Threshold resummation for $W$-boson pair production at NNLO+NNLL

We present results for threshold resummation of the invariant mass distribution, for on-shell production of a pair of $W$-bosons at next-to-next-to-leading order + next-to-next-to-leading logarithmic (NNLO+NNLL) accuracy in QCD. Owing to its sensitivity to the self-interactions between gauge bosons, this process is important to investigate at the energies of the Large Hadron Collider (LHC). We achieve this resummation by exploiting the factorization properties of the soft and virtual parts of the partonic cross-section. Our analysis has been carried out for the invariant mass distribution up to $Q$ = 2500 GeV. At this highest $Q$ we find that, for 13.6 TeV LHC, the NNLL resummation enhances the NNLO cross-sections by about $6.3%$ and reduces the conventional scale uncertainties from 6.8% at NNLO to 4.1% at NNLO+NNLL. We also estimate the intrinsic uncertainties due to the non-perturbative parton distribution functions at the highest perturbative order, for both fixed-order and resummed results, to be around 3% for $Q \sim$ 2000 GeV.

💡 Research Summary

The paper presents a comprehensive study of threshold resummation for on‑shell W‑boson pair production at the Large Hadron Collider (LHC) and future high‑energy hadron colliders, achieving next‑to‑next‑to‑leading order (NNLO) combined with next‑to‑next‑to‑leading logarithmic (NNLL) accuracy in QCD. The authors start by motivating the importance of precise predictions for the W⁺W⁻ channel: it probes the self‑interactions of electroweak gauge bosons, contributes to the determination of the W‑mass, and serves as a background for many beyond‑the‑Standard‑Model (BSM) searches, especially in the high invariant‑mass (Q) region where new physics may appear.

After reviewing the extensive literature on fixed‑order calculations (LO, NLO, NNLO) and previous resummation efforts (transverse‑momentum, jet‑veto, partial N³LL), the authors focus on the threshold limit z → 1, where the partonic center‑of‑mass energy is just enough to produce the W‑pair. In this regime, soft‑gluon emissions generate large logarithms of the form lnⁿ(1−z)/(1−z) that spoil the convergence of the perturbative series. The cross‑section is decomposed into a universal soft‑virtual (SV) piece and a regular piece. The SV component contains all singular behavior and can be expressed in Mellin (N) space as a product of a non‑logarithmic constant g₀ and an exponential of the logarithmic function Ψ_sv(N). The latter resums the large logarithms ln N to all orders, with known universal functions g₁, g₂, g₃ for NNLL accuracy. The constant g₀ is expanded in the strong coupling a_s and includes process‑dependent coefficients g₀₁ (one‑loop virtual) and g₀₂ (two‑loop virtual and one‑loop squared contributions). The authors compute g₀₁ using in‑house FORM routines and obtain g₀₂ by combining one‑loop, one‑loop‑squared, and two‑loop amplitudes, the latter extracted from the public VVamp library.

To avoid the Landau pole in the inverse Mellin transform, the minimal prescription is employed, with a contour defined by N = c + x e^{iφ} (c = 1.9, φ = 3π/4). The resummed result in z‑space is then matched to the fixed‑order NNLO calculation to prevent double counting, yielding the final NNLO+NNLL prediction.

Numerically, the study uses the NNPDF30 4‑flavour PDFs (LO, NLO, NNLO sets) and sets the renormalisation and factorisation scales equal to the invariant mass Q. The central values for the electroweak parameters are m_W = 80.385 GeV, sin²θ_W = 0.222897, α ≈ 1/132.2332, and G_F = 1.166379×10⁻⁵ GeV⁻². The analysis covers Q up to 2500 GeV for √S = 13.6 TeV LHC, as well as projections for the High‑Luminosity LHC and a 100 TeV FCC‑hh.

The key findings are:

1. At the highest studied invariant mass (Q = 2500 GeV), NNLL resummation increases the NNLO cross‑section by about 6.3 %.
2. The conventional scale uncertainty, evaluated by varying μ_R = μ_F between Q/2 and 2Q, drops from 6.8 % at NNLO to 4.1 % after NNLL resummation.
3. PDF‑induced intrinsic uncertainties are estimated to be roughly 3 % for Q ≈ 2000 GeV, both for fixed‑order and resummed results.
4. The improvement in theoretical precision is comparable to, or better than, similar resummation studies for Z‑pair and Drell‑Yan production, indicating that the methodology is robust across electroweak diboson processes.

The authors conclude that threshold resummation at NNLO+NNLL substantially reduces the dominant theoretical uncertainties in the high‑mass tail of the W‑pair invariant‑mass spectrum. This enhanced precision is crucial for future BSM searches that rely on accurate background modeling in the TeV regime. They also suggest extensions such as incorporating electroweak Sudakov logarithms, exploring alternative PDF sets, and applying the same framework to other processes (e.g., ZZ, WZ) to further solidify the theoretical foundation for precision collider phenomenology.