Refining two-loop corrections to trilinear Higgs couplings in the Two-Higgs-Doublet Model

This paper presents new calculations of leading two-loop corrections to trilinear Higgs couplings ($\lambda_{hhh}$ and $\lambda_{hhH}$) within the Two-Higgs-Doublet Model, addressing renormalization in the alignment limit and analyzing their phenomenological impact on di-Higgs production at current and future colliders.

The Gist

Imagine the universe is built on a giant, invisible trampoline. In the center of this trampoline sits a heavy ball called the Higgs boson. This ball is special because it gives other particles their "weight" (mass). But to understand how the universe works, physicists need to know exactly how this ball behaves when it bounces, stretches, or interacts with itself.

One of the most important things to measure is how the Higgs ball interacts with two other Higgs balls at the same time. This is called the "trilinear coupling." Think of it like measuring the tension in a spring: if you pull two springs together, how hard do they push back?

The Problem: The "Rough Draft"

For a long time, physicists had a "rough draft" calculation for this tension. They knew the basic rules (the Standard Model), but they realized that in more complex universes (like the Two-Higgs-Doublet Model, or 2HDM), there are extra, invisible springs and balls floating around. These extras create "noise" or "ripples" that mess up the measurement.

In the past, scientists calculated these ripples once (one-loop). But sometimes, the ripples are so huge that the first calculation isn't enough. It's like trying to predict the weather by looking at the sky for one hour; you might miss a storm brewing two hours later. To get a truly accurate prediction, you need to look deeper and calculate the "ripples of the ripples" (two-loop corrections).

The Solution: The "Master Builders"

The authors of this paper, Johannes, Felix, and Alain, are like master builders who went back to the blueprints. They didn't just look at the main Higgs ball; they looked at how it interacts with a "partner" ball (called $H$) in this complex 2HDM universe.

They did two main things:

1. They built a better calculator: They used two different, highly sophisticated mathematical methods (like using both a ruler and a laser measure) to calculate the tiny, second-order ripples. They checked their work against each other to make sure the numbers were perfect.
2. They fixed the "Alignment": Imagine trying to balance a stack of plates. If the stack is slightly tilted, it might look fine at first, but eventually, it falls. In physics, there's a special state called the "alignment limit" where the Higgs behaves exactly like the Standard Model predicts. The authors figured out how to mathematically "re-align" the stack perfectly, even when the tiny ripples try to knock it over. This is a crucial fix because previous calculations sometimes ignored this tilt.

What They Found: The "Surprise"

They tested their new, super-accurate math on two different scenarios:

• Scenario A (The Quiet Neighbor): In one case, the extra particles were heavy and far away. Here, the new math showed that the old "rough draft" was actually pretty good. The extra ripples didn't change much.
• Scenario B (The Chaotic Party): In the second case, the extra particles were closer in weight to the Higgs. Here, the new math revealed a huge difference. The "ripples of the ripples" changed the predicted tension by a significant amount (up to 37%!).

Why Should You Care? The "Collision Course"

Why does this matter? Because physicists are about to smash particles together at the Large Hadron Collider (LHC) and future machines to create pairs of Higgs bosons. It's like firing two cannonballs at each other to see what happens when they collide.

The authors showed that if you use the old, rough math, you might predict the collision will happen in one way. But with their new, precise math, the collision looks different:

• The "shape" of the collision changes.
• The "resonance" (a moment where the collision gets extra loud, like a singer hitting a high note that shatters a glass) shifts slightly.

If the scientists at the LHC use the old math, they might look for the Higgs in the wrong place or miss it entirely. By using the new "two-loop" corrections, they can tune their detectors to the exact right frequency to catch these elusive particles.

The Bottom Line

Think of this paper as upgrading the GPS for a road trip to the edge of the universe.

• Old GPS: "Turn left in 10 miles." (It gets you close, but you might miss the exit).
• New GPS (This Paper): "Turn left in 10.04 miles, accounting for the wind and the curve of the road." (It gets you exactly where you need to be).

By refining these calculations, the authors have given experimentalists a sharper, more reliable map to explore the hidden secrets of the universe, ensuring that when they find new physics, they know exactly what they are looking at.

Technical Summary

1. Problem Statement

The precise determination of Higgs self-couplings, particularly the trilinear coupling $\lambda_{hhh}$, is critical for understanding Electroweak Symmetry Breaking (EWSB) and probing physics Beyond the Standard Model (BSM). In models with extended scalar sectors, such as the Two-Higgs-Doublet Model (2HDM), large radiative corrections can significantly alter these couplings from their Standard Model (SM) values.

While one-loop corrections have been extensively studied, they can be large, raising concerns about the perturbative stability of theoretical predictions. Furthermore, previous calculations of two-loop corrections in the 2HDM (e.g., Refs. [18, 19]) relied on simplifications, such as neglecting deviations from the "alignment limit" at the tree level or using specific renormalization schemes (like $\overline{\text{MS}}$) that complicate the physical interpretation of parameters. Additionally, the trilinear coupling $\lambda_{hhH}$ (involving the heavy CP-even Higgs $H$), which is relevant for Higgs pair production ($gg \to hh$), had not been fully computed at the two-loop level with a consistent on-shell renormalization scheme.

2. Methodology

The authors perform a comprehensive calculation of leading two-loop corrections to the trilinear couplings $\lambda_{hhh}$ and $\lambda_{hhH}$ in the CP-conserving 2HDM.

