Radiative corrections to decays of the 125 GeV Higgs boson in the complex Higgs triplet model

This paper calculates full one-loop radiative corrections to the decays of the 125 GeV Higgs boson within the complex Higgs triplet model, demonstrating that characteristic deviations in decay rates—such as enhanced $WW^*/ZZ^*$ channels and suppressed $\gamma\gamma$ modes—distinguish this framework from the Standard Model and other extensions, offering testable signatures for future high-luminosity colliders.

The Gist

Imagine the universe as a giant, complex machine. For decades, physicists have had a blueprint for how this machine works called the Standard Model. In 2012, they found a crucial missing gear in this blueprint: the Higgs boson, a particle that gives other particles their mass. This gear weighed in at 125 GeV (a specific unit of energy).

However, the Standard Model blueprint has some gaps. It doesn't explain things like why neutrinos have mass or what dark matter is. To fix these gaps, scientists propose adding new parts to the machine. One popular idea is the Complex Higgs Triplet Model (CHTM).

Think of the Standard Model's Higgs field as a single, simple spring. The CHTM suggests that instead of just one spring, there is a whole spring-loaded toolkit containing not just one, but several new types of springs: some neutral, some singly charged, and some doubly charged (like a battery with two extra terminals).

The Problem: The Blueprint is "Off"

In the Standard Model, the relationship between the W and Z particles (the messengers of the weak nuclear force) is perfectly balanced, like a scale reading exactly 1.0. But in this new "Triplet Toolkit" model, adding those extra springs naturally tips the scale. The balance (called the rho parameter) becomes slightly off from 1.0.

To make this model work with the real world, the scientists have to tune the "triplet spring" to be very weak (a tiny "vacuum expectation value"). But even with this tiny setting, the math gets messy. When you try to calculate how the 125 GeV Higgs boson (the main gear we found) decays or breaks apart, the extra springs create "noise" in the calculations. This noise depends on how you look at the math (a problem called gauge dependence), making the results unreliable.

The Solution: The "Pinch" Technique

The authors of this paper are like master mechanics who decided to fix the blueprint's calculation errors. They developed a new way to do the math using a method called the Pinch Technique.

Imagine you are trying to measure the weight of a heavy box, but the scale is wobbling because of wind (the "gauge dependence"). The Pinch Technique is like building a wind tunnel around the scale and canceling out the wind perfectly. By "pinching" out specific parts of the calculation (like squeezing a tube to stop the air), they removed the wobbling noise. This allowed them to get a clean, stable, and gauge-independent result for the first time in this specific model.

What They Found: The "Signature" of the New Toolkit

Once the math was clean, they calculated how the 125 GeV Higgs boson would decay (break apart) in this new model compared to the old Standard Model. They looked at two main scenarios for the new "triplet" particles:

1. The Heavy Scenario: The doubly charged particles are the heaviest in the group.
2. The Light Scenario: The doubly charged particles are the lightest.

The Key Discovery:
They found that if the new toolkit exists, the Higgs boson would behave in a very specific, unique way that is different from other theories:

• The "Positive Push": In the "Heavy Scenario," the Higgs boson would decay into pairs of W and Z particles (the weak force messengers) more often than the Standard Model predicts. It's like the Higgs is slightly more eager to break into these specific pieces.
• The "Negative Pull": At the same time, the Higgs would decay into two photons (light particles) less often than expected (by about 20%).
• The "Self-Interaction" Explosion: The way the Higgs talks to itself (a property called self-coupling) could change by a massive 100%.

Why This Matters

The paper argues that these specific patterns are like a fingerprint. If future giant microscopes (like the High-Luminosity LHC or future "Higgs factories") measure the Higgs boson and find:

1. A slight increase in W/Z decays,
2. A significant drop in photon decays,
3. And a huge change in how the Higgs interacts with itself,

...then we can confidently say, "Aha! The universe isn't using the simple Standard Model spring; it's using the Complex Triplet Toolkit!"

The authors also built a computer program (an update to their existing H-COUP tool) that allows other scientists to run these precise calculations. They emphasize that while the Standard Model is great, finding these specific deviations would be the "smoking gun" proof that the universe is more complex and colorful than we currently know.

Technical Summary

Technical Summary: Radiative Corrections to Decays of the 125 GeV Higgs Boson in the Complex Higgs Triplet Model

Problem Statement
The Standard Model (SM) successfully describes electroweak physics but fails to explain phenomena such as neutrino masses. The Complex Higgs Triplet Model (CHTM) extends the SM Higgs sector by introducing a complex scalar triplet field with hypercharge $Y=1$. This extension facilitates the Type-II seesaw mechanism for neutrino mass generation. A defining feature of the CHTM is that it predicts the electroweak $\rho$ parameter deviates from unity at the tree level ($\rho \neq 1$), unlike the SM or models with only doublets and singlets. This deviation necessitates a distinct framework for the renormalization of electroweak parameters. While previous studies have addressed the renormalization of the Higgs sector in the CHTM, they lacked a complete set of radiative corrections for the decays of the 125 GeV Higgs boson ($h$), specifically omitting gauge-independent treatments of scalar mixing angles and full one-loop corrections to Yukawa interactions and three-body decays.

