⚛️ phenomenology

CP violation in H^\pm \to W^\pm Z: A physical approach for the 2HDM

This paper investigates CP violation in the $H^\pm \to W^\pm Z$ decay within the two-Higgs-Doublet model by expressing amplitudes in physical couplings, confirming previous interference results while identifying additional CP-violating sources from internal bosonic and fermionic loop interferences that manifest as charge asymmetry in inclusive decays.

Original authors: Wafaa Khater, Odd Magne Ogreid, Per Osland, Margarida Nesbitt Rebelo

Published 2026-02-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wafaa Khater, Odd Magne Ogreid, Per Osland, Margarida Nesbitt Rebelo

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 Big Picture: A Dance of Particles

Imagine the universe is a giant ballroom. In the center of the room, there is a special dancer called the Charged Higgs ( $H^\pm$ ). This dancer is unstable and wants to leave the floor. When they do, they split into two other dancers: a W boson and a Z boson.

The main question this paper asks is: Does the universe treat the "positive" dancer ( $H^+$ ) exactly the same as the "negative" dancer ( $H^-$ )?

In a perfectly symmetrical world, they would perform the exact same dance moves and leave the floor at the exact same speed. But if the universe has a hidden "handedness" (called CP Violation), the positive dancer might spin slightly faster or slower than the negative one. This paper investigates exactly how and why that might happen.

The Setting: The Two-Higgs-Doublet Model (2HDM)

The Standard Model of physics is like a basic recipe with one type of flour. But this paper explores a more complex recipe called the Two-Higgs-Doublet Model (2HDM), which has two types of flour (two Higgs fields). This extra ingredient opens the door for new, weird interactions that could break the symmetry between the positive and negative dancers.

The Mechanism: How the Dance Happens

When the Charged Higgs decays, it doesn't just vanish instantly. It goes through a "loop" process. Think of this like a relay race where the baton is passed through a series of invisible runners before reaching the finish line.

The paper breaks these runners into two teams:

The Boson Team: These are runners made of force-carrying particles (like other Higgs bosons, Ws, and Zs).
The Fermion Team: These are runners made of matter particles (like top quarks, bottom quarks, and tau leptons).

The paper calculates the "amplitude" (the strength and direction) of the dance for both teams.

The Twist: The "Phase" and the "Interference"

To get a difference between the positive and negative dancers, the math needs to get "complex" (in the mathematical sense, involving imaginary numbers). This happens when the energy of the decaying particle is high enough to create a "real" loop of particles inside the dance.

The paper finds three ways the symmetry can be broken:

Boson vs. Boson Interference: Sometimes, two different "Boson Team" runners interfere with each other. If they have different "phases" (like two waves crashing at slightly different times), they can create a ripple that makes the positive dancer behave differently than the negative one.
- Analogy: Imagine two drummers playing the same beat. If one is slightly out of sync, the rhythm changes. If the universe is "handed," the positive dancer hears a different rhythm than the negative one.
Fermion vs. Fermion Interference: Similarly, the "Fermion Team" runners can interfere with each other. However, the paper notes this effect is usually very small because the heavy top quark dominates, and the lighter particles (like the bottom quark or tau lepton) are too weak to make a big splash on their own.
The Big Clash (Boson vs. Fermion): The most interesting part is when the Boson Team and the Fermion Team dance together. They interfere with each other. The paper confirms previous findings that this clash creates a charge asymmetry.

The "Alignment" Limit: When Things Get Quiet

There is a special scenario called the "Alignment Limit." This is when the new, heavy Higgs particles behave almost exactly like the one Higgs particle we already know from the Standard Model.

The Paper's Finding: If we are in this "Alignment Limit" and we ignore the Fermion Team (the matter particles), the dance becomes perfectly symmetrical again. The positive and negative dancers move at the exact same speed.
The Catch: This only happens if the other invisible particles in the loop are too heavy to be created. If they are light enough, or if we include the Fermion Team (the quarks and leptons), the symmetry breaks, and the asymmetry returns.

The "Other Channels" Rule (CPT Theorem)

The paper mentions a fundamental law of physics called the CPT Theorem. It says that if you add up every single way a positive particle can decay, it must equal the total ways a negative particle can decay.

The Metaphor: If the positive dancer leaves the ballroom slightly faster through the front door ( $W^\pm Z$ ), they must leave slightly slower through the back door (other decay channels) to balance the books.
The paper shows that if you block the "back doors" (by making the other particles too heavy to exit), the asymmetry in the front door disappears. This proves that the asymmetry isn't magic; it's just the particles shifting their energy to different exit routes.

Summary of the Authors' Claims

New Discovery: While previous studies focused on the clash between Bosons and Fermions, this paper highlights that Bosons can also fight among themselves to create asymmetry, and Fermions can fight among themselves (though this is a small effect).
The "Alignment" Caveat: In the specific case where the new physics looks very much like the Standard Model (Alignment), the Boson-only contribution to the asymmetry vanishes unless the other particles are light enough to create complex loops.
General Applicability: The results apply to any version of the 2HDM, whether the CP violation is "built-in" (explicit) or "spontaneous" (arising from how the vacuum settles).

In short: The paper maps out the complex choreography of a decaying particle, showing that the universe can be "handed" in multiple ways—through the interference of force particles, matter particles, or a mix of both—creating a measurable difference between matter and antimatter decays.

Technical Summary: CP Violation in $H^\pm \to W^\pm Z$ within the 2HDM

Problem Statement
The paper investigates the process of charged Higgs decay, $H^\pm \to W^\pm Z$ , within the framework of the Two-Higgs-Doublet Model (2HDM). While the tree-level coupling for this vertex vanishes in models with identical scalar representations (a consequence of custodial symmetry considerations and gauge invariance), the process occurs at the one-loop level. The central problem addressed is the quantification of Charge-Parity (CP) violation in this decay, manifested as a charge asymmetry $\delta$ between the decay rates of $H^+$ and $H^-$ . The authors aim to express these amplitudes transparently in terms of physical couplings and masses, rather than unphysical potential parameters, to provide a general understanding of CP violation mechanisms in extended scalar sectors.

