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One-Loop QCD Corrections to $\bar{u}d \rightarrow t\bar{t}W$ at $\mathcal{O}(\varepsilon^2)$

Matteo Becchetti, Maximilian Delto, Sara Ditsch, Philipp Alexander Kreer, Mattia Pozzoli, Lorenzo Tancredi

TL;DR

This work computes the one-loop QCD corrections to ttW production retaining terms up to $ε^2$ in dimensional regularization, providing a crucial stepping stone toward exact NNLO predictions. The authors perform a tensor decomposition into a minimal form-factor basis and construct a canonical master-integral basis, expressing results in terms of Chen iterated integrals and, for weight up to two, logarithms and dilogarithms, with a semi-numerical strategy for higher weights via generalized power-series expansion. They derive an explicit analytic representation at weight two and develop a comprehensive numerical framework to evaluate the full weight-four function basis, validated against independent results and implemented in an open ancillary package. The approach illuminates the structure of massive one-loop amplitudes in multi-leg processes, informs the treatment of potential elliptic integrals at two loops, and provides a solid platform for advancing toward exact NNLO ttW phenomenology at the LHC.

Abstract

We present a computation of the one-loop QCD corrections to top-quark pair production in association with a $W$ boson, including terms up to order $\varepsilon^2$ in dimensional regularization. Providing a first glimpse into the complexity of the corresponding two-loop amplitude, this result is a first step towards a description of this process at next-to-next-to-leading order (NNLO) in QCD. We perform a tensor decomposition and express the corresponding form factors in terms of a basis of independent special functions with compact rational coefficients, providing a structured framework for future developments. In addition, we derive an explicit analytic representation of the form factors, valid up to order $\varepsilon^0$, expressed in terms of logarithms and dilogarithms. For the complete set of special functions required, we obtain a semi-numerical solution based on generalized power series expansion.

One-Loop QCD Corrections to $\bar{u}d \rightarrow t\bar{t}W$ at $\mathcal{O}(\varepsilon^2)$

TL;DR

This work computes the one-loop QCD corrections to ttW production retaining terms up to in dimensional regularization, providing a crucial stepping stone toward exact NNLO predictions. The authors perform a tensor decomposition into a minimal form-factor basis and construct a canonical master-integral basis, expressing results in terms of Chen iterated integrals and, for weight up to two, logarithms and dilogarithms, with a semi-numerical strategy for higher weights via generalized power-series expansion. They derive an explicit analytic representation at weight two and develop a comprehensive numerical framework to evaluate the full weight-four function basis, validated against independent results and implemented in an open ancillary package. The approach illuminates the structure of massive one-loop amplitudes in multi-leg processes, informs the treatment of potential elliptic integrals at two loops, and provides a solid platform for advancing toward exact NNLO ttW phenomenology at the LHC.

Abstract

We present a computation of the one-loop QCD corrections to top-quark pair production in association with a boson, including terms up to order in dimensional regularization. Providing a first glimpse into the complexity of the corresponding two-loop amplitude, this result is a first step towards a description of this process at next-to-next-to-leading order (NNLO) in QCD. We perform a tensor decomposition and express the corresponding form factors in terms of a basis of independent special functions with compact rational coefficients, providing a structured framework for future developments. In addition, we derive an explicit analytic representation of the form factors, valid up to order , expressed in terms of logarithms and dilogarithms. For the complete set of special functions required, we obtain a semi-numerical solution based on generalized power series expansion.

Paper Structure

This paper contains 11 sections, 82 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Conventions, Kinematics and Color
  3. Form Factor Decomposition
  4. Renormalization and Infrared Structure
  5. One-Loop Calculation
  6. Canonical Integrals in Terms of Special Functions
  7. Analytic Solution up to Weight Two
  8. Numerical Evaluation Beyond Weight Two
  9. Results
  10. Details on the Implementation and Numerical Performance
  11. Conclusions

Figures (1)

  • Figure 1: Diagrammatic representation of the three integral topologies. Dashed lines denote massless propagators and external legs originating from massless quarks and gluons. Orange lines correspond to propagators of mass $m_t$, and the green line corresponds to an external leg with mass $m_W$.