equation

hep-ph/9910563 SLAC–PUB–8294

DFTT 53/99

October, 1999

New Color Decompositions for Gauge Amplitudes

at Tree and Loop Level

Vittorio Del Duca

Istituto Nazionale di Fisica Nucleare

Sezione di Torino

via P. Giuria, 1

10125 - Torino, Italy

Lance Dixon

Stanford Linear Accelerator Center

Stanford University

Stanford, CA 94309, USA

Fabio Maltoni

Dipartimento di Fisica Teorica

Università di Torino

via P. Giuria, 1

10125 - Torino, Italy

Recently, a color decomposition using structure constants was introduced for purely gluonic tree amplitudes, in a compact form involving only the linearly independent subamplitudes. We give two proofs that this decomposition holds for an arbitrary number of gluons. We also present and prove similar decompositions at one loop, both for pure gluon amplitudes and for amplitudes with an external quark-antiquark pair.

Submitted to Nuclear Physics B

Address after November 1, 1999: Theory Division, CERN, CH 1211 Geneve 23, Switzerland Research supported by the US Department of Energy under grant DE-AC03-76SF00515

## 1 Introduction

The computation of multi-parton scattering amplitudes in QCD is essential for quantitative predictions of multi-jet cross-sections at high-energy colliders. The Feynman diagrams for such an amplitude can generate a thicket of complicated color algebra, tangled together with expressions composed of kinematic invariants. An extremely useful way to disentangle the color and kinematic factors is via color decompositions [1, 2, 3, 4, 5] in terms of Chan-Paton factors [6], or traces of matrices in the fundamental representation, . The traces are multiplied by purely kinematical coefficients, called partial amplitudes or subamplitudes. Originally motivated by the representation of gluon amplitudes in open string theory, such decompositions have been widely studied and applied at both the tree level [4] and the loop level [7]. Two major advantages of the trace-based color decompositions are that (a) subamplitudes are gauge-invariant, and (b) an important ‘color-ordered’ class of them receive contributions only from diagrams with a particular cyclic ordering of the external partons; these subamplitudes therefore have much simpler kinematic properties than the full amplitude.

Although the standard color decompositions are very effective, in some cases they are not quite optimal. For example, the decomposition of the purely gluonic tree amplitude is overcomplete. Explicitly, the -gluon amplitude is written in terms of single-trace color structures, as

where are the subamplitudes. The cyclic invariance of the trace has been used to fix its first entry, and the sum is over the set of non-cyclic permutations of elements, , where . However, the subamplitudes appearing in the equation are not all independent. Besides cyclic and reflection invariances, which they inherit from the traces, subamplitudes also obey a decoupling identity (also called a dual Ward identity, or subcyclic identity) [2, 8],

This identity can be derived by setting equal to the unit matrix and collecting terms containing the same trace; this gives the amplitude for a ‘photon’ and gluons, which must vanish. More general identities can be derived by assigning the generators for the external gluons to commuting subgroups such as [9]. Kleiss and Kuijf found a linear relation between subamplitudes [10] (see section 2), which is consistent with all of these identities, and which reduces the number of linearly independent subamplitudes from to .

In contrast, the conventional color factors for gluonic Feynman diagrams are composed of structure constants . In this case, the and generalized decoupling identities are all manifest; for example, the generator commutes with all other generators, so all structure constants containing it are zero. On the other hand, a particular string of contracted structure constants will not appear ab initio with a gauge-invariant kinematic coefficient. To pass from the -based decomposition to the trace-based decomposition, one substitutes and simplifies and collects the various traces. In the process, gauge invariance of the partial amplitudes is gained, but properties such as decoupling are no longer manifest. Another disadvantage of trace-based decompositions at tree level is that the color structure of amplitudes in the multi-Regge kinematics is obscured [11].

