The role of the top Yukawa coupling in triple Higgs production at the LHC

This paper demonstrates that while varying the top Yukawa coupling within experimentally allowed ranges significantly alters the total cross-section for triple Higgs production at the LHC, it has minimal impact on the invariant mass distribution of the three-Higgs system and other global observables.

The Gist

Imagine the universe is a giant, complex machine, and the Higgs boson is the "glue" that gives other particles their mass. Physicists at the Large Hadron Collider (LHC) are trying to figure out exactly how this glue works. They do this by smashing protons together to create Higgs bosons and watching how they behave.

Usually, they look for one or two Higgs bosons appearing at a time. But this paper is about something much rarer and more difficult: creating three Higgs bosons at once (Triple Higgs production).

Here is the simple breakdown of what the author, Luca Panizzi, discovered, using some everyday analogies.

1. The "Heavyweight Champion" Problem

In the Standard Model (our current best theory of physics), the Higgs boson interacts with other particles. The heavier the particle, the stronger the Higgs "hugs" it. The Top Quark is the heaviest particle we know, so it has the strongest hug. This strength is called the Top Yukawa coupling.

Think of the Top Quark as a heavyweight champion boxer. Because he is so heavy, he is the most likely person to knock over the furniture (create new particles) when he moves around.

For a long time, scientists assumed this boxer's strength was exactly what the rulebook (the Standard Model) said it was. But, just like a boxer might train differently or have a secret technique, the Top Quark's strength might be slightly different than we think.

2. The "Recipe" Analogy

Creating three Higgs bosons is like baking a very complicated cake.

• The ingredients are the particles and forces involved.
• The Top Quark is a key ingredient (like a specific type of flour).
• The Higgs Self-Interaction is the recipe itself (how much sugar and eggs to mix).

The author asked: "What happens to our cake if we change the amount of 'Top Quark flour' slightly, even within the range of what we've already measured?"

3. The Big Discovery: The "Volume Knob"

The paper found that changing the strength of the Top Quark's interaction is like turning a volume knob on a stereo.

• If the Top Quark is slightly stronger (louder volume): The number of triple-Higgs events (the "songs" playing) explodes. The total amount of "cake" produced increases dramatically. In fact, if the Top Quark is just 20% stronger than expected, we could see 255% more triple-Higgs events! This could make the difference between never seeing the event and seeing it clearly at the High-Luminosity LHC.
• If the Top Quark is slightly weaker (quieter volume): The number of events drops drastically. It becomes almost impossible to see the triple Higgs bosons, making the search much harder.

The Key Takeaway: The amount of triple Higgs bosons we see depends heavily on how strong the Top Quark is. If we don't know the Top Quark's strength precisely, we can't accurately predict how many triple Higgs bosons we should find.

4. The Shape of the Cake (Kinematics)

Here is the surprising part. While the volume (total number of events) changes wildly, the shape of the cake stays the same.

Imagine you have a pile of three Higgs bosons. You can measure their total weight (invariant mass) or how fast they are flying apart (transverse momentum).

• The Finding: Whether the Top Quark is strong or weak, the pattern of how these three bosons fly apart looks almost identical. It's like turning up the volume on a song: the music gets louder, but the melody and rhythm don't change.

This is important because it means scientists can't easily tell why they are seeing more events just by looking at the shapes of the data. They have to know the Top Quark's strength first to interpret the data correctly.

5. Why This Matters

Think of the Higgs boson's self-interactions (how Higgs talks to Higgs) as a secret code. Scientists want to crack this code to understand the history of the universe (specifically, the "electroweak phase transition" right after the Big Bang).

• The Problem: If we assume the Top Quark's strength is exactly what the rulebook says, but it's actually slightly different, we might misread the secret code. We might think the Higgs self-interaction is weird when it's actually just the Top Quark being "stronger."
• The Solution: Before we can claim we've found new physics or measured the Higgs self-interactions perfectly, we need to measure the Top Quark's strength with extreme precision.

Summary

This paper is a warning and a guide for future experiments:

1. Don't ignore the Top Quark: Small changes in the Top Quark's strength cause huge changes in how many triple Higgs bosons are created.
2. The "Shape" is stable: Changing the Top Quark doesn't change the way the particles move, just how many of them appear.
3. Precision is key: To find the "secret code" of the Higgs boson, we need to know the Top Quark's strength better than we do right now.

In short: You can't measure the Higgs boson's secrets accurately until you know exactly how strong the Top Quark is.

Technical Summary

Here is a detailed technical summary of the paper "The role of the top Yukawa coupling in triple Higgs production at the LHC" by Luca Panizzi.

1. Problem Statement

The determination of Higgs boson parameters, particularly the self-interaction couplings (trilinear $\lambda_3$ and quadrilinear $\lambda_4$), is a primary goal of high-energy physics to understand the nature of the electroweak phase transition and potential new physics. While the production of two ($HH$) and three ($HHH$) Higgs bosons at the LHC is sensitive to these couplings, existing analyses often assume the Standard Model (SM) value for the top Yukawa coupling ($y_t$).

However, the top quark is a prime candidate for new physics due to its large mass, which could receive contributions from new sectors (e.g., vector-like partners or new scalars), potentially decoupling the top mass ($m_t$) from the Yukawa coupling ($y_t$) such that $y_t \neq y_t^{SM} = \sqrt{2}m_t/v$. The author argues that the dependence of triple Higgs production on $y_t$ has been neglected in previous searches, despite $y_t$ appearing in multiple topologies (pentagon and box diagrams) with high powers (e.g., $y_t^6$, $y_t^5$) in the squared amplitudes.

2. Methodology

To quantify the impact of varying $y_t$ on the $HHH$ cross-section and kinematic distributions, the author employs a deconstruction method previously used for di-Higgs studies.

