Three-loop corrections to $gg\to ZH$ in the large top quark mass limit

TL;DR

The study addresses the need for higher-order precision in $gg\to ZH$ by computing three-loop virtual corrections in the large-$m_t$ limit. It employs a hard-mass expansion to systematically derive analytic expressions for all six form factors up to ${\cal O}(1/m_t^4)$, including both reducible and irreducible diagrams, and implements these results in the C++ library ggxy for fast numerical evaluation. The work provides IR- and UV-renormalized results and offers a benchmark for future NNLO developments, potentially enabling approximate NNLO predictions through reweighting with NLO data. The combination of detailed asymptotic techniques and ready-to-use code facilitates improved predictions in the boosted regime and establishes a foundation for more complete NNLO treatments of $ZH$ production.

Abstract

We compute three-loop virtual corrections to the associated production of a Higgs boson with a $Z$ boson in the large-$m_t$ limit. We describe in detail the application of the asymptotic expansion and provide, for all form factors, analytic results for the first three terms in the $1/m_t$ expansion. We also provide numerical routines implemented in the C++ library ggxy.

Figures (5)

• Figure 1: One-, two- and three-loop sample Feynman diagrams contributing to $gg\to ZH$. Solid thin (thick) lines denote massless (massive) quarks. Scalars and gluons are represented by dashed and curly lines, respectively. In the triangle diagrams either a $Z$ boson or Goldstone boson mediates the coupling to the Higgs and $Z$ boson in the final state. At two- and three-loop order both one-particle reducible and one-particle irreducible diagrams have to be considered. These Feynman diagrams were drawn with the help of the FeynGame program Bundgen:2025utt.
• Figure 2: Graphical representation of the asymptotic expansion applied to the diagrams on the left. On the right we show only the co-subgraphs. The corresponding subgraphs are one-, two- and three-loop vacuum integrals which are inserted in the effective vertices represented by the blobs.
• Figure 3: Squared amplitude at LO and NLO as a function of $\sqrt{s}$ for $p_T=10$ GeV, showing different expansion depths in $1/m_t^2$. Exact one-loop results and the results from the forward limit Davies:2025out at two loops are shown in black as a reference. The lower panel displays the relative difference to the reference values.
• Figure 4: Squared amplitude at NNLO as a function of $\sqrt{s}$ for $p_T=10$ GeV, showing different expansion depths in $1/m_t^2$. In the lower panel the relative difference to the best available approximation ($1/m_t^4$) is shown.
• Figure 5: Squared amplitude at LO, NLO and NNLO as a function of $\sqrt{s}$ for $p_T=10$ GeV normalized to the LO one. Left plot shows the results at the order $1/m_t^0$ and the right plot up to $1/m_t^4$.