The trace-based decomposition is also not optimal at the loop level. At one loop, the standard color decomposition [5] for -gluon amplitudes includes double trace structures, , in addition to single traces of the type that appear at tree level. The subamplitudes multiplying the single trace structures give the leading contributions in the large limit, and they are color-ordered. The double-trace subamplitudes, corresponding to subleading-in- contributions, can be written in terms of permutations of the leading (color-ordered) subamplitudes [12] in a formula reminiscent of the tree-level Kleiss-Kuijf relation (see section 3). Because the permutation formula is rather complicated, its implementation in a numerical program can be rather slow. Similar formulae hold for one-loop amplitudes with two external quarks and gluons [13].

Recently, a new color decomposition for the -gluon tree amplitude, in terms of structure constants rather than traces, has been presented [14], whose kinematic coefficients are just color-ordered subamplitudes. Because of this property, the new decomposition retains all the advantages of trace-based decompositions, yet avoids the disadvantages mentioned above. In particular, the tree-level -gluon decomposition is automatically given in terms of the independent subamplitudes, in a form that is very convenient for analyzing the high-energy limit.

The purpose of this paper is to derive the new tree-level decomposition in two different ways, and to present similar -based color decompositions at one loop, whose kinematic coefficients are again color-ordered subamplitudes. The leading and subleading-in- contributions combine neatly into one expression in the new decompositions. Also, gluons circulating in the loop are put on the same footing as fermions in the loop.

Parton-level cross-sections require amplitude squares or interferences which are summed over all external color labels. We provide general expressions for color-summed cross-sections in terms of the new color decompositions. In appendix A we give explicit evaluations for quantities encountered in cross-section computations through .

The paper is organized as follows. In section 2 we give two proofs of the new color decomposition for the -gluon tree amplitude conjectured in ref. [14], and provide accordingly the square of the tree amplitude summed over colors. In section 3 we present (and prove) new color decompositions for one-loop -gluon amplitudes and one-loop amplitudes with an external quark-antiquark pair plus gluons. In addition, we compute the color-summed interference terms between tree amplitudes and one-loop amplitudes, and the square of one-loop amplitudes, which are relevant respectively for next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) calculations of jet production rates. The color-summed interference terms and squares are given in terms of color matrices; appendix A provides many of those required for jet rate computations up to . Finally, we outline how a new color decomposition for one-loop amplitudes with external photons, gluons and a quark-antiquark pair can be obtained from the amplitudes with only gluons and a pair. In section 4 we summarize our present understanding of the color decomposition of tree-level and one-loop amplitudes, and comment briefly on possible extensions to multi-loop amplitudes.

## 2 Tree Color Decomposition

The new color decomposition for the -gluon tree amplitude
is^{1}^{1}1We choose the normalization of the fundamental representation
matrices as . Hence our structure
constants are larger than the conventional ones by a factor
of . [14]

(2.1) | |||||

where is an generator in the adjoint representation. This color decomposition is analogous to the standard decomposition for the tree amplitude with a quark-antiquark pair and gluons [3, 4],

(2.2) |

The only difference between the two is the representation used for the color matrices, namely the adjoint representation for the -gluon amplitude and the fundamental representation for the amplitude containing a pair.

Eq. (2.1) is written in terms of the subamplitudes where legs and are adjacent. This is precisely the basis of linearly independent subamplitudes which is singled out by the Kleiss-Kuijf relation. In fact, we shall show that eq. (2.1) is equivalent to the Kleiss-Kuijf relation [10], which can be written as,

(2.3) |

Here , is the number of elements in the set , the set is with the ordering reversed, and is the set of ‘ordered permutations’ (also called mergings [10]) of the elements of that preserve the ordering of the within and of the within , while allowing for all possible relative orderings of the with respect to the .

We wish to show that the new color decomposition (2.1) is equivalent to inserting the Kleiss-Kuijf relation (2.3) into the standard color decomposition (1). We first substitute for the structure constants appearing in eq. (2.1), , and use the identity

(2.4) |

We want to identify all terms that contain a trace of the form

(2.5) |

because these should give rise to . Since there are commutators, there are terms on the right-hand side of eq. (2.4), but only of them have exactly matrices appearing to the right of . The indices on the matrices to the right of must come in reversed order compared to how they appear in the string, but they can appear in that string in any relative order with respect to the that end up to the left of . Thus, for any ordered permutation , the subamplitude appears in the new decomposition (2.1) accompanied by the desired trace (2.5); a relative sign of comes from the commutators. Collecting all such ordered permutations, we obtain the Kleiss-Kuijf formula for , eq. (2.3), thus establishing its equivalence to the new color decomposition (2.1).

