.kt[a__,b__]:>kt[a//Simplify,b]),{k[2 (-2+n+Δ),x] kt[2 (-1+l+n+Δ),1-z],k[2 (-3+n+Δ),x] kt[2 (l+n+Δ),1-z],k[2 (-1+n+Δ),x] kt[2 (l+n+Δ),1-z],k[2 (-2+n+Δ),x] kt[2 (1+l+n+Δ),1-z],k[2 (l+n+Δ),x] kt[2 (-3+n+Δ),1-z],k[2 (-1+l+n+Δ),x] kt[2 (-2+n+Δ),1-z],k[2 (1+l+n+Δ),x] kt[2 (-2+n+Δ),1-z],k[2 (l+n+Δ),x] kt[2 (-1+n+Δ),1-z]}])(((x-z))^3)//Simplify;
RHSb[2]=(List@@Collect[(RHSb[1]/.k[a__,b__]:>k[a//Simplify,b]/.kt[a__,b__]:>kt[a//Simplify,b]),{k[2 (-2+n+Δ),1-z] kt[2 (-1+l+n+Δ),x],k[2 (-3+n+Δ),1-z] kt[2 (l+n+Δ),x],k[2 (-1+n+Δ),1-z] kt[2 (l+n+Δ),x],k[2 (-2+n+Δ),1-z] kt[2 (1+l+n+Δ),x],k[2 (l+n+Δ),1-z] kt[2 (-3+n+Δ),x],k[2 (-1+l+n+Δ),1-z] kt[2 (-2+n+Δ),x],k[2 (1+l+n+Δ),1-z] kt[2 (-2+n+Δ),x],k[2 (l+n+Δ),1-z] kt[2 (-1+n+Δ),x]}])(((x-z))^3)//Simplify;

1.4   Recursion Relation

We derive the recursion relation as explained in Sec. 2.1 of the paper, first multiply the crossing equation by a product of hypergeometrics, then use the orthogonality relation
(Subscript[δ, m,m'])=∮(dz/(2π i))(z^(m-m'-1))(Subscript[k, 2m+4])(z)(((Subscript[k, -2m'-2])(z))^( )) and the inner product:  (Subscript[I, m,m'])=∮(dz/(2π i))((((1-z))^(m-Δ+3))/(z^(m'-Δ+5)))(Subscript[kt, 2m])(z)(((Subscript[k, -2m'])(z))^( ))

LHSb[3]=LHSb[2]Bb[n,l](((k[-2 q,x])/((x^((5-Δ)+q)) ))*((k[-2 p,1-z])/((((1-z))^((5-Δ)+p)) )));
RHSb[3]=RHSb[2]Bb[n,l](((k[-2q,x])/((x^((5-Δ)+q)) ))*((k[-2p,1-z])/(((1-z))^((5-Δ)+p))));
LHSb[4]=Assuming[n∈Integers&&l∈Integers,Simplify[(LHSb[3]/.k[a__,c__] k[b__,c__]->(δ[((a-4)/2),((b+2)/(-2))]((((c^(((a-4)/2)-((b+2)/(-2))-1))))^(-1)))/.kt[a__,c__]k[b__,c__]->(Iib[(a/2),-(b/2),Δ](((((((1-c))^((a/2)-Δ+3))/(c^(-(b/2)-Δ+5)))))^(-1)))//PowerExpand//Simplify)/.(((1-c__))^(a__))->(((-1))^a)(((-1+c))^a)//Expand]]/.δ[a_,b_]:>δ[a//Expand,b//Expand];
RHSb[4]=Assuming[n∈Integers&&l∈Integers,Simplify[(RHSb[3]/.k[a__,c__] k[b__,c__]->(δ[((a-4)/2),((b+2)/(-2))]((((c^(((a-4)/2)-((b+2)/(-2))-1))))^(-1)))/.kt[a__,c__]k[b__,c__]->(Iib[(a/2),-(b/2),Δ](((((((1-c))^((a/2)-Δ+3))/(c^(-(b/2)-Δ+5)))))^(-1)))//PowerExpand//Simplify)/.(((1-c__))^(a__))->(((-1))^a)(((-1+c))^a)//Expand]]/.(((2-2 c__))^(-l))->(((-2))^(-l))(((-1+c))^(-l))/.δ[a_,b_]:>δ[a//Expand,b//Expand];
LHSb[5]=((List@@Collect[LHSb[4]/.List->Plus,{δ[-2-q,2-n-Δ],δ[-2-q,1-n-Δ],δ[-2-q,-n-Δ]δ[-2-q,-2-l-n-Δ],δ[-2-q,-1-l-n-Δ], δ[-2-q,-l-n-Δ]}])//Expand//Simplify)/.(((-2))^(-l))->(2^(-l))/.(((-(1/2)))^(2+l)) ->(2^(-2-l))/.(((-2))^(-l))->(2^(-l))/.(((-1))^l)->1/.(((-1))^(2-l))->1/.(((-1))^(1+l))->-1/.(((-1))^(1-2 l))->-1/.(((-1))^(-2 l))->1/.(((-1))^(1-l))->-1/.p->p+(Δ-4)/.q->q+(Δ-4);    
(*shift (p,q) by +(Δ-4) to get minimal case for (p,q)=(0,0)*)
RHSb[5]=((List@@Collect[RHSb[4]/.List->Plus,{δ[-2-p,2-n-Δ],δ[-2-p,1-n-Δ],δ[-2-p,-n-Δ]δ[-2-p,-2-l-n-Δ],δ[-2-p,-1-l-n-Δ], δ[-2-p,-l-n-Δ]}])//Expand//Simplify)/.(((-1))^(1+l))->-1/.(((-1))^(1-2 l))->-1/.(E^(-I l π))->1/.(E^(I l π))->1/.(((-1))^l)->1/.(((-1))^(2-l))->1/.(((-2))^(-l))->(2^(-l))/.(((-1))^(-2 l))->1/.(((-1))^(1-l))->-1/.p->p+(Δ-4)/.q->q+(Δ-4);

Collect[LHSb[5]/.δ[a_,b_]:>δ[-a-Δ+2,-b-Δ+2]//Expand,{Iib[l+n+Δ,-4+p+Δ,Δ]  δ[q,2+n],Iib[-1+l+n+Δ,-4+p+Δ,Δ]δ[q,1+n],Iib[1+l+n+Δ,-4+p+Δ,Δ]δ[q,1+n],Iib[l+n+Δ,-4+p+Δ,Δ]  δ[q,n], Iib[-2+n+Δ,-4+p+Δ,Δ]  δ[q,4+l+n],Iib[-3+n+Δ,-4+p+Δ,Δ] δ[q,3+l+n],Iib[-1+n+Δ,-4+p+Δ,Δ] δ[q,3+l+n],Iib[-2+n+Δ,-4+p+Δ,Δ]  δ[q,2+l+n]},FullSimplify]/.List->Plus

Collect[RHSb[5]/.δ[a_,b_]:>δ[-a-Δ+2,-b-Δ+2]//Expand,{Iib[l+n+Δ,-4+q+Δ,Δ] δ[p,2+n],Iib[-1+l+n+Δ,-4+q+Δ,Δ] δ[p,1+n],Iib[1+l+n+Δ,-4+q+Δ,Δ] δ[p,1+n],Iib[l+n+Δ,-4+q+Δ,Δ] δ[p,n],Iib[-2+n+Δ,-4+q+Δ,Δ] δ[p,4+l+n],Iib[-3+n+Δ,-4+q+Δ,Δ] δ[p,3+l+n],Iib[-1+n+Δ,-4+q+Δ,Δ] δ[p,3+l+n],Iib[-2+n+Δ,-4+q+Δ,Δ] δ[p,2+l+n]},FullSimplify]/.List->Plus

1.5   Solve for Anomalous Dimensions (Subscript[γ, n,l]) for General Spin-L Truncation for General (Subscript[Δ, 0])

Find Solutions as Lists of (Subscript[γ, n,l])-Values

Recursion relation we derived above (for general (Subscript[Δ, 0]) and we have (Subscript[Σ, n=0]^∞)(Subscript[Σ, l=0]^L) with l even):

LSb[0]=(-1-l) Bb[n,l] Iib[l+n+Δ,-4+p+Δ,Δ] δ[q,n]+(3+l) Bb[n,l] Iib[-1+l+n+Δ,-4+p+Δ,Δ] δ[q,1+n]+((1+l) (((l+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[1+l+n+Δ,-4+p+Δ,Δ] δ[q,1+n])/(4 (-1+2 l+2 n+2 Δ) (1+2 l+2 n+2 Δ) (l+2 (-1+n+Δ)))-((3+l) (((-2+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[l+n+Δ,-4+p+Δ,Δ] δ[q,2+n])/(4 (-5+2 n+2 Δ) (-3+2 n+2 Δ) (l+2 (-1+n+Δ)))-(3+l) Bb[n,l] Iib[-2+n+Δ,-4+p+Δ,Δ] δ[q,2+l+n]+(1+l) Bb[n,l] Iib[-3+n+Δ,-4+p+Δ,Δ] δ[q,3+l+n]+((3+l) (((-2+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[-1+n+Δ,-4+p+Δ,Δ] δ[q,3+l+n])/(4 (-5+2 n+2 Δ) (-3+2 n+2 Δ) (l+2 (-1+n+Δ)))-((1+l) (((l+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[-2+n+Δ,-4+p+Δ,Δ] δ[q,4+l+n])/(4 (-1+2 l+2 n+2 Δ) (1+2 l+2 n+2 Δ) (l+2 (-1+n+Δ)));
RSb[0]=-(1+l) Bb[n,l] Iib[l+n+Δ,-4+q+Δ,Δ] δ[p,n]+(3+l) Bb[n,l] Iib[-1+l+n+Δ,-4+q+Δ,Δ] δ[p,1+n]+((1+l) (((l+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[1+l+n+Δ,-4+q+Δ,Δ] δ[p,1+n])/(4 (-1+2 l+2 n+2 Δ) (1+2 l+2 n+2 Δ) (l+2 (-1+n+Δ)))-((3+l) (((-2+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[l+n+Δ,-4+q+Δ,Δ] δ[p,2+n])/(4 (-5+2 n+2 Δ) (-3+2 n+2 Δ) (l+2 (-1+n+Δ)))-(3+l) Bb[n,l] Iib[-2+n+Δ,-4+q+Δ,Δ] δ[p,2+l+n]+(1+l) Bb[n,l] Iib[-3+n+Δ,-4+q+Δ,Δ] δ[p,3+l+n]+((3+l) (((-2+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[-1+n+Δ,-4+q+Δ,Δ] δ[p,3+l+n])/(4 (-5+2 n+2 Δ) (-3+2 n+2 Δ) (l+2 (-1+n+Δ)))-((1+l) (((l+n+Δ))^2) (l+2 (-2+n+Δ)) Bb[n,l] Iib[-2+n+Δ,-4+q+Δ,Δ] δ[p,4+l+n])/(4 (-1+2 l+2 n+2 Δ) (1+2 l+2 n+2 Δ) (l+2 (-1+n+Δ)));

Solve for anomalous dimensions for general spin-L truncation and general scaling dimension (Subscript[Δ, 0]):

(*truncate spin to L and evaluate the Kronecker-deltas*)
LHS1b[L_,Δ1_]/;(L//EvenQ):=Module[{L1=L,Δ2=Δ1},
lhs[1]=(Sum[LSb[0]/.l->l1,{l1,0,L1,2}]/.δ[a__,b__]->δ1[Hold[((Solve[a-b==0,n])//Flatten)[[1]][[2]]]]//Release)/.List->Plus;
lhs[2,Δ2]=(DeleteCases[Table[Coefficient[lhs[1],δ1[i+q]]/.n->i+q,{i,-L1-4,0}]//Flatten,0]/.Δ->Δ2//Simplify);
lhs[2,Δ2]
]
RHS1b[L_,Δ1_]/;(L//EvenQ):=Module[{L1=L,Δ2=Δ1},
rhs[1]=Sum[RSb[0]/.l->l1,{l1,0,L1,2}]/.δ[a__,b__]->δ1[Hold[((Solve[a-b==0,n])//Flatten)[[1]][[2]]]]//Release;
rhs[2,Δ2]=(DeleteCases[Table[Coefficient[rhs[1],δ1[i+p]]/.n->i+p,{i,-L1-4,0}]//Flatten,0]/.Δ->Δ2//Simplify);
rhs[2,Δ2]
]
(*minimal value of (p,q) is (0,0), also evaluate inner products Ib*)
LHSb[L_,p1_,q1_,Δ_]:=(Series[LHS1b[L,Δ]/.List->Plus/.p->p1/.q->q1-ϵ,{ϵ,0,0}]/.Derivative[1,0][Bb][a__,b__]->0//Normal)/.Iib->IIb//Expand
RHSb[L_,p1_,q1_,Δ_]:=(Series[RHS1b[L,Δ]/.List->Plus/.p->p1-ϵ/.q->q1,{ϵ,0,0}]/.Derivative[1,0][Bb][a__,b__]->0//Normal)/.Iib->IIb//Expand


(* We implement the algorithm described in appendix C of the paper by defining a function Constb[L] which gives all the free parameters for any spin truncation L (notably all (Subscript[γ, n,l ])with 2n≤l≤L), we then write down the equations Eqb[L,p,q,Δ] for every p≥1 and 0≤q≤Min[p-1,L/2], replace (L+2)(L+4)/8 anomalous dimensions by the constants and solve these equations for the remaining  (Subscript[γ, n,l]) ×(Subscript[((A^((0)))), n,l])'s in terms of the constants,
AnomDimb[L,Max] then gives all  (Subscript[γ, n,l]) ×(Subscript[((A^((0)))), n,l]) with l≤min(2n-2,L) in terms of the free parameters, the solutions will be given up to n=Max for any integer Max*)

Constb[L_]/;(L//EvenQ):=Table[{Bb[n,l]->cb[n,l]},{n,0,(L/2)},{l,2n,L,2}]//Flatten
Eqb[L_,p_,q_,Δ_]:=(LHSb[L,p,q,Δ]-RHSb[L,p,q,Δ])/.Constb[L]   (*we start at (p,q)=(1,0) *)

