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<p>We perform an all-order analysis of double-logarithmic corrections to the so-called soft-overlap contribution to heavy-to-light transition form factors at large hadronic recoil. Specifically, we study <i>B</i><sub><i>c</i></sub> → <i>η</i><sub><i>c</i></sub> transitions within a perturbative non-relativistic framework, treating both the bottom and charm quarks as heavy with the hierarchy <i>m</i><sub><i>b</i></sub> ≫ <i>m</i><sub><i>c</i></sub> ≫ Λ<sub>QCD</sub>. Our diagrammatic analysis shows that double-logarithmic corrections arise from two distinct sources: exponentiated soft-gluon effects described by standard Sudakov factors, and rapidity-ordered soft-quark configurations, leading to implicit integral equations, which so far have only been studied in the context of energetic muon-electron backward scattering. We find that the all-order structure of the double logarithms is governed by a novel type of coupled integral equations, which encode the non-trivial interplay between these two effects. Whereas a closed-form solution to these equations is currently unknown, we present useful iteration formulas, and derive the asymptotic behaviour of the soft-overlap form factor for infinitely large recoil energies, showing that the Sudakov suppression is somewhat weakened by the intertwined soft-quark and soft-gluon corrections. In a broader context, our findings shed light onto the physical origin and mathematical structure of endpoint divergences arising from soft-collinear factorization and the related Feynman mechanism for power-suppressed hard exclusive processes.</p>

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Bc → ηc form factors at large recoil: interplay of soft-quark and soft-gluon dynamics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月11日

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  • Guido Bell,
  • Philipp Böer,
  • Thorsten Feldmann,
  • Dennis Horstmann,
  • Vladyslav Shtabovenko

摘要

We perform an all-order analysis of double-logarithmic corrections to the so-called soft-overlap contribution to heavy-to-light transition form factors at large hadronic recoil. Specifically, we study Bc → ηc transitions within a perturbative non-relativistic framework, treating both the bottom and charm quarks as heavy with the hierarchy mb ≫ mc ≫ ΛQCD. Our diagrammatic analysis shows that double-logarithmic corrections arise from two distinct sources: exponentiated soft-gluon effects described by standard Sudakov factors, and rapidity-ordered soft-quark configurations, leading to implicit integral equations, which so far have only been studied in the context of energetic muon-electron backward scattering. We find that the all-order structure of the double logarithms is governed by a novel type of coupled integral equations, which encode the non-trivial interplay between these two effects. Whereas a closed-form solution to these equations is currently unknown, we present useful iteration formulas, and derive the asymptotic behaviour of the soft-overlap form factor for infinitely large recoil energies, showing that the Sudakov suppression is somewhat weakened by the intertwined soft-quark and soft-gluon corrections. In a broader context, our findings shed light onto the physical origin and mathematical structure of endpoint divergences arising from soft-collinear factorization and the related Feynman mechanism for power-suppressed hard exclusive processes.