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Two-loop helicity amplitudes for $gg\to W^+W^-$ using high-energy and forward expansions

This paper presents next-to-leading order QCD corrections for the process $gg \to W^+W^-$ with off-shell $W$ bosons and virtual top quarks by combining high-energy, forward-scattering, and large-top-mass expansions to achieve precise results across the entire phase space.

Original authors: Mateusz Czaja, Ben Rott, Matthias Steinhauser, Daniel Stremmer

Published 2026-10-21

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CC BY 4.0

Original authors: Mateusz Czaja, Ben Rott, Matthias Steinhauser, Daniel Stremmer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the heart of the Large Hadron Collider, where protons smash together at speeds nearing that of light, physicists are constantly hunting for the subtle signatures of nature's fundamental forces. Among the most important processes they study is the creation of pairs of W bosons, heavy particles that carry the weak nuclear force responsible for radioactive decay. While these particles are often produced through the collision of quarks, a smaller but significant fraction arises from the collision of gluons, the particles that bind quarks together inside protons. This gluon-driven process is a crucial background for finding the Higgs boson, and measuring it with extreme precision is essential for testing the Standard Model of particle physics. However, calculating the exact behavior of these collisions is notoriously difficult because it involves complex quantum loops where heavy top quarks briefly appear and disappear, creating a mathematical maze that has resisted exact solutions for years.

A team of researchers at the Karlsruhe Institute of Technology in Germany has now navigated this maze by developing a new way to calculate these interactions. Instead of trying to solve the entire problem at once, which is computationally impossible with current methods, they broke the problem down into manageable pieces by looking at the collision from two different perspectives. They calculated what happens when the collision energy is very high, and separately, what happens when the particles scatter at very small angles, a scenario known as forward scattering. By stitching these two different views together, they created a complete and highly accurate map of the entire collision process, covering all possible outcomes without needing to rely on purely numerical guesses.

The researchers focused specifically on the contribution of the top quark, the heaviest known elementary particle, which plays a dominant role in these quantum loops. Previous attempts to understand this process relied entirely on numerical methods that required fixing specific values for the particle masses and energies at the very beginning of the calculation. This made the results rigid and difficult to adapt to different scenarios. The new work produces a flexible set of formulas where the results depend explicitly on the input parameters, allowing physicists to plug in any value they need. This flexibility is vital because it allows the results to be easily integrated into the software tools that experimentalists use to compare their data with theoretical predictions.

To achieve this, the team had to overcome significant hurdles in the mathematics of particle physics. The presence of both massive top quarks and massless bottom quarks in the same loop diagrams created a unique set of mathematical challenges that had not been solved before. The researchers developed a sophisticated technique to expand the complex equations in the high-energy limit, where the collision energy is much larger than the mass of the top quark, and in the forward limit, where the particles barely change direction. They combined these expansions with a method to handle the large mass of the top quark, using the results to cross-check each other. The final output is a set of semi-analytical results that cover the entire range of possible collision outcomes, from low energies to the highest energies achievable at the collider.

The team demonstrated that their combined approach yields precise results across the entire phase space. By comparing their calculations against exact numerical results for specific test cases, they showed that their method agrees with the truth to within one percent or better. This level of precision is sufficient for the rigorous demands of modern particle physics. The researchers also showed that their method captures subtle physical effects, such as the sudden changes in behavior that occur when the collision energy is just enough to create a pair of top quarks. These features, which appear as sharp kinks in the data, are naturally included in their formulas, eliminating the need for laborious numerical scans around these critical points.

The significance of this work extends beyond just one specific calculation. The results are designed to be implemented into the C++ library known as ggxy, which is used by the broader community to simulate gluon-induced processes. This will allow experimentalists to include the top-quark contributions in their analyses of W boson pair production with unprecedented accuracy. The authors note that while the mathematical expressions they generated are large, they possess a simple structure that makes them efficient to evaluate on computers. This efficiency is key, as it enables the rapid generation of predictions needed to interpret the vast amounts of data collected by the Large Hadron Collider.

Ultimately, this research provides a robust tool for understanding the subtle interplay of forces at the smallest scales. By combining different mathematical expansions, the team has filled a gap in our theoretical knowledge, offering a clear and flexible description of how gluons transform into W boson pairs. Their work ensures that as experimentalists continue to refine their measurements of these rare events, they will have the precise theoretical guidance needed to distinguish between known physics and potential new discoveries. The ability to calculate these processes with such detail and flexibility marks a significant step forward in the ongoing effort to map the fundamental laws of the universe.

Technical Summary: Two-loop helicity amplitudes for $gg \to W^+W^-$ using high-energy and forward expansions

Problem Statement
The production of $W^+W^-$ pairs at the Large Hadron Collider (LHC) is a precision process, with experimental uncertainties as low as 3.1%. While the dominant production channel ( $q\bar{q}$ annihilation) is known up to next-to-next-to-leading order (NNLO), the gluon fusion channel ( $gg \to W^+W^-$ ) contributes approximately 5% to the total cross-section but can reach up to 25% in fiducial regions with specific kinematic cuts. Next-to-leading order (NLO) QCD corrections for gluon-induced processes are typically large (50–100%). Previous calculations for the massless contributions exist, but the top-quark-induced contributions, which account for roughly 12–15% of the massless contribution, require precise NLO corrections. Existing calculations for the top-quark contribution (e.g., Ref. [23]) rely exclusively on numerical methods, necessitating fixed input values early in the calculation and lacking explicit dependence on all kinematic parameters. Furthermore, previous studies often assumed on-shell $W$ bosons, whereas phenomenological applications require off-shell $W$ bosons to include decay effects.

