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Precision tests of third-generation four-quark operators: $gg \to h$ and $h \to \gamma \gamma$

This paper computes the two-loop contributions to Higgs production via gluon-gluon fusion and decay into two photons arising from third-generation four-quark operators in the SMEFT, performed in the broken phase with full mass dependence to derive finite matching corrections, renormalization group effects, and new two-loop anomalous dimensions.

Original authors: Ulrich Haisch, Marco Niggetiedt

Published 2026-09-24

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CC BY 4.0

Original authors: Ulrich Haisch, Marco Niggetiedt

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 universe is built on a set of fundamental rules that dictate how particles interact, a framework physicists call the Standard Model. At the heart of this model sits the Higgs boson, a particle discovered over a decade ago that gives mass to other particles. While the Higgs boson has been studied extensively, its behavior remains a primary window into the unknown. Scientists suspect that the Standard Model is incomplete, hiding a deeper layer of reality that involves heavier, unseen particles. To find these hidden pieces without building a machine powerful enough to create them directly, researchers look for subtle deviations in how the Higgs boson is made and how it decays. They treat the Higgs like a sensitive instrument; if invisible, heavy particles are influencing it, they leave tiny fingerprints in the data, much like a heavy footstep might slightly alter the path of a rolling ball.

In a recent study, physicists Ulrich Haisch and Marco Niggetiedt from the Max Planck Institute for Physics in Germany have sharpened this instrument. They focused on a specific type of potential new physics: interactions involving the heaviest known particles, the top and bottom quarks. These heavy particles are thought to be the most likely candidates to interact with new, unseen forces. The researchers calculated how these heavy quarks, if they were part of a larger, hidden set of rules, would influence two specific processes: the creation of a Higgs boson when two gluons collide, and the decay of a Higgs boson into two photons. These calculations were not simple; they required looking at the problem with extreme precision, accounting for complex interactions that happen twice in a row, known as two-loop effects.

The team performed these calculations by breaking down the problem into manageable pieces, using advanced mathematical tools to track how the heavy quarks move and interact within the quantum foam of the vacuum. They found that the presence of these specific heavy-quark interactions would change the rate at which Higgs bosons are produced and how often they turn into light. Crucially, they determined exactly how these changes depend on the masses of the particles involved. Their work revealed that the production of the Higgs boson from colliding gluons is significantly more sensitive to these new interactions than the decay into photons. This is because the decay into light is dominated by a different, well-understood process involving the W boson, which acts as a shield, drowning out the subtle signals from the heavy quarks. In contrast, the production process offers a clearer view of these heavy-quark effects.

The researchers also discovered that the way these calculations are performed depends on the specific mathematical rules used to handle a property called "handedness" in particle physics. They showed that using one set of rules versus another leads to different numerical predictions for how the signals change. This finding is important because it tells future researchers that they must be consistent in their methods to avoid confusion. By providing precise formulas for these effects, the study allows scientists to compare their experimental data from the Large Hadron Collider directly against these theoretical predictions.

When the team applied their new formulas to the actual data collected by the ATLAS experiment, they were able to set new limits on how strong these hidden interactions could be. They found that the current data does not rule out the existence of these heavy-quark interactions, but it does constrain them to be relatively weak. Interestingly, for one specific type of interaction involving the bottom quark, their study provided the tightest limit available to date, as no other method had previously been able to probe it so effectively. While these limits are not yet strong enough to confirm the existence of new physics, they represent a significant step forward in the search. The study demonstrates that by pushing the precision of theoretical calculations to the highest possible level, scientists can extract more information from the data they already have, turning the Higgs boson into an even more powerful tool for exploring the frontiers of the universe.

Technical Summary: Precision tests of third-generation four-quark operators: $gg \to h$ and $h \to \gamma\gamma$

Problem Statement
The Standard Model Effective Field Theory (SMEFT) provides a framework for describing Beyond the Standard Model (BSM) physics characterized by heavy mass scales $\Lambda$ . A specific subset of dimension-six operators involving only third-generation quark fields (four-quark operators) remains weakly constrained. Direct probes via processes like $t\bar{t}t\bar{t}$ , $t\bar{t}b\bar{b}$ , or $b\bar{b}b\bar{b}$ suffer from large experimental uncertainties (e.g., $\sim 30\%$ for $t\bar{t}t\bar{t}$ ) and rely on quadratic terms in the $1/\Lambda^2$ expansion, raising concerns regarding SMEFT power counting. Consequently, indirect tests via Higgs physics observables have attracted attention. However, previous calculations regarding the impact of third-generation four-quark operators on Higgs production ( $gg \to h$ ) and decay ( $h \to \gamma\gamma$ ) exhibited discrepancies and lacked a complete treatment of two-loop effects, including full mass dependence and renormalization group (RG) evolution.

Methodology
The authors compute the two-loop contributions to the $gg \to h$ and $h \to \gamma\gamma$ amplitudes arising from insertions of the third-generation four-quark operators defined in the Warsaw basis. The calculation is performed in the broken phase of the theory, retaining the full dependence on the Higgs mass ( $m_h$ ) and heavy-quark masses ( $m_t, m_b$ ).

