The triple collinear limit of one - loop QCD amplitudes ∗

We consider the singular behaviour of one-loop QCD matrix elements when several external partons become simultaneously parallel. We present a new factorization formula that describes the singular collinear behaviour directly in colour space. The collinear singularities are embodied in process-independent splitting matrices that depend on the momenta, flavours, spins and colours of the collinear partons. We give the general structure of the infrared and ultraviolet divergences of the one-loop splitting matrices. We also present explicit one-loop results for the triple collinear splitting, q → qQ̄Q, of a quark and a quark–antiquark pair of different flavours. The one-loop triple collinear splitting is one of the ingredients that can be used to compute the evolution of parton distributions at the next-to-next-to-leading order in QCD perturbation theory. CERN–TH/2003-206 December 2003 Work supported in part by EC 5th Framework Programme under contract number HPMF-CT-200000989. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] The high precision of experiments at past (LEP), present (HERA, Tevatron) and future (LHC, ee linear colliders) particle colliders demands a corresponding precision in theoretical predictions. As for perturbative QCD predictions, this means calculations beyond the next-to-leading order (NLO) in the strong coupling αS. Recent years have witnessed much progress in this field (see Ref. [1] and references therein). In particular, a great deal of work has been devoted to study the properties of QCD scattering amplitudes in the infrared (soft and collinear) region [2]–[12]. The understanding of the infrared singular behaviour of multiparton QCD amplitudes is a prerequisite for the evaluation of infrared-finite cross sections (and, more generally, infraredand collinear-safe QCD observables) at the next-to-next-to-leading order (NNLO) in perturbation theory [1]. The information on the infrared properties of the amplitudes have also been exploited to compute large (logarithmically enhanced) perturbative terms and to resum them to all perturbative orders [13]. The investigation of these properties is also valuable for improving the physics content of Monte Carlo event generators (see e.g. Ref. [14]). In addition, the results of these studies prove to be useful beyond the strict QCD context, since they can provide hints on the structure of highly symmetric gauge theories at infinite orders in the perturbative expansion (see e.g. Ref. [15]). In this paper we consider the collinear limit of multiparton QCD amplitudes at oneloop order. We present a general factorization formula, which is valid directly in colour space, and discuss some properties of its infrared-divergent contributions. These results apply to the multiple collinear limit of an arbitrary number of QCD partons. As an example of application beyond the double collinear limit, we present the result of the explicit evaluation of a triple collinear configuration of three quarks. Besides its interest within the general framework outlined above, our study of the one-loop triple collinear limit has specific relevance to the NNLO calculation of the Altarelli–Parisi kernels that control the scale evolution of parton densities and fragmentation functions [16]. This formidable NNLO computation is being completed [17] by using traditional methods. Kosower and Uwer [18] have proposed to exploit collinear factorization at the amplitude level as an alternative method to perform the NNLO calculation of the Altarelli–Parisi kernels. To this purpose, the one-loop triple collinear splitting is one of the necessary ingredients. Two other ingredients are the tree-level quadruple collinear splitting [11] and the two-loop double collinear splitting. A detailed discussion of the multiple collinear limit and of the results presented in this letter will appear in a forthcoming paper [19]. We consider a generic scattering process involving final-state QCD partons (massless quarks and gluons) with momenta p1, p2, . . . Non-QCD partons (γ , Z,W, . . .) are always understood. The corresponding matrix element is denoted by M1212 a1,a2,... (p1, p2, . . .) , (1) where {c1, c2, . . .}, {s1, s2, . . .} and {a1, a2, . . .} are respectively colour, spin and flavour indices. To take into account the colour and spin structures, we use the notation of Refs. [3, 20]. We introduce an orthonormal basis {|c1, c2, . . .〉 ⊗ |s1, s2, . . .〉} in colour + spin space, in such a way that the matrix element in Eq. (1) can be written as M1212 a1,a2,... (p1, p2, . . .) ≡ ( 〈c1, c2, . . .| ⊗ 〈s1, s2, . . .| ) |Ma1,a2,...(p1, p2, . . .)〉 . (2) Thus |Ma1,a2,...(p1, p2, . . .)〉 is a vector in colour + spin space. It is important to specify