Chasing the two-Higgs-doublet model via electroweak corrections at $e^+e^-$ colliders

This paper demonstrates that precision measurements of Higgs boson production with neutrino pairs at future $e^+e^-$ colliders, incorporating next-to-leading-order electroweak corrections, can reveal two-Higgs-doublet model effects even in the Higgs alignment limit.

The Gist

Imagine the universe is a giant, complex machine, and for decades, scientists have been trying to figure out how it works. The "Standard Model" is the instruction manual they've been using. It's been incredibly successful, but we know it's missing some pages. It doesn't explain things like dark matter or why the universe is made of stuff instead of nothing.

To find the missing pages, scientists are looking for "New Physics." One popular idea for these missing pages is the Two-Higgs-Doublet Model (2HDM). Think of the Standard Model as having one "Higgs field" (a kind of invisible energy field that gives particles mass). The 2HDM suggests there are actually two of these fields dancing together.

The Problem: The "Perfect Imposter"

Here's the tricky part: In many versions of this 2HDM, the extra Higgs fields hide so well that the one we already discovered (the 125 GeV Higgs) looks exactly like the one predicted by the old Standard Model. It's like an imposter who has memorized your voice, your walk, and your handwriting perfectly. If you just look at the surface (what scientists call "Leading Order" or tree-level calculations), you can't tell the difference.

The Solution: Listening for the "Static"

This paper is about a clever way to catch this imposter. The authors suggest we don't just look at the main picture; we need to listen for the "static" or the background noise.

In physics, when particles smash together, there are tiny, subtle corrections to the math called Electroweak Corrections. These are like the tiny ripples on a pond after a stone is thrown, or the slight distortion in a phone call when the signal is weak.

The authors calculated these tiny ripples for a specific event: smashing electrons and positrons together to create a Higgs boson and a pair of invisible neutrinos ($e^+e^- \to h \nu\bar{\nu}$).

The Analogy: The Orchestra vs. The Soloist

Imagine you are at a concert.

• The Standard Model is a solo violinist playing a perfect note.
• The 2HDM is that same violinist, but they are secretly accompanied by a second, invisible violinist (the extra Higgs fields).
• Leading Order (LO) is just hearing the main melody. If the second violinist is playing in perfect harmony, you can't tell they are there.
• Next-to-Leading Order (NLO) is listening to the entire acoustic environment. The authors found that even if the second violinist is trying to hide, the way the sound waves bounce off the walls (the quantum corrections) creates a unique "echo" or "reverberation" that is slightly different from a solo performance.

What They Found

The researchers used powerful computer simulations (like a super-advanced video game engine for particle physics) to predict what would happen at future particle colliders (like the FCC-ee or ILC).

1. The "Alignment" Trap: Even in the scenario where the extra Higgs fields are perfectly hidden (the "alignment limit"), the tiny quantum ripples (NLO corrections) revealed a difference.
2. The Size of the Clue: They found that the presence of these extra fields changes the probability of this event happening by about 2% to 7%.
   • To put that in perspective: If you were betting on a coin flip, and someone told you the odds were actually 48% or 52% instead of 50%, that's a huge difference in the world of high-precision physics.
3. Energy Matters: They found that running the collider at higher energies (like 550 GeV) makes these "echoes" even clearer, acting like turning up the volume on the distortion.

Why This Matters

Currently, the Large Hadron Collider (LHC) is like a sledgehammer; it smashes things apart to see what's inside. It's great for finding big, heavy new particles. But if the new particles are light or very well-hidden, the sledgehammer might miss them.

Future colliders ($e^+e^-$ machines) are like precision scalpels. They don't just smash; they measure with extreme accuracy. This paper shows that even if the "new physics" is hiding perfectly in the main data, it will leak out through these tiny quantum corrections.

In short: The authors proved that by measuring the "static" in the signal with extreme precision, we can detect the presence of a second Higgs field even if it's trying to be invisible. It's a new window into the universe that doesn't require finding a new heavy particle, but rather noticing that the old one behaves just a tiny bit differently than we thought.

Technical Summary

1. Problem Statement

While the Standard Model (SM) Higgs boson has been discovered, fundamental issues such as the origin of baryon asymmetry and dark matter remain unsolved, necessitating physics Beyond the Standard Model (BSM). The Two-Higgs-Doublet Model (2HDM) is a minimal and well-motivated extension of the SM.

• The Challenge: In the "alignment limit" (where the discovered 125 GeV Higgs boson behaves exactly like the SM Higgs at tree level), direct searches for new scalars or deviations in Higgs couplings often yield null results or are suppressed.
• The Gap: While the on-shell Higgsstrahlung process ($e^+e^- \to Zh$) has been extensively studied at Next-to-Leading Order (NLO) in BSM theories, the off-shell single Higgs production process ($e^+e^- \to h\nu\bar{\nu}$) has not received a complete NLO electroweak (EW) study in BSM contexts.
• The Goal: To determine if precision measurements of $e^+e^- \to h\nu\bar{\nu}$ at future colliders (FCC-ee, CEPC, ILC, CLIC) can probe 2HDM effects, specifically in the alignment limit, by including NLO EW corrections.

2. Methodology

The authors performed a comprehensive NLO EW calculation for the process $e^+e^- \to h\nu\bar{\nu}$ in both the SM and the 2HDM.

