(* ::Package:: *)

(* ::Title:: *)
(*This file contains the orthonormal multiplet basis for 4 gluons*)


(* ::Subsection:: *)
(*The vectors are named and ordered as:*)


(* ::Text:: *)
(*All non-charge conjugation invariant vectors are included in *)
(*CompleteNormalized4gBasis, whereas Normalized4gBasis only contains *)
(*vectors which are (all but 10 and 10bar) or can be combined *)
(*into charge conjugation invariant states (10+10bar)*)


Normalized4gBasis={Vector11,Vector8s8s,Vector8a8a,Vector2727,Vector1010,Vector10bar10bar,Vector00};

CompleteNormalized4gBasis={Vector11,Vector8s8s,Vector8s8a,Vector8a8s,Vector8a8a,Vector2727,Vector1010,Vector10bar10bar,Vector00};
CMNMBq0g4=CompleteNormalized4gBasis;


(* ::Subsection:: *)
(*The explicit forms of the vectors are:*)



Subscript[Vector11, {G1_, G3_, G2_, G4_}] = 
    (Superscript[\[CapitalDelta], {G1, G3}]*Superscript[\[CapitalDelta], 
       {G2, G4}])/(-1 + Nc^2);
 
Subscript[Vector8a8a, {G1_, G3_, G2_, G4_}] = 
    -Superscript[o, {G1, G3, G2, G4}]/(2*TR*Sqrt[Nc^2*(-1 + Nc^2)*TR^2]) + 
     Superscript[o, {G1, G3, G4, G2}]/(2*TR*Sqrt[Nc^2*(-1 + Nc^2)*TR^2]) + 
     Superscript[o, {G1, G2, G4, G3}]/(2*TR*Sqrt[Nc^2*(-1 + Nc^2)*TR^2]) - 
     Superscript[o, {G1, G4, G2, G3}]/(2*TR*Sqrt[Nc^2*(-1 + Nc^2)*TR^2]);
 
Subscript[Vector8a8s, {G1_, G3_, G2_, G4_}] = 
    Superscript[o, {G1, G3, G2, G4}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]) + 
     Superscript[o, {G1, G3, G4, G2}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]) - 
     Superscript[o, {G1, G2, G4, G3}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]) - 
     Superscript[o, {G1, G4, G2, G3}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]);
 
Subscript[Vector8s8a, {G1_, G3_, G2_, G4_}] = 
    -Superscript[o, {G1, G3, G2, G4}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]) + 
     Superscript[o, {G1, G3, G4, G2}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]) - 
     Superscript[o, {G1, G2, G4, G3}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]) + 
     Superscript[o, {G1, G4, G2, G3}]/(2*TR*Sqrt[(4 - 5*Nc^2 + Nc^4)*TR^2]);
 
Subscript[Vector8s8s, {G1_, G3_, G2_, G4_}] = 
    Superscript[o, {G1, G3, G2, G4}]/
      (2*TR*Sqrt[((-4 + Nc^2)^2*(-1 + Nc^2)*TR^2)/Nc^2]) + 
     Superscript[o, {G1, G3, G4, G2}]/
      (2*TR*Sqrt[((-4 + Nc^2)^2*(-1 + Nc^2)*TR^2)/Nc^2]) + 
     Superscript[o, {G1, G2, G4, G3}]/
      (2*TR*Sqrt[((-4 + Nc^2)^2*(-1 + Nc^2)*TR^2)/Nc^2]) + 
     Superscript[o, {G1, G4, G2, G3}]/
      (2*TR*Sqrt[((-4 + Nc^2)^2*(-1 + Nc^2)*TR^2)/Nc^2]) - 
     (2*TR*Superscript[\[CapitalDelta], {G1, G3}]*
       Superscript[\[CapitalDelta], {G2, G4}])/
      (Nc*Sqrt[((-4 + Nc^2)^2*(-1 + Nc^2)*TR^2)/Nc^2]);
 
Subscript[Vector1010, {G1_, G3_, G2_, G4_}] = 
    Superscript[o, {G1, G3, G2, G4}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     Superscript[o, {G1, G3, G4, G2}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     Superscript[o, {G1, G2, G3, G4}]/(2*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     Superscript[o, {G1, G2, G4, G3}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) + 
     Superscript[o, {G1, G4, G3, G2}]/(2*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) + 
     Superscript[o, {G1, G4, G2, G3}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     (Superscript[\[CapitalDelta], {G1, G4}]*Superscript[\[CapitalDelta], 
        {G3, G2}])/(2*Sqrt[4 - 5*Nc^2 + Nc^4]) + 
     (Superscript[\[CapitalDelta], {G1, G2}]*Superscript[\[CapitalDelta], 
        {G3, G4}])/(2*Sqrt[4 - 5*Nc^2 + Nc^4]);
 
