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Implications of portal vector-like lepton on associated Higgs production at a multi-TeV muon collider

This paper investigates a portal vector-like lepton extension of the Standard Model, demonstrating that a multi-TeV muon collider can significantly enhance the associated production of the Higgs boson with a dark photon ( $h\gamma_d$ ) compared to the standard $hZ$ channel, thereby offering a promising avenue to probe dark matter phenomenology, constrain dark photon parameters, and address the muon $g-2$ anomaly.

Original authors: Krishna Tewary, Sanjoy Biswas, Shivam Verma

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Krishna Tewary, Sanjoy Biswas, Shivam Verma

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

Imagine the universe as a giant, complex machine built from a standard set of Lego bricks. Physicists call this the "Standard Model." For decades, this model has worked perfectly, but there are a few loose screws and missing pieces that don't quite fit. One of the biggest mysteries is "Dark Matter"—an invisible substance that holds galaxies together but refuses to show up in our detectors.

This paper proposes a new, slightly larger Lego brick to fix these issues: a Portal Vector-Like Lepton (pVLL). Think of this not just as a new brick, but as a secret tunnel connecting our visible world to a hidden "Dark Sector."

Here is the story of the paper, broken down into simple concepts:

1. The Secret Tunnel (The Portal)

In our world, we have particles like electrons and muons (a heavier cousin of the electron). The paper suggests there is a hidden, heavy "twin" muon that lives in the Dark Sector.

The Analogy: Imagine you have a normal muon (a regular citizen) and a heavy, invisible twin (the pVLL). They are like twins who look similar but have different passports.
The Connection: These twins can swap places or "mix" with each other. This mixing creates a bridge (a portal) that allows our visible muons to talk to the invisible Dark Sector.

2. The Invisible Messenger (The Dark Photon)

The Dark Sector has its own force carrier, called a Dark Photon.

The Analogy: If the Dark Sector is a secret society, the Dark Photon is their messenger bird. It flies around, but it's invisible to us unless it interacts with our world through that secret tunnel (the pVLL).
The Problem: Usually, this tunnel is very narrow. It's hard for our muons to talk to the Dark Photon.
The Twist: The paper discovers a "non-decoupling" effect. This is a fancy way of saying that even if the heavy twin is massive (like a giant), the connection to the Dark Photon doesn't get weak. In fact, the heavier the twin, the stronger the "whisper" between our muon and the Dark Photon becomes.

3. The Big Experiment: The Muon Collider

To test this, the authors imagine a future machine called a Muon Collider.

The Analogy: Think of the Large Hadron Collider (LHC) as a giant, chaotic demolition derby where cars crash into each other. A Muon Collider is like a precision billiard table where two smooth, heavy balls (muons) are smashed together at incredible speeds.
The Goal: Smash these muons together to see what flies out. Usually, we expect to see a Higgs Boson (the "God particle" that gives mass) and a Z-boson (a known force carrier).

4. The Surprise Discovery: Higgs + Invisible Ghost

The paper predicts something exciting: When we smash muons together, we might see a Higgs Boson appear alongside an invisible Dark Photon instead of the usual Z-boson.

The Magic Trick: Because of the "non-decoupling" effect mentioned earlier, the rate at which we see "Higgs + Dark Photon" could be 1 to 100 times higher than the standard "Higgs + Z-boson" rate.
Why it matters: It's like expecting to find a silver coin in a jar, but instead, you find a gold coin appearing 50 times more often than expected. This huge spike in numbers would be a smoking gun for new physics.

5. Solving Two Mysteries at Once

The paper shows this single idea (the pVLL) solves two problems:

Dark Matter: The heavy twin helps Dark Matter particles annihilate (destroy each other) into muons. This explains why we don't see too much Dark Matter in the universe today.
The Muon Mystery: There is a known discrepancy in how muons wiggle (the "g-2" measurement). The heavy twin and the Dark Photon explain this wiggle perfectly.

6. The Detective Work (How to Find It)

Since the Dark Photon is invisible, it leaves no trace in the detector. It just disappears, taking energy with it.

