Recursive Approach to One-loop QCD Matrix Elements

Recently, a “weak-weak” duality, between N = 4 super Yang-Mills and a topological string theory propagating in twistor space, has been proposed [1] implying an identical perturbative S-matrix for the two theories. The existence of a duality between the two theories implies a surprising structure within the S-matrix of gauge theory. This has inspired considerable progress in computing scattering amplitudes. The generalisation of these ideas combined with ideas from the unitarity method [2,3] has led to new ideas in computing one-loop gluon scattering amplitudes [4,5,6] in theories with less than maximal or no supersymmetry such as massless QCD. In this talk we discuss and review this work with particular reference to the results for oneloop QCD amplitudes [4,6]. The particular approach that we describe is recursive and our aim is to establish recursion relations where an n-point one-loop amplitude is obtained from expressions for lower-point amplitudes, bypassing the need for performing any loop integrations. As yet, this approach only works in cases where certain criteria on the unitarity cuts are satisfied. But in the cases where the criteria are satisfied, it is a particularly effective. The duality is most obvious if we express the amplitude in terms of fermionic “twistor” variables. We can achieve this by replacing everywhere the massless momentum paȧ by λaλ̄ȧ where paȧ = (σ )aȧpμ. The external polarisation vectors can also be defined in terms of spinor variables [7] using the spinor-helicity notation. This talk is primarily about loop calculations, however, there are two twistor inspired techniques for computing tree amplitudes which we wish to discuss. First there is the MHV-vertex construction by Cachazo, Svrcek and Witten (CSW) [8] and secondly there is the recursion relations by Britto, Cachazo, Feng and Witten (BCFW) [9]. In the MHV vertex approach, amplitudes are obtained by sewing together “MHV vertices”. A n-point MHV vertex has exactly two gluons of negative helicity and all remaining helicities positive. Amplitudes with more negative helicities, for example, next-to-MHV or ‘NMHV” amplitudes, are in the CSW formalism constructible from products of MHV vertices. The forms of these vertices are those of the Parke-Taylor amplitudes [10] where a specific off-shell continuation is employed for the internal particle lines. The CSW amplitude construction is explicitly asymmetric in gluon helicity. This formalism is a remarkable rewriting of perturbation theory. It has been extended to a variety of cases beyond that of gluon scattering [11]. The MHV amplitude has been shown to extend to one-loop amplitudes within supersymmetric theories [12] although application of these rules still requires integration and an extension to non-supersymmetric theories proves more difficult. The BCFW recursion relations [9] rely on the analytic structure of the amplitude after it has been continued to a function in the complex plane A(z). This continuation is a shift in the (spino-