⚛️ phenomenology

One-loop contributing for $A \rightarrow Z Z γ$ in the Two-Higgs-Doublet Model with Vector-Like Quarks

This paper presents the first calculation of one-loop decay widths and branching ratios for the process $A \rightarrow ZZ\gamma$ within the Two-Higgs-Doublet Model extended by Vector-Like Quarks, demonstrating that these contributions can reach $10^{-5}-10^{-4}$ and modify total rates by approximately 5%, potentially becoming experimentally accessible at future multi-TeV muon colliders.

Original authors: Dzung Tri Tran, Vo Quoc Phong, Khiem Hong Phan

Published 2026-10-14

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CC BY 4.0

Original authors: Dzung Tri Tran, Vo Quoc Phong, Khiem Hong Phan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is built on a foundation of particles and forces, but for decades, physicists have suspected that the visible world is only a fraction of the whole. At the heart of this mystery lies the Higgs field, an invisible energy that gives mass to other particles. In 2012, scientists confirmed the existence of the Higgs boson, a ripple in this field, which validated the Standard Model of particle physics. However, the Standard Model is incomplete; it cannot explain gravity, dark matter, or why there is more matter than antimatter in the universe. To fill these gaps, theorists have proposed extensions, such as the Two-Higgs-Doublet Model. This idea suggests that instead of just one Higgs field, there are two, creating a richer family of particles. Among these new particles is a heavy, invisible cousin of the known Higgs, called a pseudoscalar. Scientists are eager to find it, but it is elusive, often decaying into other particles so quickly that it leaves no direct trace. To catch it, researchers must look for the rare, specific patterns of energy it leaves behind when it falls apart.

In a recent study, a team of physicists from Vietnam explored one of these rare patterns: the decay of a heavy pseudoscalar particle into two Z bosons and a single photon. The Z boson is a heavy carrier of the weak nuclear force, and the photon is the particle of light. This specific combination of three particles is extremely difficult to produce, making it a pristine laboratory for testing new physics. The researchers calculated how often this event should happen in their proposed models, paying close attention to a hypothetical new type of matter called Vector-Like Quarks. Unlike the quarks that make up protons and neutrons, these new particles would have unique properties that allow them to interact with the Higgs field in ways that standard quarks cannot. By running complex simulations, the team mapped out exactly how these new quarks would influence the decay, calculating the probability of the event occurring under various conditions.

The results of these calculations reveal that while the event is rare, it is not impossible. In the scenarios the team examined, the chance of this decay happening ranges from one in one hundred thousand to one in ten thousand. This frequency is low, but high enough that it could be detected if the right conditions are met. The study also looked at the spin, or polarization, of the Z bosons produced in the decay. They found that the process is heavily dominated by a specific orientation where the Z bosons are aligned in a particular way, while other orientations are so rare they are almost non-existent. This distinction is crucial because it gives experimentalists a clear signal to look for, helping them separate the new physics from the background noise of ordinary particle collisions.

A key finding of the paper is the role of the Vector-Like Quarks. When these hypothetical particles are included in the calculations, they change the total rate of the decay by about five percent. While five percent might sound small, in the world of high-energy physics, where measurements are often precise to the fraction of a percent, this is a significant shift. It suggests that if these new quarks exist, they would leave a measurable fingerprint on the decay process. The researchers noted that this effect is most pronounced when the new quarks are relatively light, but even as they get heavier, their influence does not vanish completely. Instead, they subtly alter the balance of the decay, offering a way to probe their existence even if they are too heavy to be created directly in current particle accelerators.

The study also examined how the likelihood of this decay changes as the mass of the heavy pseudoscalar particle varies. They found that the probability peaks when the mass of the decaying particle is around 600 gigaelectronvolts, a specific energy scale that is within the reach of future experiments. At this mass, the branching ratio—the fraction of particles that take this specific decay path—reaches its highest point. As the mass increases further, the probability drops slightly, but the signal remains strong enough to be of interest. The team also looked at how the energy of the resulting particles is distributed. They discovered that the energy spectrum shows distinct peaks corresponding to the masses of other known particles in the model, acting like a signature that confirms the underlying theory.

Looking ahead, the researchers point to the potential of future particle colliders, specifically multi-teraelectronvolt muon colliders. These proposed machines would collide muons, which are heavier cousins of electrons, at energies far beyond what is currently possible. The high intensity of these future collisions would generate enough data to spot the rare events the team calculated. If such a collider is built, the specific decay pattern of a heavy pseudoscalar into two Z bosons and a photon could serve as a smoking gun for the existence of Vector-Like Quarks and the extended Higgs sector. The work provides a clear roadmap for what to look for, transforming a theoretical possibility into a concrete target for experimental verification.

Ultimately, this research bridges the gap between abstract theory and experimental reality. By calculating the precise rates and patterns of a rare decay, the team has provided a tool for physicists to test the limits of our current understanding of the universe. The presence of Vector-Like Quarks, if confirmed, would not only solve the mystery of the heavy pseudoscalar's behavior but also open a new window into the fundamental structure of matter. The findings suggest that the universe may be more complex than the Standard Model allows, hiding new particles in the subtle deviations of rare events. As technology advances, the search for these deviations will continue, with this study offering a precise guide for the next generation of discovery.

Technical Summary: One-Loop Contributions to $A \to ZZ\gamma$ in the 2HDM with Vector-Like Quarks

Problem Statement
The nature of the Higgs sector remains an open question in particle physics, motivating the study of Beyond the Standard Model (BSM) scenarios such as the Two-Higgs-Doublet Model (2HDM). While the decay of a CP-odd Higgs boson ( $A$ ) into various final states has been investigated, the specific one-loop induced decay process $A \to ZZ\gamma$ has not been systematically studied within the 2HDM or its extensions. This paper addresses the calculation of this rare decay channel, specifically examining how the inclusion of Vector-Like Quarks (VLQs) propagating in the loop modifies the decay rates and branching ratios. The study focuses on the Type-II 2HDM, a framework relevant to supersymmetric extensions like the MSSM, extended to include VLQs in various gauge-covariant multiplet representations.

