Higgs pair production in gluon fusion to higher orders in Higgs Effective Field Theory

This paper investigates Higgs pair production via gluon fusion within the Higgs Effective Field Theory (HEFT) framework, demonstrating that a consistent power counting at next-to-leading order requires the inclusion of higher-dimensional operators and necessitates a critical re-evaluation of the kinematic benchmark scenarios used in experimental di-Higgs searches.

The Gist

Imagine the universe is built like a giant, complex Lego set. For decades, scientists have been using a specific instruction manual called the Standard Model to understand how the pieces fit together. One of the most important pieces in this set is the Higgs boson, a particle that gives other particles their mass.

Usually, scientists study these Lego pieces one at a time. But this paper is about what happens when you try to snap two Higgs bosons together at the same time. This is called "Higgs pair production." It's incredibly rare—like trying to catch two specific grains of sand falling from the sky at the exact same moment. Because it's so rare, it's hard to study, but it offers a unique chance to see if the "instruction manual" is complete or if there are hidden rules we haven't discovered yet.

Here is a breakdown of what the authors did, using simple analogies:

1. The Two Instruction Manuals: SMEFT vs. HEFT

The paper compares two different ways of writing the "instruction manual" for the universe:

• SMEFT (The Strict Manual): This version assumes the universe follows very rigid, linear rules. If you change one rule, it affects everything else in a predictable, straight line.
• HEFT (The Flexible Manual): This is a more general version. It allows the rules to be "curved" or non-linear. Think of it like the difference between a straight ruler (SMEFT) and a flexible rubber band (HEFT). In the flexible version, the rules for how Higgs bosons interact can be completely different from the strict version, even at the most basic level.

The authors chose to study the HEFT (Flexible Manual) because it allows them to test if the universe is actually "rigid" or "flexible."

2. The "Power Counting" Problem

When you try to calculate what happens in these particle collisions, you have to add up millions of tiny possibilities (like adding up the weight of every single grain of sand in a beach).

• The Old Way: Previous studies only looked at the "biggest" contributions (the heaviest grains of sand) and added a little bit of correction for the smaller ones.
• The New Way (This Paper): The authors realized that if you want to be truly accurate with the "Flexible Manual," you can't just look at the big grains. You have to include higher-order rules (smaller, more complex interactions) that were previously ignored.

They used a system called "Power Counting" to decide which rules to include. It's like a budget: "We have enough energy to calculate up to this level of complexity, so we must include these specific extra rules to stay within our budget." They found that to get the math right, they had to include new, complex interactions that involve extra "glue" (gluons) and "springs" (derivatives) between the particles.

3. The "Shape" of the Collision

When two Higgs bosons are created, they fly apart with a certain speed and energy. Scientists look at the invariant mass distribution, which is basically a histogram showing how often pairs are created at different energy levels.

• The Clustering Game: The authors asked: "If we change the rules in our Flexible Manual, does the shape of this histogram change in a way we can actually see?"
• They used a computer algorithm (like a smart sorting machine) to group thousands of possible scenarios into "clusters."
• The Result: They found that for the most common scenarios, the existing experimental "buckets" (clusters) used by scientists are actually doing a great job. They cover almost everything.
• The Surprise: However, they found a few very rare, weird scenarios where the histogram looked totally different (like a sharp spike or a flat plateau) that the old buckets didn't catch. These are like "ghost shapes" that only appear if you include the new, complex rules they discovered.

4. The "Angle" Test

Besides the energy, scientists also look at the angle at which the particles fly apart.

• In the standard model, this angle is usually flat and boring (like a calm lake).
• The authors checked if their new, complex rules would make the lake ripple. They found that while the rules can create ripples, the ripples are currently too small to see with our current "telescopes" (experimental uncertainty). To see these ripples, we would need to make our measurements about 10% more precise.

5. The "Positivity" Rule

The authors also applied a logical check called Positivity Bounds.