• Lagrangian Deconstruction: The process is modeled by introducing deviations ($\delta$) from SM values for the Higgs potential couplings ($\lambda_3, \lambda_4$) and the top Yukawa coupling ($y_t$). The Lagrangian terms are written as:
  $$\mathcal{L} \supset (\lambda^{SM} + \delta\lambda_3)vh^3 + \frac{1}{4}(\lambda^{SM} + \delta\lambda_4)h^4 + (y_t^{SM} + \delta y_t)h\bar{t}t + \text{h.c.}$$
  These are mapped to coupling modifiers: $\kappa_{3,4} = 1 + \delta\lambda_{3,4}/\lambda^{SM}$ and $\kappa_t = 1 + \delta y_t/y_t^{SM}$.

• Cross-Section Expansion: The total triple-Higgs cross-section ($\sigma_{HHH}$) is expressed as a weighted sum of reduced cross-sections ($\hat{\sigma}_{ijk}$) corresponding to unique products of coupling deviations:
  $$\sigma_{HHH} = \sum_{i=0}^{6} \sum_{j=0}^{4} \sum_{k=0}^{2} (\delta y_t)^i (\delta \lambda_3)^j (\delta \lambda_4)^k \hat{\sigma}_{ijk}$$
  Here, indices $i, j, k$ represent the number of vertices associated with each coupling in the squared amplitude.

• Simulation Strategy:

  • Tools: Simulations were performed using MG5_aMC@NLO with a custom UFO model where each coupling deviation corresponds to a distinct coupling order.
  • Setup: Calculations were done at 13 TeV using NNPDF4.0 PDFs at Leading Order (LO).
  • Efficiency: Instead of simulating every possible coupling combination, only 40 unique Monte Carlo (MC) samples were generated (one for each valid term in the expansion). These samples were then re-weighted semi-analytically to reconstruct cross-sections and distributions for any combination of $\kappa_t, \kappa_3, \kappa_4$.
  • Neglect of $y_b$: Bottom Yukawa variations were tested and found to have negligible effects ($<0.05\%$) on the cross-section, so they were fixed to SM values.

3. Key Contributions

• Quantification of $y_t$ Sensitivity: The study provides the first dedicated quantification of how variations in the top Yukawa coupling within experimental $2\sigma$windows affect the$HHH$ production rate and the derived constraints on Higgs self-couplings.
• Efficient Re-weighting Framework: Demonstrates the utility of the deconstruction method for triple-Higgs processes, allowing for rapid exploration of a multi-dimensional parameter space ($\kappa_t, \kappa_3, \kappa_4$) without generating new event samples for every point.
• Kinematic Analysis: Investigates whether $y_t$ variations alter the shape of kinematic distributions (invariant mass, transverse momentum sums) or merely rescale the total rate.

4. Results

A. Impact on Total Cross-Section and Constraints

• Cross-Section Sensitivity: The $HHH$ cross-section is highly sensitive to $\kappa_t$.
  • Increasing $\kappa_t$ by 20% ($\kappa_t = 1.2$) increases the cross-section by +255% (from 0.034 fb to 0.12 fb).
  • Decreasing $\kappa_t$ by 20% ($\kappa_t = 0.8$) reduces the cross-section by -80% (to 0.007 fb).
• Constraints on Self-Couplings: Variations in $y_t$ significantly modify the exclusion contours for $\kappa_3$ and $\kappa_4$.
  • Larger $y_t$ values tighten the bounds on $\kappa_3$ and $\kappa_4$, shrinking the allowed parameter space toward the SM point.
  • The uncertainty in $y_t$ has an impact comparable to applying higher-order QCD corrections (K-factors), suggesting that $y_t$ precision is critical for interpreting $HHH$ limits.

B. Kinematic Distributions

• Shape Invariance: The study analyzed normalized differential distributions for:
  • Invariant mass of the three-Higgs system ($M_{inv}^{HHH}$).
  • Scalar sum of transverse momenta ($\sum |p_T^H|$).
  • Reconstructed hadronic observables (after decay and jet reconstruction).
• Finding: While the normalization (total rate) changes drastically with $\kappa_t$, the shapes of the distributions remain largely unchanged.
  • There is a mild tendency for distributions to rise slightly faster for larger $\kappa_t$, but the effect is not significant.
  • The shapes are dominated by the Higgs self-couplings ($\kappa_3, \kappa_4$), not $y_t$.
  • This implies that $y_t$ variations act primarily as a rescaling factor rather than a shape modifier at LO.

5. Significance and Conclusions

• Critical for Future Searches: The precision of the top Yukawa coupling determination is paramount for assessing the observability of the $HHH$ process at the High-Luminosity LHC (HL-LHC). A 20% deviation in $y_t$ could change the expected event yield from negligible to potentially observable (>300 events in 3 ab$^{-1}$).
• Interpretation of Limits: Current limits on Higgs self-couplings derived from $HHH$ searches are highly dependent on the assumed value of $y_t$. Future analyses must treat $y_t$ as a free parameter or include its uncertainty to avoid biased constraints on $\kappa_3$ and $\kappa_4$.
• Search Strategy: Since $y_t$ variations do not significantly alter kinematic shapes, search strategies relying on shape analysis to distinguish new physics in the Higgs potential may need to account for $y_t$ degeneracies primarily through rate-based constraints.

In summary, the paper establishes that the top Yukawa coupling is a dominant systematic uncertainty in triple Higgs production, capable of altering the predicted signal rate by factors of several, while leaving the kinematic topology largely intact. This necessitates a simultaneous fit of $y_t$ and Higgs self-couplings in future LHC analyses.