Triple (and quadruple) soft-gluon radiation in QCD hard scattering

Published for SISSA by Springer Received: August 15, 2019 Accepted: December 4, 2019 Published: January 20, 2020 Stefano Catani, Dimitri Colferai and Alessandro Torrini INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia, Università di Firenze, Via Sansone 1, Sesto Fiorentino I-50019, Italy E-mail: , , Abstract: We consider the radiation of three soft gluons in a generic process for multiparton hard scattering in QCD. In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current. We compute the tree-level current for triple soft-gluon emission from both massless and massive hard partons. The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations. We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current. The radiation of one and two soft gluons leads to colour dipole correlations. Triple soft-gluon radiation produces in addition colour quadrupole correlations between the hard partons. We examine the soft and collinear singularities of the squared current in various energy ordered and angular ordered regions. We discuss some features of soft radiation to all-loop orders for processes with two and three hard partons. Considering triple soft-gluon radiation from three hard partons, colour quadrupole interactions break the Casimir scaling symmetry between quarks and gluons. We also present some results on the radiation of four soft gluons from two hard partons, and we discuss the colour monster contribution and its relation with the violation (and generalization) of Casimir scaling. We also compute the first correction of O(1/Nc2 ) to the eikonal formula for multiple soft-gluon radiation with strong energy ordering from two hard gluons. Keywords: Perturbative QCD, Scattering Amplitudes ArXiv ePrint: 1908.01616 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2020)118 JHEP01(2020)118 Triple (and quadruple) soft-gluon radiation in QCD hard scattering Contents 2 2 Soft factorization and soft-gluon currents 2.1 Soft factorization of scattering amplitudes 2.2 Tree-level current: single and double gluon emission 4 4 7 3 Tree-level soft current for triple gluon emission 9 4 Squared amplitudes and currents 4.1 Tree-level squared current: single and double gluon emission 13 14 5 Tree-level squared current for triple gluon emission 5.1 General structure 5.2 Dipole correlation 5.3 Quadrupole correlation 5.4 Collinear singularities 18 18 19 22 24 6 Processes with soft gluons and three hard partons 6.1 All-order features 6.2 Soft-gluon radiation at the tree level 28 28 30 7 Processes with soft gluons and two hard partons 7.1 All-order features 7.2 Soft-gluon radiation at the tree level 7.3 Quadruple soft-gluon radiation at the tree level 35 35 39 41 8 Generating functional and exponentiation 49 9 Summary 52 A Current conservation 53 B Colour quadrupole operators 55 C Momentum dependence of quadrupole and dipole correlations for triple soft-gluon radiation 60 –1– JHEP01(2020)118 1 Introduction 1 Introduction –2– JHEP01(2020)118 The first two operational runs of proton-proton collisions at the LHC have produced a large amount of high-precision data on hard-scattering final states. Similar data are expected in the next phases of the LHC. The high-precision LHC data demand for a corresponding accuracy in theoretical predictions. Such theoretical accuracy is required both to test our present understanding of the Standard Model and to discover and investigate (probably tiny) signals of new physics phenomena. In the context of QCD, one way to increase the theoretical accuracy consists in performing calculations at higher perturbative orders in the QCD coupling αS . The LHC physics program has moved the present frontier of perturbative calculations to the next-to-next-tonext-to-leading order (N3 LO). During the last few years, much effort has been devoted to high-order perturbative computations and much progress has been already achieved at the N3 LO frontier. We limit ourselves to explicitly mentioning few examples among very many others. The total (partonic) cross section for Higgs boson production in hadron-hadron collisions is known up to N3 LO [1–3]. Substantial advances have been achieved toward the complete N3 LO calculation [4, 5] of the evolution kernels of the parton distribution functions. The structure of the infrared (IR) divergences of multileg QCD scattering amplitudes has been explicitly computed at the three-loop level [6]. A relevant feature of QCD scattering amplitudes is the presence of singularities in soft and collinear regions of the phase space, and the corresponding presence of IR divergences in virtual radiative corrections at the loop level. The theoretical study of these aspects of the scattering amplitudes is relevant ‘per se’ in QCD and, more generally, in perturbative gauge field theories. We know that soft/collinear singularities and IR divergences have a process-independent structure, and they are controlled by universal factorization formulae. In the computation of physical observables, phase space and loop singularities cancel between themselves, but much technical effort is required to achieve the cancellations, and the effort highly increases at higher perturbative orders. The explicit knowledge of soft/collinear factorization of scattering amplitudes at O(αS ) has been essential to devise fully general (observable-independent and process-independent) methods to carry out next-to-leading order (NLO) QCD calculations (see, e.g., refs. [7–10]). Analogously, the knowledge of soft/collinear factorization at O(αS2 ) [11–21] is exploited to develop methods (see, e.g., a list of references in section 1.2.4 of ref. [22]) at the next-to-next-to-leading order (NNLO). Soft/collinear factorization formulae at O(αS3 ) can be used in the context of N3 LO calculations. The perturbative radiative corrections to hard-scattering observables in kinematical regions close to the exclusive boundary of the phase space are affected by large logarithmic contributions. These large contributions have to be computed at sufficiently-high perturbative orders and, possibly, resummed to all orders (see, e.g., the list of references in refs. [23, 24]). The large logarithms arise from the unbalance between loop and real radiative corrections in the soft and collinear regions of the phase space. The explicit knowledge of soft/collinear factorization at O(αS3 ) gives information that is necessary in resummed calculations at the next-to-next-to-next-to-leading logarithmic accuracy. Independently of –3– JHEP01(2020)118 resummation, the explicit calculation of large logarithmic terms can be used to obtain approximated fixed-order results. Indeed, we note that the first approximated N 3 LO result for Higgs boson production [1] was o (...truncated)