Abstract
We present a new algorithm to construct a purely four dimensional representation of higher-order perturbative corrections to physical cross-sections at next-to-leading order (NLO). The algorithm is based on the loop-tree duality (LTD), and it is implemented by introducing a suitable mapping between the external and loop momenta of the virtual scattering amplitudes, and the external momenta of the real emission corrections. In this way, the sum over degenerate infrared states is performed at integrand level and the cancellation of infrared divergences occurs locally without introducing subtraction counter-terms to deal with soft and final-state collinear singularities. The dual representation of ultraviolet counter-terms is also discussed in detail, in particular for self-energy contributions. The method is first illustrated with the scalar three-point function, before proceeding with the calculation of the physical cross-section for $\gamma^* \to q \bar{q}(g)$, and its generalisation to multi-leg processes. The extension to next-to-next-to-leading order (NNLO) is briefly commented.
Highlights
There are several variants of the subtraction method at next-to-leading order (NLO) and beyond [3,4,5,6,7,8,9,10,11,12,13], which involve treating separately real and virtual contributions
We present a new algorithm to construct a purely four dimensional representation of higher-order perturbative corrections to physical cross-sections at next-to-leading order (NLO)
In other words, working in the context of dimensional regularisation (DREG) [28,29,30,31] with d = 4 − 2 the number of space-time dimensions, the mapped real-virtual contributions do not lead to -poles, which implies that the limit → 0 can safely be considered
Summary
We review the main ideas behind the LTD method. The LTD theorem [14] establishes a direct connection among loop and phase-space integrals. Since LTD is derived through the application of the Cauchy’s residue theorem, the remaining d − 1 dimensional integration is performed over the forward on-shell hyperboloids defined by the solution of GF (qi)−1 = 0 with qi,0 > 0 Notice that these on-shell hyperboloids degenerate to light-cones when internal particles are massless. The dual representation shown in eq (2.2) is built by adding all possible single-cuts of the original loop diagram. In this procedure, the propagator associated with the cut line is replaced by eq (2.6) whilst the remaining uncut Feynman propagators are promoted to dual ones. Together with the positive energy constraint imposed by the delta distribution in eq (2.6), it restricts the possible situations compatible with a sequential decay of on-shell physical particles
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