The inclusive Higgs boson cross-section in gluon-gluon fusion in soft-virtual approximation at fourth order in QCD

We present precise results for the inclusive Higgs boson cross-section in gluon-gluon fusion at the LHC considering state-of-the-art fourth-order results in perturbative QCD arising from the dominant soft and virtual gluon emissions. Utilizing four-loop QCD results for the gluon-form factor, the splitting function and related anomalous dimensions, we study the effects of threshold enhanced soft gluon emissions and estimate their impact on the total cross-section at the fourth order. Our study highlights the role of these higher-order contributions in improving the perturbative convergence and in significantly reducing the renormalization and factorization scale uncertainties. The results provide strong evidence for the perturbative stability and reliability of Higgs boson cross-section predictions at the LHC, thereby reinforcing the robustness of theoretical inputs in precision Higgs phenomenology. We also provide cross-section predictions using a large set of available parton distribution functions and show that, together with the value of the strong coupling $α_s(m_Z)$, they cause the largest residual uncertainty for the Higgs boson cross-section in gluon-gluon fusion.

💡 Research Summary

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The paper presents a state‑of‑the‑art calculation of the inclusive Higgs boson production cross‑section via gluon‑gluon fusion at the LHC, incorporating the dominant soft‑gluon and virtual contributions at fourth order in perturbative QCD (α_s⁴). Using the heavy‑top effective theory (HEFT) in the infinite‑top‑mass limit, the authors express the cross‑section as a convolution of parton distribution functions (PDFs) with a coefficient function that is expanded in powers of α_s. The fourth‑order soft‑virtual (SV) contribution is obtained by combining several recent four‑loop results: the gluon form factor (providing pure virtual corrections), the four‑loop gluon splitting function (governing collinear emissions), and the four‑loop cusp and regular anomalous dimensions. These ingredients allow the authors to reconstruct the threshold‑enhanced logarithms in Mellin‑N space, which are then transformed back to the physical z‑space (z = m_H²/ŝ) to yield the SV terms proportional to δ(1−z) and plus‑distributions.

The calculation adopts a central renormalisation and factorisation scale μ_c = m_H/2, a choice known to improve perturbative convergence. Scale uncertainties are estimated with the standard seven‑point variation (μ_R, μ_F ∈ {½,1,2} μ_c, subject to ½ ≤ μ_R/μ_F ≤ 2). The authors evaluate the impact of PDFs by employing a wide range of modern sets accessed via LHAPDF: ABMP16, ABMPtt, CT18, MSHT20, NNPDF40, PDF4LHC21 (all at NNLO) and approximate N³LO PDFs (MSHT20an3lo, NNPDF40an3lo). Two scenarios for the strong coupling are considered: (i) each PDF’s own α_s(m_Z) value (ABMP sets have lower α_s ≈ 0.115, others use 0.118), and (ii) a common fixed α_s = 0.118 with a ±1σ variation (0.1165–0.1195).

Numerical results for √S = 13.6 TeV, m_H = 125 GeV, and m_t = 172.5 GeV show that the N⁴LO SV correction changes the total cross‑section by only about –0.1 % relative to the N³LO result, indicating excellent perturbative stability. More importantly, the scale uncertainty is reduced from roughly ±3.5 % at N³LO to about ±1.5 % at N⁴LO SV, i.e. a factor of two improvement. PDF uncertainties remain at the 1–2 % level for each set, but the spread among central predictions of different PDFs can reach up to 7 %. The dominant residual uncertainty stems from the value of α_s(m_Z); its 1σ variation induces an uncertainty of ≈ 4 % on the cross‑section, which dominates the total theoretical error budget.

The authors conclude that while the soft‑virtual approximation at N⁴LO provides a highly reliable estimate of the Higgs production rate and dramatically reduces scale‑related ambiguities, the precision of the prediction is now limited by the knowledge of PDFs and the strong coupling constant. They advocate for the computation of the full N⁴LO corrections (including non‑soft contributions) and for further improvements in PDF determinations and α_s measurements. Such advances will be essential for exploiting the full potential of the LHC and future colliders in precision Higgs physics and for probing possible deviations from the Standard Model.