AnomDimb[L_,Δ_,Max_]:=AnomDimb[L,Δ,Max]=Module[{L1=L,Δ1=Δ,max=Max},
EqListb=Table[Eqb[L1,i,j,Δ1]/.Bb->γ,{i,1,max},{j,0,Min[i-1,(L1/2)]}];
(Solve[EqListb==0]//Flatten)/.γ->Bb]

ListDim1b[L_,Δ_,Max_]:=Table[AnomDimb[L,Δ,Max][[i]][[1]],{i,1,Length[AnomDimb[L,Δ,Max]]}]    (*List of (Subscript[γ, n,l]) (× (A^((0)))) that we solved for, up to n=Max*)
ListDim2b[L_,Δ_,Max_]:=Table[AnomDimb[L,Δ,Max][[i]][[2]],{i,1,Length[AnomDimb[L,Δ,Max]]}]    (*List of these (Subscript[γ, n,l]) (× (A^((0)))) in terms of (1/8)(L+2)(L+4) constants*)

max=35;            (*max=35 is enough to get analytic solutions for first few L truncations*)

Resultsb[L_,Δ_]:=Resultsb[L,Δ]=AnomDimb[L,Δ,max]        (*gives results in form of replacement rules*)
Dimsb[L_,Δ_]:=ListDim1b[L,Δ,max]                                                (*gives a list of (Subscript[γ, n,l]) (×(A^((0)))) that we solved for*)
Solsb[L_,Δ_]:=ListDim2b[L,Δ,max]                                                (*gives the (Subscript[γ, n,l])(×(A^((0)))) from previous line in terms of the (1/8)(L+2)(L+4) free parameters*)

ConstLabelb[L_]:=Table[Constb[L][[i]][[2]]/.cb[a_,b_]->{a,b},{i,1,Length[Constb[L]]}]
SolNumb[L_]:=Length[ConstLabelb[L]]   
SolNum2[L_]:=(1/8)(L+2)(L+4)


Posb[L_,Δ_,l1_]:=Position[Dimsb[L,Δ]/.Bb[a_,l1]->c[l1],c[l1]]//Flatten       (*gives list of all positions of (Subscript[γ, n,l])(×(A^((0)))) with l=l1 *)
PartSolb[L_,Δ_,l_]:=Part[Solsb[L,Δ],Posb[L,Δ,l]]                                                           (*gives all solutions for (Subscript[γ, n,l])(×(A^((0)))) for a specific l *)

LabelPosb[L_,l_]:=Position[ConstLabelb[L]/.{a_,l}->c[l],c[l]]//Flatten;
LabelPosMinb[L_,l_]:=LabelPosb[L,l][[1]];

CoeffPartSolb[L_,Δ_,l_,constlabel_]/;(1≤constlabel≤Length[Constb[L]]):=Coefficient[PartSolb[L,Δ,l],Constb[L][[constlabel]][[2]]]      (*find part of solution that is multiplied by a specific free parameter*)
(*to find analytic expressions of (Subscript[γ, n,l]) (×(A^((0)))), write the solution like ~ f1[n] cb[0,0]+f2[n] cb[0,2]+f3[n] cb[1,2].. (schematically)*)

write recursion relation for L=0  in terms of p:

LHSbB[L_,p,q1_,Δ_]:=(Series[LHS1b[L,Δ]/.List->Plus/.q->q1-ϵ,{ϵ,0,0}]/.Derivative[1,0][Bb][a__,b__]->0//Normal)//Expand
RHSbB[L_,p,q1_,Δ_]:=(Series[RHS1b[L,Δ]/.List->Plus/.q->q1,{ϵ,0,0}]/.Derivative[1,0][Bb][a__,b__]->0//Normal)//Expand

-LHSbB[0,p,0,Δ]

Collect[-RHSbB[0,p,0,Δ],{Iib[-6+p+Δ,-4+Δ,Δ] Bb[-4+p,0],Iib[-6+p+Δ,-4+Δ,Δ] Bb[-3+p,0],Iib[-4+p+Δ,-4+Δ,Δ] Bb[-3+p,0],Iib[-4+p+Δ,-4+Δ,Δ] Bb[-2+p,0],Iib[-2+p+Δ,-4+Δ,Δ] Bb[-2+p,0],Iib[-2+p+Δ,-4+Δ,Δ] Bb[-1+p,0],Iib[p+Δ,-4+Δ,Δ]γ[-1+p,0] ,Iib[p+Δ,-4+Δ,Δ]Bb[p,0] },FullSimplify]

We now want to find analytic solutions:

Analytic Solutions

Compute

Before we can calculate analytic anomalous dimensions for any spin truncation L and any scaling dimension (Subscript[Δ, 0]),
we have to evaluate Resultsb[L,(Subscript[Δ, 0])] for the cases we want to find the analytic expressions for. Resultsb[L,(Subscript[Δ, 0])] gives a list of anomalous dimensions (Subscript[γ, n,l]) ×(Subscript[((A^((0)))), n,l]).

Resultsb[0,4];    
Resultsb[0,6];

Resultsb[2,4];
Resultsb[2,6];

Resultsb[4,4];

Checks

We check if the anomalous dimensions given by Resultsb[L,Δ] indeed satisfy that Eqb[L,p,q,Δ] =LHSb[L,p,q,Δ]-RHSb[L,p,q,Δ]=0, where LHSb and RHSb are the left-hand and right-hand side  of the recursion relation derived above

{Eqb[0,4,3,4],Eqb[0,7,8,4],Eqb[0,8,2,4],Eqb[0,12,4,4]}/.Resultsb[0,4]//Simplify

{Eqb[0,4,3,6],Eqb[0,7,8,6],Eqb[0,8,2,6],Eqb[0,12,4,6]}/.Resultsb[0,6]//Simplify

{Eqb[2,7,8,4],Eqb[2,8,2,4],Eqb[2,12,4,4]}/.Resultsb[2,4]//Simplify

{Eqb[2,7,8,6],Eqb[2,8,2,6],Eqb[2,12,4,6]}/.Resultsb[2,6]//Simplify

{Eqb[4,7,8,4],Eqb[4,8,2,4],Eqb[4,12,4,4]}/.Resultsb[4,4]//Simplify

Find Analytic Solutions

AnalyticDimb[L,(Subscript[Δ, 0])] then gives analytic expressions for (Subscript[γ, n,l]) for any L and (Subscript[Δ, 0]) as a list with a separate expression for any l=0,2,...L

AnalyticDimb[L1_,Δ1_]/;(L1//EvenQ):=AnalyticDimb[L1,Δ1]=Module[{L=L1,Δ=Δ1},
Do[
minlab=LabelPosMinb[L,l];
If[CoeffPartSolb[L,Δ,l,minlab]≠{},
{
AΓpt[L,Δ,l,minlab]=(FindSequenceFunction[CoeffPartSolb[L,Δ,l,minlab],n])/.n->n-(-1+minlab)//FullSimplify;  
Γptb[L,Δ,l,minlab]=AΓpt[L,Δ,l,minlab]/Afreeb[Δ,n,l]//FullSimplify;       
},Γptb[L,Δ,l,minlab]=0];  
(*AΓpt denotes the part of the anomalous dimension multiplying the free parameter with label constlabel, it is multiplied by the free OPE coefficient Afreeb we solved for above*)
(*to get Γpt (anomalous dimensions): divide by Afreeb*)

Do[If[CoeffPartSolb[L,Δ,l,constlabel]≠{},
{
Γptb[L,Δ,l,constlabel]=(FindSequenceFunction[CoeffPartSolb[L,Δ,l,constlabel]/CoeffPartSolb[L,Δ,l,minlab],n])/.n->n-(-1+minlab)//FullSimplify;
},Γptb[L,Δ,l,constlabel]=0]
,{constlabel,minlab+1,Length[ConstLabelb[L]]}];

(*combine all parts, schematically like:~ f1[n] cb[0,0]+f2[n] cb[0,2]+f3[n] cb[1,2]..*)
Γb[L,Δ,l]=(Γptb[L,Δ,l,LabelPosMinb[L,l]](Constb[L][[LabelPosMinb[L,l]]][[2]]+Sum[Constb[L][[constlabel]][[2]] Γptb[L,Δ,l,constlabel],{constlabel,LabelPosMinb[L,l]+1,Length[Constb[L]]}]));

,{l,0,L,2}];
Table[Γb[L,Δ,l],{l,0,L,2}]
]
(*AnalyticDimb[L,Δ] gives a list of analytic (Subscript[γ, n,l]) with one expression for any l=0,2,...,L*)
(*In Γb[L,Δ,n,l] we rewrite these results as function of (L,Δ,n,l) for each l separately*)
Γb[L_,Δ_,n1_,l_]/;(l//EvenQ&&l≤L):=Γb[L,Δ,n1,l]=AnalyticDimb[L,Δ][[(l/2)+1]]/.n->n1

some examples:

AnalyticDimb[0,4]

AnalyticDimb[2,4]

AnalyticDimb[4,4]

Some Solutions

In this section we look at the lower-spin truncations in more detail

Spin-0 Truncation, General (Subscript[Δ, 0])

{Γb[0,4,n,0],Γb[0,6,n,0]}

general expression for any (Subscript[Δ, 0]):

Γspin0b[0,Δ_,n_,0]:=((1+n) (2+n) (-2+n+Δ) (-1+n+Δ) (-5+n+2 Δ) (-4+n+2 Δ) (3+4 (-2+Δ) Δ) cb[0,0])/(8 (((-2+Δ))^2) (-1+Δ) (-5+2 n+2 Δ) (-3+2 n+2 Δ) (-1+2 n+2 Δ))

comparing this result to solution from CPW expansion of Dbar[(Subscript[Δ, 0]),(Subscript[Δ, 0]),(Subscript[Δ, 0]),(Subscript[Δ, 0,])x,z] , they agree for the choice of free parameter:
cb[0,0]->-((2 ((((-1+(Subscript[Δ, 0]))!))^3) (Subscript[Δ, 0])!)/((2 (Subscript[Δ, 0]))!))

ParameterChoiceL0b={cb[0,0]->-((2 ((((-1+Δ)!))^3) Δ!)/((2 Δ)!))};

large-n behaviour: (n^3)

(Series[Γspin0b[0,Δ,n,0]/.n->(1/t),{t,0,-3}]//Normal)/.t->(1/n)

Γspin0b[0,Δ,n,0]/.ParameterChoiceL0b/.Δ->4

Checks

Table[{Eqb[0,5,4,Δ],Eqb[0,8,4,Δ],Eqb[0,2,0,Δ],Eqb[0,15,9,Δ]}/.Bb[a_,b_]:>Afreeb[Δ,a,b]Γspin0b[0,Δ,a,b],{Δ,4,12}]

Eqb[0,0,2,Δ]/.Bb[a_,b_]:>Afreeb[Δ,a,b]Γspin0b[0,Δ,a,b]//FullSimplify

Spin-2 Truncation, General (Subscript[Δ, 0])

Γb[2,4,n,0]

Γb[2,6,n,0]

Solutions:
Can write spin-2 solutions in terms of spin-0 solutions as   Γspin2b[2,Δ,n,0]=Γspin0b[0,Δ,n,0] +f(1) cb[0,2]+f(2) cb[1,2], deduce formulas for  general Δ:

Γspin2b[2,Δ_,n_,0]:=((Γspin0b[0,Δ,n,0])/(cb[0,0]))(cb[0,0]+cb[0,2]Γspin2pt1b[Δ,n]+cb[1,2]Γspin2pt2b[Δ,n])

Γspin2b[2,Δ_,n_,2]:=-Γspin0b[0,Δ,n,0](((-1+n) (3+n) (4+n) (n+Δ) (1+n+Δ) (-1+2 Δ) (((1+2 Δ))^2) (3+2 Δ) (-3+n+2 Δ) (-2+n+2 Δ) (n+2 Δ) (cb[0,2]-(4 n Δ (-1+n+2 Δ) (15+16 Δ+4 (Δ^2)) cb[1,2])/((-1+n) (((2+Δ))^2) (n+2 Δ) (-1-Δ+4 (Δ^2)+4 (Δ^3)))))/(4 (-1+Δ) (Δ^4) (((1+Δ))^2) (1+2 n+2 Δ) (3+2 n+2 Δ) (3+4 (-2+Δ) Δ) cb[0,0]))

Γspin2pt1b[Δ_,n_]:=((n (((1+2 Δ))^2) (3+2 Δ) (-3+n+2 Δ))/((-1+Δ) (Δ^4) (((1+Δ))^2) (-3+2 Δ) (-7+2 n+2 Δ) (1+2 n+2 Δ)))(5 (n^6)+15 (n^5) (-3+2 Δ)+(n^3) (-3+2 Δ) (29-22 Δ+78 (Δ^2))+(n^4) (127-161 Δ+89 (Δ^2))+((6 (-1+Δ) (Δ^2) (-7+2 Δ) (-1+5 Δ+12 (Δ^2)+4 (Δ^3)))/(1+2 Δ))+n (-3+2 Δ) (-6-33 Δ-215 (Δ^2)-14 (Δ^3)+48 (Δ^4))+2 (n^2) (-39+117 Δ-107 (Δ^2)-143 (Δ^3)+82 (Δ^4)))
Γspin2pt2b[Δ_,n_]:=((n (1+2 Δ) (((3+2 Δ))^2) (5+2 Δ) (-3+n+2 Δ))/((3-2 Δ) (-1+Δ) (Δ^3) (((1+Δ))^3) (((2+Δ))^2) (-1+2 Δ) (-7+2 n+2 Δ) (1+2 n+2 Δ)))(20 (n^6)+60 (n^5) (-3+2 Δ)+24 (-1+Δ) (Δ^2) (1+Δ) (-7+2 Δ) (-1+2 Δ)+4 (n^3) (-3+2 Δ) (29-98 Δ+78 (Δ^2))+(n^4) (508+4 Δ (-199+89 Δ))+4 n (-3+2 Δ) (-6+Δ (21+Δ (-41+4 Δ (-25+12 Δ))))+8 (n^2) (-39+Δ (-27+2 Δ (104+Δ (-131+41 Δ)))))

large-n behaviour:  ~ (n^9)    (Note that in L-0 truncation: (n^3)- difference is 6 as expected (see paper) ), we also expect a second solution with (n^7) (difference to spin-0 being 4), this is satisfied for the following choice of the free parameters: 
cb[1,2] ->(((1+Δ) (((2+Δ))^2) (-1+2 Δ) (1+2 Δ) )/(4 Δ (3+2 Δ) (5+2 Δ)))cb[0,2]

