Jet grooming algorithms are widely used in experimental analyses at hadron colliders to remove contaminating radiation from within jets. While the algorithms perform a great service to the experiments, their intricate algorithmic structure and multiple parameters has frustrated precision theoretic understanding. In this paper, we demonstrate that one particular groomer called soft drop actually makes precision jet substructure easier. In particular, we derive a factorization formula for a large class of soft drop jet substructure observables, including jet mass. The essential observation that allows for this factorization is that, without the soft wide-angle radiation groomed by soft drop, all singular contributions are collinear. The simplicity and universality of the collinear limit in QCD allows us to show that to all orders, the normalized differential cross section has no contributions from non-global logarithms. It is also independent of process, up to the relative fraction of quark and gluon jets. In fact, soft drop allows us to define this fraction precisely. The factorization theorem also explains why soft drop observables are less sensitive to hadronization than their ungroomed counterparts. Using the factorization theorem, we resum the soft drop jet mass to next-to-next-to-leading logarithmic accuracy. This requires calculating some clustering effects that are closely related to corresponding effects found in jet veto calculations. We match our resummed calculation to fixed order results for both e + e − → dijets and pp → Z + j events, producing the first jet substructure predictions (groomed or ungroomed) to this accuracy for the LHC. arXiv:1603.09338v2 [hep-ph] 11 Jul 2016 Contents 1. Recluster the jet with a sequential k T -type [45-47] jet algorithm. This produces an infrared and collinear (IRC) safe branching history of the jet. The k T clustering metric 1 While we will not do it in this paper, one could use the results of Ref. [36] which calculates the anomalous dimension of the soft function for event-wide (recoil-free) angularities [37-40] or energy correlation functions with arbitrary angular exponent. This would enable us to extend our results to the case with α = 2. 2 The jet mass has been calculated at NNLL using other methods [41-43] as has 2-subjettiness [44]. However, without grooming the jets, there are non-global logarithms which are not resummed (and which may or may not be quantitatively important) and uncontrollable sensitivity to pileup (which is very quantitatively important).

We develop further an approach to computing energy-energy correlations (EEC) directly from finite correlation functions. In this way, one completely avoids infrared divergences. In maximally supersymmetric Yang-Mills theory (N ¼ 4 sYM), we derive a new, extremely simple formula relating the EEC to a triple discontinuity of a four-point correlation function. We use this formula to compute the EEC in N ¼ 4 sYM at next-to-next-to-leading order in perturbation theory. Our result is given by a twofold integral representation that is straightforwardly evaluated numerically. We find that some of the integration kernels are equivalent to those appearing in sunrise Feynman integrals, which evaluate to elliptic functions. Finally, we use the new formula to provide the expansion of the EEC in the back-to-back and collinear limits.

We derive and solve renormalization group equations that allow for the resummation of subleading power rapidity logarithms. Our equations involve operator mixing into a new class of operators, which we term the "rapidity identity operators", that will generically appear at subleading power in problems involving both rapidity and virtuality scales. To illustrate our formalism, we analytically solve these equations to resum the power suppressed logarithms appearing in the back-to-back (double light cone) limit of the Energy-Energy Correlator (EEC) in N = 4 super-Yang-Mills. These logarithms can also be extracted to O(α 3 s) from a recent perturbative calculation, and we find perfect agreement to this order. Instead of the standard Sudakov exponential, our resummed result for the subleading power logarithms is expressed in terms of Dawson's integral, with an argument related to the cusp anomalous dimension. We call this functional form "Dawson's Sudakov". Our formalism is widely applicable for the resummation of subleading power rapidity logarithms in other more phenomenologically relevant observables, such as the EEC in QCD, the p T spectrum for color singlet boson production at hadron colliders, and the resummation of power suppressed logarithms in the Regge limit.

Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This novel technique is based on a method due to Höschele et al. and relies only on the knowledge of a single integral of uniform transcendental weight. As a corollary, the algorithm can also be used to test the uniform transcendentality of a given integral. We discuss the application to several cutting-edge examples, including non-planar four-loop HQET and non-planar twoloop five-point integrals. A Mathematica implementation of our algorithm is made available together with this paper.

Infrared divergences have long been heralded to cancel in sufficiently inclusive crosssections, according to the famous Kinoshita-Lee-Nauenberg theorem which mandates an initial and final state sum. While well-motivated, this theorem is much weaker than necessary: for finiteness, one need only sum over initial or final states. Moreover, the cancellation generically requires the inclusion of the forward scattering process. We provide a number of examples showing the importance of this revised understanding: in e + e − → Z at next-to-leading order, one can sum over certain initial and final states with an arbitrary number of extra photons, or only over final states with a finite number of photons, if forward scattering is included. For Compton scattering, infrared finiteness requires the indistinguishability of hard forward-scattered electrons and photons. This implies that in addition to experimental limits on the energy and angular resolution, there must also be an experimental limit on the momentum at which electric charge can be observed. Similar considerations are required to explain why the rate for γγ to scatter into photons alone is infrared divergent but the rate for γγ to scatter into photons or charged particles is finite. This new understanding sheds light on the importance of including degenerate initial states in physical predictions, the relevance of disconnected Feynman diagrams, the importance of dressing initial or final-state charged particles, and the quest to properly define the S matrix.