Wigner 6 𝑗 symbols with gluon lines: completing the set of 6j symbols required for color decomposition

We construct a set of Wigner 6 𝑗 symbols with gluon lines (adjoint representations) in closed form, expressed in terms of similar 6 𝑗 symbols with quark lines (fundamental representations). Together with Wigner 6 𝑗 symbols with quark lines, this gives a set of 6 𝑗 symbols sufficient for treating QCD color structure for any number of external particles, in or beyond perturbation theory. This facilitates a complete treatment of QCD color structure in terms of orthogonal multiplet bases, without the need of ever explicitly constructing the corresponding bases. We thereby open up for a completely representation theory based treatment of SU(N) color structure, with the potential of significantly speeding up the color structure treatment.

A problem with the trace and color-flow bases is that they are only orthogonal in the limit  → ∞, and in fact overcomplete for many particles; for high multiplicities they are severely overcomplete 20 , with a dimension that scales as the factorial of the number of gluons plus quark-antiquark pairs.If one does not want to exploit sampling over different color structures 1 , like done in for example the CVolver program 15,17,21 , this gives rise to a major bottle neck for the squaring of the color structure, which then scales as a factorial square.
It appears appealing to explore minimal orthogonal bases.This is accomplished by multiplet bases 20,[22][23][24][25][26][27][28][29][30][31][32] , which rely on the Clebsch-Gordan decomposition of the involved particle representation for constructing orthogonal bases.Examples of multiplet bases can be found in Refs.22-25, and a general construction in Refs.26,31.However, it is possible to do better than that: Any color structure can be decomposed into a multiplet basis without explicitly constructing this basis, by making use of the group invariant Wigner 6  symbols (here 6 s for short, also known as 6  coefficients, Racah coefficients, or Racah  coefficients, up to signs), along with Wigner 3  coefficients and dimensions of representations.The problem of decomposing the color structure is then essentially reduced to the problem of finding a sufficient set of 6  symbols for the color decomposition in question.Some work in this direction has been pursued in Refs.20,32, where symmetry is exploited to recursively calculate a set of 6  symbols applicable for processes with a limited number of partons.Other recent work obtains SU(3) 6  symbols numerically, by first calculating SU(3) Clebsch-Gordan coefficients 33,34 .For SU (2), the problem is addressed in Ref. 35.
In a recent paper 36 , we started to explore a third avenue, namely to recursively derive 6  symbols in terms of other 6  symbols and dimensions of representations.We there derived closed forms of a set of 6  symbols characterized by having quarks (fundamental representa-tions) in opposing positions.In the present paper, we complete this set of 6  symbols with symbols where two of the opposing representations are quarks or gluons (adjoint representations), and 6 s where one vertex only contains fundamental and adjoint representations, whereas the other representations are arbitrary.As we will see, this class of 6 s define a complete set for decomposing any color structure appearing in the standard model.
We lay out the basics of SU() color calculations using the birdtrack method in Sec.II.
In Sec.III we go through a general procedure for decomposing the color structure.This allows us to identify a set of 6  symbols that is sufficient to decompose any color structure to any order in perturbation theory.While one of the necessary classes of 6  symbols is calculated in Ref. 36, the remaining ones are calculated in Sec.V, after a careful discussion on how to define vertices in Sec.IV.Finally, we make concluding remarks in Sec.VI.