(Series[{Γspin2b[2,Δ,n,0],Γspin2b[2,Δ,n,2]}/.n->(1/t),{t,0,-9}]//Normal)/.t->(1/n)//Simplify

Solve[((((2+Δ))^2) (-1-Δ+4 (Δ^2)+4 (Δ^3)) cb[0,2]-4 Δ (15+16 Δ+4 (Δ^2)) cb[1,2])==0,cb[1,2]]//Flatten//Factor

(n^7)-solution:

(Series[{Γspin2b[2,Δ,n,0],Γspin2b[2,Δ,n,2]}/.{cb[1,2]->((1+Δ) (((2+Δ))^2) (-1+2 Δ) (1+2 Δ) cb[0,2])/(4 Δ (3+2 Δ) (5+2 Δ))}/.n->(1/t),{t,0,-7}]//Normal)/.t->(1/n)//Simplify

match solutions to those of CPW expansion: one result from 4-derivative solution (~(n^7)) and one from 6-derivative solution (~(n^9)) 
(n^7) solutions from CPW of:  ((Subscript[F, 4])^(spin-2))(u,v)=(1+u+v)Dbar[Δ+1,Δ+1,Δ+1,Δ+1](u,v)
(n^9)  solutions from CPW of :  ((Subscript[F, 6])^(spin-2))(u,v) =Dbar[Δ+2,Δ+1,Δ+2,Δ+1](u,v)+Dbar[Δ+1,Δ+2,Δ+1,Δ+2](u,v)+(u^2)Dbar[Δ+2,Δ+2,Δ+1,Δ+1](u,v)+u Dbar[Δ+2,Δ+2,Δ+1,Δ+1](u,v)+(v^2)Dbar[Δ+1,Δ+2,Δ+2,Δ+1](u,v)+v Dbar[Δ+2,Δ+1,Δ+1,Δ+2](u,v)

ParameterChoiceL2Sol4b={cb[0,0]->-((4 (((Δ!))^3) (1+Δ)!)/((2+2 Δ)!)),cb[0,2]->-((2 Δ! ((((1+Δ)!))^2) (2+Δ)!)/(3 (1+2 Δ) (4+2 Δ)!)),cb[1,2]->-(((1+Δ) (-1+2 Δ) (-1+Δ)! ((((2+Δ)!))^2) (3+Δ)!)/(3 (3+2 Δ) (6+2 Δ)!))};

ParameterChoiceL2Sol6b={cb[0,0]->-((4 (((Δ!))^2) ((((1+Δ)!))^2))/((2+2 Δ)!)),cb[0,2]->-((2 (2+3 Δ) Δ! ((((1+Δ)!))^2) (2+Δ)!)/(3 (1+2 Δ) (4+2 Δ)!)),cb[1,2]->-(((1+Δ) (-2+7 Δ+6 (Δ^2)) (-1+Δ)! ((((2+Δ)!))^2) (3+Δ)!)/(3 (3+2 Δ) (6+2 Δ)!))};

(Series[({Γspin2b[2,Δ,n,0],Γspin2b[2,Δ,n,2]}/.ParameterChoiceL2Sol4b)/.n->(1/t),{t,0,-7}]//Normal//Expand//Simplify)/.t->(1/n)//Expand//Simplify

(Series[({Γspin2b[2,Δ,n,0],Γspin2b[2,Δ,n,2]}/.ParameterChoiceL2Sol6b)/.n->(1/t),{t,0,-9}]//Normal//Expand//Simplify)/.t->(1/n)

gives expected large-n behaviour

Checks:

Table[Γspin2b[2,Δ,n,0],{Δ,4,10,2}]-Table[Γb[2,Δ,n,0],{Δ,4,10,2}]//Simplify

Table[Γspin2b[2,Δ,n,2],{Δ,4,10,2}]-Table[Γb[2,Δ,n,2],{Δ,4,10,2}]//Simplify

Table[{Eqb[2,5,4,Δ],Eqb[2,8,4,Δ],Eqb[2,2,0,Δ],Eqb[2,15,9,Δ]}/.Bb[a_,b_]:>Afreeb[Δ,a,b]Γspin2b[2,Δ,a,b],{Δ,4,12}]//Simplify

Spin-4 Truncation ((Subscript[Δ, 0])=4)

could look at any value of (Subscript[Δ, 0]) and find general expression as well, as an example look at (Subscript[Δ, 0])=4 here

Γb[4,4,n,0]

Γb[4,4,n,2]

Γb[4,4,n,4]/.GammaReplace

Γspin4b[4,4,n_,0]:=(1/(96 (3+2 n) (5+2 n) (7+2 n)))35 (1+n) (((2+n))^2) (((3+n))^2) (4+n) (cb[0,0]+(297 n (5+n) (14944+n (5+n) (7814+n (5+n) (532+5 n (5+n)))) cb[0,2])/(32000 (1+2 n) (9+2 n))+(143 n (5+n) (-165975699456+n (5+n) (-45499938048+n (5+n) (21506397312+n (5+n) (4872020768+n (5+n) (431742944+65 n (5+n) (232500+n (5+n) (2356+7 n (5+n)))))))) cb[0,4])/(526848000 (-1+2 n) (1+2 n) (9+2 n) (11+2 n))-(1573 n (5+n) (2016+n (5+n) (1062+n (5+n) (76+n (5+n)))) cb[1,2])/(168000 (1+2 n) (9+2 n))-(31603 n (5+n) (843042816+n (5+n) (-869294592+n (5+n) (-353008512+n (5+n) (134612832+n (5+n) (22247576+5 n (5+n) (178860+n (5+n) (2106+7 n (5+n)))))))) cb[1,4])/(36415733760 (-1+2 n) (1+2 n) (9+2 n) (11+2 n))+(785213 (-1+n) n (5+n) (6+n) (-145152000+n (5+n) (-30064896+n (5+n) (21308544+n (5+n) (2899616+n (5+n) (107916+5 n (5+n) (264+n (5+n))))))) cb[2,4])/(614515507200 (-1+2 n) (1+2 n) (9+2 n) (11+2 n)))
Γspin4b[4,4,n_,2]:=-(1/(819200 (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n)))693 (-1+n) (1+n) (((2+n))^2) (((3+n))^3) (((4+n))^3) (((5+n))^2) (6+n) (8+n) (cb[0,2]+(13 n (7+n) (880548+13 n (7+n) (25750+n (7+n) (1532+7 n (7+n)))) cb[0,4])/(277830 (1+2 n) (13+2 n))-((572 n (7+n) cb[1,2])/(2835 (-1+n) (8+n)))-(14365 n (7+n) (92772+n (7+n) (33030+n (7+n) (1402+7 n (7+n)))) cb[1,4])/(96018048 (1+2 n) (13+2 n))+(71383 n (7+n) (13500+n (7+n) (4722+n (7+n) (178+n (7+n)))) cb[2,4])/(324060912 (1+2 n) (13+2 n)))
Γspin4b[4,4,n_,4]:=(1859 (-2+n) (-1+n) (1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (7+n) (10+n) (11+n) (cb[0,4]-((425 n (9+n) cb[1,4])/(1728 (-10+9 n+(n^2))))+((27455 n (9+n) cb[2,4])/(530712 (-2+n) (11+n)))) (7+n)! (2 (7+n))! (8+n)! Gamma[9+n] Gamma[3+2 n])/(56448000 (14+2 n)! (16+2 n)! Gamma[2+n] Gamma[3+n] Gamma[8+n])

checks:

{Eqb[4,5,4,4],Eqb[4,8,4,4],Eqb[4,2,0,4]}/.Bb[a_,b_]:>Afreeb[4,a,b]Γspin4b[4,4,a,b]//Simplify

Summary: L=0, 2 Solutions for General (Subscript[Δ, 0])

Γspin”L”[L-maximal spin in truncation, (Subscript[Δ, 0]),n,l]

Γspin0b[0,Δ_,n_,0]:=((1+n) (2+n) (-2+n+Δ) (-1+n+Δ) (-5+n+2 Δ) (-4+n+2 Δ) (3+4 (-2+Δ) Δ) cb[0,0])/(8 (((-2+Δ))^2) (-1+Δ) (-5+2 n+2 Δ) (-3+2 n+2 Δ) (-1+2 n+2 Δ))

Γspin2b[2,Δ_,n_,0]:=((Γspin0b[0,Δ,n,0])/(cb[0,0]))(cb[0,0]+cb[0,2]Γspin2pt1b[Δ,n]+cb[1,2]Γspin2pt2b[Δ,n])

Γspin2b[2,Δ_,n_,2]:=-Γspin0b[0,Δ,n,0](((-1+n) (3+n) (4+n) (n+Δ) (1+n+Δ) (-1+2 Δ) (((1+2 Δ))^2) (3+2 Δ) (-3+n+2 Δ) (-2+n+2 Δ) (n+2 Δ) (cb[0,2]-(4 n Δ (-1+n+2 Δ) (15+16 Δ+4 (Δ^2)) cb[1,2])/((-1+n) (((2+Δ))^2) (n+2 Δ) (-1-Δ+4 (Δ^2)+4 (Δ^3)))))/(4 (-1+Δ) (Δ^4) (((1+Δ))^2) (1+2 n+2 Δ) (3+2 n+2 Δ) (3+4 (-2+Δ) Δ) cb[0,0]))

Γspin2pt1b[Δ_,n_]:=((n (((1+2 Δ))^2) (3+2 Δ) (-3+n+2 Δ))/((-1+Δ) (Δ^4) (((1+Δ))^2) (-3+2 Δ) (-7+2 n+2 Δ) (1+2 n+2 Δ)))(5 (n^6)+15 (n^5) (-3+2 Δ)+(n^4) (127-161 Δ+89 (Δ^2))+(n^3) (-3+2 Δ) (29-22 Δ+78 (Δ^2))+2 (n^2) (-39+117 Δ-107 (Δ^2)-143 (Δ^3)+82 (Δ^4))+n (-3+2 Δ) (-6-33 Δ-215 (Δ^2)-14 (Δ^3)+48 (Δ^4))+((6 (-1+Δ) (Δ^2) (-7+2 Δ) (-1+5 Δ+12 (Δ^2)+4 (Δ^3)))/(1+2 Δ)))
Γspin2pt2b[Δ_,n_]:=((n (1+2 Δ) (((3+2 Δ))^2) (5+2 Δ) (-3+n+2 Δ))/((3-2 Δ) (-1+Δ) (Δ^3) (((1+Δ))^3) (((2+Δ))^2) (-1+2 Δ) (-7+2 n+2 Δ) (1+2 n+2 Δ)))(20 (n^6)+60 (n^5) (-3+2 Δ)+(n^4) 4 (127-199 Δ+89 (Δ^2))+4 (n^3) (-3+2 Δ) (29-98 Δ+78 (Δ^2))+8 (n^2) (-39+Δ (-27+2 Δ (104+Δ (-131+41 Δ))))+4 n (-3+2 Δ) (-6+Δ (21+Δ (-41+4 Δ (-25+12 Δ))))+24 (-1+Δ) (Δ^2) (1+Δ) (-7+2 Δ) (-1+2 Δ))


(*choices of free parameters to match the results to solutions from CPW expansion*)

ParameterChoiceL0b={cb[0,0]->-((2 ((((-1+Δ)!))^3) Δ!)/((2 Δ)!))};

ParameterChoiceL2Sol4b={cb[0,0]->-((4 (((Δ!))^3) (1+Δ)!)/((2+2 Δ)!)),cb[0,2]->-((2 Δ! ((((1+Δ)!))^2) (2+Δ)!)/(3 (1+2 Δ) (4+2 Δ)!)),cb[1,2]->-(((1+Δ) (-1+2 Δ) (-1+Δ)! ((((2+Δ)!))^2) (3+Δ)!)/(3 (3+2 Δ) (6+2 Δ)!))};

ParameterChoiceL2Sol6b={cb[0,0]->-((4 (((Δ!))^2) ((((1+Δ)!))^2))/((2+2 Δ)!)),cb[0,2]->-((2 (2+3 Δ) Δ! ((((1+Δ)!))^2) (2+Δ)!)/(3 (1+2 Δ) (4+2 Δ)!)),cb[1,2]->-(((1+Δ) (-2+7 Δ+6 (Δ^2)) (-1+Δ)! ((((2+Δ)!))^2) (3+Δ)!)/(3 (3+2 Δ) (6+2 Δ)!))};

1.6   OPE Coefficients (Subscript[((A^((1)))), n,l])

Analytic Formulas for (Subscript[γ, n,l]) (L=0,2)

evaluate this to compute first order OPE coefficients

(*results from above*)

Afreeb[Δ_,n_,l_]:=((2+l)Pochhammer[1+n,Δ-3] Pochhammer[3+l+n,Δ-3](2Δ-3+l+2 n) (2Δ-2+l+2 n) (Δ-3+n)! (Δ-1+ (l+n))! Gamma[2Δ-5+n] Gamma[2Δ-3+l+n])/((1/2) ((Gamma[-2+Δ])^2) ((Gamma[Δ])^2) (2Δ-6+2 n)! (2Δ-2+2 (l+n))!)  