Subscript[Vector10bar10bar, {G1_, G3_, G2_, G4_}] = 
    Superscript[o, {G1, G3, G2, G4}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     Superscript[o, {G1, G3, G4, G2}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) + 
     Superscript[o, {G1, G2, G3, G4}]/(2*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     Superscript[o, {G1, G2, G4, G3}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     Superscript[o, {G1, G4, G3, G2}]/(2*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) + 
     Superscript[o, {G1, G4, G2, G3}]/(2*Nc*Sqrt[4 - 5*Nc^2 + Nc^4]*TR^2) - 
     (Superscript[\[CapitalDelta], {G1, G4}]*Superscript[\[CapitalDelta], 
        {G3, G2}])/(2*Sqrt[4 - 5*Nc^2 + Nc^4]) + 
     (Superscript[\[CapitalDelta], {G1, G2}]*Superscript[\[CapitalDelta], 
        {G3, G4}])/(2*Sqrt[4 - 5*Nc^2 + Nc^4]);
 
Subscript[Vector2727, {G1_, G3_, G2_, G4_}] = 
    -(Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]*Superscript[o, {G1, G3, G2, G4}])/
      (2*Nc^2*(-6 + Nc + 4*Nc^2 + Nc^3)*TR^2) - 
     (Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]*Superscript[o, {G1, G3, G4, G2}])/
      (2*Nc^2*(-6 + Nc + 4*Nc^2 + Nc^3)*TR^2) + 
     Superscript[o, {G1, G2, G3, G4}]/(2*Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]*
       TR^2) - (Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]*Superscript[o, 
        {G1, G2, G4, G3}])/(2*Nc^2*(-6 + Nc + 4*Nc^2 + Nc^3)*TR^2) + 
     Superscript[o, {G1, G4, G3, G2}]/(2*Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]*
       TR^2) - (Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]*Superscript[o, 
        {G1, G4, G2, G3}])/(2*Nc^2*(-6 + Nc + 4*Nc^2 + Nc^3)*TR^2) + 
     (Superscript[\[CapitalDelta], {G1, G4}]*Superscript[\[CapitalDelta], 
        {G3, G2}])/(2*Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]) + 
     (Superscript[\[CapitalDelta], {G1, G2}]*Superscript[\[CapitalDelta], 
        {G3, G4}])/(2*Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]) + 
     (Superscript[\[CapitalDelta], {G1, G3}]*Superscript[\[CapitalDelta], 
        {G2, G4}])/((1 + Nc)*(2 + Nc)*Sqrt[Nc^2*(-3 + 2*Nc + Nc^2)]);
 
Subscript[Vector00, {G1_, G3_, G2_, G4_}] = 
    -(Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]*Superscript[o, {G1, G3, G2, G4}])/
      (2*Nc^2*(6 + Nc - 4*Nc^2 + Nc^3)*TR^2) - 
     (Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]*Superscript[o, {G1, G3, G4, G2}])/
      (2*Nc^2*(6 + Nc - 4*Nc^2 + Nc^3)*TR^2) - 
     Superscript[o, {G1, G2, G3, G4}]/(2*Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]*
       TR^2) - (Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]*Superscript[o, 
        {G1, G2, G4, G3}])/(2*Nc^2*(6 + Nc - 4*Nc^2 + Nc^3)*TR^2) - 
     Superscript[o, {G1, G4, G3, G2}]/(2*Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]*
       TR^2) - (Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]*Superscript[o, 
        {G1, G4, G2, G3}])/(2*Nc^2*(6 + Nc - 4*Nc^2 + Nc^3)*TR^2) + 
     (Superscript[\[CapitalDelta], {G1, G4}]*Superscript[\[CapitalDelta], 
        {G3, G2}])/(2*Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]) + 
     (Superscript[\[CapitalDelta], {G1, G2}]*Superscript[\[CapitalDelta], 
        {G3, G4}])/(2*Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]) + 
     (Superscript[\[CapitalDelta], {G1, G3}]*Superscript[\[CapitalDelta], 
        {G2, G4}])/((-2 + Nc)*(-1 + Nc)*Sqrt[Nc^2*(-3 - 2*Nc + Nc^2)]);



(* ::Text:: *)
(*(Private note: Created using MakeTrace2gVectors.nb 14 03 10. Explaining notes added at the top 180921.)*)