Methodology
The authors calculate the NLO QCD corrections for $gg \to W^+W^-$ with two off-shell $W$ bosons, explicitly accounting for virtual top quarks in loop diagrams and treating bottom quarks as massless. The calculation relies on a semi-analytical approach combining expansions valid in different regions of phase space:

Kinematics and Amplitude Construction: The process is defined with incoming gluons ( $q_1, q_2$ ) and outgoing off-shell $W$ bosons ( $q_3, q_4$ ). The helicity amplitudes are constructed using a basis of 18 independent parity-odd tensor structures for the axial-vector contribution and standard projectors for the vector contribution. Only eight independent helicity amplitudes are computed due to symmetry relations.
High-Energy Expansion: An expansion is performed in the limit where the Mandelstam variables ( $s, t, u$ $s, t, u$ ) are much larger than the top-quark mass ( $m_t$ $m_{t}$ ), while $m_t$ $m_{t}$ remains larger than the $W$ $W$ -boson virtualities ( $q_3^2, q_4^2$ $q_{3}^{2}, q_{4}^{2}$ ).
- The calculation involves 254 master integrals, of which only 18 are already known from previous calculations, implying 236 new integrals.
- Differential equations in $m_t^2/s$ and $t/s$ are used. The $m_t \to 0$ limit is handled via the method of regions, utilizing the AsyInt package for asymptotic expansions and Mellin-Barnes (MB) representations.
- Complex MB integrals (up to three-dimensional) are evaluated numerically or expanded in specific limits ( $t \to 0$ ) and reconstructed using PSLQ algorithms and Harmonic PolyLogarithms (HPLs).
Forward Expansion: An expansion is performed around the forward-scattering limit ( $t \to 0$ $t \to 0$ ).
- Two approaches are employed: (i) expanding the integrand in $\delta q = q_1 + q_3$ (simultaneous Taylor expansion in $t, q_3^2, q_4^2$ ), and (ii) expanding the master integrals in $t \to 0$ using differential equations.
- Unlike previous calculations for neutral bosons which encountered elliptic sectors, the presence of massless bottom quarks in the loops simplifies the integral families, allowing all master integrals to be computed analytically in terms of iterated integrals.
- The "expand and match" algorithm is used to generate deep numerical expansions, which are then matched to analytical forms.
Combination and Padé Improvement: To cover the entire phase space, the results from the high-energy and forward expansions are combined. Padé approximants are applied to both expansions to improve convergence and extend the validity range. The final results are cross-checked against a large-mass expansion ( $m_t \to \infty$ ).

Key Contributions

Semi-Analytical Results: The paper provides helicity amplitudes that depend explicitly on all input parameters (kinematics and masses), contrasting with previous purely numerical approaches. This allows for flexible implementation in numerical codes (e.g., the C++ library ggxy).
Off-Shell Treatment: The calculation includes two different virtualities for the final-state $W$ bosons, facilitating the straightforward inclusion of $W$ -boson decays.
New Master Integrals: The authors compute 236 new two-loop master integrals arising from the specific topology of $gg \to W^+W^-$ (involving both massive top and massless bottom quarks), which differ from those in $gg \to HH$ or $gg \to ZZ$ .
Methodological Refinement: The work demonstrates a robust strategy for combining high-energy and forward expansions, utilizing Padé approximants to achieve high precision across the full phase space.

Results

Precision: The combined expansions yield precise results for all helicity amplitudes across the entire phase space.
- For scalar integrals, the Padé-improved expansions agree with exact numerical results (computed via AMFlow) at the 1% level or better.
- For helicity amplitudes, the agreement between the $t$ -expansion and high-energy expansion in the overlap region ( $p_T \sim 200$ GeV) exceeds five significant figures.
- The forward expansion achieves sub-per-mille accuracy in the low- $p_T$ region.
Convergence: The forward expansion for $gg \to W^+W^-$ converges less rapidly than in $gg \to HH$ due to $t$ -channel cuts through a single massive top-quark line. However, Padé acceleration extends the valid overlap region up to $p_T \approx 400$ GeV or higher.
Threshold Behavior: The semi-analytic results correctly capture non-analytic structures, such as the cusp at $\sqrt{s} = 2m_t$ , which arises from the production of virtual top-quark pairs in an S-wave state.

Significance and Claims
The authors claim that their work provides the first semi-analytical calculation of the top-quark-induced contributions to $gg \to W^+W^-$ with off-shell $W$ bosons. The explicit dependence on all parameters makes the results suitable for implementation in event generators (specifically ggxy and POWHEG), offering a significant improvement in flexibility and speed over grid-based numerical methods. The paper emphasizes that the combination of forward and high-energy expansions successfully covers the entire physical phase space for $W^+W^-$ pair production. The results serve as a necessary component for precise theoretical predictions of $W^+W^-$ production, which is a critical background for Higgs boson measurements and a probe for new physics. The authors note that the large- $m_t$ expansion, while limited in phenomenological utility due to branch cuts from massless bottom quarks, serves as a valuable cross-check for the full calculation.

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