On the evaluation of two-loop electroweak box diagrams for $e^+e^- \to HZ$ production

TL;DR

The paper tackles the challenge of computing NNLO electroweak corrections to $e^+e^- \to HZ$ by developing a numerical method for planar and non-planar two-loop box diagrams with massive propagators. It combines a dispersion relation for one sub-loop with Feynman parameterization for the other, enabling direct evaluation of tensor integrals and reducing the problem to three-dimensional numerical integrals that run in minutes on a CPU core. This approach achieves roughly $0.1\%$ relative precision and is validated through two independent implementations, addressing infrared regulation and numerical instabilities. The method provides a practical route to include dominant two-loop effects in future collider studies and can be extended to other $2\to 2$ processes, albeit with considerations for higher energies and numerical precision.

Abstract

Precision studies of the Higgs boson at future $e^+e^-$ colliders can help to shed light on fundamental questions related to electroweak symmetry breaking, baryogenesis, the hierarchy problem, and dark matter. The main production process, $e^+e^- \to HZ$, will need to be controlled with sub-percent precision, which requires the inclusion of next-to-next-to-leading order (NNLO) electroweak corrections. The most challenging class of diagrams are planar and non-planar double-box topologies with multiple massive propagators in the loops. This article proposes a technique for computing these diagrams numerically, by transforming one of the sub-loops through the use of Feynman parameters and a dispersion relation, while standard one-loop formulae can be used for the other sub-loop. This approach can be extended to deal with tensor integrals. The resulting numerical integrals can be evaluated in minutes on a single CPU core, to achieve about 0.1% relative precision.

Figures (4)

• Figure 1: Planar (left) and non-planar (right) two-loop box diagrams with top quarks in the loop. The bottom row visually illustrates the effect of introducing Feynman parameters for the top loop. If $V_{1,2}=\gamma,Z$ then $f'=e$, $q'=t$, whereas $f'=\nu_e$ and $q'=b$ for $V_{1,2}=W$.
• Figure 2: Integration contours for the dispersion relations for the one-loop scalar self-energy function $B_0$ for the cases $m_1^2,m_2^2>0$ (left) and $m_1^2<0$, $m_2^2>0$ (right). The zigzag lines denote the branch cuts, ending at the branch point $(m_1{+}m_2)^2$. The circle sections are understood to have a radius $R \to \infty$.
• Figure 3: Dependence of the $\gamma\gamma$ (left) and $\gamma Z$ (right) two-loop boxes on the photon mass $m_\gamma$. The lines depict linear fits to the data points.
• Figure 4: Dependence of various groups of two-loop box contributions on the scattering angle $\theta$.