⚛️ high-energy theory

Electroweak corrections to Higgs boson pair production: The quark channel

This paper presents the first calculation of mixed QCD-electroweak corrections to Higgs boson pair production in the quark-antiquark channel, deriving fully analytic virtual amplitudes and implementing them in the POWHEG-BOX framework to reveal significant shape modifications in differential cross sections, particularly near the production threshold.

Original authors: Marco Bonetti, Gudrun Heinrich, Philipp Rendler, William J. Torres Bobadilla

Published 2026-06-25

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CC BY 4.0

Original authors: Marco Bonetti, Gudrun Heinrich, Philipp Rendler, William J. Torres Bobadilla

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

The Big Picture: Hunting for a "Double Higgs"

Imagine the Large Hadron Collider (LHC) is a giant, high-speed billiard table where physicists smash particles together to see what happens. One of their main goals is to create two Higgs bosons at the same time (a "Higgs pair").

Why? Because the way these two particles interact tells us about a fundamental rule of the universe called the "trilinear Higgs coupling." Think of this like checking the tension of a spring. If the spring is tighter or looser than we expect, it might mean there are new, hidden rules of physics (beyond what we currently know).

The Problem: We Need a Better Map

To find these new rules, physicists need to know exactly what the "Standard Model" (our current best theory) predicts should happen. If the experiment shows something different from the prediction, that's a discovery.

However, the predictions aren't perfect yet.

The Old Way: Scientists had already calculated the effects of the "strong force" (QCD) very precisely. It's like they had a map of the terrain, but the map had some blurry spots.
The New Challenge: They needed to add the effects of the "electroweak force" (which includes electricity, magnetism, and the weak nuclear force) to clear up those blurry spots.

What This Paper Did: The "Quark Channel"

Most Higgs pairs are created when two gluons (particles that carry the strong force) smash together. Scientists have studied this "gluon channel" for a long time.

This paper focuses on a much rarer, previously ignored path: the quark channel.

The Analogy: Imagine you are trying to get to a destination. Everyone usually takes the main highway (gluons). This paper looked at a tiny, dusty backroad (quarks and antiquarks).
The Discovery: Even though this backroad is rarely used (it contributes almost nothing to the total number of Higgs pairs created), the scenery on this road is very different. When you drive this path, the shape of the journey changes significantly in specific areas.

How They Did It: The "Mathematical Blueprint"

Calculating what happens when two quarks turn into two Higgs bosons is incredibly hard. It involves complex quantum loops that look like tangled knots.

The Tangled Knots: The authors had to untangle these knots using a method called "differential equations." Imagine trying to solve a maze by writing down the rules for every turn rather than just walking through it.
The Boundary: To make sure their math was right, they checked the edges of the maze (the "large mass limit") where the math is easier, ensuring their complex solution matched up with the simple one.
The Result: They created a precise, analytical blueprint of this process. They then turned this blueprint into a computer code (POWHEG-BOX) that can simulate these collisions.

The Surprising Findings

When they ran the numbers, they found something interesting about the shape of the data, even if the total amount didn't change much.

The "Threshold" Effect: Near the minimum energy required to create the pair (the "production threshold"), the corrections from this quark channel were huge.
- The Analogy: Imagine a crowd of people trying to squeeze through a door. The main highway (gluons) flows smoothly. But on the backroad (quarks), right at the door, the crowd suddenly surges. The paper found that this "surge" increases the number of events by about 10% in that specific low-energy zone.
High Energy: At higher energies, the effect was smaller but still noticeable (about +3%).

Why It Matters

The authors conclude that while this "quark channel" is a minor player in the total count of Higgs pairs, it acts like a distortion lens. If you ignore it, your picture of the "shape" of the data (specifically the mass of the Higgs pair) will be slightly warped, especially near the threshold.

To get the most accurate measurement of the Higgs spring tension, physicists must now include this "backroad" in their calculations. It's a small detail, but in the world of high-precision physics, small details can hide big secrets.

In short: The paper mapped a rarely used path in particle physics. They found that while the path is short, the view from it changes the landscape significantly in one specific spot, and ignoring that view would make our map of the universe slightly inaccurate.