The Clue: The Higgs Boson decays into two bottom quarks, which look like a tight bundle of energy (a "jet").
The Strategy: The researchers simulate the collision and look for a specific pattern:
- A tight bundle of energy (the Higgs).
- A huge amount of "missing energy" (the invisible Dark Photon running away).
- The Filter: They use a "jet substructure" technique. Imagine looking at a bundle of sticks. If it's a normal background event, the sticks are messy. If it's a Higgs, the sticks are perfectly aligned and compressed. This filter helps them ignore the "noise" of the universe and spot the signal.

7. The Verdict

The authors ran the numbers for a future 3 TeV and 10 TeV Muon Collider.

The Result: They found that if this theory is true, the collider could rule out (or find) Dark Photons with masses up to 80 GeV (about 80 times heavier than a proton) with high confidence.
The Bottom Line: Even if the connection between our world and the Dark World is incredibly tiny (a mixing angle of 1 in 100,000), the unique "non-decoupling" nature of this heavy particle makes the signal loud enough to hear.

In summary: This paper suggests that by building a super-precise muon collider, we might finally catch a glimpse of the "Dark Sector" not by seeing the dark matter itself, but by seeing a Higgs Boson dancing with an invisible partner, a dance made possible by a heavy, secret twin particle that links our world to the dark one.

Technical Summary: Implications of Portal Vector-Like Lepton on Associated Higgs Production at a Multi-TeV Muon Collider

Problem Statement
The paper investigates the phenomenological implications of a "portal vector-like lepton" (pVLL) extension of the Standard Model (SM) at future multi-TeV muon collider facilities. Specifically, it addresses the associated production of the Higgs boson with a vector boson ($hV$, where $V = Z$ or a dark photon $\gamma_d$ ). While the $hZ$ production channel is a well-studied SM process, the authors explore how the introduction of a pVLL—a vector-like lepton that acts as a bridge between the visible SM sector and a dark sector—alters these production rates. A primary motivation is to determine if the non-decoupling nature of the pVLL interaction can significantly enhance the $h\gamma_d$ production rate, thereby offering a new probe for dark sector physics and potentially resolving anomalies such as the muon $g-2$ discrepancy and dark matter (DM) relic density issues without relying on large gauge kinetic mixing.

Methodology
The study employs a theoretical framework extending the SM gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$ with an additional abelian dark gauge group $U(1)_D$ . The model includes:

A vector-like lepton (VLL) singlet under $SU(2)_L$ with hypercharge $-1$, which mixes with the second-generation SM charged lepton (muon).
A dark photon ( $\gamma_d$ ) associated with $U(1)_D$ , acquiring mass via a dark scalar field $\Phi_d$ .
A Dirac fermion dark matter candidate ( $\chi$ ) charged under $U(1)_D$ .

The authors analyze the model by:

Theoretical Constraints: Deriving mass matrices and mixing angles ( $\theta_L, \theta_R$ ) for the muon and heavy muon ( $\mu_p$ ). They impose constraints from perturbative unitarity, electroweak precision observables (S and T parameters), lepton flavor universality, Higgs signal strengths, and LHC searches for VLLs.
Dark Matter Phenomenology: Calculating the DM relic density using micrOMEGAs interfaced with a model implemented in FeynRules. They examine how the pVLL enhances the annihilation channel $\chi\bar{\chi} \to \mu^+\mu^-$ via the modified $\mu^+\mu^-\gamma_d$ vertex, allowing for correct relic abundance even with small gauge kinetic mixing ( $\epsilon$ ).
Collider Simulation: Simulating the process $\mu^+\mu^- \to h\gamma_d$ at center-of-mass energies $\sqrt{s} = 3$ TeV and $10 $TeV using *MadGraph5_aMC@NLO*. The analysis focuses on the final state$ b\bar{b} + \text{missing energy}$ (since $\gamma_d$ is treated as invisible for light DM).
Background Suppression: Identifying dominant SM backgrounds ( $b\bar{b}\nu\bar{\nu}$ from $Zh, ZZ, h\nu\bar{\nu}, Z\nu\bar{\nu}$ ) and applying kinematic cuts. Key selection criteria include missing energy thresholds, jet substructure analysis (using MassDropTagger on Cambridge-Aachen jets with $R=1.2$ ) to identify boosted Higgs bosons, and invariant mass reconstruction of the $b\bar{b}$ system.