Γ1b[0,Δ_,n_,0]:=(((1+n) (2+n) (-2+n+Δ) (-1+n+Δ) (-5+n+2 Δ) (-4+n+2 Δ) (3+4 (-2+Δ) Δ) cb[0,0])/(8 (((-2+Δ))^2) (-1+Δ) (-5+2 n+2 Δ) (-3+2 n+2 Δ) (-1+2 n+2 Δ)))   (*renamed here to keep it general for computation of OPE coeffs, spin0 and spin2 solutions together in Γ1*)

Γ1b[0,Δ_,n_,l_]/;(l≠0):=0

(*Γ1[2,Δ_,n_,2]:=((-Γ1b[0,Δ,n,0])/( cb[0,0]))(((-1+n) (3+n) (4+n) (n+Δ) (1+n+Δ) (-1+2 Δ) (((1+2 Δ))^2) (3+2 Δ) (-3+n+2 Δ) (-2+n+2 Δ) (n+2 Δ) (cb[0,2]-((4 n Δ (-1+n+2 Δ) (15+16 Δ+4 (Δ^2)) cb[1,2])/((-1+n) (((2+Δ))^2) (n+2 Δ) (-1-Δ+4 (Δ^2)+4 (Δ^3))))))/(4 (-1+Δ) (Δ^4) (((1+Δ))^2) (1+2 n+2 Δ) (3+2 n+2 Δ) (3+4 (-2+Δ) Δ)))*)
(*expand this here to avoid poles*)

Γ1b[2,Δ_,n_,2]:=-(((24+50 n+35 (n^2)+10 (n^3)+(n^4)) (1+2 Δ) (3+2 Δ) ((n^8)+4 (n^7) (-4+3 Δ)+2 (n^6) (49-81 Δ+31 (Δ^2))+4 Δ (((2-3 Δ+(Δ^2)))^2) (15-Δ-12 (Δ^2)+4 (Δ^3))+10 (n^5) (-28+81 Δ-69 (Δ^2)+18 (Δ^3))+(n^4) (329-1788 Δ+2729 (Δ^2)-1602 (Δ^3)+321 (Δ^4))+2 (n^3) (28+699 Δ-2220 (Δ^2)+2400 (Δ^3)-1095 (Δ^4)+180 (Δ^5))+(n^2) (-428+606 Δ+2065 (Δ^2)-5370 (Δ^3)+4655 (Δ^4)-1764 (Δ^5)+248 (Δ^6))+2 n (120-582 Δ+593 (Δ^2)+612 (Δ^3)-1585 (Δ^4)+1182 (Δ^5)-388 (Δ^6)+48 (Δ^7))) (-2 Δ (((2+Δ))^2) (-1-Δ+4 (Δ^2)+4 (Δ^3)) cb[0,2]+(n^2) ((((2+Δ))^2) (-1-Δ+4 (Δ^2)+4 (Δ^3)) cb[0,2]-4 Δ (15+16 Δ+4 (Δ^2)) cb[1,2])+n (-1+2 Δ) ((((2+Δ))^2) (-1-Δ+4 (Δ^2)+4 (Δ^3)) cb[0,2]-4 Δ (15+16 Δ+4 (Δ^2)) cb[1,2])))/(32 (((-2+Δ))^2) (((-1+Δ))^2) (Δ^4) (((1+Δ))^3) (((2+Δ))^2) (-5+2 n+2 Δ) (-3+2 n+2 Δ) (-1+2 n+2 Δ) (1+2 n+2 Δ) (3+2 n+2 Δ)))

Γ1b[2,Δ_,n_,0]:=((Γ1b[0,Δ,n,0])/(cb[0,0]))(cb[0,0]+cb[0,2]Γspin2pt1b[Δ,n]+cb[1,2]Γspin2pt2b[Δ,n])
Γspin2pt1b[Δ_,n_]:=((n (((1+2 Δ))^2) (3+2 Δ) (-3+n+2 Δ))/((-1+Δ) (Δ^4) (((1+Δ))^2) (-3+2 Δ) (-7+2 n+2 Δ) (1+2 n+2 Δ)))(5 (n^6)+15 (n^5) (-3+2 Δ)+(n^3) (-3+2 Δ) (29-22 Δ+78 (Δ^2))+(n^4) (127-161 Δ+89 (Δ^2))+((6 (-1+Δ) (Δ^2) (-7+2 Δ) (-1+5 Δ+12 (Δ^2)+4 (Δ^3)))/(1+2 Δ))+n (-3+2 Δ) (-6-33 Δ-215 (Δ^2)-14 (Δ^3)+48 (Δ^4))+2 (n^2) (-39+117 Δ-107 (Δ^2)-143 (Δ^3)+82 (Δ^4)))
Γspin2pt2b[Δ_,n_]:=((n (1+2 Δ) (((3+2 Δ))^2) (5+2 Δ) (-3+n+2 Δ))/((3-2 Δ) (-1+Δ) (Δ^3) (((1+Δ))^3) (((2+Δ))^2) (-1+2 Δ) (-7+2 n+2 Δ) (1+2 n+2 Δ)))(20 (n^6)+60 (n^5) (-3+2 Δ)+24 (-1+Δ) (Δ^2) (1+Δ) (-7+2 Δ) (-1+2 Δ)+4 (n^3) (-3+2 Δ) (29-98 Δ+78 (Δ^2))+(n^4) (508+4 Δ (-199+89 Δ))+4 n (-3+2 Δ) (-6+Δ (21+Δ (-41+4 Δ (-25+12 Δ))))+8 (n^2) (-39+Δ (-27+2 Δ (104+Δ (-131+41 Δ)))))

Γ1b[2,Δ_,n_,l_]/;(l≠0&&l≠2):=0


(*L=4, Δ=4*)
Γ1b[4,4,n_,0]:=((4+5 n+(n^2)) (((6+5 n+(n^2)))^2) (24580620288000 (-99-40 n+392 (n^2)+160 (n^3)+16 (n^4)) c[0,0]+11 n (5+n) (20739898368 (-164384-130890 n+608922 (n^2)+513165 (n^3)+193379 (n^4)+41095 (n^5)+5073 (n^6)+400 (n^7)+20 (n^8)) cb[0,2]+13 (46656 (-165975699456-227499690240 n+492159994752 (n^2)+824066569120 (n^3)+656747294912 (n^4)+336178346020 (n^5)+119252837368 (n^6)+30432405755 (n^7)+5695321069 (n^8)+790521875 (n^9)+82493125 (n^10)+6584825 (n^11)+392015 (n^12)+15925 (n^13)+455 (n^14)) cb[0,4]-1609445376 (-22176-18090 n+81682 (n^2)+70745 (n^3)+27887 (n^4)+6395 (n^5)+893 (n^6)+80 (n^7)+4 (n^8)) cb[1,2]-149175 (843042816-4346472960 n-9694507392 (n^2)+13296518880 (n^3)+23647688888 (n^4)+15937667980 (n^5)+6430967982 (n^6)+1762998395 (n^7)+348369451 (n^8)+50979375 (n^9)+5608675 (n^10)+469025 (n^11)+28905 (n^12)+1225 (n^13)+35 (n^14)) cb[1,4]+219640 (870912000+176186880 n-3912666624 (n^2)-1090305600 (n^3)+1542972440 (n^4)+1497210220 (n^5)+674372598 (n^6)+194615525 (n^7)+39891745 (n^8)+6053025 (n^9)+693121 (n^10)+60575 (n^11)+3915 (n^12)+175 (n^13)+5 (n^14)) cb[2,4]))))/(67421129932800 (-1+2 n) (1+2 n) (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n))
Γ1b[4,4,n_,2]:=-((11 (6+7 n+(n^2)) (((10+7 n+(n^2)))^2) (((12+7 n+(n^2)))^3) (12962436480 (-104-133 n+177 (n^2)+56 (n^3)+4 (n^4)) cb[0,2]+13 n (7+n) (46656 (-7044384-12582164 n+6798226 (n^2)+9037392 (n^3)+3214549 (n^4)+527800 (n^5)+45942 (n^6)+2548 (n^7)+91 (n^8)) cb[0,4]-201180672 (13+28 n+4 (n^2)) cb[1,2]-85 (-8+7 n+(n^2)) (1755 (92772+231210 n+101728 (n^2)+22029 (n^3)+2431 (n^4)+147 (n^5)+7 (n^6)) cb[1,4]-2584 (13500+33054 n+13444 (n^2)+2835 (n^3)+325 (n^4)+21 (n^5)+(n^6)) cb[2,4]))))/(168552824832000 (1+2 n) (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n)))
Γ1b[4,4,n_,4]:=(1859 (-2+n) (-1+n) (1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (7+n) (10+n) (11+n) (cb[0,4]-((425 n (9+n) cb[1,4])/(1728 (-10+9 n+(n^2))))+((27455 n (9+n) cb[2,4])/(530712 (-2+n) (11+n)))) (7+n)! (2 (7+n))! (8+n)! Gamma[9+n] Gamma[3+2 n])/(56448000 (14+2 n)! (16+2 n)! Gamma[2+n] Gamma[3+n] Gamma[8+n])

Γ1b[4,Δ_,n_,l_]/;(l≠0&&l≠2&&l≠4):=0

(*choices of free parameters to match the results from recursion and CPW expansion*)

ParameterChoiceL0b={cb[0,0]->-((2 ((((-1+Δ)!))^3) Δ!)/((2 Δ)!))};

ParameterChoiceL2Sol4b={cb[0,0]->-((4 (((Δ!))^3) (1+Δ)!)/((2+2 Δ)!)),cb[0,2]->-((2 Δ! ((((1+Δ)!))^2) (2+Δ)!)/(3 (1+2 Δ) (4+2 Δ)!)),cb[1,2]->-(((1+Δ) (-1+2 Δ) (-1+Δ)! ((((2+Δ)!))^2) (3+Δ)!)/(3 (3+2 Δ) (6+2 Δ)!))};

ParameterChoiceL2Sol6b={cb[0,0]->-((4 (((Δ!))^2) ((((1+Δ)!))^2))/((2+2 Δ)!)),cb[0,2]->-((2 (2+3 Δ) Δ! ((((1+Δ)!))^2) (2+Δ)!)/(3 (1+2 Δ) (4+2 Δ)!)),cb[1,2]->-(((1+Δ) (-2+7 Δ+6 (Δ^2)) (-1+Δ)! ((((2+Δ)!))^2) (3+Δ)!)/(3 (3+2 Δ) (6+2 Δ)!))};

OPE Coefficients

find ((Subscript[A, n,l])^((1))) from ((Subscript[A, n,l])^((1)))=(1/2)(∂_n)(((Subscript[A, n,l])^((0)))(Subscript[γ, n,l]))

lvalues[L_]:=Table[i,{i,0,L,2}]
A1b[L_,Δ_,n_,l_]/;(MemberQ[lvalues[L],l]):=(((D[(Afreeb[Δ,r,l] Γ1b[L,Δ,r,l]),r])/.r->n//FullSimplify))
A1b[L_,Δ_,n_,l_]/;(FreeQ[lvalues[L],l]):=0

OPECoeffsb[L_,Δ_,Max_]:=Table[A1b[L,Δ,i,j],{i,0,Max},{j,0,Max}]//Flatten

OPECoeffsb[0,4,5]

OPECoeffsb[0,4,5]/.ParameterChoiceL0b/.Δ->4

OPECoeffsb[2,6,2]

OPECoeffsb[2,6,2]/.ParameterChoiceL2Sol4b/.Δ->6

OPECoeffsb[2,6,2]/.ParameterChoiceL2Sol6b/.Δ->6

OPECoeffsb[4,4,2]

2. (2,0) Theory

Can evaluate ‘Definitions’ cell and go directly to ‘Solve for Anomalous Dimensions (Subscript[γ, n,l]) for General Spin-L Truncation’

2.1   Definitions

(*Note that we will use the variables (x,z) instead of (z,zb) compared to the paper*)
(*conformal blocks, given as (Subscript[((G^DO)), Δ,l]) in appendix A of the paper:*)
G[Δ_,l_,x_,z_,Δ12_,Δ34_]:=(F[0,0,x,z,Δ,l,Δ12,Δ34]-((l+3)/(l+1))F[-1,1,x,z,Δ,l,Δ12,Δ34]-((Δ-4)/(Δ-2))(((Δ+l-Δ12)(Δ+l+Δ12)(Δ+l+Δ34)(Δ+l-Δ34))/(16(Δ+l-1)(((Δ+l))^2)(Δ+l+1)))F[1,1,x,z,Δ,l,Δ12,Δ34]+(((Δ-4)(l+3))/((Δ-2)(l+1)))(((Δ-l-Δ12-4) (Δ-l+Δ12-4) (Δ-l+Δ34-4) (Δ-l-Δ34-4))/(16(Δ-l-5)(((Δ-l-4))^2)(Δ-l-3)))F[0,2,x,z,Δ,l,Δ12,Δ34]+2(Δ-4)(l+3)((Δ12 Δ34)/((Δ+l)(Δ+l-2)(Δ-l-4)(Δ-l-6)))F[0,1,x,z,Δ,l,Δ12,Δ34])
F[a_,b_,x_,z_,Δ_,l_,Δ12_,Δ34_]:=(((((x z))^((1/2)(Δ-l)))/(((x-z))^3))((((((-1)/2)x))^l)(x^(a+3))(z^b) f[(1/2)(Δ+l-Δ12)+a,(1/2)(Δ+l+Δ34)+a,Δ+l+2a,x] f[(1/2)(Δ-l-Δ12)-3+b,(1/2)(Δ-l+Δ34)-3+b,Δ-l-6+2b,z]-((((((-1)/2)z))^l)(z^(a+3))(x^b) f[(1/2)(Δ+l-Δ12)+a,(1/2)(Δ+l+Δ34)+a,Δ+l+2a,z] f[(1/2)(Δ-l-Δ12)-3+b,(1/2)(Δ-l+Δ34)-3+b,Δ-l-6+2b,x])))//Simplify        (*f are 2F1 hypergeometrics*)

(*We add a 's' to function and variable names in the following, s for the supersymmetric theory*)

Bs[a_,l_]/;(a<0):=0          (*Bs=anomalous dimension × (A^((0))), s for Susy*)

(*conventions for writing the hypergeometrics*)
η[β_,z_]:=Hypergeometric2F1[(β/2),(β/2)-1,β,z]
ηt[β_,z_]:=((Gamma[β])/(Gamma[(β/2)]Gamma[(β/2)-1]))Hypergeometric2F1[(β/2)+1,(β/2),2,z]

(*definition of inner product we will use for derivation of recursion relation*)
IIs[m_,m1_]:=Residue[((((1-x))^(m-3))/(x^(m1-1)))ηt[2m,x] η[-2m1,x],{x,0}]

(*(A^((0))), is derived in following section*)

Afrees[n_,l_]:=(((2+l) (9+l+2 n) (10+l+2 n) Gamma[4+n] Gamma[5+n] Gamma[6+l+n] Gamma[7+l+n])/(72 Gamma[2 (5+l+n)] Gamma[6+2 n]))            (*for l...even, l...odd: Afrees=0*)