• Imagine you are building a bridge. Physics has a rule that says the bridge must be stable and can't collapse backward in time.
• They proved that for their new, complex rules to make sense in the real world, certain numbers in their equations must be positive (or follow a specific relationship). If they aren't, the theory breaks the laws of physics (causality). This acts as a filter to remove impossible scenarios.

Summary

In short, this paper is a theoretical upgrade for how we predict what happens when two Higgs bosons collide.

1. They updated the math to include more complex, "hidden" interactions that were previously ignored.
2. They checked if these new interactions create new, detectable patterns in the data.
3. They found that while the current experimental methods are very good at catching the most common patterns, there are a few rare, exotic patterns they might be missing.
4. They also showed that looking at the angles of the collision is currently too difficult to be useful, but looking at the energy distribution is the best way to find new physics.

The paper doesn't claim to have found new particles yet; rather, it provides a better, more complete map for future experiments to use when they finally catch those rare Higgs pairs.

Technical Summary

Technical Summary: Higgs Pair Production in Gluon Fusion to Higher Orders in Higgs Effective Field Theory

Problem Statement
Higgs pair production ($pp \to hh$) serves as a critical probe for the structure of the Higgs sector, specifically testing whether electroweak symmetry breaking is realized linearly (as in the Standard Model Effective Field Theory, SMEFT) or non-linearly (as in Higgs Effective Field Theory, HEFT). While SMEFT correlations among couplings persist at dimension-six, HEFT allows for fully de-correlated couplings at leading order. Previous calculations of Higgs pair production in gluon fusion within HEFT have largely relied on the leading-order (LO) HEFT Lagrangian supplemented by Next-to-Leading Order (NLO) QCD corrections. However, a consistent power counting in HEFT dictates that including NLO QCD corrections necessitates the inclusion of higher-dimensional operators (specifically those with chiral dimensions $N_\chi = 2$ and $N_\chi = 4$) to maintain consistency in the expansion. The authors identify a gap in the literature where these specific higher-order operators, which can arise at tree-level or lower loop orders in the EFT expansion, have not been systematically included alongside NLO QCD corrections in a consistent power counting framework.

Methodology
The authors employ the HEFT framework, utilizing the chiral dimension power counting scheme proposed in Ref. [52]. The methodology involves:

1. Lagrangian Construction: The authors construct the HEFT Lagrangian up to NLO and NNLO in the chiral expansion. They define the Lagrangian as $\mathcal{L}_{\text{HEFT}} = \mathcal{L}_{\text{LO}} + \mathcal{L}_{\text{NLO}} + \mathcal{L}_{\text{NNLO}} + \dots$. They explicitly include operators with chiral dimensions $N_\chi = 2$ and $N_\chi = 4$ (denoted as $\delta\mathcal{L}$) which were previously omitted in studies focusing solely on the $\kappa$-formalism ($\mathcal{L}_\kappa$). These operators involve couplings of gluon field strength tensors to Higgs bosons and derivatives of the Higgs field.
2. Power Counting Scheme: The authors adopt the counting scheme $N_{s,M}^{\text{HEFT}}$, which includes the number of strong coupling constants ($g_s$) in the counting. This scheme requires that increasing the loop order of a process (e.g., adding NLO QCD corrections) must be accompanied by the inclusion of diagrams with lower loop orders but higher chiral dimension vertices. They truncate the cross-section contributions at $N_{\text{trunc}}^{\text{HEFT}} = 10$.
3. Calculation: The calculation focuses on the gluon-fusion process $gg \to hh$, neglecting bottom quark loops. The amplitude is decomposed into two projection operators. The authors compute the differential cross-section $d\sigma/dm_{hh}$ as a linear combination of kinematic coefficients ($A_i$) multiplied by various combinations of Wilson coefficients ($c_i$). They include contributions from $N_\chi=0$, $N_\chi=2$, and $N_\chi=4$ operators.
4. Positivity Bounds: To ensure compatibility with causality and unitarity, the authors derive positivity bounds on the Wilson coefficients $b_g^{(1)}$ and $b_g^{(2)}$ using the optical theorem and forward scattering amplitudes of auxiliary states.
5. Phenomenological Analysis:
   • Clustering: A clustering algorithm based on a $\chi^2$-test is employed to categorize invariant mass ($m_{hh}$) distributions. The algorithm compares normalized differential distributions against a set of benchmark clusters derived from the $\mathcal{L}_\kappa$ Lagrangian (as used in experimental analyses).
   • Validation Scans: The authors perform validation scans with $10^8$ samples to test the coverage of existing benchmark scenarios when the new $\delta\mathcal{L}$ operators are included, varying theoretical uncertainties and parameter ranges.
   • Angular Observables: The impact of the new operators on the scattering angle ($|\cos \theta^*|$) and transverse momentum ($p_{T,h}$) distributions is analyzed.
   • SMEFT Comparison: A parallel analysis is performed for SMEFT at dimension-six (linear order) to contrast the kinematic diversity between the two EFT frameworks.

