The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to d-dimensional d log-form integrands. In this work, we explore the generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct d-dimensional d log-form integrands, and discuss how to convert them to Feynman integrals. We describe the technical details of our method, in particular how to deal with the difficulties encountered in the construction procedure. Our method provides a constructive approach to the problem of finding canonical bases of Feynman integrals, and we demonstrate its applicability to complicated scattering amplitudes involving multiple physical scales.

We consider the two-loop corrections to the HW+W− vertex at order ααs. We construct a canonical basis for the two-loop integrals using the Baikov representation and the intersection theory. By solving the ϵ-form differential equations, we obtain fully analytic expressions for the master integrals in terms of multiple polylogarithms, which allow fast and accurate numeric evaluation for arbitrary configurations of external momenta. We apply our analytic results to the decay process H → νeeW, and study both the integrated and differential decay rates. Our results can also be applied to the Higgs production process via W boson fusion.

The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to d-dimensional d log-form integrands. In this work, we introduce the concept of generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct d-dimensional d logform integrands, and discuss how to convert them to Feynman integrals. We describe the technical details of our method, in particular how to deal with the difficulties encountered in the construction procedure. Our method provides a constructive approach to the problem of finding canonical bases of Feynman integrals, and we demonstrate its applicability to complicated scattering amplitudes involving multiple physical scales.

In this paper, we present the universal structure of the alphabet of one-loop Feynman integrals. The letters in the alphabet are calculated using the Baikov representation with cuts. We consider both convergent and divergent cut integrals, and find that letters in the divergent cases can be easily obtained from the convergent cases by taking certain limits. The letters are written as simple expressions in terms of various Gram determinants. The knowledge of the alphabet makes it easy to construct the canonical differential equations of the $d\log$ form, and helps to bootstrap the symbols of the solutions. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

The associated production of a Higgs boson and a top-quark pair is important for probing the Yukawa coupling of the top quark, and calls for better theoretical modeling. In this paper, we calculate the two-loop infrared divergences in $$ t\overline{t}H $$ t t ¯ H production at hadron colliders. To do that we compute the one-loop amplitudes to higher orders in the dimensional regulator ϵ. Numeric results for the infrared poles are given as a reference at several representative phase-space points. The result in this work serves as a part of the ongoing efforts towards the $$ t\overline{t}H $$ t t ¯ H cross sections at the next-to-next-to-leading order.