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We derive the leading non-global logarithms (NGLs) of ratios of jet massesm_{1,2} and a jet energy veto \Lambda due to soft gluons splitting into regionsin and out of jets. Such NGLs appear in any exclusive jet cross section withmultiple jet measurements or with a veto imposed on additional jets. Here, weconsider back-to-back jets of radius R produced in e^+e^- collisions, foundwith a cone or recombination algorithm. The leading NGLs are of the form\alpha_s^2 \ln^2(\Lambda/m_{1,2}) or \alpha_s^2\ln^2(m_1/m_2). Theircoefficients depend both on the algorithm and on R. We consider cone, \kt,anti-\kt, and Cambridge-Aachen algorithms. In addition to determining the fullalgorithmic and R dependence of the leading NGLs, we derive new relations amongtheir coefficients. We also derive to all orders in \alpha_s a factorized formfor the soft function S(k_L,k_R,\Lambda) in the cross section\sigma(m_1,m_2,\Lambda) in which dependence on each of the global logs of\mu/k_L, \mu/k_R and \mu/\Lambda determined by the renormalization group areseparated from one another and from the non-global logs. The same kind of softfunction, its associated non-global structure, and the algorithmic dependencewe derive here will also arise in exclusive jet cross sections at hadroncolliders, and must be understood and brought under control to achieve precisetheoretical predictions.

Double Non-Global Logarithms In-N-Out of Jets