Methodology
The authors perform a complete one-loop calculation of the decay amplitude for $A \to ZZ\gamma$ using the 't Hooft–Feynman gauge. The methodology involves:

Model Framework: The study utilizes the CP-conserving Type-II 2HDM extended with Vector-Like Quarks. The scalar potential is defined by seven independent parameters ( $M_h, M_H, M_A, M_{H^\pm}, \tan\beta, \sin(\beta-\alpha), m_{12}^2$ ). The VLQ sector is analyzed primarily through the doublet $XT$ scenario, though singlet and triplet representations are discussed. The mixing between SM quarks and VLQs is parameterized by mixing angles ( $\theta_{L,R}$ ) and phases, with specific constraints applied to ensure consistency with flavor physics and electroweak precision observables (EWPOs).
Diagrammatic Analysis: The calculation includes all one-loop Feynman diagrams contributing to the process, categorized into three topologies:
1. Triangle and Self-Energy Diagrams: These involve reducible contributions from intermediate scalar poles ( $\phi^* \to Z\gamma$ , where $\phi = h, H$ ) and box diagrams. The loops contain charged fermions (SM quarks and VLQs), $W^\pm$ gauge bosons, charged Goldstone bosons ( $G^\pm$ ), ghosts ( $u^\pm$ ), and charged Higgs bosons ( $H^\pm$ ).
2. Box Diagrams: These involve four external legs. Due to CP invariance in the scalar sector, contributions from gauge and charged Higgs bosons in the box loops vanish; thus, the dominant box contributions arise from the heaviest charged fermions (top quark and VLQs).
Computational Tools: The calculation employs the Passarino-Veltman (PV) reduction scheme to decompose tensor integrals into scalar functions ( $A_0, B_0, C_0, D_0$ ). The authors utilize a suite of computational tools: FeynArts for diagram generation, FeynCalc and Package-X for algebraic manipulation and UV divergence cancellation, LoopTools and Collier for numerical evaluation of loop functions, and Vegas for multidimensional phase-space integration.
Renormalization and Gauge Invariance: The calculation is performed within the on-shell renormalization scheme for gauge and fermion sectors and an improved on-shell scheme for the scalar sector. Gauge invariance is verified via the Ward identity ( $k_3^\mu \mathcal{M}_{\alpha\beta\mu} = 0$ ), ensuring the cancellation of ultraviolet divergences.

Key Contributions

First Systematic Study: This work presents the first systematic calculation of the one-loop decay $A \to ZZ\gamma$ in the 2HDM and its extension with VLQs.
Polarization Analysis: The authors calculate partial decay widths and branching ratios for both unpolarized final states and specific polarization configurations of the $Z$ bosons (longitudinal $\lambda=0$ and transverse $\lambda=L, R$ ).
VLQ Phenomenology: The paper quantifies the specific impact of VLQs (specifically the $XT$ doublet scenario) on the decay rates, distinguishing between light-light, heavy-heavy, and light-heavy interaction contributions.
Kinematic Regimes: The analysis covers both kinematic regimes where the CP-odd Higgs mass $M_A$ is above the threshold for on-shell intermediate scalars ( $M_A \ge M_\phi + M_Z$ ) and below it ( $M_A < M_\phi + M_Z$ ), utilizing the Narrow Width Approximation (NWA) where applicable.

Results

Branching Ratios: Within the viable parameter space of the 2HDM Type-II, the branching ratios for $A \to ZZγ$ are found to range from $10^{-5}$ to $10^{-4}$ . The rates peak at $M_A \approx 600$ GeV and decrease for higher masses due to the behavior of the total decay width.
VLQ Impact: The inclusion of Vector-Like Quarks modifies the total decay rates by approximately 5%. This enhancement is most pronounced at lower $M_A$ and decreases as the VLQ mass ( $m_T$ ) increases, consistent with decoupling behavior. The effect is slightly more significant for lower values of $\tan\beta$ ( $\tan\beta=2$ ) compared to higher values ( $\tan\beta=6$ ).
Polarization Dependence: The longitudinal polarization mode ( $\lambda_1 = \lambda_2 = 0$ ) dominates the decay, yielding branching ratios roughly an order of magnitude larger than mixed modes and many orders of magnitude larger than purely transverse modes. The transverse modes are heavily suppressed ( $O(10^{-14})$ to $O(10^{-11})$ ).
Invariant Mass Distributions: The differential decay rates with respect to the invariant mass of the $Z\gamma$ system ( $m_{Z\gamma}$ ) exhibit distinct features, including peaks corresponding to the light Higgs pole ( $M_h \approx 125$ GeV) and the heavy Higgs resonance ( $M_H \approx 500$ GeV). The inclusion of VLQs shifts the differential rates upward across the energy spectrum without altering the fundamental distribution shapes.

Significance and Claims
The paper claims that the $A \to ZZ\gamma$ channel serves as a complementary probe for extended Higgs sectors and the presence of Vector-Like Quarks. While the branching ratios are small, the authors suggest that with the high integrated luminosities expected at future multi-TeV muon colliders, these loop-induced effects could become experimentally accessible. The work provides a theoretical baseline for searching for these rare decays and quantifies the potential sensitivity to new physics in the form of VLQs, noting that current constraints from flavor physics and EWPOs allow for the parameter space explored. The authors do not claim immediate experimental discovery but rather establish the theoretical feasibility and magnitude of the effect for future high-energy facilities.

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