Key Contributions and Results

Non-Decoupling Effect and Enhanced Production:
The paper demonstrates a crucial non-decoupling effect arising from the model's charge assignments. While the right-handed muon ( $\mu_R$ ) and the heavy muon ( $\mu_{pR}$ ) share identical $SU(2)_L \times U(1)_Y$ charges (preventing off-diagonal $Z$ couplings), the muon is neutral under $U(1)_D$ . Consequently, a rotation to the mass basis induces an off-diagonal $\mu_R - \mu_{pR} - \gamma_d$ coupling.
- Even with a tiny left-handed mixing angle ( $\sin\theta_L \sim 10^{-5}$ ), the right-handed mixing angle $\sin\theta_R$ can be sizable ( $\sim 0.1$ ) for heavy muon masses in the TeV range.
- This leads to a substantial enhancement in the $\mu^+\mu^- \to h\gamma_d$ cross-section. The ratio of the BSM $h\gamma_d$ rate to the SM $hZ$ rate can exceed unity by a factor of 1 to 100 within the perturbative unitarity limit, whereas the $hZ$ rate remains close to the SM prediction.
Dark Matter and Muon $g-2$ Compatibility:
The authors identify regions of parameter space consistent with the observed DM relic abundance ( $\Omega h^2 \approx 0.12$ ) and the muon $g-2$ anomaly.
- The pVLL facilitates DM annihilation into muon pairs via the modified vertex, resolving the issue of overabundant relic density that typically plagues models with small kinetic mixing.
- The model remains consistent with direct detection limits (e.g., LZ, XENONnT) because the contribution to DM-nucleon scattering arises only at the one-loop level via vacuum polarization, unlike tree-level constraints in models with large kinetic mixing.
Collider Analysis and Exclusion Limits:
A detailed analysis of the $b\bar{b} + \text{missing energy}$ channel was performed.
- Signal vs. Background: The signal events are characterized by highly boosted Higgs bosons, resulting in a single collimated jet (in the LQ1CJ channel) with distinct substructure, whereas SM backgrounds often produce multi-jet final states or different missing energy distributions.
- Exclusion Sensitivity: Assuming an integrated luminosity of $1 \text{ ab}^{-1}$ at $\sqrt{s} = 3$ TeV, the study establishes $2\sigma$ exclusion limits. For a heavy muon mass of $\sim 3$ TeV, the analysis can exclude dark photon masses ( $m_{\gamma_d}$ ) up to 80 GeV (for $g_d = 0.05$ ) and 76 GeV (for $g_d = 0.5$ ) in the exactly one central-jet plus missing energy channel.
- At $\sqrt{s} = 10$ TeV with $10 \text{ ab}^{-1}$ , the sensitivity extends to higher dark photon masses and allows for constraints on the dark gauge coupling $g_d$ down to $\sim 0.15$ for $m_{\gamma_d} \le 90$ GeV.

Significance
The paper claims that the $\mu^+\mu^- \to h\gamma_d$ process offers a "promising channel" for probing Higgs interactions and new physics at future muon colliders. Its primary significance lies in:

Independence from Kinetic Mixing: The ability to probe the dark photon mass and gauge coupling is largely independent of the gauge kinetic mixing parameter ( $\epsilon$ ), which is often tightly constrained by low-energy experiments.
Enhanced Sensitivity: The non-decoupling nature of the pVLL interaction allows for significant signal rates even with very small mixing angles ( $\sin\theta_L \sim 10^{-6}$ ), making the channel sensitive to a wide range of parameter space that might be inaccessible to other probes.
Dual Utility: The process serves a dual purpose: it provides a potential explanation for the muon $g-2$ anomaly and DM relic density while simultaneously offering a high-precision test of Higgs physics (potentially improving measurement precision by $\sim 10\%$ compared to the $hZ$ channel) and setting stringent limits on dark sector parameters.

The authors conclude that the Higgs production in association with a dark photon at future muon collider facilities provides a sensitive probe for a large region of the $g_d$ – $m_{\gamma_d}$ and $m_{\mu_p}$ – $m_{\gamma_d}$ parameter planes, effectively addressing the muon $g-2$ and DM relic density puzzles within a unified framework.