Key Contributions

• Consistent Power Counting Implementation: The paper demonstrates that a consistent HEFT power counting with NLO QCD corrections necessitates the inclusion of $N_\chi=2$ and $N_\chi=4$ operators.
• Derivation of Positivity Bounds: New analytic constraints on the signs and magnitudes of the Wilson coefficients $b_g^{(1)}$ and $b_g^{(2)}$ are derived.
• Extension of Benchmark Scenarios: The study extends the kinematic benchmark scenarios used in experimental non-resonant di-Higgs searches by incorporating the effects of higher-dimensional operators.
• Quantitative Assessment of Coverage: The authors quantify how well existing experimental benchmark clusters cover the full HEFT parameter space when higher-order operators are included.

Results

• Invariant Mass Distributions: The inclusion of the new operators ($\delta\mathcal{L}$) introduces novel kinematic features in the $m_{hh}$ distribution, such as pronounced peaks near the production threshold ($2m_h$) followed by rapid flattening, or plateau-like structures.
• Cluster Coverage: Despite the introduction of these new shapes, the authors find that the existing 12-cluster benchmark scenarios (derived from $\mathcal{L}_\kappa$) cover approximately 94% to 99% of the parameter space under full theoretical uncertainty assumptions. The coverage drops to roughly 70-80% only when theoretical uncertainties are artificially halved, revealing rare parameter choices that produce distributions outside existing categories.
• SMEFT vs. HEFT: In contrast to HEFT, the SMEFT analysis at linear order (dimension-six) shows significantly less kinematic diversity. Four clusters were sufficient to cover 99.98% of the SMEFT parameter space, highlighting the greater flexibility of the HEFT framework.
• Angular Observables: The $|\cos \theta^*|$ distribution, which is flat in the SM and LO HEFT, shows sensitivity to the new operators (specifically $b_c$, $b_g^{(1)}$, and $b_g^{(2)}$). However, the deviations are small. The authors estimate that current uncertainties are too large to distinguish these modifications; a reduction in uncertainty by approximately 10% would be required to probe these deviations effectively.

Significance and Claims
The paper claims that while the inclusion of higher-order HEFT operators introduces qualitatively new kinematic features in Higgs pair production, these features are rare within the parameter space. Consequently, the current experimental benchmark scenarios used for non-resonant di-Higgs searches remain robust and provide very good coverage. The study serves as a "prime example" of the power counting method proposed in Ref. [52], illustrating that consistent EFT expansions require the simultaneous inclusion of higher-loop QCD corrections and higher-dimensional operators. The authors conclude that while the existing benchmarks are sufficient for current sensitivity, a global approach incorporating multiple observables (including angular distributions) will be essential for the robust identification of EFT contributions as experimental precision improves. The work does not propose new experimental searches but rather re-assesses the theoretical completeness of the current search strategies.