We discuss consistent power counting for integrating soft and collinear degrees of freedom over arbitrary regions of phase space in the soft-collinear effective theory, and illustrate our results at one-loop with several jet algorithms: JADE, Sterman-Weinberg and k ? . Consistently applying soft-collinear effective theory power counting in phase space, along with nontrivial zero-bin subtractions, prevents double counting of final states. The resulting phase space integrals over soft and collinear regions are individually ultraviolet divergent, but the phase space ultraviolet divergences cancel in the sum. Whether the soft and collinear contributions are individually infrared safe depends on the jet definition. We show that while this is true at one-loop for JADE and Sterman-Weinberg, the k ? algorithm does not factorize into individually infrared safe soft and collinear pieces in dimensional regularization. We point out that this statement depends on the ultraviolet regulator, and that in a cutoff scheme the soft functions are infrared safe.

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We present a complete analysis of the cosmological constraints on decaying dark matter. Previous analyses have used the cosmic microwave background and Type Ia supernova. We have updated them with the latest data as well as extended the analysis with the inclusion of Lyman-α forest, large scale structure and weak lensing observations. Astrophysical constraints are not considered in the present paper. The bounds on the lifetime of decaying dark matter are dominated by either the late-time integrated Sachs-Wolfe effect for the scenario with weak reionization, or CMB polarization observations when there is significant reionization. For the respective scenarios, the lifetimes for decaying dark matter are Γ −1 100 Gyr and (f Γ) −1 5.3 × 10 8 Gyr (at 95.4% confidence level), where the phenomenological parameter f is the fraction of the decay energy deposited in baryonic gas. This allows us to constrain particle physics models with dark matter candidates through investigation of dark matter decays into Standard Model particles via effective operators. For decaying dark matter of ∼ 100 GeV mass, we found that the size of the coupling constant in the effective dimension-4 operators responsible for dark matter decay has to generically be 10 −22 . We have also explored the implications of our analysis for representative models in theories of gauge-mediated supersymmetry breaking, minimal supergravity and little Higgs.

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