QCD scattering amplitudes in the high-energy limit

Abstract

In perturbative quantum field theory, logarithmic divergences generically occur whenever there is a large disparity between physical scales. In this thesis we are interested in the so-called high-energy logarithms, L, that arise in quantum chromodynamics (QCD). These logarithms occur when the centre-of-mass energy of a collision is much larger than the transverse scales of the produced particles. Even if we are within the perturbative regime, where the strong coupling constant, αs, is much less than unity, at sufficiently large centre-of-mass energy the combination αsL may be of order unity. These terms therefore need to be summed to all orders in αs to ensure the stability of perturbative predictions. In contrast to a fixed-order expansion, where accuracy is stated in terms of leading order (LO), next-to-leading order (NLO) etc., we can also consider an expansion of the amplitude in terms of leading logarithmic (LL) terms, next-to-leading logarithmic (NLL) terms etc.

The first all-orders summation of high-energy logarithms in QCD was obtained via the LL BFKL equation. This programme was then extended to NLL accuracy. There has been a recent renewal of interest in the high energy asymptotics of QCD, and in particular there has been a recent effort to extend the BFKL framework to next-to-next-to-leading logarithmic (NNLL) accuracy. A necessary ingredient is the one-loop central emission vertex (CEV) for two gluons which are not strongly ordered in rapidity. In the first part of this thesis we consider the one-loop six-gluon amplitude in N = 4 super Yang-Mills (SYM) theory in a central next-to-multi-Regge kinematic (NMRK) limit. In this limit, the dispersive part of the amplitude factorises and we extract the one-loop two-gluon CEV for any helicity configuration within this theory. This is a component of the twogluon CEV in QCD. We show that the absorptive part of the one-loop six-gluon amplitude does not contribute to NNLL accuracy in this limit.

The primary motivation of the work presented in this thesis is the inclusion of such logarithmic terms to all orders in αs for stable and high-precision predictions for the Large Hadron Collider (LHC). At the LHC, the centre-of-mass energy is large, but finite, and so some of the asymptotic approximations of the high-energy literature are too severe to be applicable to our current experiments. The High Energy Jets (HEJ) framework was developed to solve these issues. In particular, it uses Monte Carlo event generation for phase-space integration, which allows the use of arbitrary kinematic cuts, the freedom to use more involved factorised expressions to approximate the hard matrix element, and the ability to match to full fixed-order results. The factorised expressions that enter into the kernel of the BFKL equation have been re-derived with minimal approximations in order to capture significantly more of the fixed-order physics. In the second part of this thesis we derive two new factorised expressions, which were the last two-parton components required to bring the HEJ framework to NLL accuracy.

Prior to the work presented here, HEJ has been matched to LO fixed-order results, and incorporated LL resummation, as well as providing a NLL description of some key processes. In the final part of this thesis we present the method by which HEJ was further matched to a NLO fixed-order calculation, namely the production of a W boson in association with two jets.