Hence we can derive the new -gluon color decomposition via its connection to the Kleiss-Kuijf relation. The latter relation was checked up to in ref. [10]. It can be proved for all using the same techniques that were used to prove an analogous one-loop formula [12], eq. (3.2) below. Consider first the set of color-ordered Feynman diagrams with only three-point vertices, which are ‘multi-peripheral’ with respect to and , by which we mean that all other external legs connect directly to the line extending from to ; i.e. there are no non-trivial trees branching off of this line. Label the legs on one side of the – line by , and those on the other side by . These diagrams contribute to both sides of the Kleiss-Kuijf relation (2.3) in the proper way: There is no relative ordering requirement on the with respect to the on the left-hand side of the relation, and the ordering is summed over on the right-hand side; the sign factor comes from the antisymmetry of color-ordered three-point vertices (in Feynman gauge).

Next consider diagrams with non-trivial trees attached to the – line. They work in exactly the same way as the multi-peripheral diagrams if the leaves (external legs) of each tree all belong to the same set, either or . However, if a tree contains leaves from both sets, then the diagram does not appear on the left-hand side of eq. (2.3), so one must show that it cancels out of the permutation sum on the right-hand side. For trees with only three-point vertices, this can be done using the antisymmetry of the vertices. For other cases the cancellations are slightly more complicated; one way of establishing them is via the generalized decoupling identities [9] mentioned in the introduction [12].

There is another way to prove eq. (2.1). For this argument it is convenient to use a graphical notation for color factors made out of structure constants, in which is represented by a three-vertex and (an index contraction) is represented by a line [1]. Then the color factors appearing in eq. (2.1) are associated with the multi-peripheral color diagrams shown in fig. 1. The color factor for a generic -gluon Feynman diagram is not of this form. However, we can use the Jacobi identity

(2.6) |

shown graphically in fig. 2, to put it into this form [1]. Fig. 3 represents the color factor for a generic Feynman diagram graphically, as a tree structure with only three-point vertices. It also shows how it can be simplified into multi-peripheral form by repeated application of the Jacobi identity. (If the line running from to is in the fundamental representation, then the same steps, but using , lead directly to the color decomposition (2.2) for a quark-antiquark pair and gluons.)

So far we have established that a color decomposition of the form (2.1) exists, but we have not yet shown that the kinematic coefficients are equal to the partial amplitudes . However, we can do this rather simply by equating the new decomposition (but with unknown kinematic coefficients) to the standard one, eq. (1), and contracting both sides with , i.e. summing over all , , for some permutation . Because the kinematic coefficients contain no dependence, we may retain only the leading contraction terms as . But it is well known that single traces for different cyclic orderings are orthogonal at large [4]. Similarly, fig. 4 shows that only one of the multi-peripheral color factors in eq. (2.1) survives the contraction, because the other permutations give rise to nonplanar, and hence -suppressed, color topologies. Thus the contraction selects a unique term from either side of the equation, corresponding to the same ordering of the color indices , and with the same weight at large , namely . This proves that the coefficients in the new color decomposition (2.1) are indeed the subamplitudes .

Because we showed that the new color decomposition is equivalent to the Kleiss-Kuijf relation, our second proof of the new decomposition can alternatively be viewed as a proof of the Kleiss-Kuijf relation. Interestingly, this proof does not rely on any kinematic properties of color-ordered Feynman diagrams; only group-theoretic properties such as the Jacobi identity and the large- behavior of color traces were used.