2.2   Free Disconnected Contribution (Subscript[((A^((0)))), n,l])

Fs[x_,z_]:=A(1+(1/(((x z))^2))+(1/((((1-x)(1-z)))^2)))                    (*prepotential corresponding to free disconnected part of 4-pt correlator*)

Gs[x_,z_]=((1/(x-z))(Fs[x,z]-(A/(((x z))^2)))//Simplify)/.A->1     (*perform CPW expansion of this to find leading contribution to OPE coefficients*)

max=5;
Κ[n_]:=If[n==0,1,0]    (*note: extra term for n=0*)
G1s[x_,z_]:=Sum[C[m,n](-Κ[n]((1+m)/(3+m)) (1/(x z))(  (z^(3+m)) Hypergeometric2F1[3+m,4+m,8+2 m,z]-(x^(3+m)) Hypergeometric2F1[3+m,4+m,8+2 m,x])+(1/((3+m) (1+n) (x^5) (z^5)))((((-1))^(1+m-n)) (2^(m-n)) (1+m-n) (((x-z))^3) G[8+m+n,m-n,x,z,0,-2]/.f->Hypergeometric2F1)) ,{n,0,max+1},{m,n,max+1}];
EqGs=Series[G1s[x,z]-(((x-z))^2)Gs[x,z],{x,0,max},{z,0,max}]//Normal;

(*The free OPE coefficients are calculated from the CPW expansion and given by CoeffLists in terms of m and n*)
(*m and n are related to the scaling dimension Δ and the spin l as: Δ=2n+l+8 and l=m-n *)
CoeffLists=SolveAlways[EqGs==0,{x,z}]//Flatten

CoeffLists/.C[m_,n_]->C[n,m-n]       (*rewrite in terms of n and spin l=m-n*)

compare to solutions from reference 1712.08570

AfreeRef[n_,l_]:=((((2+l) (5+l+2 n) (6+l+2 n) Gamma[2+n] Gamma[3+n] Gamma[4+l+n] Gamma[5+l+n]))/((72 Gamma[2 (3+l+n)] Gamma[2+2 n])))

Table[{C[n,l],AfreeRef[n,l]},{n,0,4},{l,0,4,2}]   (*agrees with results found above*)

due to our conventions that n≥0, we need to shift n-> n+2

AfreeRef[n,l]/.n->n+2//Simplify//Factor

Result:

Afrees[n_,l_]:=(((2+l) (9+l+2 n) (10+l+2 n) Gamma[4+n] Gamma[5+n] Gamma[6+l+n] Gamma[7+l+n])/(72 Gamma[2 (5+l+n)] Gamma[6+2 n]))

2.3   Crossing Equation

we can write the supersymmetric conformal blocks in terms of the bosonic ones as: (((((-1))^(1+l))(2^l)(1+l))/((3+n)(5+l+n)))((((x-z))^3)/(((x z))^5))G[2 n+l+12,l,x,z,0,-2]

(*LHSs[i]/RHSs[i] denote the left-hand/right-hand side of the crossing equation*)

LHSs[1]=Coefficient[Coefficient[D[(1/((x-z)))(((((-1))^(1+l)) (2^l) (1+l))/((3+n) (5+l+n)))((((x-z))^3)/(((x z))^5))G[2 n+l+12,l,x,z,0,-2]/.f[a__,b__,c__,z]->ht[c,1-z](1-z) Log[1-z]/.f[a__,b__,c__,x]->h[c,x],n],Log[x z]],Log[1-z]]/.h[a__,b__]:>h[a//Expand,b//Expand]/.ht[a__,b__]:>ht[a//Expand,b//Expand];
RHSs[1]=Coefficient[Coefficient[D[(1/((-x+z)))(((((-1))^(1+l)) (2^l) (1+l))/((3+n) (5+l+n)))((((-x+z))^3)/((((1-x)(1-z)))^5))G[2n+l+12,l,1-x,1-z,0,-2]/.f[a__,b__,c__,1-x]->ht[c,x]x Log[x]/.f[a__,b__,c__,1-z]->h[c,1-z],n],Log[(-1+x)(-1+z)]],Log[x]]/.h[a__,b__]:>h[a//Expand,b//Expand]/.ht[a__,b__]:>ht[a//Expand,b//Expand];


LHSs[2]=(List@@Collect[LHSs[1],{h[8+2 n,x] ht[10+2 l+2 n,1-z],h[6+2 n,x] ht[12+2 l+2 n,1-z],h[10+2 n,x] ht[12+2 l+2 n,1-z],h[8+2 n,x] ht[14+2 l+2 n,1-z],h[12+2 l+2 n,x] ht[6+2 n,1-z],h[10+2 l+2 n,x] ht[8+2 n,1-z],h[14+2 l+2 n,x] ht[8+2 n,1-z],h[12+2 l+2 n,x] ht[10+2 n,1-z]}])(x-z)//Simplify;
RHSs[2]=(List@@Collect[RHSs[1],{h[6+2 n,1-z] ht[12+2 l+2 n,x],h[12+2 l+2 n,1-z] ht[6+2 n,x],h[8+2 n,1-z] ht[10+2 l+2 n,x],h[10+2 l+2 n,1-z] ht[8+2 n,x],h[8+2 n,1-z] ht[14+2 l+2 n,x],h[14+2 l+2 n,1-z] ht[8+2 n,x],h[10+2 n,1-z] ht[12+2 l+2 n,x],h[12+2 l+2 n,1-z] ht[10+2 n,x]}])(x-z)//Simplify;

2.4   Recursion Relation

We derive the recursion relation as explained in Sec. 3.1 of the paper, first multiply the crossing equation by a product of hypergeometrics, then use the orthogonality relation
(Subscript[δ, m,m'])=∮(dz/(2π i))((z^(m-m'-1))/((1-z)))(Subscript[h, 2m+4])(z)(((Subscript[h, -2m'-2])(z))^( )) and the inner product:  (Subscript[I, m,m'])=∮(dz/(2π i))((((1-z))^(m-3))/(z^(m'-1)))(Subscript[ht, 2m])(z)(((Subscript[h, -2m'])(z))^( ))

LHSs[3]=LHSs[2]Bs[n,l](((h[-2q,x])/((x^q)(1-x)))*((h[-2p,1-z])/((((1-z))^p) z)));
RHSs[3]=RHSs[2]Bs[n,l](((h[-2q,x])/((x^q)(1-x)))*((h[-2p,1-z])/((((1-z))^p)z)));
LHSs[4]=Assuming[n∈Integers&&l∈Integers,Simplify[(LHSs[3]/.h[a__,c__] h[b__,c__]->(δ[((a-4)/2),((b+2)/(-2))](((((c^(((a-4)/2)-((b+2)/(-2))-1))/(1-c))))^(-1)))/.ht[a__,c__]h[b__,c__]->(Iis[(a/2),-(b/2)](((((((1-c))^((a/2)-3))/(c^(-(b/2)-1)))))^(-1)))//PowerExpand//Simplify)/.(((1-c__))^(a__))->(((-1))^a)(((-1+c))^a)//Expand]];
RHSs[4]=Assuming[n∈Integers&&l∈Integers,Simplify[(RHSs[3]/.h[a__,c__] h[b__,c__]->(δ[((a-4)/2),((b+2)/(-2))](((((c^(((a-4)/2)-((b+2)/(-2))-1))/(1-c))))^(-1)))/.ht[a__,c__]h[b__,c__]->(Iis[(a/2),-(b/2)](((((((1-c))^((a/2)-3))/(c^(-(b/2)-1)))))^(-1)))//PowerExpand//Simplify)/.(((1-c__))^(a__))->(((-1))^a)(((-1+c))^a)//Expand]];
(*shift (p,q) by +2 to get minimal case for (p,q)=(0,0)*)
LHSs[5]=((List@@Collect[LHSs[4]/.List->Plus,{δ[1+n,-1+q],δ[2+n,-1+q],δ[3+n,-1+q],δ[3+l+n,-1+q],δ[4+l+n,-1+q],δ[5+l+n,-1+q]}])//Expand//Simplify)/.q->q+2/.p->p+2;
RHSs[5]=((List@@Collect[RHSs[4]/.List->Plus,{δ[1+n,-1+p],δ[2+n,-1+p],δ[3+n,-1+p],δ[3+l+n,-1+p],δ[4+l+n,-1+p],δ[5+l+n,-1+p]}])//Expand//Simplify)/.q->q+2/.p->p+2;

Collect[LHSs[5]/.List->Plus//Expand,{δ[1+n,1+q],Iis[5+l+n,2+p] δ[2+n,1+q],Iis[7+l+n,2+p] δ[2+n,1+q],δ[3+n,1+q],δ[3+l+n,1+q],Iis[3+n,2+p] δ[4+l+n,1+q],Iis[5+n,2+p] δ[4+l+n,1+q],Iis[4+n,2+p] δ[5+l+n,1+q]},FullSimplify]

Collect[RHSs[5]/.List->Plus//Expand,{δ[1+n,1+p],Iis[5+l+n,2+q] δ[2+n,1+p],Iis[7+l+n,2+q] δ[2+n,1+p],δ[3+n,1+p],δ[3+l+n,1+p],Iis[3+n,2+q] δ[4+l+n,1+p],Iis[5+n,2+q] δ[4+l+n,1+p],Iis[4+n,2+q] δ[5+l+n,1+p]},FullSimplify]

2.5   Solve for Anomalous Dimensions (Subscript[γ, n,l]) for General Spin-L Truncation

Find Solutions as Lists of (Subscript[γ, n,l])-Values

Recursion Relation we derived above (we have (Subscript[Σ, n=0]^∞)(Subscript[Σ, l=0]^L) with l even):

LSs[0]=(((1+l) Bs[n,l] Iis[6+l+n,2+p] δ[1+n,1+q])/((3+n) (5+l+n)))-(((3+l) Bs[n,l] Iis[5+l+n,2+p] δ[2+n,1+q])/((3+n) (5+l+n)))-(((1+l) (7+l+n) (8+l+2 n) Bs[n,l] Iis[7+l+n,2+p] δ[2+n,1+q])/(4 (3+n) (11+2 l+2 n) (13+2 l+2 n) (l+2 (5+n))))+(((3+l) (5+n) (8+l+2 n) Bs[n,l] Iis[6+l+n,2+p] δ[3+n,1+q])/(4 (5+l+n) (7+2 n) (9+2 n) (l+2 (5+n))))+(((3+l) Bs[n,l] Iis[4+n,2+p] δ[3+l+n,1+q])/((3+n) (5+l+n)))-(((1+l) Bs[n,l] Iis[3+n,2+p] δ[4+l+n,1+q])/((3+n) (5+l+n)))-(((3+l) (5+n) (8+l+2 n) Bs[n,l] Iis[5+n,2+p] δ[4+l+n,1+q])/(4 (5+l+n) (7+2 n) (9+2 n) (l+2 (5+n))))+(((1+l) (7+l+n) (8+l+2 n) Bs[n,l] Iis[4+n,2+p] δ[5+l+n,1+q])/(4 (3+n) (11+2 l+2 n) (13+2 l+2 n) (l+2 (5+n))));
RSs[0]=(((1+l) Bs[n,l] Iis[6+l+n,2+q] δ[1+n,1+p])/((3+n) (5+l+n)))-(((3+l) Bs[n,l] Iis[5+l+n,2+q] δ[2+n,1+p])/((3+n) (5+l+n)))-(((1+l) (7+l+n) (8+l+2 n) Bs[n,l] Iis[7+l+n,2+q] δ[2+n,1+p])/(4 (3+n) (11+2 l+2 n) (13+2 l+2 n) (l+2 (5+n))))+(((3+l) (5+n) (8+l+2 n) Bs[n,l] Iis[6+l+n,2+q] δ[3+n,1+p])/(4 (5+l+n) (7+2 n) (9+2 n) (l+2 (5+n))))+(((3+l) Bs[n,l] Iis[4+n,2+q] δ[3+l+n,1+p])/((3+n) (5+l+n)))-(((1+l) Bs[n,l] Iis[3+n,2+q] δ[4+l+n,1+p])/((3+n) (5+l+n)))-(((3+l) (5+n) (8+l+2 n) Bs[n,l] Iis[5+n,2+q] δ[4+l+n,1+p])/(4 (5+l+n) (7+2 n) (9+2 n) (l+2 (5+n))))+(((1+l) (7+l+n) (8+l+2 n) Bs[n,l] Iis[4+n,2+q] δ[5+l+n,1+p])/(4 (3+n) (11+2 l+2 n) (13+2 l+2 n) (l+2 (5+n))));

Solve for anomalous dimensions for general spin-L truncation:

(*truncate spin to L and evaluate the Kronecker-deltas*)
LHS1s[L_]/;(L//EvenQ):=Module[{L1=L},
lhs[1]=Sum[LSs[0]/.l->l1,{l1,0,L1,2}]/.δ[a__,b__]->δ1[Hold[((Solve[a-b==0,n])//Flatten)[[1]][[2]]]]//Release;
lhs[2]=(DeleteCases[Table[Coefficient[lhs[1],δ1[i+q]]/.n->i+q,{i,-4-L1,0}]//Flatten,0]//Simplify);
lhs[2]
]
RHS1s[L_]/;(L//EvenQ):=Module[{L1=L},
rhs[1]=Sum[RSs[0]/.l->l1,{l1,0,L1,2}]/.δ[a__,b__]->δ1[Hold[((Solve[a-b==0,n])//Flatten)[[1]][[2]]]]//Release;
rhs[2]=(DeleteCases[Table[Coefficient[rhs[1],δ1[i+p]]/.n->i+p,{i,-4-L1,0}]//Flatten,0]//Simplify);
rhs[2]
]
(*minimal value of (p,q) is (0,0), also evaluate inner products Is*)
LHSs[L_,p1_,q1_]:=(Series[LHS1s[L]/.List->Plus/.p->p1/.q->q1-ϵ,{ϵ,0,0}]/.Derivative[1,0][Bs][a__,b__]->0//Normal)/.Iis->IIs//Expand
RHSs[L_,p1_,q1_]:=(Series[RHS1s[L]/.List->Plus/.p->p1-ϵ/.q->q1,{ϵ,0,0}]/.Derivative[1,0][Bs][a__,b__]->0//Normal)/.Iis->IIs//Expand