• Framework: The calculation is performed in the Higgs basis, where the SM-like doublet $\Phi_{SM}$ contains the vacuum expectation value (VEV) $v$, and the second doublet $\Phi_{BSM}$ has a vanishing tree-level VEV ($v_{BSM}=0$). The alignment limit is defined by $\Lambda_6 \to 0$.
• Approximations:
  • External momenta are set to zero (vanishing limit).
  • The light Higgs mass is set to zero ($m_h \to 0$).
  • The calculation is performed in the gaugeless limit (neglecting gauge boson contributions).
  • Subleading contributions from light scalars are neglected.
• Computational Approaches: To ensure robustness, the authors employ two independent methods:
  1. Diagrammatic Approach: Using FeynArts (with a SARAH-generated model file), FeynCalc, and Tarcer to generate and reduce genuine two-loop and subloop diagrams to master integrals.
  2. Effective Potential Approach: Calculating the effective potential up to the leading two-loop level involving BSM scalars and the top quark, then taking field derivatives with respect to $h$ and $H$.
  • Cross-check: Both methods yield identical results.
• Renormalization Scheme: The authors implement a full On-Shell (OS) renormalization scheme, which is a significant improvement over previous $\overline{\text{MS}}$-based calculations.
  • Parameters Renormalized: Tadpoles ($T_h, T_H$), masses ($m_h, m_H, m_A, m_{H^\pm}$), VEVs ($v, v_{BSM}$), and quartic couplings ($M_{22}^2, \Lambda_6, \Lambda_7$).
  • Alignment Condition: A key technical innovation is the definition of the counterterm for $\Lambda_6$ (and consequently the mixing angle $\alpha$) by demanding the cancellation of off-diagonal contributions to the CP-even scalar mass matrix at zero external momentum. This ensures the alignment limit is preserved order-by-order in perturbation theory.
  • $\Lambda_7$ Renormalization: Defined via an OS condition on the $\lambda_{HAA}$ vertex.

3. Key Contributions

• First Full OS Renormalization of Alignment: The paper presents the first calculation of two-loop corrections to trilinear couplings in the 2HDM where the alignment condition is renormalized on-shell up to the two-loop level. This resolves ambiguities regarding the definition of the mixing angle $\alpha$ (or $\Lambda_6$) in previous works.
• Inclusion of $\lambda_{hhH}$: The authors provide the first leading two-loop results for the coupling $\lambda_{hhH}$, which is crucial for resonant Higgs pair production.
• Validation of Perturbative Convergence: By comparing one-loop and two-loop results, the study tests the stability of the perturbative expansion in scenarios with large BSM corrections.
• Phenomenological Implementation: The results are integrated into the public code anyHH to predict differential distributions for Higgs pair production at the (HL-)LHC.

4. Results

The authors analyze two benchmark scenarios with $\tan\beta = 2$ and a heavy Higgs mass scale around 600 GeV:

• Scenario 1 ($M=m_H=600$ GeV, varying $m_A$):
  • In this case, the finite part of the $\Lambda_6$ counterterm has a negligible impact because the dominant contribution is proportional to $(M^2 - m_H^2)$, which vanishes.
  • The two-loop corrections to $\lambda_{hhH}$ become significant for large mass splittings ($m_A - m_H \gtrsim 300$ GeV), causing deviations from the one-loop prediction. These are identified as a new class of contributions rather than a breakdown of perturbation theory.
• Scenario 2 ($M=600$ GeV, varying $m_H=m_A=m_{H^\pm}$):
  • Here, $M \neq m_H$, making the finite part of the $\Lambda_6$ counterterm significant. The proper OS treatment reduces the magnitude of the two-loop corrections to $\kappa_\lambda$ (the modifier of $\lambda_{hhh}$).
  • For $\lambda_{hhH}$, one-loop contributions are large. The two-loop corrections reduce the magnitude of the loop effects, demonstrating perturbative convergence within the allowed parameter space.

Phenomenological Impact on $gg \to hh$:

• Differential Distributions: The inclusion of two-loop corrections does not qualitatively change the shape of the $m_{hh}$ distribution but has a notable quantitative impact.
  • The two-loop corrections increase the BSM deviation in $\kappa_\lambda$, shifting the location of maximal destructive interference (between box and triangle diagrams) to higher $m_{hh}$ values.
  • The resonant peak structure (dip-peak) around $m_H$ is affected, particularly in Scenario 2 where $\lambda_{hhH}$ decreases by $\sim 30\%$ at two loops.
• Total Cross-Sections:
  • Scenario 1: Two-loop corrections reduce the total cross-section by $\sim 20\%$ compared to the one-loop result.
  • Scenario 2: The reduction is even more pronounced, at $\sim 37\%$.
  • These reductions are driven by changes in the interference pattern near the threshold and the modification of the resonant peak.

5. Significance

This work is pivotal for the precision physics program at future colliders (HL-LHC, ILC, CLIC, FCC):

1. Theoretical Reliability: It confirms that despite large one-loop corrections in extended scalar sectors, the perturbative series remains convergent when proper renormalization (specifically OS alignment) is applied.
2. Experimental Interpretation: The inclusion of two-loop corrections is essential for correctly interpreting experimental bounds on $\kappa_\lambda$ and $\lambda_{hhH}$. Ignoring these terms could lead to incorrect exclusions or inclusions of BSM parameter space.
3. Methodological Benchmark: The rigorous treatment of the alignment limit renormalization sets a new standard for calculations in the 2HDM and similar BSM models, ensuring that theoretical predictions match the precision expected from future high-luminosity data.

In conclusion, the paper demonstrates that while perturbative convergence is maintained, the dominant two-loop BSM corrections are numerically significant and must be included to achieve the precision required for future Higgs coupling measurements.