II. REDUCING SU(𝑁) COLOR STRUCTURE IN BIRDTRACK NOTATION
In this section we briefly outline how to calculate SU() invariants, using the birdtrack method, and assuming knowledge of a sufficient set of Wigner 3 2 and Wigner 6  symbols.
It is worth remarking that while our discussion focuses on SU(), in particular SU(3), this reduction method is applicable for any Lie group.For a full, comprehensive introduction to the birdtrack formalism, we refer to Ref.As we are interested in fully color summed (averaged) color structures, every color structure can be seen as a fully connected graph of SU() representations, for example as in Fig. 1.This entails of course triplet and octet representations but also the higher dimensional irreducible representations (irreps), used in the construction of multiplet bases, or appearing during the calculations.In the end, we want to calculate a scalar product in color space, for example between a Feynman diagram and basis vector in an orthogonal multiplet basis.
Generally, the color structure then consists of a fully connected graph.The graph contains FIG.1: Example of a fully contracted color structure with triplet and and adjoint representations, contracted with a set of general representations, denoted by double lines.
These may come from a multiple basis vector, or arise previous contraction steps.Note that "bubbles", in green, can be contracted away using Eq. ( 9), and that vertex corrections (in blue) can be removed using Eq. ( 5).To address the remaining color structure, the completeness relation Eq. ( 2) is in general needed.loops of various length, for example, we may encounter , where the double lines denote any irrep of SU(), and in general should be supplied with representation labels and arrow directions, which we suppress here for readability.
While short loops of length up to three can be immediately removed (see below), the fallback method to handle long loops is to split them up to shorter loops by repeated insertion of the completeness relation where   denotes the dimension of the irrep , appearing in the Clebsch-Gordan decomposition, and where the denominator is a Wigner 3  symbol.
Tracing both sides of this equation, and using it is clear that the completeness relation implies     =    , as anticipated.
Applying the completeness relation (2) to two of the representations in Eq. ( 1), marked in red below, schematically results in where we now have a "vertex correction" loop with three internal representations.This loop can be removed using the Wigner 6  symbols ( The sum above runs over instances  of the irrep  in  ⊗ , for example the two octets in 8 ⊗ 8 = 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕ 10 ⊕ 27.In this paper, every encountered vertex will contain at least one fundamental or adjoint representation, implying that most often there is only one instance, but for  ⊗ , with  being the adjoint representation for SU(), and  being an arbitrary irrep, there can be up to  − 1 representations of type  26 .We will choose the corresponding vertices to be mutually orthogonal, in the sense that Furthermore, for the 6  symbols that we derive, we choose to normalize our vertices such that a b α β γ ≡   , for all non-vanishing vertices, i.e., the 3  coefficients are normalized to one.This implies in particular that = 1, in contrast to the standard QCD normalization = 1 2 ( 2 − 1), for the generator normal-ization tr[    ] = 1 2   .We explain in App.D how to easily transform our results to any desired normalization.
Note that after having applied Eq. (5) to Eq. ( 4), we are left with a loop with one representation less, Repeatedly applying this procedure to Eq. (1), it is thus possible to reduce loops with any number of internal representations down to loops of length three (removed using Eq. ( 5)) or length two, removed using the "self energy" relation In this way, assuming the knowledge of the 6  symbols, it is possible to reduce any fully connected graph to a number.
It should be noted that the required set of 6 s depends on how the contraction is performed, and what basis vectors are used.In the present paper, we consider basis vectors of the form, where we have a backbone chain of general representations  1 ,  2 , • • • ,   , denoted by double lines with suppressed arrows, to which the external particle representations, octets, triplets and antitriples, denoted by single lines, are attached (also with suppressed arrows).To the authors knowledge all multiplet bases in the literature are of this form.One could also imagine bases where general irreps are contracted in vertices.Such bases will require 6  symbols beyond those presented here, and are therefore beyond the scope of this paper.We emphasize once more, that the philosophy underlying the present work is to avoid explicitly constructing any bases, and instead achieve a decomposition using 6 s.