Methodology
The authors perform a comprehensive one-loop calculation of the decay amplitude $\mathcal{M}(H^\pm \to W^\pm Z)$ . The methodology involves:

Physical Coupling Formalism: Instead of using Lagrangian parameters, the analysis utilizes physical couplings ( $e_i$ , $f_i$ , $q_i$ ) and Yukawa couplings ( $\tilde{\rho}$ ). The $e_i$ factors parametrize neutral scalar couplings to gauge bosons ( $H_i VV$ ), while $f_i$ parametrize the trilinear couplings between neutral and charged scalars ( $H_i H^\pm W^\mp$ ).
Amplitude Decomposition: The total amplitude is split into bosonic ( $\mathcal{M}_{boson}$ ) and fermionic ( $\mathcal{M}_{fermion}$ ) parts. The bosonic part includes pure boson loops and fermion tadpoles (due to their dependence on $f_i$ ), while the fermionic part includes quark and lepton loops.
Lorentz Structure: The amplitude is decomposed into form factors $F$ , $G$ , and $H$ . The $H$ term, representing parity violation, arises exclusively from fermionic triangle diagrams.
CP Violation Mechanism: CP violation requires complex interference. The authors identify that the overall amplitude must be complex not just due to overall coefficients, but because specific loop integrals develop imaginary parts. This occurs when the invariant mass of the decaying particle ( $m_{H^\pm}$ ) exceeds the sum of the masses of the particles in the loop.
Custodial Symmetry Analysis: The paper examines the implications of custodial symmetry (CS) on the potential, distinguishing between cases where CS enforces CP conservation and cases where a CS-symmetric potential allows for spontaneous CP violation.

Key Contributions and Results

Identification of New CP Violation Sources: While previous work (Kanemura and Mura) focused on the interference between bosonic and fermionic amplitudes, this study identifies additional sources of CP violation:
- Purely Bosonic Sector: CP violation arising from the interference among different bosonic loop diagrams (triangles, bubbles, and tadpoles).
- Purely Fermionic Sector: CP violation arising from the interference among different fermionic loop diagrams (quark and lepton sectors).
The Alignment Limit: In the alignment limit (where the lightest neutral scalar $H_1$ $H_{1}$ mimics the Standard Model Higgs, $e_1 \to v$ $e_{1} \to v$ ), the authors demonstrate that:
- If $m_{H_2} + m_{H_3} > m_Z$ , the purely bosonic loop integrals remain real, and thus, in the absence of fermionic contributions, the charge asymmetry vanishes.
- However, a charge asymmetry persists in the alignment limit if fermionic loop contributions (specifically top and bottom quark loops) are included, provided there are non-trivial relative phases in the Yukawa couplings ( $\tilde{\rho}$ ).
Role of Yukawa Couplings: The analysis covers generic Yukawa structures, allowing for both explicit and spontaneous CP violation. The asymmetry is shown to depend on the relative phases of the Yukawa couplings for up-type quarks ( $\tilde{\rho}_t$ ), down-type quarks ( $\tilde{\rho}_b$ ), and leptons ( $\tilde{\rho}_\tau$ ), as well as the relative phase between the bosonic and fermionic sectors ( $\theta_{BF}$ ).
CPT Theorem Consistency: The paper confirms that the observed charge asymmetry in $H^\pm \to W^\pm Z$ is balanced by asymmetries in other decay channels (e.g., $H^\pm \to W^\pm H_i$ , $H^\pm \to t\bar{b}$ , $H^\pm \to W^\pm \gamma$ ) to satisfy the CPT theorem, which requires total decay widths $\Gamma(H^+) = \Gamma(H^-)$ . Numerical tests confirm that if kinematically forbidden channels are artificially removed, the asymmetry vanishes.
Numerical Findings:
- Fermion tadpole contributions often dominate the decay rate but can interfere destructively with pure fermion loops.
- The asymmetry is sensitive to the "alignment deficit" (how $e_2$ and $e_3$ share the deviation from $e_1=v$ ) and the signs of the couplings $q_i$ .
- Mass degeneracy (e.g., $m_{H^\pm} = m_3$ ) does not necessarily eliminate CP violation, as other non-degenerate loops (e.g., involving $H_1$ or $H_2$ ) can still generate complex amplitudes.

Significance and Claims
The paper claims to provide a "physical approach" that clarifies the origin of CP violation in this specific decay channel. Its primary significance lies in:

Completing the Picture: It extends previous analyses by explicitly demonstrating that CP violation can arise from purely bosonic and purely fermionic interferences, not just the boson-fermion interference previously highlighted.
General Applicability: The results are formulated for generic Yukawa couplings and cover both explicit and spontaneous CP violation scenarios, making the findings applicable to a broad class of 2HDMs.
Alignment Limit Nuance: It clarifies that while the alignment limit suppresses certain CP-violating effects in the bosonic sector (under specific mass conditions), it does not eliminate the possibility of observing a charge asymmetry entirely, provided fermionic contributions are present.
Theoretical Consistency: The work reinforces the necessity of considering the full set of decay channels to satisfy CPT invariance, linking the $H^\pm \to W^\pm Z$ asymmetry to the broader phenomenology of the charged Higgs sector.

The authors conclude that such a charge asymmetry could be a generic feature of any CP-nonconserving multi-Higgs-doublet model and provide Mathematica files containing the analytic results for further exploration.