(* We implement the algorithm described in appendix C of the paper by defining a function Consts[L] which gives all the free parameters for any spin truncation L (notably all (Subscript[γ, n,l ])with 2n≤l≤L), we then write down the equations Eqs[L,p,q] for every p≥1 and 0≤q≤Min[p-1,L/2], replace (L+2)(L+4)/8 anomalous dimensions by the constants and solve these equations for the remaining  (Subscript[γ, n,l]) ×(Subscript[((A^((0)))), n,l])'s in terms of the constants,
AnomDims[L,Max] then gives all  (Subscript[γ, n,l]) ×(Subscript[((A^((0)))), n,l]) with l≤min(2n-2,L) in terms of the free parameters, the solutions will be given up to n=Max for any integer Max*)

Consts[L_]/;(L//EvenQ):=Table[{Bs[n,l]->cs[n,l]},{n,0,(L/2)},{l,2n,L,2}]//Flatten
Eqs[L_,p_,q_]:=(LHSs[L,p,q]-RHSs[L,p,q])/.Consts[L]  (*we start at (p,q)=(1,0) *)

AnomDims[L_,Max_]:=AnomDims[L,Max]=Module[{L1=L,max=Max},
EqLists=Table[Eqs[L1,i,j]/.Bs->γ,{i,1,max},{j,0,Min[i-1,(L1/2)]}];
(Solve[EqLists==0]//Flatten)/.γ->Bs]

ListDim1s[L_,Max_]:=Table[AnomDims[L,Max][[i]][[1]],{i,1,Length[AnomDims[L,Max]]}]      (*List of (Subscript[γ, n,l])(× (A^((0)))) that we solved for, up to n=Max+2*)
ListDim2s[L_,Max_]:=Table[AnomDims[L,Max][[i]][[2]],{i,1,Length[AnomDims[L,Max]]}]      (*List of these (Subscript[γ, n,l]) (× (A^((0)))) in terms of (1/8)(L+2)(L+4) constants*)

max=35;            (*max=35 is enough to get analytic solutions for first few L-truncations*)

Results[L_]:=Results[L]=AnomDims[L,max]            (*gives results in form of replacement rules*)
Dimss[L_]:=ListDim1s[L,max]                                         (*gives a list of (Subscript[γ, n,l]) (×(A^((0)))) that we solved for*)
Solss[L_]:=ListDim2s[L,max]                                         (*gives the (Subscript[γ, n,l])(×(A^((0)))) from previous line in terms of the (1/8)(L+2)(L+4) fixed anomalous dimensions*)

ConstLabels[L_]:=Table[Consts[L][[i]][[2]]/.cs[a_,b_]->{a,b},{i,1,Length[Consts[L]]}]
SolNums[L_]:=Length[ConstLabels[L]]   
SolNum2[L_]:=(1/8)(L+2)(L+4)

Poss[L_,l1_]:=Position[Dimss[L]/.Bs[a_,l1]->c[l1],c[l1]]//Flatten     (*gives list of all positions of (Subscript[γ, n,l]) (×(A^((0)))) with l=l1 *)
PartSols[L_,l_]:=Part[Solss[L],Poss[L,l]]                                                                (*gives all solutions for (Subscript[γ, n,l])(×(A^((0))))  for a specific l *)

LabelPoss[L_,l_]:=Position[ConstLabels[L]/.{a_,l}->c[l],c[l]]//Flatten;
LabelPosMins[L_,l_]:=LabelPoss[L,l][[1]];

CoeffPartSols[L_,l_,constlabel_]/;(1≤constlabel≤Length[Consts[L]]):=Coefficient[PartSols[L,l],Consts[L][[constlabel]][[2]]]       (*find part of solution that is multiplied by a specific free parameter*)
(*to find analytic expressions of (Subscript[γ, n,l]) (×(A^((0)))), write the solution like ~ f1[n] cs[0,0]+f2[n] cs[0,2]+f3[n] cs[1,2].. (schematically)*)

write recursion relation for L=0  in terms of p:

LHSsB[L_,p,q1_]:=(Series[LHS1s[L]/.List->Plus/.q->q1-ϵ,{ϵ,0,0}]/.Derivative[1,0][Bs][a__,b__]->0//Normal)//Expand
RHSsB[L_,p,q1_]:=(Series[RHS1s[L]/.List->Plus/.q->q1,{ϵ,0,0}]/.Derivative[1,0][Bs][a__,b__]->0//Normal)//Expand

LHSsB[0,p,0]

(List@@Collect[RHSsB[0,p,0],{ Iis[p,2] Bs[-4+p,0],Iis[p,2] Bs[-3+p,0],Iis[2+p,2] Bs[-3+p,0],Iis[2+p,2] Bs[-2+p,0],Iis[4+p,2] Bs[-2+p,0],Iis[4+p,2] Bs[-1+p,0], Iis[6+p,2] Bs[-1+p,0],Iis[6+p,2] Bs[p,0]},FullSimplify]//Factor)/.List->Plus

We now want to find analytic solutions:

Analytic Solutions

Compute

Before we can calculate analytic anomalous dimensions for any spin truncation L,
we have to evaluate Results[L] for the cases we want to find the analytic expressions for. Results[L] gives a list of anomalous dimensions (Subscript[γ, n,l]) ×(Subscript[((A^((0)))), n,l]).

Results[0];

Results[2];

Results[4];

Results[6];

Checks

We check if the anomalous dimensions given by Results[L] indeed satisfy that Eqs[L,p,q] =LHSs[L,p,q]-RHSs[L,p,q]=0, where LHSs and RHSs are the left-hand and right-hand side  of the recursion relation derived above

{Eqs[0,0,3],Eqs[0,7,8],Eqs[0,8,2],Eqs[0,12,4]}/.Results[0]//Simplify

{Eqs[2,7,8],Eqs[2,8,2],Eqs[2,12,4]}/.Results[2]//Simplify

{Eqs[4,7,8],Eqs[4,8,2],Eqs[4,12,4]}/.Results[4]//Simplify

{Eqs[6,7,8],Eqs[6,8,2],Eqs[6,12,4]}/.Results[6]//Simplify

Find Analytic Solutions

AnalyticDims[L] then gives analytic expressions for (Subscript[γ, n,l]) for any L  as a list with a separate expression for any l=0,2,...L

(*AnalyticDims[L] gives analytic expressions of γ for any spin-truncation *)
AnalyticDims[L1_]/;(L1//EvenQ):=AnalyticDims[L1]=Module[{L=L1},
Do[
minlab=LabelPosMins[L,l];
If[CoeffPartSols[L,l,minlab]≠{},
{
AΓpt[L,l,minlab]=(FindSequenceFunction[CoeffPartSols[L,l,minlab],n])/.n->n-(-1+minlab)//FullSimplify;
Γpts[L,l,minlab]=AΓpt[L,l,minlab]/Afrees[n,l]//FullSimplify;
},Γpts[L,l,minlab]=0];  
(*AΓpt denotes the part of the anomalous dimension multiplying the free parameter with label constlabel, it is multiplied by the free OPE coefficient Afrees we solved for above*)
(*to get Γpts (anomalous dimensions): divide by Afrees *)

Do[If[CoeffPartSols[L,l,constlabel]≠{},
{
Γpts[L,l,constlabel]=(FindSequenceFunction[CoeffPartSols[L,l,constlabel]/CoeffPartSols[L,l,minlab],n])/.n->n-(-1+minlab)//FullSimplify;
},Γpts[L,l,constlabel]=0]
,{constlabel,minlab+1,Length[ConstLabels[L]]}];

(*combine all parts, schematically like:~ f1[n] cs[0,0]+f2[n] cs[0,2]+f3[n] cs[1,2]..*)
Γs[L,l]=(Γpts[L,l,LabelPosMins[L,l]](Consts[L][[LabelPosMins[L,l]]][[2]]+Sum[Consts[L][[constlabel]][[2]] Γpts[L,l,constlabel],{constlabel,LabelPosMins[L,l]+1,Length[Consts[L]]}]));

,{l,0,L,2}];
Table[Γs[L,l],{l,0,L,2}]
]

(*AnalyticDims[L] gives a list of analytic (Subscript[γ, n,l]) with one expression for any l=0,2,...,L*)
(*In Γs[L,n,l] we rewrite these results as function of (L,n,l) for each l separately*)
Γs[L_,n1_,l_]/;(l//EvenQ&&l≤L):=Γs[L,n1,l]=AnalyticDims[L][[(l/2)+1]]/.n->n1

some examples:

AnalyticDims[0]

AnalyticDims[2]

AnalyticDims[4]

AnalyticDims[6]

Some Solutions

In this section we look at the lower-spin truncations in more detail

Spin-0 Truncation

We want to compare our solutions to those from 1712.08570 (which need to be shifted by n-> n+2 to match our conventions that n≥0)
This solution is obtained from the CPW expansion of (F^(spin-0))(u,v)=(λ^2)u v Dbar[5,7,5,5](u,v) and we expect large-n  behaviour ~(n^11)

ΓCPW0[n_,0]:=-(((1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (8+n))/(2240 (7+2 n) (9+2 n) (11+2 n)))

result from Recursion:

Γs[0,n,0]

Γspin0s[0,n_,0]:=((11 (1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (8+n) cs[0,0])/(2304000 (7+2 n) (9+2 n) (11+2 n)))

{Eqs[0,5,4],Eqs[0,8,4],Eqs[0,2,0],Eqs[0,15,9]}/.Bs[a_,b_]:>Γs[0,a,b]Afrees[a,b]//Simplify

{Eqs[0,5,4],Eqs[0,8,4],Eqs[0,2,0],Eqs[0,15,9]}/.Bs[a_,b_]:>Γspin0s[0,a,b]Afrees[a,b]//Simplify

choice of free parameters in recursion relation result to match the solutions from CPW expansion

ParameterChoiceL0s=SolveAlways[ΓCPW0[n,0]==Γspin0s[0,n,0],n]//Flatten

large-n limit:  ~(n^11) (agrees with 1712.08570)

(Series[Γspin0s[0,n,0]/.n->(1/t),{t,0,-11}]//Normal)/.t->(1/n)

Spin-2 Truncation

Γs[2,n,0]

Γs[2,n,2]

Γspin2s[2,n_,0]:=((Γspin0s[0,n,0])/(cs[0,0]))(cs[0,0]+cs[0,2]Γspin2Pt1s[n]+cs[1,2]Γspin2Pt2s[n])

Γspin2s[2,n_,2]:=((-Γspin0s[0,n,0])/(cs[0,0]))((845 (-1+n) (5+n) (6+n) (8+n) (((9+n))^2) (10+n) (12+n) )/(4064256 (13+2 n) (15+2 n) ))(cs[0,2]-((51 n (11+n) cs[1,2])/(364 (-1+n) (12+n))))

Γspin2Pt1s[n_]:=((325 n (9+n) (1512620+1251666 n+385476 (n^2)+64233 (n^3)+6201 (n^4)+351 (n^5)+13 (n^6)))/(1016064 (5+2 n) (13+2 n)))

Γspin2Pt2s[n_]:=-((1105 n (9+n) (441700+370494 n+117468 (n^2)+20601 (n^3)+2157 (n^4)+135 (n^5)+5 (n^6)))/(9483264 (5+2 n) (13+2 n)))

save results for calculation of corrected OPE coefficients (expand to avoid poles)

Γ1s[2,n_,0]:=(11 (8+9 n+(n^2)) (((5040+8028 n+5104 (n^2)+1665 (n^3)+295 (n^4)+27 (n^5)+(n^6)))^2) (28449792 (65+36 n+4 (n^2)) cs[0,0]+65 n (9+n) (140 (1512620+1251666 n+385476 (n^2)+64233 (n^3)+6201 (n^4)+351 (n^5)+13 (n^6)) cs[0,2]-51 (441700+370494 n+117468 (n^2)+20601 (n^3)+2157 (n^4)+135 (n^5)+5 (n^6)) cs[1,2])))/(65548320768000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n))
Γ1s[2,n_,2]:=-((143 (10+11 n+(n^2)) (((30+11 n+(n^2)))^3) (((12096+17688 n+10078 (n^2)+2871 (n^3)+433 (n^4)+33 (n^5)+(n^6)))^2) (364 (-12+11 n+(n^2)) cs[0,2]-51 n (11+n) cs[1,2]))/(52438656614400 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n)))

large-n limit: ~(n^17)  (Note that in L-0 truncation: (n^11)- difference is 6 as expected (see paper) ), we also expect a second solution with (n^15)  (difference to spin-0 being 4), this is satisfied for the following choice of the free parameters: 
cs[1,2] -> (364/51)cs[0,2]

(Series[{Γspin2s[2,n,0],Γspin2s[2,n,2]}/.n->(1/t),{t,0,-17}]//Normal)/.t->(1/n)

Solve[((1859 cs[0,2])/2996494663680)-((2431 cs[1,2])/27967283527680)==0]//Flatten

(n^15)-solution

(Series[{Γspin2s[2,n,0],Γspin2s[2,n,2]}/.cs[1,2]->(364/51)cs[0,2]/.n->(1/t),{t,0,-15}]//Normal)/.t->(1/n)

match solutions to those of CPW expansion: one result from 4-derivative solution (~(n^15)) and one from 6-derivative solution (~(n^17)) 
(n^15) solutions from CPW of:  ((Subscript[F, 4])^(spin-2))(u,v)=2(λ^2)uv (Dbar[6,7,7,6](u,v)+Dbar[7,6,7,6](u,v)+Dbar[7,7,6,6](u,v))
(n^17)  solutions from CPW of :  ((Subscript[F, 6])^(spin-2))(u,v)=6(λ^2)uv Dbar[7,7,7,7](u,v)

ParameterChoiceL2Sol4s={cs[0,0]->-(360000/1001),cs[0,2]->(80640/1859),cs[1,2]->(752640/2431)};  (*find this below*)
ParameterChoiceL2Sol6s={cs[0,0]->(3888000/1001),cs[0,2]->-(241920/1859),cs[1,2]->-(451584/221)};