• Theoretical Framework:

  • Model: A CP-conserving 2HDM with a softly broken $Z_2$ symmetry. The study covers all four Yukawa types (Type I, II, X, and Y).
  • Parameters: The extended Higgs sector is defined by mixing angles ($\alpha, \beta$), the alignment parameter ($c_{\beta\alpha} = \cos(\beta-\alpha)$), the ratio of vacuum expectation values ($t_\beta = \tan\beta$), the quartic coupling $\lambda_5$, and the masses of the additional scalars ($m_H, m_A, m_{H^\pm}$).
  • Renormalization: The authors employed on-shell renormalization schemes for mixing angles (using the background-field approach to estimate scheme uncertainties) and the $\overline{\text{MS}}$ scheme for $\lambda_5$.

• Computational Tools:

  • Generator: The automated NLO framework of Whizard (v2.8/3.0) was used for event generation and integration.
  • Amplitude Providers: Interfaces to OpenLoops2 (for SM) and Recola2 (for 2HDM) were utilized to compute one-loop amplitudes.
  • Infrared Regularization: Massive electron beams were used to regulate initial-state collinear singularities, while soft singularities were handled via the Frixione-Kunszt-Signer subtraction method.
  • Validation: The SM results were validated against Ref. [37] (agreement at 0.2% level at $\sqrt{s}=500$ GeV), and 2HDM results were cross-checked with HAWK and Ref. [45] (agreement at the permille level).

• Parameter Scan:

  • Approximately 100,000 parameter points were generated using ScannerS and HiggsTools, constrained by theoretical limits (perturbative unitarity, vacuum stability) and experimental data (LEP, LHC, flavor physics).
  • The study focused on center-of-mass energies $\sqrt{s} = 365$ GeV and $550$ GeV, where the $WW$-fusion channel dominates.

3. Key Contributions

• First Complete NLO EW Study: This is the first full NLO EW calculation for $e^+e^- \to h\nu\bar{\nu}$ in the 2HDM, filling a gap in the literature where only SM calculations or partial SUSY corrections existed.
• Demonstration of Alignment Limit Sensitivity: The paper proves that even when the tree-level Higgs couplings are indistinguishable from the SM (alignment limit, $c_{\beta\alpha} \approx 0$), NLO EW corrections involving new particles induce observable deviations.
• Differential Analysis: The study utilizes differential cross-sections to disentangle the $Zh$ (Higgsstrahlung) and $WW$-fusion contributions, showing that the $WW$-fusion channel is particularly sensitive to 2HDM effects at high energies.

4. Key Results

• Total Cross Section Deviations:

  • At Leading Order (LO), the deviation between the 2HDM and SM is negligible (permille level) and vanishes in the alignment limit.
  • At NLO EW, significant deviations appear.
    • General 2HDM: Deviations reach $-6\%$ to $-7\%$ for $|c_{\beta\alpha}| < 0.1$.
    • Alignment Limit: Even with $c_{\beta\alpha} = 0$, deviations reach $-1.7\%$ to $-2.1\%$ (depending on energy and $\lambda_5$).
  • Energy Dependence: The effect is slightly more pronounced at $\sqrt{s} = 550$ GeV compared to 365 GeV. The $WW$-fusion dominance at higher energies enhances sensitivity.

• Differential Distributions:

  • The ratio of 2HDM to SM cross sections is relatively flat across the Higgs momentum spectrum but shows distinct behavior near the $Zh$ resonance peak.
  • The separation of channels allows for simultaneous probing of different new physics mechanisms.

• Parameter Dependence:

  • The deviations are not driven solely by the mixing angle $c_{\beta\alpha}$ or the self-coupling $\lambda_5$.
  • Large deviations in the "small $|\lambda_5|$" branch are driven by a combination of mixing angles, top-Yukawa corrections, and mass effects ($m_\phi$), highlighting the complexity of UV-complete theories that cannot be easily captured by Effective Field Theory (EFT) alone.

• Experimental Feasibility:

  • Future $e^+e^-$ colliders aim for cross-section measurement uncertainties of 0.3% – 0.6%.
  • Since the predicted NLO deviations ($\sim 2\%$ in alignment, up to $7\%$ otherwise) exceed these experimental uncertainties, the effect is experimentally observable.
  • Theoretical uncertainties (renormalization scheme dependence) are estimated at $\sim 0.7-0.8\%$, which is sub-dominant to the expected experimental precision.

5. Significance

• Complementarity to LHC: While the LHC struggles to probe 2HDM in the alignment limit due to suppressed production rates and model-dependent Yukawa couplings, $e^+e^-$ colliders offer a unique "indirect" probe. The $e^+e^- \to h\nu\bar{\nu}$ process remains sensitive to new physics even when direct production of heavy scalars is kinematically forbidden or suppressed.
• NLO as a Discovery Tool: The study underscores that NLO corrections are not just refinements but essential discovery tools in BSM physics. Relying solely on LO predictions would lead to the false conclusion that the 2HDM is indistinguishable from the SM in the alignment limit.
• Validation of Precision Physics: The results validate the strategy of future Higgs factories to search for new physics through high-precision measurements of Higgs production rates, specifically leveraging the $WW$-fusion channel at high energies.

In conclusion, the paper establishes that precision measurements of $e^+e^- \to h\nu\bar{\nu}$ at future colliders, when analyzed with full NLO EW accuracy, provide a powerful and necessary window to detect 2HDM physics, even in scenarios where the Higgs boson appears Standard-Model-like.