III. A SUFFICIENT SET OF 6 𝑗 SYMBOLS FOR DECOMPOSING COLOR STRUCTURE
In this section we identify a sufficient set of 6  symbols for decomposing color structure into the orthogonal basis vectors in Eq. ( 10) to any order in perturbation theory.We start out with considering tree-level color structures, and return to higher orders later.
Again, letting single lines schematically denote triplet, octet or singlet representations (i.e. representations of the external particles) and letting double lines denote the general irreps encountered in the basis vectors, the fully connected graph for a tree-level color structure contracted with a basis vector, will always contain at least two loops of the form where the characterizing feature is that there is only one vertex from the initial color structure (the gray blob, representing a quark-gluon or triple-gluon vertex).Typically the color structure will contain many color structures that are trivial to contract using Eq. ( 5) and Eq. ( 9) , but we here consider a worst, general case.To reduce loops of this type, the completeness relation, Eq. ( 2), and the vertex correction relation, Eq. ( 5), can be applied to the two red representations below Repeating this procedure will eventually result in a vertex correction containing the gray blob.(For the loop in the above example, this step would need to be repeated two more times.)The vertex correction with the gray blob gives for some representations ,  and .This last step removes two vertices, one gray blob, i.e., a vertex from the initial color structure and one vertex between arbitrary representations in the basis vector, Eq. ( 10).As every contracted loop removes one vertex from the basis vector and one from the color structure to be decomposed, the resulting graph is topologically equivalent to a graph for a tree-level color structure with one less external patron.After a loop of the form of Eq. ( 11) has been contracted, there must thus exist at least two loops of the type in Eq. ( 11) in the resulting color structure by the above argument.Hence any tree-level color structure can be completely contracted by repeatedly contracting loops of the form of Eq. ( 11).
Only treating loops of the form of Eq. ( 11) is thus sufficient for tree-level color structures.
We now address the situation where the color structure to be decomposed itself contains loops.It is then not always possible to choose color loops of the form in Eq. ( 10).At one-loop this happens for diagrams where all external partons form a single loop (For all other one-loop color structures there is at least one vertex with two uncontracted indices, implying that a loop of the from in Eq. ( 11) can be found, such that it is possible to contract loops as in Eq. ( 12).)For color structures of the form of Eq. ( 14), there always exists loops of the form Similarly to the loop in Eq. ( 11), the steps detailed in Eq. ( 12) remain valid.However, at the end, instead of contracting a loop of the form in Eq. ( 13), a loop with four vertices is encountered, Treating a loop like this removes two vertices from the color structure, and none from the basis vector.Since a one-loop color structure has two vertices more than a tree-level color structure for the same process, the number of vertices after the contraction matches a treelevel structure.Since two legs which initially belonged to the loop of the color structure to be decomposed (the upper legs in Eq. ( 16)) now attach directly to the sequence of basis vector representations, the topology after the contraction is equivalent to that of a treelevel color structure.Note that a loop of the type in Eq. ( 15), need not be the first loop to be contracted (most one-loop diagrams contain no loop of the form in Eq. ( 14)), but such loops may at some point be encountered, and necessary to contract to continue the reduction.In this way, we can thus contract any one-loop diagram.Color structures of arbitrary order in perturbation theory can be decomposed by contracting loops similar to Eq. ( 12) and Eq. ( 16), possibly with more than two (cf.Eq. ( 16)) vertices from the initial color structure.The final steps in the contraction would then proceed as in Eq. ( 16), but with more completeness relations inserted.