Methodology
The authors perform a comprehensive calculation of the full set of radiative corrections to the decays of the SM-like Higgs boson ($h$) within the CHTM. The methodology involves:

1. Renormalization Scheme: The calculation utilizes an on-shell renormalization scheme. To address the issue of gauge dependence in the counterterms of scalar mixing angles (specifically $\alpha$ and $\beta'$), the authors employ the pinch technique. This technique extracts gauge-dependent parts from vertex, box, and external leg corrections in $f\bar{f} \to f\bar{f}$ scattering processes and adds them to the scalar self-energies, rendering the mixing angle counterterms gauge-independent.
2. Input Parameters: The electroweak sector is defined using the $G_F$ input scheme, where the Fermi constant $G_F$ replaces the $W$-boson mass as a primary input to ensure precision. The triplet vacuum expectation value (VEV), $v_\Delta$, is treated as a derived parameter linked to the mixing angle $\beta'$.
3. Calculations:
   • One-Loop EW Corrections: Full one-loop contributions from extra Higgs bosons (neutral, singly charged $H^\pm$, and doubly charged $H^{\pm\pm}$), SM fermions, and gauge bosons are calculated for two-body decays ($h \to f\bar{f}$, $h \to VV$) and three-body decays ($h \to VV^* \to V f\bar{f}$).
   • Higher-Order QCD: Higher-order QCD corrections are implemented for decays into quarks and loop-induced processes ($h \to gg, \gamma\gamma, Z\gamma$).
   • Vertex Functions: Analytic formulae for the renormalized $hf\bar{f}$, $hVV$, and $hVV'$ vertices are derived, including the necessary counterterms and tadpole contributions in the alternative tadpole scheme.
4. Numerical Analysis: The authors scan the CHTM parameter space under theoretical constraints (vacuum stability, tree-level unitarity, perturbativity) and experimental constraints (electroweak precision observables, direct searches for additional Higgs bosons, and Higgs signal strength measurements). They compare the CHTM predictions against the SM, Two-Higgs-Doublet Models (2HDMs), the Higgs Singlet Model (HSM), and the Inert Doublet Model (IDM).

Key Contributions
This work provides several novel contributions to the theoretical understanding of the CHTM:

• Gauge-Invariant Renormalization: For the first time in the CHTM, a gauge-invariant renormalization scheme is constructed by eliminating the gauge dependence of scalar mixing angle counterterms using the pinch technique.
• Analytic Formulae: The paper provides explicit analytic expressions for the renormalized $hf\bar{f}$ vertex function, a component previously unaddressed in full one-loop calculations for this model.
• Full Decay Rates: It presents the first calculation of full one-loop corrections to Higgs boson decay rates in the CHTM, encompassing not only two-body decays but also three-body decays ($h \to VV^*$).
• Implementation: These analytical results have been implemented into the H-COUP program (version 3 and beyond), enabling systematic numerical evaluations and comparisons across different extended Higgs models.

Results
The numerical analysis reveals distinct patterns of deviation from SM predictions, particularly in scenarios where the triplet VEV $v_\Delta$ is of the order of 1–10 GeV:

• Decoupling Behavior: In the decoupling limit (where additional Higgs masses are large), the deviations vanish as expected. However, for lighter spectra, significant deviations emerge.
• Heaviest $H^{\pm\pm}$ Scenario (HS): When the doubly charged Higgs is the heaviest among the extra scalars, the decay rates for $h \to WW^*$ and $h \to ZZ^*$ can be a few percent larger than the SM value. This positive deviation is a characteristic signature distinguishing the CHTM from other models like 2HDMs or HSMs, where such deviations are typically smaller or negative.
• Correlations:
  • In the HS, the positive deviation in $h \to VV^*$ is accompanied by a significant negative deviation in the $h \to \gamma\gamma$ decay rate (up to $\sim -20\%$) and a large positive deviation in the Higgs self-coupling (up to $\sim 100\%$).
  • In the Lightest $H^{\pm\pm}$ Scenario (LS), the $h \to \gamma\gamma$ decay rate can exhibit a positive deviation (up to $\sim 10\%$), which is rare in other extended models.
• Impact of NLO Corrections: Including Next-to-Leading Order (NLO) electroweak corrections generally shifts the predicted deviations in the negative direction by approximately 1% for $h \to VV^*$ and $h \to f\bar{f}$, and by $\sim -10\%$ for $h \to \gamma\gamma$. Despite this shift, the characteristic correlation patterns (e.g., positive $VV^*$ deviations coupled with negative $\gamma\gamma$ deviations in the HS) remain robust and distinguishable from other models.

Significance
The paper claims that these characteristic patterns of deviations serve as a "smoking gun" for identifying the CHTM. The authors argue that the specific correlations between the decay rates of $h \to VV^*$, $h \to \gamma\gamma$, and the Higgs self-coupling allow the CHTM to be distinguished from other extended Higgs models (such as 2HDMs, HSM, and IDM) that might otherwise appear similar at the tree level. These predictions are expected to be testable at the High-Luminosity LHC (HL-LHC) and future Higgs factories (e.g., ILC, CEPC, FCC-ee), where precision measurements of Higgs couplings will reach the necessary sensitivity to detect these few-percent level deviations. The work establishes a rigorous theoretical foundation for interpreting future high-precision Higgs data within the context of the Complex Higgs Triplet Model.