Bootstrapping rapidity anomalous dimension for transverse-momentum resummation

Abstract: Soft function relevant for transverse-momentum resummation for Drell-Yan or Higgs production at hadron colliders are computed through to three loops in the expansion of strong coupling, with the help of bootstrap technique and supersymmetric decomposition. The corresponding rapidity anomalous dimension is extracted. An intriguing relation between anomalous dimensions for transverse-momentum resummation and threshold resummation is found.

• The paper presents a novel bootstrap technique to compute the rapidity anomalous dimension at three loops, addressing key challenges in QCD resummation.
• It combines supersymmetric decomposition, SCET, and rapidity RG methods to effectively resum large logarithms, achieving N³LL + NNLO accuracy.
• The study provides a robust framework for higher-order QCD computations, deepening understanding of soft function interactions and non-Abelian dynamics.

Bootstrapping Rapidity Anomalous Dimension for Transverse-Momentum Resummation: A Scholarly Analysis

The paper "Bootstrapping rapidity anomalous dimension for transverse-momentum resummation" by Ye Li and Hua Xing Zhu presents a comprehensive study on the computation of the soft function pertinent to transverse-momentum resummation in Quantum ChromoDynamics (QCD), specifically targeting Drell-Yan and Higgs production processes at hadron colliders. This research addresses the challenges posed by large logarithms generated when the transverse momentum ($q_T$) is small compared to the invariant mass $Q$ of the system, an area of vast intrigue within QCD due to perturbative and non-perturbative effects.

Technical Approach and Methodology

The authors employ a combination of bootstrap techniques and supersymmetric decomposition to compute the soft function through three loops. By leveraging the Soft-Collinear Effective Theory (SCET) and the rapidity Renormalization Group (RG) method, the research adequately resums these large logarithms to all orders. The factorization representation highlights the decomposition into hard functions, TMD beam functions, and TMD soft functions. The novel approach introduces a rapidity regulator as proposed by Neill and Li, allowing for resummation of rapidity logarithms crucial to achieving analytical precision across QCD analysis.

The fully differential soft function undergoes explicit computation via dimension analysis, and a strategy akin to bootstrapping scattering amplitudes—developed from harmonic polylogarithms (HPL)—is used to resolve the complexity inherent in the perturbative expansion. Through systematic reductions using Integration-By-Parts identities, the authors adeptly reconstruct the three-loop soft function coefficients.

Numerical Results and Computational Assertions

The paper presents well-founded computational assertions, notably calculating the rapidity anomalous dimension up to three loops, a significant achievement. Utilizing an empirical ansatz and brute-force calculations, the researchers ensure the congruence and comprehensive resolution of multi-loop terms. Results include specific numerical values for the rapidity anomalous dimension $r_2$, alongside critical relations connecting it to the threshold soft function and the QCD beta function. These observations provide essential insights into the resummation and factorization properties within QCD. Furthermore, the precise numerical results for Drell-Yan and Higgs processes are displayed, highlighting the expansive scale reaching N$^3$LL + NNLO accuracy.

Implications and Future Developments

The paper's outcomes have both practical and theoretical implications. Practically, they provide a methodologically robust framework for higher-order calculations in QCD that enhance the precision of predictions in collider physics, directly impacting experimental analysis at LHC or future colliders. Theoretically, it mobilizes further in-depth analysis of transcendent relations within QCD resummation, motivating the scrutiny of non-Abelian structures and soft function interactions.

Looking forward, this research lays foundational work conducive to exploration in aspects such as higher-order rapidity renormalization effects and the potential generalization of $\tau$-subtraction methods to N$^3$LO calculations. Such endeavors could redefine existing paradigms in perturbative QCD.

In summary, Li and Zhu offer a meticulous study that elucidates the complexities of transverse momentum resummation through innovative computational methods within QCD. It is a significant contribution that promises to influence both the strategic and operational facets of future theoretical and applied research in particle physics.