Antenna Factorization of Gauge-Theory Amplitudes

I derive a single factorization formula which summarizes all soft and collinear divergences of a tree-level gauge theory amplitude. The singular factor squared is in a certain sense the generalization of the Catani–Seymour dipole factorization formula. Submitted to Physical Review D Laboratory of the Direction des Sciences de la Matière of the Commissariat à l’Energie Atomique of France. The factorization properties of gauge-theory amplitudes in infrared-singular limits have played an important role not only in our conceptual understanding of infrared safety, but also in explicit calculations at next-to-leading order in perturbative QCD. These infrared singularities may be classified into two types: soft and collinear. The former arises when a gluon four-momentum vanishes, ks → 0; the latter when the momenta of two massless particles become proportional, ka → z(ka + kb), kb → (1 − z)(ka + kb). The properties of non-Abelian gauge-theory amplitudes in these limits are easiest to understand in the context of a color decomposition [1]. For tree-level all-gluon amplitudes in an SU(N) gauge theory the color decomposition has the form, A n ({ki, λi, ai}) = ∑ σ∈Sn/Zn Tr(T aσ(1) · · ·T σ(n))A n (σ(11 , . . . , nn)) , (1) where Sn/Zn is the group of non-cyclic permutations on n symbols, and j λj denotes the j-th momentum and helicity. As is by now standard, I use the normalization Tr(T T ) = δ. One can write analogous formulæ for amplitudes with quark-antiquark pairs or uncolored external lines. The color-ordered or partial amplitude An is gauge invariant, and has simple factorization properties in both the soft and collinear limits, A n (. . . , a λa , bb , . . .) a‖b −−−→ ∑ λ=± Split −λ (a λa , bb)A n−1(. . . , (a + b) , . . .) , A n (. . . , a, s λs , b, . . .) ks→0 −−−→ Soft(a, ss , b)A n−1(. . . , a, b, . . .) . (2) The collinear splitting amplitude, squared and summed over helicities, gives the usual unpolarized Altarelli–Parisi splitting function [2]. While the complete amplitude also factorizes in the collinear limit, the same is not true of the soft limit; the eikonal factors get tangled up with the color structure. It is for this reason that the color decomposition is useful. Its use, and the simple factorization properties of the partial amplitudes, allowed Giele and Glover, in their pioneering paper [3], to derive simple and universal functions summarizing the integration over soft and collinear radiation. Of course, it is possible, as was done by later authors [4,5], to square the amplitude first, and tangle up the color factors with the eikonal factors, thereby recovering earlier forms of these two factorizations [2,6], but this obscures the clean structure in eqn. (2). The color decomposition (1) is also the form in which one writes tree-level amplitudes in an open string theory, and the gauge-theory partial amplitudes may be computed as the infinite-tension 2 limit of the corresponding string amplitudes. These are given by the Koba–Nielsen formula [7], A n = τ n−4 2 ∫ xa<x1<xb···<xn−2 n−3 ∏ i=1 dxi (xb − xa)(xn−2 − xa)(xn−2 − xb) × ∏ i 6=j∈E |xj − xi| ki·kj/2 exp   ∑ i 6=j 1 2 εi · εj (xj − xi) + τki · εj (xi − xj)   ∣∣∣∣∣ multilinear , (3) which I have written in a form that will be useful for the derivation to be presented in this paper. In this equation, {ki, εi} are the momenta and polarization vectors of the gluons; E denotes the set of all external legs, {a, 1, b, 2, . . . , n− 2}; τ is the square root of the inverse string tension; and the subscript ‘multilinear’ tells us to extract the terms in the exponential linear in each of the polarization vectors. Three of the xi may be fixed at will; it will be convenient to choose xa = 0 and xb = 1. The gauge theory limit is given by τ → 0. This formula gives an explicit form for the n-gluon amplitude, and examining its limits is thus a convenient method of deriving [8] factorization formulæ such as those in eqn. (2). More recently, Catani and Seymour [5] wrote down a dipole factorization formula which gives a single function capturing the singular behavior of the squared matrix element in both the soft and collinear limits. Of course, since it is at the level of the amplitude squared rather than the amplitude, and since it does not take advantage of the color decomposition, it is not quite a true factorization but is still tangled up with the color algebra. The purpose of this note is to derive a single function unifying the soft and collinear factors in eqn. (2) at the level of color-ordered amplitudes. This will provide a true factorization. I will discuss its connection with the Catani– Seymour dipole factorization formula later. Before embarking upon the derivation, I note that the Koba–Nielsen form is valid whatever the dimensions of the external momenta and polarization vectors, so long as the former are onshell, and the latter transverse. It thus allows for a treatment corresponding to any of the possible variants of dimensional regularization: the conventional scheme (CDR) [9], the original ’t Hooft– Veltman scheme (HV) [10], dimensional reduction (DR) [11], or the four-dimensional helicity scheme (FDH) [12]. The forms of the collinear and soft factorization functions suggest that we can associate the singular limits with a trio (a, 1, b) of color-ordered partons: either a soft parton with two hard neighbors, or a pair of collinear partons neighboring another hard parton. This in turn suggests generalizing the derivation in ref. [8] to extract all singular behavior in either of the two invariants sa1 or s1b. This will summarize the singular behavior whenever the trio of momenta become degenerate without either a or b becoming soft, that is as ∆(a, 1, b)/sab → 0. (∆(a, 1, b) is the Gram determinant of the (a, 1, b) system.) 3 If we examine equation (3) we see that singularities in these two invariants can arise only from the integration region x1 ≃ xa = 0 or x1 ≃ xb = 1, and then only in those terms which have a lone inverse power of x1 or 1− x1 coming from the expansion of the exponential. (The integrals of x or (1− x) are finite by analytic continuation.) Separating out all terms containing the integration variable x1, we obtain