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

TL;DR

The paper addresses precision predictions for boosted, off-shell top-quark pair production in $e^+e^-$ annihilation with a semileptonic top decay in the small-$M_{j_b}$ region. It develops a factorization framework using SCET and bHQET, introducing a novel ultra-collinear-soft function $S_{ucs}$ that captures non-factorizable soft radiation across production and decay, and computes its $\mathcal{O}(\alpha_s)$ corrections with three-loop anomalous dimensions available for consistency checks. The factorization combines double-hemisphere invariant-mass factorization and endpoint semileptonic heavy-meson decays, yielding a master cross section with RG evolution for all components and a nonperturbative hemisphere soft function $S_{hemi}$ that can be constrained from $e^+e^-$ event shapes. A fixed-order NLO study illustrates the qualitative impact on the $M_{j_b\ell}$ spectrum and endpoint shifts, while outlining the path to NNLL or NLL' resummation needed for precision phenomenology, including nonperturbative and hadronization effects. The framework provides a principled approach to top-mass sensitive observables and offers a benchmark for MC generators and future lepton-collider studies, with potential applicability to LHC-like top-mass extractions once narrowed-$b$-jets and extended observables are incorporated.

Abstract

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.

Figures (7)

• Figure 1: Graphical illustration of all distinct radiation modes relevant for the leading power 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 illustrated modes are large-angle hemisphere soft (green), $n$-ultra-collinear (blue), $\bar{n}$-ultra-collinear (orange) and $n'$-hard-collinear (red). The total hadronic momentum $H^\mu$ in the top hemisphere, which excludes the leptonic momentum $q^\mu = p_\ell^\mu + p_{\nu_\ell}^\mu$, is indicated by the upper gray ellipse. The total top hemisphere momentum is $M_t^\mu=H^\mu+q^\mu$. The total antitop hemisphere momentum $M_{\bar{t}}^\mu$ is indicated by the lower gray ellipse.
• Figure 2: Example of the cancellation of soft gluon attachments to the decay products.
• Figure 3: Matching factors, factorization functions and RG evolution as shown in the factorization formula (\ref{['eq:dhmdecayfinalrge']}) for boosted top-antitop pair production with a semileptonic top quark decay in the top mass sensitive endpoint region, and where the hemisphere masses $M_t$ and $M_{\bar{t}}$ are in the resonance region. The $b$-jet is defined as the top hemisphere without the lepton momenta. The evolution of $U_{H_Q}$, $U_{H_m}$ and $U_{H_d}$ are local, while those in $U_J$, $U_{J_B}$, $U_S$ and $U_\mathrm{shape}$ involve convolutions. The global renormalization scale $\mu$, to which all matching factors and functions appearing below the hard scattering scale $Q$ evolve, may be set to any other scale below $m_t$ due to renormalization consistency.
• Figure 4: One-loop diagrams for the computation of the NLO QCD corrections to the ultra-collinear-soft function.
• Figure 5: Left panel: Tree-level (blue) and ${\cal O}(\alpha_s)$ fixed order evaluation of the factorization theorem for the $b$-jet lepton invariant mass $M_{j_b\ell}$ distribution of Eq. (\ref{['eq:NLOfactorization']}) for $Q=700$ GeV, $m_t=173$ GeV and $\Gamma_t=1.42$ GeV and cuts on the top and antitop hemisphere invariant masses $|M_{t,\bar{t}}-m_t|<10$ GeV in the pole mass scheme and without gap subtraction. The $W$-boson is treated in the narrow width limit $\Gamma_W\to 0$, and we used the tree-level (partial) widths $\Gamma_{W,{\rm lep}}^{\rm tree}=G_F M_W^3/(6\pi\sqrt{2})$ and $\Gamma_t^{\rm tree}=G_F m_t^3/(8\pi\sqrt{2})(1-M_W^2/m_t^2)^2(1+2M_W^2/m_t^2)$ for normalization. At ${\cal O}(\alpha_s)$ we used the renormalization scales $\mu=M_W$ (red), $m_t$ (green) and $2m_t$ (yellow). The right panel shows all distributions normalized to unity to visualize the impact of the ${\cal O}(\alpha_s)$ corrections on the shape of the distribution.