Precision predictions for trilinear scalar couplings and Higgs pair production in models with extended scalar sectors

This paper summarizes recent theoretical advances in calculating precise predictions for trilinear scalar couplings and Higgs pair production within extended scalar sector models, which are essential for reconstructing the Higgs potential and probing physics beyond the Standard Model at the (HL-)LHC.

The Gist

Imagine the universe is built on a giant, invisible landscape called the "Higgs potential." Think of this landscape like a bowl or a valley. The shape of this bowl determines how particles get their mass and how the universe behaved right after the Big Bang.

The author of this paper, J. Braathen, is a scientist trying to figure out the exact shape of this bowl. Why? Because if the bowl looks different than what we expect (the Standard Model), it means there is new, hidden physics waiting to be discovered.

Here is a breakdown of the paper's main points using simple analogies:

1. The Goal: Mapping the Invisible Bowl

To understand the shape of this "bowl," scientists need to measure specific points on its surface. One of the most important points is how the Higgs particle interacts with itself.

• The Analogy: Imagine the Higgs particle is a ball rolling in the bowl. The "trilinear coupling" (a fancy math term) is like measuring how hard the ball pushes against the sides of the bowl when it bumps into itself.
• The Problem: In the old, simple version of physics (the Standard Model), we know exactly how hard that push should be. But in newer, more complex theories (BSM models), the bowl might have extra bumps or curves. This changes the "push."
• The Paper's Contribution: The author has built better "rulers" (mathematical tools) to measure this push with extreme precision, including corrections that account for tiny, invisible quantum effects.

2. The Tools: "anyH3" and "anyHH"

To do this measuring, the author developed two digital tools (software) that act like high-tech surveying equipment.

• anyH3: Think of this as a tool that measures the "push" (the trilinear coupling) inside the bowl. It can handle any shape of the bowl, even if the bowl has extra hidden layers (extended scalar sectors).
• anyHH: This tool simulates what happens when two Higgs particles are created at the same time (like smashing two balls together). It calculates how often this happens and what the resulting pattern looks like.
• The Innovation: These tools are "automated." Instead of a scientist spending years doing math by hand for every new theory, these tools can instantly calculate the results for any new model the scientist wants to test.

3. The Discovery: Why "Loop Corrections" Matter

The paper shows that if you only use the basic, simple math (called "tree-level"), you might get the wrong answer. You need to include "loop corrections."

• The Analogy: Imagine you are trying to predict the path of a boat in a river.
  • Tree-level: You only look at the current and the wind.
  • Loop corrections: You also account for the tiny eddies, the wake from other boats, and the friction of the water against the hull.
• The Result: In the paper's examples, ignoring these tiny "eddies" (quantum loops) completely changed the prediction.
  • In one scenario, the simple math said, "We can't tell the difference between the new theory and the old one."
  • But when the author added the "loop corrections," the prediction changed drastically. Suddenly, the new theory looked very different from the old one, making it easy to spot.
  • The "Flip": In some cases, adding these corrections didn't just change the size of the effect; it flipped the sign (like turning a hill into a valley). This changed the entire shape of the signal scientists would see in their detectors.

4. The Big Picture

The paper argues that to find new physics at the Large Hadron Collider (LHC), we cannot rely on rough estimates. We need these super-precise, automated calculations.

• If we use the old, rough math, we might miss a new discovery or think we found one when we didn't.
• By using the new tools (anyH3 and anyHH) and including the complex "loop" corrections, scientists can accurately predict what the detectors should see if the universe has an "extended" Higgs sector.

In summary: The author has built better, automated calculators to measure the shape of the universe's energy landscape. They proved that if you ignore the tiny, complex quantum details (the "loops"), your map of the landscape will be wrong, and you might miss the discovery of a lifetime.

Technical Summary

1. Problem Statement

Reconstructing the shape of the Higgs potential is a primary objective for current and future colliders (such as the HL-LHC) to understand the electroweak phase transition and search for Beyond-the-Standard-Model (BSM) physics.

• The Core Challenge: The trilinear self-coupling of the Higgs boson ($\lambda_{hhh}$) and general trilinear scalar couplings ($\lambda_{ijk}$) dictate the form of the Higgs potential. In models with extended scalar sectors, radiative corrections from BSM scalars can induce significant deviations in these couplings from their Standard Model (SM) values.
• The Gap: Current experimental limits on the ratio $\kappa_\lambda \equiv \lambda_{hhh}/\lambda_{hhh}^{SM}$ are derived from di-Higgs production searches. However, interpreting these limits requires precise theoretical predictions. Existing analyses often rely on tree-level couplings or incomplete loop corrections, which can lead to erroneous conclusions regarding the exclusion or discovery of BSM scenarios. Specifically, the interference between resonant and non-resonant production diagrams is highly sensitive to the precise values of these couplings.

2. Methodology and Tools

The paper focuses on the automation of high-precision calculations for trilinear couplings and Higgs pair production within the anyBSM framework.

