Hindawi Publishing Corporation Advances in High Energy Physics Volume 2012, Article ID 379736, 10 pages doi:10.1155/2012/379736 Research Article Soft Collinear Degeneracies in an Asymptotically Free Theory Martin Lavelle, 1 David McMullan, 1 and Tom Steele 2 1 School of Computing and Mathematics, University of Plymouth, Plymouth PL4 8AA, UK 2 Department of Physics & Engineering Physics, University of Saskatchewan, Saskatoon, SK, Canada S7N 5E2 Correspondence should be addressed to David McMullan, dmcmullan@plymouth.ac.uk Received 7 July 2011; Accepted 10 November 2011 Academic Editor: Anastasios Petkou Copyright q 2012 Martin Lavelle et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In asymptotically free theories with collinear divergences it is sometimes claimed that these diver- gences are cancelled if one sums over initial and final state degenerate cross-sections and uses an off-shell renormalisation scheme. We show for scalar φ 3 theory in six dimensions that there are further classes of soft collinear divergences and that they are not cancelled. Furthermore, they yield a nonconvergent series of terms at a fixed order of perturbation theory. Similar effects in gauge theories are also summarised. 1. Introduction Infrared IR divergences plague the extraction of physical cross-sections in gauge theories. In practice most calculations are restricted to infrared safe quantities. However, according to the Lee-Nauenberg LN theorem 1, by summing over all degenerate states including both final and initial state radiation one should obtain an IR finite sufficiently inclusive cross- section. This should be contrasted with the Bloch-Nordsieck BN trick 2 where one just sums over final state radiation. It should be noted that the BN trick breaks down in theories with collinear divergences, for example, QCD or QED with massless fermions. It is worth noting that one can have large collinear-type logarithms even for massive particles in some kinematic regions. These should also be removed by a proper treatment of the infrared. The indistinguishable processes which are summed over depend upon experimental resolutions. In this paper we will consider two such thresholds. There is an energy resolution, Δ, such that particles with an energy less than this cannot be detected. Such undetected par- ticles are called soft. There is also an angular resolution, δ, such that massless particles within a cone cannot be distinguished from each other. Note that such collinear particles are not nec- essarily soft, they can carry a significant fraction of the energy in the jet, that is, have energy