Abstract
We compute the 1-jettiness soft function for the decay of a heavy quark into a light quark jet plus colorless particles at three-loop order in soft-collinear effective theory. The 1-jettiness measurement fixes the total small light-cone momentum component of the soft radiation with respect to the jet direction. This soft function is a universal ingredient to the factorization of heavy-to-light quark decays in the limit of small 1-jettiness. Our three-loop result is required for resummation at the N3LL′ level, e.g. near the endpoint in the photon energy spectrum of the B → Xsγ decay. It is also a necessary ingredient for future calculations of fully-differential heavy-to-light quark decay rates at N3LO using the N -jettiness subtraction method, e.g. for semileptonic top decays. Using our result for the soft anomalous dimension we confirm predictions on the universal infrared structure of QCD scattering amplitudes with a massive external quark at three loops.
Highlights
∆ ΛQCD nonperturbative effects are sizable and one can further factorize S into a purely perturbative ‘heavy-to-light soft function’ Shl and a nonperturbative shape function F [8]: S(ω, μ) = dω Shl(ω − ω, μ) F (ω )
Our three-loop result is required for resummation at the N3LL level, e.g. near the endpoint in the photon energy spectrum of the B → Xsγ decay
The current state of the art for such predictions includes resummation at the primed next-to-next-to-leading logarithmic (NNLL ) level [8, 16], where the NNLL expression [17] is augmented with the next-to-next-to-leading order (NNLO) corrections to the factorization functions at their matching scales, i.e. to H(μH ), J(μJ ), Shl(μS)
Summary
The 1-jettiness soft function for heavy-to-light decays is defined by the vacuum matrix element. Where averaging over color and spin of the external HQET b-quark states is understood, the latter are normalized such that bv|hv(0)hv(0)|bv = 1, hv is the HQET heavy quark field with velocity v, and Dμ = ∂μ + igAμ(0) This b-quark matrix element was calculated to O(αs) in refs. In the underlying full QCD processes the external heavy and light quark lines are correspondingly crossed from initial to final state or vice versa. Some of these soft functions are e.g. relevant for s- and t-channel single top production as well as charm production in deep-inelastic neutrino scattering (‘light-to-heavy DIS’). There is, to the best of our knowledge, no simple argument why this equality should hold at three loops and beyond, not even between heavy-to-light decay and light-to-heavy DIS soft functions. A dedicated three-loop analysis along the lines of ref. [30] would require to derive the analytic structure for the relevant two-loop single-emission and one-loop double-emission heavy-light soft currents, which is beyond the scope of this work
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