The decomposition (2.1) has a color-ladder structure which naturally arises in the configurations where the gluons are strongly ordered in rapidity, i.e. in the multi-Regge kinematics [15]. (Indeed, this is how eq. (2.1) was first discovered [14].) In contrast, some laborious work is necessary to obtain the color-ladder structure from the trace-based decomposition (1) in the multi-Regge kinematics [11]. Let gluons be the particles produced in a scattering where gluons 1 and are the incoming particles. The available final-state phase space may be divided into simplices, in each of which the produced gluons are ordered in rapidity. Let us take the simplex defined by the rapidity ordering , and consider its sub-simplex with strong rapidity ordering . Then the crucial point to note is that the dominant subamplitudes in eq. (1) in the multi-Regge limit are , where and are both increasing sequences, whose union is ( is in reversed order). In other words, contains the sequence . If is the number of elements in , then for each there are such choices of , . Summing over , there are orderings in total [16]. Using the identity (2.4) and the fact that in the multi-Regge kinematics

(2.7) |

one obtains the color-ladder structure of eq. (2.1), with just one allowed string of structure constants . The procedure can be repeated for all of the simplices, thus generating the strings of ’s of eq. (2.1).

The tree-level partonic scattering cross-section is given by the square of the tree amplitude , summed over all colors. This expression can be written either of two ways,

(2.8) | |||||

(2.9) |

where eq. (2.8) is obtained from eq. (1), while eq. (2.9) is obtained from eq. (2.1), and the subscript on now refers to the subamplitude evaluated for the permutation in or , respectively. In eq. (2.8), the color matrix is

(2.10) |

whereas in eq. (2.9) the matrix is

(2.11) |

In refs. [10, 17], the Kleiss-Kuijf relation was used to calculate from . From eq. (2.11) we see that it can be calculated directly in terms of structure constants. In appendix A we give for .

Although the expression (2.9) seems to have fewer terms than eq. (2.8), the sparseness of the matrices is also important for determining which expression has the fastest numerical evaluation [17]. For an evaluation that is good only to leading order in — the Leading Color Approximation, or LCA — it is best to use eq. (2.8) and the large- orthogonality of different single traces, to obtain

(2.12) |

where

(2.13) |

Up to , the LCA is exact, and so eq. (2.8) is also superior to eq. (2.9) for an exact evaluation. For , where the LCA has subleading corrections, it is best to take the leading terms from eq. (2.8), and the subleading terms from eq. (2.9) (these are the terms proportional to in eq. (A.3)) [17]. For , ref. [17] employed the linear dependences in the Kleiss-Kuijf relation to find an even more compact form for the subleading terms than using eq. (2.9). (See ref. [18] for another approach to numerically evaluating multi-parton amplitudes beyond the LCA.)

## 3 One-Loop Color Decompositions

The standard color decomposition for one-loop -gluon amplitudes in gauge theory with flavors of quarks is [5]

(3.1) | |||||

where are the subamplitudes, is the subset of that leaves the double trace structure invariant, and is the greatest integer less than or equal to . The superscript denotes the spin of the particle circulating in the loop. The subamplitudes are color-ordered, and give rise to the leading contributions to the cross-section in the large- limit.

The subleading subamplitudes can be obtained from the leading ones through the permutation sum [12]

(3.2) |

where , , and is the set of all permutations of with held fixed that preserve the cyclic ordering of the within and of the within , while allowing for all possible relative orderings of the with respect to the .

In a dual representation of one-loop amplitudes, the subleading subamplitudes come from the annulus diagram where gluons belonging to and are emitted from the inner and outer boundaries of the annulus, respectively. Thus their color properties are very similar to those of the tree subamplitudes if legs and are ‘sewn together’, i.e. contracted with . Indeed, with this interpretation, eq. (3.2) has a very similar structure to the Kleiss-Kuijf relation (2.3). These remarks suggest that the new tree-level color decomposition should have a one-loop analog, where the strings of structure constants that appear are ‘ring diagrams’ instead of multi-peripheral diagrams, as depicted in fig. 5a.

The new one-loop -gluon color decomposition, expressed in terms of adjoint generator matrices , is

(3.3) |

Here is the group of non-cyclic permutations, and is the reflection: . Thus the number of linearly independent subamplitudes is .