(Series[({Γspin2s[2,n,0],Γspin2s[2,n,2]}/.ParameterChoiceL2Sol4s)/.n->(1/t),{t,0,-15}]//Normal//Expand//Simplify)/.t->(1/n)

(Series[({Γspin2s[2,n,0],Γspin2s[2,n,2]}/.ParameterChoiceL2Sol6s)/.n->(1/t),{t,0,-17}]//Normal//Expand//Simplify)/.t->(1/n)

gives expected large-n behaviour

checks:

{Eqs[2,5,4],Eqs[2,8,4],Eqs[2,2,0],Eqs[2,15,9]}/.Bs[a_,b_]:>Γspin2s[2,a,b]Afrees[a,b]//Simplify

compare to results from 1712.08570 (our solutions can be written as superpositions of the solutions found in the reference by CPW expansion) ,  shift n by +2 again to match our conventions

(*4-derivative solutions*)
ΓCPW1[n_,0]:=(((1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (8+n) (-54000+75150 n+91537 (n^2)+32337 (n^3)+5644 (n^4)+513 (n^5)+19 (n^6)))/(483840 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n)))
ΓCPW1[n_,2]:=(((1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^2) (((8+n))^2) (((9+n))^2) (10+n))/(1935360 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n)))
(*6-derivative solutions*)
ΓCPW2[n_,0]:=(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (((8+n))^2) (243000+226800 n+82872 (n^2)+16461 (n^3)+1927 (n^4)+135 (n^5)+5 (n^6)))/(1612800 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n)))
ΓCPW2[n_,2]:=-(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^2) (((8+n))^2) (((9+n))^2) (((10+n))^2))/(6451200 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n)))

Series[{ΓCPW1[n,0],ΓCPW1[n,2],ΓCPW2[n,0],ΓCPW2[n,2]}/.n->(1/t),{t,0,-17}]

ParameterChoiceL2Sol4s=SolveAlways[Γspin2s[2,n,0]==ΓCPW1[n,0],n]//Flatten

ParameterChoiceL2Sol6s=SolveAlways[Γspin2s[2,n,0]==ΓCPW2[n,0],n]//Flatten

SolveAlways[Γspin2s[2,n,2]==ΓCPW1[n,2],n]      (*agrees with ParameterChoiceL2Sol4s*)

SolveAlways[Γspin2s[2,n,2]==ΓCPW2[n,2],n]       (*agrees with ParameterChoiceL2Sol6s*)

Spin-4 Truncation

Γs[4,n,0]//Expand//Simplify

Γs[4,n,2]//Expand//Simplify

Γs[4,n,4]//Expand//Simplify

Γ1s[4,n_,0]:=((8+9 n+(n^2)) (((5040+8028 n+5104 (n^2)+1665 (n^3)+295 (n^4)+27 (n^5)+(n^6)))^2) (219856933654364160 (2925+3960 n+1736 (n^2)+288 (n^3)+16 (n^4)) cs[0,0]+n (9+n) (70323821568000 (68067900+110779290 n+68456876 (n^2)+21774285 (n^3)+4133337 (n^4)+495963 (n^5)+38025 (n^6)+1872 (n^7)+52 (n^8)) cs[0,2]+17 (191276800 (35895900902400+82368108247680 n+83168945615232 (n^2)+49970484234144 (n^3)+20076461206768 (n^4)+5698197667620 (n^5)+1180647168136 (n^6)+182605424607 (n^7)+21404941333 (n^8)+1921766751 (n^9)+132975989 (n^10)+7049781 (n^11)+276743 (n^12)+7497 (n^13)+119 (n^14)) cs[0,4]-1506939033600 (19876500+32573430 n+20390644 (n^2)+6637869 (n^3)+1308573 (n^4)+166131 (n^5)+13713 (n^6)+720 (n^7)+20 (n^8)) cs[1,2]-931 (36465 (-2466858240000+16424775129600 n+37954159727040 (n^2)+32604146013312 (n^3)+15596862761736 (n^4)+4854007913004 (n^5)+1062534461038 (n^6)+170945416161 (n^7)+20748442193 (n^8)+1927561077 (n^9)+137799609 (n^10)+7520067 (n^11)+302651 (n^12)+8379 (n^13)+133 (n^14)) cs[1,4]-97888 (-1208042035200-1120257630720 n+466800310080 (n^2)+1013043651840 (n^3)+593622219416 (n^4)+200414553156 (n^5)+45789681578 (n^6)+7586501787 (n^7)+945423515 (n^8)+90286263 (n^9)+6648115 (n^10)+374193 (n^11)+15529 (n^12)+441 (n^13)+7 (n^14)) cs[2,4])))))/(46050034103605002240000 (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n))
Γ1s[4,n_,2]:=-(((10+11 n+(n^2)) (((30+11 n+(n^2)))^3) (((12096+17688 n+10078 (n^2)+2871 (n^3)+433 (n^4)+33 (n^5)+(n^6)))^2) (571381050240000 (-1020+407 n+521 (n^2)+88 (n^3)+4 (n^4)) cs[0,2]+17 n (11+n) (191276800 (-419107680+39225384 n+249821756 (n^2)+106998034 (n^3)+20747701 (n^4)+2176306 (n^5)+133144 (n^6)+5236 (n^7)+119 (n^8)) cs[0,4]-4709184480000 (85+44 n+4 (n^2)) cs[1,2]-931 (-12+11 n+(n^2)) (36465 (44923960+34369632 n+9221460 (n^2)+1285559 (n^3)+98667 (n^4)+4389 (n^5)+133 (n^6)) cs[1,4]-97888 (2382120+1769504 n+456588 (n^2)+63085 (n^3)+4985 (n^4)+231 (n^5)+7 (n^6)) cs[2,4]))))/(575625426295062528000000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n)))
Γ1s[4,n_,4]:=(323 (12+13 n+(n^2)) (((1680+1066 n+251 (n^2)+26 (n^3)+(n^4)))^3) (((23760+32916 n+17404 (n^2)+4485 (n^3)+595 (n^4)+39 (n^5)+(n^6)))^2) (171142400 (420-572 n+125 (n^2)+26 (n^3)+(n^4)) cs[0,4]-931 n (13+n) (36465 (-30+13 n+(n^2)) cs[1,4]-5152 (-14+13 n+(n^2)) cs[2,4])))/(1151250852590125056000000 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n) (19+2 n))

checks:

{Eqs[4,5,4],Eqs[4,8,4],Eqs[4,2,0],Eqs[4,15,9]}/.Bs[a_,b_]:>Γs[4,a,b]Afrees[a,b]//Simplify

Large-n limit:

Series[{Γ1s[4,n,0],Γ1s[4,n,2],Γ1s[4,n,4]}/.n->(1/t),{t,0,-23}]

Solutions from 1712.085770:

shift n -> n+2 again, found from CPW of 8,10,12 - derivative  solutions

ΓSp4CPWSol8[n_,0]:=(((1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (8+n) P12[n])/(141926400 (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n)))
ΓSp4CPWSol8[n_,2]:=(((1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^2) (((8+n))^2) (((9+n))^2) (10+n) P6[n])/(354816000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n)))
ΓSp4CPWSol8[n_,4]:=(((1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^3) (((8+n))^3) (((9+n))^2) (((10+n))^2) (((11+n))^2) (12+n))/(2128896000 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n) (19+2 n)))

ΓSp4CPWSol10[n_,0]:=(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (((8+n))^2) Q12[n])/(2554675200 (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n)))
ΓSp4CPWSol10[n_,2]:=-(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^2) (((8+n))^2) (((9+n))^2) (((10+n))^2) Q6[n])/(6386688000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n)))
ΓSp4CPWSol10[n_,4]:=-(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^3) (((8+n))^3) (((9+n))^2) (((10+n))^2) (((11+n))^2) (((12+n))^2))/(12773376000 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n) (19+2 n)))

ΓSp4CPWSol12[n_,0]:=-(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (((8+n))^2) R14[n])/(66421555200 (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n)))
ΓSp4CPWSol12[n_,2]:=(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^2) (((8+n))^2) (((9+n))^2) (((10+n))^2) R8[n])/(166053888000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n)))
ΓSp4CPWSol12[n_,4]:=-(((((1+n))^2) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^3) (((6+n))^3) (((7+n))^3) (((8+n))^3) (((9+n))^2) (((10+n))^2) (((11+n))^2) (((12+n))^2) (299+39 n+3 (n^2)))/(332107776000 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n) (19+2 n)))

P6[n_]:=(11(n^6)+363 (n^5)+5051 (n^4)+37917 (n^3)+153558 (n^2)+280940 n+170500)
P12[n_]:=(161 (n^12)+8694 (n^11)+211587 (n^10)+3066120 (n^9)+29230411 (n^8)+189615150 (n^7)+828090989 (n^6)+2269349028 (n^5)+2904406372 (n^4)-3232338192 (n^3)-20688563520 (n^2)-34461028800 n-20357568000)
Q6[n_]:=(80850+55440 n+45454 (n^2)+24651 (n^3)+5053 (n^4)+429 (n^5)+13 (n^6))
Q12[n_]:=(132024816000+267821769600 n+234512496000 (n^2)+123273763272 (n^3)+44802597060 (n^4)+11932653618 (n^5)+2376089599 (n^6)+354407346 (n^7)+39283491 (n^8)+3183030 (n^9)+182109 (n^10)+6750 (n^11)+125 (n^12))

R8[n_]:=(100900800+98618520 n+36901558 (n^2)+7700055 (n^3)+1253086 (n^4)+176781 (n^5)+17215 (n^6)+924 (n^7)+21 (n^8))
R14[n_]:=(-13730580864000-29569786646400 n-27618687080640 (n^2)-15225402164760 (n^3)-5572035205164 (n^4)-1410251843358 (n^5)-246044122769 (n^6)-27741698829 (n^7)-1457742654 (n^8)+102444075 (n^9)+29166024 (n^10)+3022785 (n^11)+184970 (n^12)+6615 (n^13)+105 (n^14))

large-n limit:

Series[{ΓSp4CPWSol8[n,0],ΓSp4CPWSol8[n,2],ΓSp4CPWSol8[n,4]}/.n->(1/t),{t,0,-19}]

Series[{ΓSp4CPWSol10[n,0],ΓSp4CPWSol10[n,2],ΓSp4CPWSol10[n,4]}/.n->(1/t),{t,0,-21}]

Series[{ΓSp4CPWSol12[n,0],ΓSp4CPWSol12[n,2],ΓSp4CPWSol12[n,4]}/.n->(1/t),{t,0,-23}]

ParameterChoiceL4Sol8s={SolveAlways[Γ1s[4,n,0]==ΓSp4CPWSol8[n,0],n],SolveAlways[Γ1s[4,n,2]==ΓSp4CPWSol8[n,2],n],SolveAlways[Γ1s[4,n,4]==ΓSp4CPWSol8[n,4],n]}//Flatten

ParameterChoiceL4Sol10s={SolveAlways[Γ1s[4,n,0]==ΓSp4CPWSol10[n,0],n],SolveAlways[Γ1s[4,n,2]==ΓSp4CPWSol10[n,2],n],SolveAlways[Γ1s[4,n,4]==ΓSp4CPWSol10[n,4],n]}//Flatten

ParameterChoiceL4Sol12s={SolveAlways[Γ1s[4,n,0]==ΓSp4CPWSol12[n,0],n],SolveAlways[Γ1s[4,n,2]==ΓSp4CPWSol12[n,2],n],SolveAlways[Γ1s[4,n,4]==ΓSp4CPWSol12[n,4],n]}//Flatten

Series[({{Γ1s[4,n,0],Γ1s[4,n,2],Γ1s[4,n,4]}/.ParameterChoiceL4Sol8s,{Γ1s[4,n,0],Γ1s[4,n,2],Γ1s[4,n,4]}/.ParameterChoiceL4Sol10s,{Γ1s[4,n,0],Γ1s[4,n,2],Γ1s[4,n,4]}/.ParameterChoiceL4Sol12s}//Simplify)/.n->(1/t),{t,0,-23}]

gives expected large-n limit

Summary: L=0, 2 ,4 Solutions

Γspin0s[0,n_,0]:=((11 (1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (8+n) cs[0,0])/(2304000 (7+2 n) (9+2 n) (11+2 n)))

Γspin2s[2,n_,0]:=((Γspin0s[0,n,0])/(cs[0,0]))(cs[0,0]+cs[0,2]Γspin2Pt1s[n]+cs[1,2]Γspin2Pt2s[n])

Γspin2s[2,n_,2]:=((-Γspin0s[0,n,0])/(cs[0,0]))((845 (-1+n) (5+n) (6+n) (8+n) (((9+n))^2) (10+n) (12+n) )/(4064256 (13+2 n) (15+2 n) ))(cs[0,2]-((51 n (11+n) cs[1,2])/(364 (-1+n) (12+n))))

Γspin2Pt1s[n_]:=((325 n (9+n) (1512620+1251666 n+385476 (n^2)+64233 (n^3)+6201 (n^4)+351 (n^5)+13 (n^6)))/(1016064 (5+2 n) (13+2 n)))

Γspin2Pt2s[n_]:=-((1105 n (9+n) (441700+370494 n+117468 (n^2)+20601 (n^3)+2157 (n^4)+135 (n^5)+5 (n^6)))/(9483264 (5+2 n) (13+2 n)))