In the above color decomposition procedure, we can identify a minimal set of necessary 6  symbols, namely those appearing in the different steps above, Eqs.(12-13) and Eq. ( 16).
Keeping in mind that the single lines above denote adjoint or fundamental representations, we conclude that the 6 s we are after can be divided into the cases in table I.
We note that the 6 s of type (0) in table I are known from Ref. 36.In this article we address the computation of the remaining 6 s.Before taking on this task, we must, however, be careful with how we define the vertices for the cases where we have more than one vertex between the same set of representations, which can happen for vertices with gluons, as discussed below Eq. ( 5).We will normalize all our vertices such that all non-vanishing 3  symbols are equal to 1, as already mentioned following Eq.( 6).Readers who prefer to work with different normalizations are referred to App.D for a simple transformation rule.
For each instance of the irrep  in the complete reduction of  ⊗  we have to construct where  thus runs from 1 to the multiplicity of  in  ⊗ .Essentially, we construct these vertices by splitting the gluon line into a  q-pair.More precisely, we consider all diagrams where  enumerates admissible intermediate irreps according to a scheme to be explained below, and construct the desired vertices as linear combinations of these diagrams.In general, these vertices then take the form with coefficients     ∈ C, which will actually turn out to be real-valued functions of .We distinguish the two cases  ≠  and  = .
If  ≠  then there is only a non-zero vertex (17) if  can be found in the Clebsch-Gordan decomposition of  ⊗ .This means that we can obtain  from  by adding a box in one row and subsequently removing a box in a different row (possibly after first adding a column of length ) 26 .In this case there is a unique intermediate irrep  1 in diagram (18), representing the intermediate step in this process after adding a box but before removing the other box.
Hence, for such  ≠ , we find where the constant has to be chosen such that the normalization condition for the corre- is fulfilled.After a few steps, spelled out in App.A, we get from Eq. (A8) For  =  there can be up to  − 1 vertices of type (17) In this case there exist  + 1 different admissible irreps   rendering the diagrams non-zero, as discussed in Appendix A, where we also show that all vertices in Eq. ( 23) are linear combinations of theses diagrams, i.e.
Hence, we can obtain a set of orthonormal vertices (23) by applying the Gram-Schmidt algorithm to the set of diagrams (24) with admissible intermediate irreps   .
In order to obtain a unique result when carrying out Gram-Schmidt we have to decide how to sort the diagrams (24).To this end, note that an admissible   is obtained by adding an extra box to .We say that if in   this extra box is added further down compared to where it was added in   .We then sort the birdtrack diagrams (24) in increasing order.Hence, the first birdtrack diagram in our list is always the diagram with intermediate irrep  1 which is obtained by adding a box to the first row of .In Appendix A we show that the last diagram in this list is always a linear combinations of the first  diagrams and can thus be omitted.The sum in Eq. ( 25) hence runs from 1 to .
In order to carry out Gram-Schmidt we only need to know the scalar products between all diagrams (24), which are calculated in App.A, Eq. (A6)-(A8).We denote this The scalar products    also depend on the irrep  but we do not display this dependence in our notation since in the following    for different irreps  never appear alongside each other in our equations.
We explicitly state the formulae for the first two vertices, which are the only vertices in the physically relevant case  = 3, Further vertices, which only exist for  > 3, are calculated by straightforwardly continuing Gram-Schmidt.In App.B we illustrate the explicit vertex construction with a few examples.