• anyH3 (Trilinear Couplings):

  • Originally designed for full one-loop calculations of $\lambda_{hhh}$, the tool has been extended to compute any trilinear coupling ($\lambda_{ijk}$) at the one-loop level for models defined by UFO files.
  • It incorporates complete two-loop results for trilinear and quartic self-couplings of arbitrary scalars in general renormalizable theories.
  • The methodology utilizes symmetry properties of Feynman diagrams to simplify calculations and applies effective field theory (EFT) techniques for scenarios with large BSM deviations to include higher-order corrections.

• anyHH (Higgs Pair Production):

  • A new module within the anyBSM framework designed to calculate total cross-sections and differential invariant mass distributions ($m_{hh}$) for processes like $gg \to h_i h_j$.
  • Key Features:
    • Includes all non-resonant and resonant contributions (for any number of resonant scalars).
    • Implements complete one-loop corrections to all relevant trilinear scalar couplings.
    • Accounts for one-loop propagator corrections in $s$-channel diagrams.
    • Handles the full momentum dependence of trilinear couplings.
  • Limitation: Currently restricted to models without additional colored particles.

3. Key Contributions

1. Automation of Precision Calculations: The development and extension of anyH3 and anyHH allow for the systematic inclusion of one-loop and two-loop radiative corrections in BSM models with extended scalar sectors, moving beyond tree-level approximations.
2. Demonstration of Loop Correction Impact: The paper provides concrete evidence that loop corrections to trilinear couplings are not merely small perturbations but can drastically alter phenomenological predictions.
3. Momentum Dependence Analysis: The study investigates the impact of including momentum-dependent form factors in trilinear couplings versus using fixed values.
4. Two-Loop Integration: The work demonstrates the interfacing of two-loop corrected couplings with the Higgs pair production calculation, showing the necessity of higher-order terms for reliable predictions.

4. Key Results

The paper presents benchmark scenarios in the Real Singlet Extension of the SM (RxSM) and the Two-Higgs-Doublet Model (2HDM) to illustrate the magnitude of these effects:

• Significant Cross-Section Variations:

  • In the RxSM benchmark, including one-loop corrections to $\lambda_{ijk}$ reduced the total cross-section from 19.6 fb (tree-level) to 9.8 fb (one-loop).
  • In the 2HDM alignment limit, the total cross-section approximately doubled when moving from tree-level to one-loop couplings due to changes in the interference pattern.

• Alteration of Kinematic Distributions:

  • Interference Patterns: Loop corrections modify the interference between box and triangle diagrams. For instance, in the RxSM, the sign flip of the $\lambda_{hhH}$ coupling changed the resonance structure around the heavy Higgs mass ($m_H$) from a "peak-dip" to a "dip-peak" structure.
  • Resonance Shifts: In the 2HDM, loop-induced non-zero values for $\lambda_{hhH}$ (which is zero at tree level in the alignment limit) introduced a resonant peak around $m_{hh} \approx m_H$ that was absent in tree-level predictions.

• Impact on Discovery Potential ($Z_{hh}$):

  • The statistical significance ($Z_{hh}$) of distinguishing a BSM scenario from the SM is highly sensitive to the order of perturbation theory used.
  • RxSM Example: Using tree-level couplings yielded $Z_{hh} = 0.1$ (no sensitivity), whereas using one-loop corrected couplings yielded $Z_{hh} = 9.4$ (well above the $5\sigma$ discovery threshold). This implies that neglecting loop corrections could lead to missing a discovery or falsely excluding a valid model.

• Two-Loop Effects:

  • While two-loop corrections did not qualitatively change the shape of differential distributions, they had a significant numerical impact. In the 2HDM example, two-loop corrections further increased the deviation in $\kappa_\lambda$ and reduced the total cross-section by an additional ~37% compared to the one-loop result, primarily due to a ~30% decrease in the magnitude of $\lambda_{hhH}$.

• Momentum Dependence:

  • Including the momentum dependence of couplings in the 2HDM scenario resulted in a ~20% reduction in the total cross-section but had a minimal effect on the shape of the differential $m_{hh}$ distribution.

5. Significance

• Necessity of Precision: The results conclusively demonstrate that tree-level predictions are insufficient for interpreting HL-LHC data on Higgs pair production. The inclusion of at least one-loop, and ideally two-loop, BSM radiative corrections is mandatory for reliable phenomenology.
• Reconstruction of the Potential: Accurate reconstruction of the Higgs potential requires precise knowledge of $\lambda_{ijk}$. The tools presented (anyH3, anyHH) provide the necessary infrastructure to map experimental limits back to the fundamental parameters of BSM theories.
• Discovery vs. Exclusion: The dramatic shift in statistical significance ($Z_{hh}$) based on the order of calculation highlights that current experimental limits may be misinterpreted if theoretical predictions lack the required precision. This work establishes a new standard for theoretical inputs in di-Higgs analyses.