Endpoint Factorization for Semileptonic Decays of Boosted and Resonant Off-Shell Top Quarks with a Large-Radius Bottom Jet

We derive a factorization formula for boosted double resonant top-antitop pair production in $e^+e^-$ annihilation with a semileptonic top quark decay in the phase space region where the $b$-jet invariant mass is small. The decaying top quark state is defined through invariant mass measurements on the final states in the top and antitop hemispheres, and the $b$-jet is defined from clustering all hadrons in the top hemisphere. The factorization does not rely on the narrow width limit and accounts for the QCD off-shell and interference effects. The approach employs Soft-Collinear-Effective Theory and boosted Heavy-Quark-Effective-Theory and relies on a combination of factorization theorems known from $e^+e^-$ dijet production and inclusive semileptonic heavy meson endpoint decays. The result provides a first principles treatment of the dominant hadronization effects, which can be determined from $e^+e^-$ event shapes. In the factorization a new distribution function arises, called the ultra-collinear-soft (ucs) function, which encodes the Fermi motion of the decaying top quark within the state defined from the invariant mass measurement. The ucs function is a differential generalization of the inclusive bHQET jet function and shares properties of the shape function in semileptonic heavy meson decays. In frames where the top quark is very slow, it describes the coherent soft radiation arising from top production, propagation and decay, and encodes all effects that are non-factorizable from the perspective of the NW limit. Its form and renormalization depend on two light-cone momenta related to the top-jet and $b$-jet directions and their relative angle. Due to the large top quark width, the ucs function can be computed perturbatively, and we determine the QCD corrections at ${\cal O}(α_s)$. The anomalous dimension is known to three loops.

💡 Research Summary

The paper presents a first‑principles factorization theorem for boosted top‑antitop pair production in electron‑positron annihilation when one top decays semileptonically and the associated $b$‑jet is defined with a small invariant mass. Unlike the traditional narrow‑width (NW) approximation, the authors do not treat the top quark as an on‑shell asymptotic particle; instead they keep the full complex pole mass and therefore incorporate off‑shell and non‑factorizable QCD effects that arise from the overlap of production‑ and decay‑stage radiation.

The theoretical framework combines Soft‑Collinear Effective Theory (SCET) with boosted Heavy‑Quark Effective Theory (bHQET). Four distinct momentum modes are identified: (i) ultra‑collinear‑soft radiation that is soft in the top rest frame (called “top‑ultra‑collinear”), (ii) the analogous mode for the antitop, (iii) large‑angle soft radiation sensitive to the hemisphere boundary, and (iv) hard‑collinear radiation that builds the $b$‑jet. Modes (ii)–(iv) factorize cleanly into separate functions, while mode (i) couples production and decay and is responsible for the so‑called non‑factorizable contributions.

To capture mode (i) the authors introduce a new distribution, the ultra‑collinear‑soft (ucs) function. The ucs function is a differential generalization of the inclusive bHQET jet function and shares many properties with the shape function that appears in inclusive semileptonic $B$‑meson decays. Physically it describes the “Fermi motion” of the decaying top quark inside the jet defined by the hemisphere invariant‑mass measurement. Because the top width $\Gamma_t\sim1.4,$GeV is large, the ucs function can be computed perturbatively; the authors evaluate it at ${\cal O}(\alpha_s)$ and note that its anomalous dimension is known to three loops, enabling NNLL (or NLL′) resummation.

The final factorized cross‑section for the triple‑differential observable $d^3\sigma/(dM_t^2,dM_{\bar t}^2,dX)$ (where $X$ denotes any observable built from the $b$‑jet and the lepton momenta) reads schematically

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