V. FORMULAE FOR GLUON 6 𝑗 SYMBOLS
We will now describe how the calculation of the different classes of 6  symbols proceeds.
Our main tool will be the repeated insertion of vertex corrections, and the Fierz identity, Eq. (A7), to decompose gluon lines.We will go through the basic idea and steps for the simpler cases here, but defer the details of longer calculations to App.C for the sake of readability.
Case 1: 6 s with a quark-gluon vertex We here consider the 6  symbol which contains a  q vertex (in its center).We proceed to calculate this by expanding the gluon vertex into a vertex correction where we have assumed  ≠  and used Eq.(A8) from App. A, which builds on the Fierz identity, Eq. (A7).
In the case  = , again using Eq.(A8), for the first two vertices  = 1, 2 we obtain and ( 31) Note that the last expression vanishes for  = 1.For  > 3, there might, as described, be more vertices which then are treated similarly.
where the 6  symbols with two quark lines are given in closed form in Ref. 36.
where a minus sign next to a vertex indicates that the lines are connected to this vertex in opposite order, i.e.
see also appendix C of Ref. 36.For the above vertices with quarks this only makes a difference for the antisymmetic vertex of  ⊗ . 36Our final result for the  -vertex is while we obtain for the -vertex,

VI. CONCLUSIONS AND OUTLOOK
In the present paper we have shown how to calculate a set of Wigner 6  coefficients with adjoint representations.Together with a set of previously derived 6 s 36 , this set constitutes a complete set of 6 s required to decompose any color structure, to any order into orthogonal multiplet bases, cf.Eq. ( 10).
This opens up for the usage of orthogonal representation theory based color bases also for processes with high multiplicities, including the analysis of evolution equations in color space 40 .
We note, however, that the present work does not close the research area of representation theory based treatment of color structure.In particular, more general 6  symbols are required for fully general multiplet bases (with vertices between general representations).
We believe that this can be addressed with similar methods.
the complete reduction of  ⊗ , multiply with a quark-gluon vertex (Lie algebra generators), and contract the quark and antiquark lines, yielding The l.h.s.vanishes since the SU() generators are traceless, i.e. we have found a non-trivial vanishing linear combination of the diagrams in Eq. ( 24).
Next we show that all vertices in Eq. ( 23) are linear combinations of the diagrams in Eq. (24).To this end, consider a gluon exchange between  and a quark line, and insert two completeness relations: Due to Schur's Lemma the middle segment can only be non-zero if   and   are equivalent.
If two irreps in the complete reduction of  ⊗ are equivalent then they are the same, i.e.
with some constant    .Substituting into Eq.(A3), multiplying with a quark-gluon vertex, and contracting the quark and antiquark line we find The quark loop on the l.h.s.can be traded for a factor of ( 2 − 1) −1 (recall that we set all 3  symbols equal to 1), and by defining     =     2   ( 2 − 1) we obtain Eq. ( 25), as claimed in Sec.IV.Now we can even take advantage of the linear dependence (A2) of the vertex correction diagrams (24).Equation (A2) tells us that any one of the  + 1 diagrams (24) can be expressed as a linear combination of the other  diagrams, since none of the coefficients in Eq. (A2) vanishes.In Sec.IV we order the diagrams in a unique way and determine the orthonormal vertices in Eq. ( 23) by means of the Gram-Schmidt algorithm.Since the last vertex correction diagram is guaranteed to be a linear combination of the first  diagrams, we can always terminate Gram-Schmidt before using the last diagram, i.e the vertices (23)   are actually linear combinations of the first  vertex correction diagrams (24).
Finally, we explicitly determine the scalar products between all vertex correction diagrams (18).The result is the main ingredient for the Gram-Schmidt process in Sec.IV.
The square diagram can be evaluated by invoking the Fierz identity (or adjoint representation projector, also equivalent to the completeness relation for  ⊗ q), which with unit 3  symbols takes the form Inserting this gives for the scalar product and (B5) are related to  and  by a unitary transformation, and vice versa, facilitating easy conversion.
The coefficients of this unitary transformation are determined by scalar products between the two sets of vertices, and these scalar products can be evaluated by calculations similar to Eqs. (A6)-(A8).

Appendix C: Details of 6 𝑗 derivations
We here give, in full detail, the intermediate steps for the derivation in Sec.V.

Derivation for case 2
We here derived the form of the 6  coefficients in Eq. (33).In essence the vertices involving gluons are expressed in terms of vertex corrections, after which the Fierz identity, Eq. (A7), is applied, and vertex corrections are removed using Eq. ( 5) We remark that the result looks very similar to the result for case 2, but that it is now expressed in terms of the 6 s from case 2.

Derivation for case 4
Again the gluon vertices are expressed in terms of vertex corrections with quarks, both in the triple-gluon vertices and in the vertices with the general representations.This gives for the antisymmetric (  ) triple-gluon vertex

Case 2 :
6 s with a gluon line opposing a quark lineIn this case we have one quark and one gluon line attaching to different vertices.Rewriting the gluon vertices in terms of vertex corrections and invoking the Fierz identity, we then find, after a few steps spelled out in App.C

Case 3 :Case 4 :
6 s with two opposing gluon lines Case 3 can be addressed with a similar strategy as the other cases.Our result, for which we demonstrate all intermediate steps in App.C, reads 6 s with three-gluon verticesThe class of the 6  symbols with three gluons consists of two three-gluon vertices, which typically is taken to be proportional to    and   .We will illustrate the case of    first.In particular, we use the definition of the    -vertex in terms of traces, and then insert vertex corrections, identity, Eq. (A7), we can remove all internal gluon lines and the results are expressed in terms of a number of different diagrams which reduce to 3  symbols, dimensions and traces over quark lines, see Appendix C. A single non-trivial diagram remains, which can be expressed as