The terms proportional to in eq. (3.3) follow from eq. (3.1) using only the reflection identity satisfied by the : . To prove that the remaining terms in eq. (3.3) are correct, we follow closely the second proof of eq. (2.1). Using the Jacobi identity, the generic one-loop color factor can be turned into ring diagrams. The manipulations are exactly the same as those shown in fig. 3, once legs and are joined together to form a loop. This establishes eq. (3.3), but with an unknown coefficient for . Next we equate this expression to the standard color decomposition (3.1), and contract both sides with . Again a unique term survives on each side of the equation, at leading order in , and this establishes that the coefficient is correctly given by . (If we had not removed the reflections from eq. (3.3), then both a ring diagram and its reflection would have contributed to the contraction; this would have led to a factor of in the coefficient of .)

Eq. (3.3) puts the gluon and fermion loops manifestly on the same footing. The contributions of the subleading-color subamplitudes are neatly packaged into the ring-diagram color structures. In addition, all of the generalized decoupling identities are automatically incorporated. Finally, eq. (3.3) shows explicitly the difference between the tree-level and the one-loop color decomposition of the gluon sector for any number of external legs. For instance, if we take the amplitude for a Higgs boson plus three gluons, both eq. (1) and the gluon-loop contribution to eq. (3.1) have as a color factor ; the actual difference in the color structure is concealed in the different properties of the subamplitudes and . In contrast, eq. (2.1) and eq. (3.3) show explicitly the difference in color structure.

One-loop amplitudes contribute to NLO QCD cross-sections through their interference with tree amplitudes. Carrying out the color-summed NLO interference terms for amplitudes color-decomposed as in eqs. (2.1) and (3.3) is straightforward:

where

(3.5) |

with the permutation in and the permutation in .

NNLO production rates include the virtual contributions,

(3.6) |

For the square of one-loop -gluon amplitudes on the right-hand side of eq. (3.6), we obtain

(3.7) | |||||

where

(3.8) |

is given in eq. (2.10), and is the permutation in . In appendix A we give the explicit values for the color matrices , , , and that are required for cross-section computations up to .

A similar analysis can be applied to one-loop amplitudes with an external quark-antiquark pair plus gluons. These amplitudes have the standard color decomposition [13],

(3.9) |

where the color structures are defined by

(3.10) |

and is the subgroup of that leaves invariant.

The leading-color subamplitudes in eq. (3.9) can be expressed in terms of color-ordered (‘primitive’) amplitudes as

(3.11) | |||||

whereas the subleading contributions are given by the permutation formula [13],

(3.12) | |||||

where , , with held fixed. In eqs. (3.11) and (3.12), the quantities and are defined in terms of color-ordered Feynman diagrams, much like the -gluon subamplitudes . However, because an external pair is present now, one also has to specify which way the line running from the antiquark to the quark turns when it enters the loop, left or right; this accounts for the additional index, or . Again the index denotes the spin of a particle circulating around the loop, for those graphs in the primitive amplitude where the external line does not enter the loop. The reflection identity,

(3.13) |

relates the left and right subamplitudes.

There are two types of subleading-color contributions in eq. (3.12), distinguished by whether they contain a factor of or not. Although both types are generated by cyclicly ordered permutations, their origins are quite different. In particular, the subleading terms arise from the term in the Fierz identity,

(3.14) |

where , are strings of generator matrices .
Such manifest terms are not generated by strings of structure
constants. (In fact, terms are generally suppressed when
s are present, because factors come from explicitly
projecting out the factor in , while
structure constants accomplish this projection automatically.) Because of
this fact, the freedom to include structure constants in a color
decomposition does not seem to lead to any simpler representation of the
subleading terms in eq. (3.12), and we shall be content to
simplify the no- terms^{2}^{2}2
Since the terms in the subamplitudes arise from
projecting out a factor, they can also be viewed as subamplitudes
for a fictitious theory containing
two fermion representations, say (the external
pair) and (the fermion in the loop).
This description makes it a bit more transparent how the color structure
multiplying these subamplitudes corresponds to their definition.
But it does not lead to a distinct color decomposition..

The new color decomposition for one-loop amplitudes with an external pair is

where for the product of generators in the adjoint representation reduces to the identity, ; the color structures are defined in eq. (3.10); and