Technical Summary: Electroweak Corrections to Higgs Boson Pair Production in the Quark Channel

Problem and Motivation
Constraining the trilinear Higgs coupling ( $\lambda$ ) is a primary objective of the High Luminosity LHC phase. While current constraints allow for significant deviations from the Standard Model (SM), precise theoretical predictions are required to interpret experimental data. Although Next-to-Leading Order (NLO) QCD corrections for Higgs boson pair production via gluon fusion have been extensively studied (reducing scale uncertainties to $\mathcal{O}(3\%)$ ), electroweak (EW) corrections are becoming increasingly relevant as QCD uncertainties diminish. Previous studies have shown EW corrections can reach $\mathcal{O}(30\%)$ in differential distributions. However, the specific contribution of the quark–antiquark ( $q\bar{q}$ ) initiated channel to NLO EW corrections in Higgs pair production had not been previously calculated. This work addresses that gap, focusing on the process $q\bar{q} \to HH$ .

Methodology
The authors present a fully analytic calculation of the mixed QCD-electroweak corrections for the $q\bar{q} \to HH$ channel, considering massless quarks from the first two generations ( $u, d, s, c$ ) and a diagonal CKM matrix. The calculation proceeds as follows:

Amplitude Generation: Leading Order (LO) and real-emission amplitudes are generated using GoSam-3. The focus of the analytic work is the two-loop virtual corrections. Feynman diagrams are generated with QGRAF and translated into analytic expressions using FORM.
Gauge and Regularization: Calculations are performed in both unitary and Feynman gauges to ensure agreement. Naive Dimensional Regularization (NDR) is employed for $\gamma_5$ matrices. Since all $\gamma_5$ matrices appear on open fermion lines, no traces containing $\gamma_5$ in $D$ dimensions are required, avoiding NDR ambiguities. The amplitudes are decomposed into left- and right-handed components, reducing the matrix elements to single form factors.
Master Integrals and Differential Equations:
- Integration-by-Parts (IBP) relations are applied using Reduze 2 and Kira 3 to reduce integrals to a set of 51 master integrals.
- These integrals belong to two top-level sectors: a double-box topology (45 integrals) and a second topology (6 integrals).
- The integrals are solved using the method of canonical differential equations. A canonical basis is constructed using DlogBasis and a loop-by-loop approach. The differential equations are brought into the $d\log$ form: $d\mathbf{J} = \varepsilon \sum A_i d\log(\alpha_i) \mathbf{J}$ .
- Boundary conditions are determined by matching the expressions to the large mass expansion limit ( $s, t, u, m_H^2 \ll m_{W/Z}^2$ ).
Numerical Evaluation:
- The master integrals are expressed in terms of iterated integrals over logarithmic kernels and further rotated into a basis of graded transcendental functions using FiniteFlow.
- Numerical evaluation in the physical region is performed using DiffExp. The integration path is split: one segment connects the large mass limit to a reference point (cross-checked with AMFlow), and a second segment explores the physical region from this reference point.
- To facilitate implementation in the POWHEG-BOX framework, an interpolation grid of approximately 20,000 phase-space points is constructed based on the variables $\beta$ and $\cos\theta$ .
Renormalization and Subtraction: UV poles are removed via renormalization of the strong coupling $\alpha_s$ in the $\overline{MS}$ scheme. IR poles are handled using the Catani subtraction scheme.

Key Results
The study calculates differential distributions at a center-of-mass energy of $\sqrt{S} = 13.6$ TeV. The findings indicate:

Total Cross Section: The contribution of the $q\bar{q}$ channel to the total cross section is negligible.
Differential Distributions: The corrections have a significant impact on the shapes of differential distributions, particularly near the production threshold.
- Invariant Mass ( $m_{HH}$ ): Near the threshold, the NLO corrections to the $q\bar{q}$ channel exhibit a relative enhancement of approximately +10% compared to the LO gluon-fusion channel. This is primarily driven by real-emission contributions, which soften the spectrum and populate the threshold region more heavily. At larger invariant masses, the corrections remain positive but decrease to approximately +3%.
- Transverse Momentum ( $p_T$ ): The $p_T$ distribution of a single Higgs boson shows a similar pattern: an enhancement at low $p_T$ (driven by real emissions) and a moderate increase in the high- $p_T$ tail.
Precision: The analytic amplitudes can be evaluated with a precision of 16 significant digits in approximately one minute per phase-space point.

Significance and Claims
The authors claim this work represents the first study of EW effects in Higgs boson pair production initiated by a quark–antiquark pair. While the channel does not significantly alter the total cross section, the paper emphasizes that it leads to significant distortions of differential distributions close to the production threshold. The results are implemented in the POWHEG-BOX framework to enable phenomenological studies. The authors note that future work will involve the public release of the implementation and a detailed study of the interplay between additional EW and QCD contributions.