Boosted Higgs-strahlung off a $W$ boson at next-to-next-to-next-to-leading order in QCD

Problem
The associated production of a Higgs boson with a charged weak boson ($pp \to W^+H$), known as Higgs-strahlung, is a critical process for scrutinizing electroweak symmetry breaking and probing physics beyond the Standard Model, particularly in the boosted regime where the Higgs boson possesses large transverse momentum ($p_T^H$). While inclusive total cross-section predictions for this process are known up to next-to-next-to-next-to-leading order (N3LO) in perturbative Quantum Chromodynamics (QCD), differential predictions have historically been limited to next-to-next-to-leading order (NNLO). The increasing experimental precision anticipated at the High-Luminosity LHC (HL-LHC) necessitates theoretical predictions at higher orders, especially within specific fiducial regions of phase space. Previous studies indicated that the perturbative series for the boosted regime exhibits poor convergence up to NNLO, rendering N3LO predictions essential for reliable phenomenology.

Methodology
The authors present the first fully differential calculation of the $pp \to W^+( \to e^+ \nu_e)H$ process at N3LO in QCD. The calculation is performed within the $q_T$-slicing framework, which decomposes the radiation phase space using a slicing cut ($q_T^{\text{cut}}$) on the transverse momentum of the color-singlet system ($WH$).

• Below the cut ($q_T < q_T^{\text{cut}}$): The singular contribution is computed using the N3LO expansion of the leading-power resummation formula for the transverse-momentum spectrum, implemented via the RadISH framework.
• Above the cut ($q_T > q_T^{\text{cut}}$): The regular contribution is evaluated using an NNLO calculation for the $pp \to W^+H + 1$ jet process, computed with the NNLOJET framework using the antenna subtraction method.
• Top-Yukawa Contributions: The calculation includes contributions where the Higgs boson couples to a top-quark loop. Purely virtual corrections are treated in the heavy-top-quark limit, while full top-mass dependence is retained for contributions involving additional real emissions. These corrections, starting at $\mathcal{O}(y_t \alpha_s^2)$, are computed independently and added to the cross-section.
• Setup: The study considers proton-proton collisions at $\sqrt{s} = 13.6$ TeV using the NNPDF40 an3lo parton density set. Two kinematic setups are analyzed: a "boosted" setup defined by $p_T^H \geq 250$ GeV or $400$ GeV, and a "STXS" setup adding a requirement of $p_T^W \geq 250$ GeV.

Key Contributions

1. First N3LO Differential Calculation: This work provides the first fully differential N3LO QCD prediction for Higgs-strahlung off a $W$ boson in the boosted regime.
2. Validation of Methodology: The authors validate their approach by reproducing the known inclusive N3LO cross-section results from literature, confirming the absence of linear power corrections in their slicing procedure and establishing $q_T^{\text{cut}} = 5$ GeV as an optimal value.
3. Handling of Power Corrections: The study systematically addresses linear power corrections of $\mathcal{O}(q_T^{\text{cut}}/Q)$ in symmetric cut configurations by calculating them explicitly and adding them to the below-cut contribution, ensuring the residual dependence remains quadratic.
4. Top-Yukawa Integration: The inclusion of numerically relevant top-Yukawa-induced contributions at $\mathcal{O}(y_t \alpha_s^2)$ is integrated into the N3LO framework.

Results

• Perturbative Convergence: In the boosted regime, the perturbative series up to NNLO shows poor convergence, with neither NLO nor NNLO values lying within the scale-variation bands of the preceding orders. The inclusion of N3LO corrections restores convergence.
• Cross-Section Corrections: The N3LO corrections increase the NNLO prediction by approximately $+1.5\%$ to $+2\%$ in the boosted setup. In the STXS setup, the correction is roughly $+1.5\%$.
• Uncertainty Reduction: The residual scale dependence at N3LO is reduced to below the percent level (specifically $0.5\% - 0.8\%$ depending on the setup), marking a significant improvement over previous orders.
• Differential Distributions: The $p_T^H$ spectrum shows that N3LO corrections generally increase the NNLO result by $1.5\% - 2\%$ across the boosted range. In the STXS setup, the corrections induce shape distortions, with a small negative correction near $p_T^H \approx 250$ GeV and a larger positive correction ($2-3\%$) in the high-$p_T$ tail.
• Scale vs. Slicing Uncertainty: The uncertainty associated with the $q_T$-slicing procedure remains well below the scale-variation band throughout the displayed kinematic range, demonstrating excellent numerical control.

Significance
The paper claims this calculation constitutes a milestone for the Higgs precision program. By achieving percent-level QCD precision and significantly reducing the residual dependence on perturbative scales, the results provide state-of-the-art predictions necessary for current and future boosted-Higgs analyses at the LHC and HL-LHC. The authors emphasize that the N3LO corrections in the boosted regime cannot be approximated by scaling inclusive results, highlighting the necessity of fully differential computations. These predictions strengthen the viability of Higgs-strahlung as a precision probe of Standard Model dynamics and potential new physics, bringing theoretical precision closer to the experimental capabilities expected from the HL-LHC. The authors note that further investigation into electroweak and mixed QCD+EW effects is required to fully meet the percent-level target for Higgs precision phenomenology.