Γspin4s[4,n_,0]:=((8+9 n+(n^2)) (((5040+8028 n+5104 (n^2)+1665 (n^3)+295 (n^4)+27 (n^5)+(n^6)))^2) (219856933654364160 (2925+3960 n+1736 (n^2)+288 (n^3)+16 (n^4)) cs[0,0]+n (9+n) (70323821568000 (68067900+110779290 n+68456876 (n^2)+21774285 (n^3)+4133337 (n^4)+495963 (n^5)+38025 (n^6)+1872 (n^7)+52 (n^8)) cs[0,2]+17 (191276800 (35895900902400+82368108247680 n+83168945615232 (n^2)+49970484234144 (n^3)+20076461206768 (n^4)+5698197667620 (n^5)+1180647168136 (n^6)+182605424607 (n^7)+21404941333 (n^8)+1921766751 (n^9)+132975989 (n^10)+7049781 (n^11)+276743 (n^12)+7497 (n^13)+119 (n^14)) cs[0,4]-1506939033600 (19876500+32573430 n+20390644 (n^2)+6637869 (n^3)+1308573 (n^4)+166131 (n^5)+13713 (n^6)+720 (n^7)+20 (n^8)) cs[1,2]-931 (36465 (-2466858240000+16424775129600 n+37954159727040 (n^2)+32604146013312 (n^3)+15596862761736 (n^4)+4854007913004 (n^5)+1062534461038 (n^6)+170945416161 (n^7)+20748442193 (n^8)+1927561077 (n^9)+137799609 (n^10)+7520067 (n^11)+302651 (n^12)+8379 (n^13)+133 (n^14)) cs[1,4]-97888 (-1208042035200-1120257630720 n+466800310080 (n^2)+1013043651840 (n^3)+593622219416 (n^4)+200414553156 (n^5)+45789681578 (n^6)+7586501787 (n^7)+945423515 (n^8)+90286263 (n^9)+6648115 (n^10)+374193 (n^11)+15529 (n^12)+441 (n^13)+7 (n^14)) cs[2,4])))))/(46050034103605002240000 (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n))
Γspin4s[4,n_,2]:=-(((10+11 n+(n^2)) (((30+11 n+(n^2)))^3) (((12096+17688 n+10078 (n^2)+2871 (n^3)+433 (n^4)+33 (n^5)+(n^6)))^2) (571381050240000 (-1020+407 n+521 (n^2)+88 (n^3)+4 (n^4)) cs[0,2]+17 n (11+n) (191276800 (-419107680+39225384 n+249821756 (n^2)+106998034 (n^3)+20747701 (n^4)+2176306 (n^5)+133144 (n^6)+5236 (n^7)+119 (n^8)) cs[0,4]-4709184480000 (85+44 n+4 (n^2)) cs[1,2]-931 (-12+11 n+(n^2)) (36465 (44923960+34369632 n+9221460 (n^2)+1285559 (n^3)+98667 (n^4)+4389 (n^5)+133 (n^6)) cs[1,4]-97888 (2382120+1769504 n+456588 (n^2)+63085 (n^3)+4985 (n^4)+231 (n^5)+7 (n^6)) cs[2,4]))))/(575625426295062528000000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n)))
Γspin4s[4,n_,4]:=(323 (12+13 n+(n^2)) (((1680+1066 n+251 (n^2)+26 (n^3)+(n^4)))^3) (((23760+32916 n+17404 (n^2)+4485 (n^3)+595 (n^4)+39 (n^5)+(n^6)))^2) (171142400 (420-572 n+125 (n^2)+26 (n^3)+(n^4)) cs[0,4]-931 n (13+n) (36465 (-30+13 n+(n^2)) cs[1,4]-5152 (-14+13 n+(n^2)) cs[2,4])))/(1151250852590125056000000 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n) (19+2 n))


(*choices of free parameters to match recursion relation results to solutions from CPW in 1712.08570*)
ParameterChoiceL0s={cs[0,0]->-(7200/77)};
ParameterChoiceL2Sol4s={cs[0,0]->-(360000/1001),cs[0,2]->(80640/1859),cs[1,2]->(752640/2431)};
ParameterChoiceL2Sol6s={cs[0,0]->(3888000/1001),cs[0,2]->-(241920/1859),cs[1,2]->-(451584/221)};

ParameterChoiceL4Sol8s={cs[0,0]->-(1468800/143),cs[0,2]->(14999040/31603),cs[0,4]->(18289152/785213),cs[1,2]->(380233728/46189),cs[1,4]->(298598400/1356277),cs[2,4]->(102643200/141151),cs[0,2]->(14999040/31603),cs[0,4]->(18289152/785213),cs[1,2]->(380233728/46189),cs[1,4]->(298598400/1356277),cs[2,4]->(102643200/141151),cs[0,4]->(18289152/785213),cs[1,4]->(298598400/1356277),cs[2,4]->(102643200/141151)};
ParameterChoiceL4Sol10s={cs[0,0]->(4233600/143),cs[0,2]->-(3951360/31603),cs[0,4]->-(36578304/785213),cs[1,2]->-(151882752/46189),cs[1,4]->-(99532800/104329),cs[2,4]->-(718502400/141151),cs[0,2]->-(3951360/31603),cs[0,4]->-(36578304/785213),cs[1,2]->-(151882752/46189),cs[1,4]->-(99532800/104329),cs[2,4]->-(718502400/141151),cs[0,4]->-(36578304/785213),cs[1,4]->-(99532800/104329),cs[2,4]->-(718502400/141151)};
ParameterChoiceL4Sol12s={cs[0,0]->(16934400/143),cs[0,2]->(189665280/31603),cs[0,4]->-(420650496/785213),cs[1,2]->(6770147328/46189),cs[1,4]->-(16970342400/1356277),cs[2,4]->-(139748716800/1834963),cs[0,2]->(189665280/31603),cs[0,4]->-(420650496/785213),cs[1,2]->(6770147328/46189),cs[1,4]->-(16970342400/1356277),cs[2,4]->-(139748716800/1834963),cs[0,4]->-(420650496/785213),cs[1,4]->-(16970342400/1356277),cs[2,4]->-(139748716800/1834963)};

2.6   OPE Coefficients (Subscript[((A^((1)))), n,l])

Analytic Formulas for (Subscript[γ, n,l]) (L=0,2,4)

Evaluate this to compute first order OPE coefficients

(*results from above*)

Γ1s[0,n_,0]:=((11 (1+n) (((2+n))^2) (((3+n))^2) (((4+n))^2) (((5+n))^2) (((6+n))^2) (((7+n))^2) (8+n) cs[0,0])/(2304000 (7+2 n) (9+2 n) (11+2 n)))
Γ1s[2,n_,0]:=(11 (8+9 n+(n^2)) (((5040+8028 n+5104 (n^2)+1665 (n^3)+295 (n^4)+27 (n^5)+(n^6)))^2) (28449792 (65+36 n+4 (n^2)) cs[0,0]+65 n (9+n) (140 (1512620+1251666 n+385476 (n^2)+64233 (n^3)+6201 (n^4)+351 (n^5)+13 (n^6)) cs[0,2]-51 (441700+370494 n+117468 (n^2)+20601 (n^3)+2157 (n^4)+135 (n^5)+5 (n^6)) cs[1,2])))/(65548320768000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n))
Γ1s[2,n_,2]:=-((143 (10+11 n+(n^2)) (((30+11 n+(n^2)))^3) (((12096+17688 n+10078 (n^2)+2871 (n^3)+433 (n^4)+33 (n^5)+(n^6)))^2) (364 (-12+11 n+(n^2)) cs[0,2]-51 n (11+n) cs[1,2]))/(52438656614400 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n)))
Γ1s[4,n_,0]:=((8+9 n+(n^2)) (((5040+8028 n+5104 (n^2)+1665 (n^3)+295 (n^4)+27 (n^5)+(n^6)))^2) (219856933654364160 (2925+3960 n+1736 (n^2)+288 (n^3)+16 (n^4)) cs[0,0]+n (9+n) (70323821568000 (68067900+110779290 n+68456876 (n^2)+21774285 (n^3)+4133337 (n^4)+495963 (n^5)+38025 (n^6)+1872 (n^7)+52 (n^8)) cs[0,2]+17 (191276800 (35895900902400+82368108247680 n+83168945615232 (n^2)+49970484234144 (n^3)+20076461206768 (n^4)+5698197667620 (n^5)+1180647168136 (n^6)+182605424607 (n^7)+21404941333 (n^8)+1921766751 (n^9)+132975989 (n^10)+7049781 (n^11)+276743 (n^12)+7497 (n^13)+119 (n^14)) cs[0,4]-1506939033600 (19876500+32573430 n+20390644 (n^2)+6637869 (n^3)+1308573 (n^4)+166131 (n^5)+13713 (n^6)+720 (n^7)+20 (n^8)) cs[1,2]-931 (36465 (-2466858240000+16424775129600 n+37954159727040 (n^2)+32604146013312 (n^3)+15596862761736 (n^4)+4854007913004 (n^5)+1062534461038 (n^6)+170945416161 (n^7)+20748442193 (n^8)+1927561077 (n^9)+137799609 (n^10)+7520067 (n^11)+302651 (n^12)+8379 (n^13)+133 (n^14)) cs[1,4]-97888 (-1208042035200-1120257630720 n+466800310080 (n^2)+1013043651840 (n^3)+593622219416 (n^4)+200414553156 (n^5)+45789681578 (n^6)+7586501787 (n^7)+945423515 (n^8)+90286263 (n^9)+6648115 (n^10)+374193 (n^11)+15529 (n^12)+441 (n^13)+7 (n^14)) cs[2,4])))))/(46050034103605002240000 (3+2 n) (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n))
Γ1s[4,n_,2]:=-(((10+11 n+(n^2)) (((30+11 n+(n^2)))^3) (((12096+17688 n+10078 (n^2)+2871 (n^3)+433 (n^4)+33 (n^5)+(n^6)))^2) (571381050240000 (-1020+407 n+521 (n^2)+88 (n^3)+4 (n^4)) cs[0,2]+17 n (11+n) (191276800 (-419107680+39225384 n+249821756 (n^2)+106998034 (n^3)+20747701 (n^4)+2176306 (n^5)+133144 (n^6)+5236 (n^7)+119 (n^8)) cs[0,4]-4709184480000 (85+44 n+4 (n^2)) cs[1,2]-931 (-12+11 n+(n^2)) (36465 (44923960+34369632 n+9221460 (n^2)+1285559 (n^3)+98667 (n^4)+4389 (n^5)+133 (n^6)) cs[1,4]-97888 (2382120+1769504 n+456588 (n^2)+63085 (n^3)+4985 (n^4)+231 (n^5)+7 (n^6)) cs[2,4]))))/(575625426295062528000000 (5+2 n) (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n)))
Γ1s[4,n_,4]:=(323 (12+13 n+(n^2)) (((1680+1066 n+251 (n^2)+26 (n^3)+(n^4)))^3) (((23760+32916 n+17404 (n^2)+4485 (n^3)+595 (n^4)+39 (n^5)+(n^6)))^2) (171142400 (420-572 n+125 (n^2)+26 (n^3)+(n^4)) cs[0,4]-931 n (13+n) (36465 (-30+13 n+(n^2)) cs[1,4]-5152 (-14+13 n+(n^2)) cs[2,4])))/(1151250852590125056000000 (7+2 n) (9+2 n) (11+2 n) (13+2 n) (15+2 n) (17+2 n) (19+2 n))

(*parameter choices to reproduce results from 1712.08570*)

ParameterChoiceL0s={cs[0,0]->-(7200/77)};
ParameterChoiceL2Sol4s={cs[0,0]->-(360000/1001),cs[0,2]->(80640/1859),cs[1,2]->(752640/2431)};
ParameterChoiceL2Sol6s={cs[0,0]->(3888000/1001),cs[0,2]->-(241920/1859),cs[1,2]->-(451584/221)};
ParameterChoiceL4Sol8s={cs[0,0]->-(1468800/143),cs[0,2]->(14999040/31603),cs[0,4]->(18289152/785213),cs[1,2]->(380233728/46189),cs[1,4]->(298598400/1356277),cs[2,4]->(102643200/141151),cs[0,2]->(14999040/31603),cs[0,4]->(18289152/785213),cs[1,2]->(380233728/46189),cs[1,4]->(298598400/1356277),cs[2,4]->(102643200/141151),cs[0,4]->(18289152/785213),cs[1,4]->(298598400/1356277),cs[2,4]->(102643200/141151)};
ParameterChoiceL4Sol10s={cs[0,0]->(4233600/143),cs[0,2]->-(3951360/31603),cs[0,4]->-(36578304/785213),cs[1,2]->-(151882752/46189),cs[1,4]->-(99532800/104329),cs[2,4]->-(718502400/141151),cs[0,2]->-(3951360/31603),cs[0,4]->-(36578304/785213),cs[1,2]->-(151882752/46189),cs[1,4]->-(99532800/104329),cs[2,4]->-(718502400/141151),cs[0,4]->-(36578304/785213),cs[1,4]->-(99532800/104329),cs[2,4]->-(718502400/141151)};
ParameterChoiceL4Sol12s={cs[0,0]->(16934400/143),cs[0,2]->(189665280/31603),cs[0,4]->-(420650496/785213),cs[1,2]->(6770147328/46189),cs[1,4]->-(16970342400/1356277),cs[2,4]->-(139748716800/1834963),cs[0,2]->(189665280/31603),cs[0,4]->-(420650496/785213),cs[1,2]->(6770147328/46189),cs[1,4]->-(16970342400/1356277),cs[2,4]->-(139748716800/1834963),cs[0,4]->-(420650496/785213),cs[1,4]->-(16970342400/1356277),cs[2,4]->-(139748716800/1834963)};

OPE Coefficients

find ((Subscript[A, n,l])^((1))) from ((Subscript[A, n,l])^((1)))=(1/2)(∂_n)(((Subscript[A, n,l])^((0)))(Subscript[γ, n,l]))

Afrees[n_,l_]:=(((2+l) (9+l+2 n) (10+l+2 n) Gamma[4+n] Gamma[5+n] Gamma[6+l+n] Gamma[7+l+n])/(72 Gamma[2 (5+l+n)] Gamma[6+2 n]))

lvalues[L_]:=Table[i,{i,0,L,2}]
A1s[L_,n_,l_]/;(MemberQ[lvalues[L],l]):=(((D[(Afrees[r,l] Γ1s[L,r,l]),r])/.r->n//FullSimplify))
A1s[L_,n_,l_]/;(FreeQ[lvalues[L],l]):=0

OPECoeffss[L_,Max_]:=Table[A1s[L,i,j],{i,-1,Max},{j,0,Max}]//Flatten

OPECoeffss[0,5]

OPECoeffss[0,5]/.ParameterChoiceL0s

OPECoeffss[2,5]/.ParameterChoiceL2Sol4s

OPECoeffss[2,5]/.ParameterChoiceL2Sol6s

OPECoeffss[4,5]/.ParameterChoiceL4Sol8s

OPECoeffss[4,5]/.ParameterChoiceL4Sol10s

OPECoeffss[4,5]/.ParameterChoiceL4Sol12s