Regularization and Renormalization: The computation employs dimensional regularization ( $d=4-2\epsilon$ ) with an anti-commuting $\gamma_5$ (NDR) scheme. A mixed renormalization scheme is adopted: operators are renormalized in the Modified Minimal Subtraction ( $\overline{\text{MS}}$ ) scheme, while heavy-quark masses and Yukawa couplings are treated in the On-Shell (OS) scheme.
Computational Tools: The calculation utilizes the Mathematica packages FeynRules, FeynArts, FormCalc, and LiteRed. FeynArts generates diagrams and amplitudes; FormCalc handles color and Dirac algebra and projects onto form factors; LiteRed reduces scalar integrals to master integrals (massive one-loop bubbles and triangles).
Topology Analysis: The two-loop amplitudes are categorized into three topological classes based on the insertion of dimension-six operators:
1. Insertion into internal propagators (canceled by OS mass counterterms).
2. Insertions requiring Yukawa coupling renormalization via mixing into dipole operators ( $Q_{tH}, Q_{bH}$ ).
3. Insertions involving mixing into chromomagnetic/electroweak dipole operators ( $Q_{tG}, Q_{bG}$ , etc.) and subsequently into Higgs-gauge operators ( $Q_{HG}, Q_{HB}$ , etc.).
Anomalous Dimensions: The renormalization of the two-loop amplitudes necessitates specific two-loop SMEFT anomalous dimensions, which are derived as a byproduct of the calculation.

Key Contributions and Results

Two-Loop Anomalous Dimensions: The authors derive the two-loop beta functions describing the mixing of third-generation four-quark operators into the Higgs-gauge operators ( $Q_{HG}, Q_{HB}, Q_{HW}, Q_{HWB}$ ). These results are presented in the NDR scheme (Eq. 4.2). The authors note that these beta functions are scheme-dependent, a feature inherent to NLO SMEFT calculations involving $\gamma_5$ and chirality-flipping operators.
Matching Conditions: The paper provides the exact two-loop matching conditions for the $gg \to h$ $g g \to h$ and $h \to \gamma\gamma$ $h \to γ γ$ form factors. These are expressed in terms of the Wilson coefficients of the third-generation four-quark operators, including finite matching corrections and logarithmic terms ( $\ln(\mu^2/m_i^2)$ $ln (μ^{2} / m_{i}^{2})$ ) arising from RG running.
- For $gg \to h$ , the form factor corrections $G^{(2)}$ are given in Eq. (5.5).
- For $h \to \gamma\gamma$ , the form factor corrections $A^{(2)}$ are given in Eq. (5.10).
- The results confirm consistency with recent works [26, 27, 29] regarding the $t\bar{t}$ contributions but differ from findings in [25], where certain topological contributions appeared to be missing.
Phenomenological Constraints: Using the derived formulas, the authors calculate the modification to the signal strengths $\delta\kappa_g$ $δ κ_{g}$ and $\delta\kappa_\gamma$ $δ κ_{γ}$ .
- Sensitivity: The $gg \to h$ process is found to be generally more sensitive to these operators than $h \to \gamma\gamma$ , as the latter is dominated by $W$ -boson loops in the SM.
- Mass Suppression: Contributions involving bottom quarks are suppressed relative to top quarks by a factor of $\sim m_b^2/m_t^2 \ln^2(m_h^2/m_b^2)$ .
- Enhancement: The operator $Q^{(1)}_{qtqb}$ shows enhanced sensitivity in $gg \to h$ due to a factor of $m_t/m_b$ in the form factors, though this enhancement is model-dependent (e.g., absent in Minimal Flavor Violation scenarios).
- Bounds: The authors derive 68% CL constraints on the Wilson coefficients using LHC Run 2 data. They find these bounds are generally weaker than those from EW precision observables, LHC top-quark data, or $h \to b\bar{b}$ decays. However, for the operator $C^{(8)}_{qb}$ , no existing constraints were known, making this the first (albeit weak) limit.
Scheme Dependence and Flat Directions: The paper highlights that combining $gg \to h$ and $h \to \gamma\gamma$ data can lift flat directions in the parameter space of $C^{(1)}_{qt}$ and $C^{(8)}_{qt}$ , but only when the NDR scheme is used. In the 't Hooft-Veltman (HV) scheme, these channels remain degenerate at linear order.

Significance and Claims
The authors position this work as a necessary precision step for global SMEFT analyses. By providing the exact two-loop matching conditions and the associated anomalous dimensions, they enable a consistent inclusion of third-generation four-quark operators in global fits. The paper emphasizes that:

The derived results allow for a cross-check of previous findings in the literature.
The scheme dependence of the results is an unavoidable feature of NLO SMEFT calculations; however, the choice of NDR combined with specific operator bases (such as the LHCTopWG basis) can simplify computations by avoiding non-zero traces involving $\gamma_5$ and the need for Wess-Zumino terms.
While the constraints derived from $gg \to h$ and $h \to \gamma\gamma$ are currently weaker than other probes for most operators, they provide the first constraints for $C^{(8)}_{qb}$ and offer a method to resolve parameter degeneracies that are scheme-dependent.

The authors conclude that their analytical and numerical expressions facilitate the integration of these specific Higgs observables into broader global SMEFT analyses, improving the robustness of indirect BSM searches.

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