and one gluon reads 1 = 1 =For 1 = (𝑁 2 − 1
Appendix B: Examples of vertex constructionillustrate how to construct vertices of type(17) using methods and results from Sec. IV.First consider an example with  ≠ .For  = and  = the unique intermediate irrep is  1 =.Then, using Eq.(22) for normalization, the unique vertex with irreps , with the smallest number of boxes for which there is more than one vertex is  =  = , i.e., the octet for  = 3 (note that this is not the adjoint representation for  ≠ 3).The admissible intermediate irreps are  1 = and  2 = .Using Eq. (29), the orthonormal vertices become  ( 2 − 1)  =  =  = (recall that the black column stands for a column with  − 1 boxes) we obtain three-gluon vertices for general .The admissible intermediate irreps are then  1 = and  2 = .Using Eq. (29), our orthonormal vertices read that for  = 3 Eqs.(B2)/(B3) and Eqs.(B4)/(B5) coincide.Instead of the latter vertices, one will likely want to use the much more common antisymmetric  and symmetric  vertices, to which our vertices are related by a unitary transformation, which we explicitly state below.Like all other vertices in this article we normalize  and  such that the corresponding 3  symbols are equal to one, i.e.D for how to easily transform results to other normalizations.The vertices (B4)

Derivation for case 3
The steps in the derivation of Eq. (34) progress similarly to those in the derivation of Eq. (33),

= 1 .
() vertex differ only by the sign of the second term.The second term above is calculated using the Fierz identity (A7), quark loop in the last diagram simply yields a factor of , and the others are easy to evaluate using the self energy relation, Eq. (9), for example          ℓ term needs to be reduced using 6  symbols, sign next to a vertex indicates that the lines are connected to this vertex in opposite order, see Eq. (37).The expressions calculated here are assembled in Eq. (38) and Eq.(39) for the antisymmetric and symmetric vertices, respectively.Appendix D: Vertex normalizations leading to non-trivial 3  symbols All explicit formulae for Wigner 6  symbols in this article, in particular the results in Sec.V, are valid for vertices normalized such that all non-vanishing 3  symbols are equal to While this normalization is convenient, it differs from normalizations typically applied in the context of QCD.We therefore here give a simple rule for how to transform any of our 6  symbols when changing the normalization of any 3  symbol.Assume we have calculated the 6  symbol we prefer this 3  symbol to be equal to  ≠ 1 we define a vertex In short: For each vertex whose 3  symbol you normalize to a number ≠ 1 multiply our 6  symbol by the square root of the value of your 3  symbol in order to obtain the value of the 6  symbol with your normalization convention.
37, a minimal introduction can be found in Appendix A of Ref. 26, whereas a more pedagogical account is written up in Ref. 38.Examples of birdtrack calculations for QCD can be found in Refs.27 and 20.

TABLE I :
The required set of 6  symbols for color decomposition into multiplet bases of the form in(10).The last two 6 s have the antisymmetric (  ) and symmetric () triple-gluon vertices in the middle respectively.
32,39ERTEX CONSTRUCTIONIn table I we sorted the 6  symbols that we are going to study in this work according to the number of gluon lines and according to the number of vertices with gluon lines that these 6  symbols contain.Before we can evaluate the 6  symbols, we have to construct all vertices with at least one gluon line.When discussing how many vertices with a given set of irreps there are, it is useful to think of the general irrep labels (for which we use Greek letters) as Young diagrams.In fact, a systematic labeling of SU() irreps applicable for arbitrary  should rather be in terms of pairs of Young diagrams.32,39However,ifweallowfor Young diagrams with columns with an -dependent number of boxes, we can replace each pair of Young diagrams by a single Young diagram32.For instance, the adjoint representation is then labeled by the Young diagram  =, where, here and in what follows, a black column always represents a column with  − 1 boxes.Hence, in the following, we can always think of irrep labels as single Young diagrams with, possibly, -dependent column lengths.
, cf.Appendix B of Ref. 26, i.e., if the multiplicity of  in the complete reduction of  ⊗  is , we have to construct the a,  = 1, . . .,  .