New Avenues of Heavy Neutral Lepton at Muon Collider

Imagine the universe as a giant, complex puzzle. For decades, physicists have been trying to figure out why neutrinos—tiny, ghost-like particles that pass through everything—have such incredibly small masses. The leading theory to explain this is called the "seesaw mechanism." Think of it like a playground seesaw: if one side (the heavy particles) is very heavy, the other side (the light neutrinos we see) must be very light to balance it out.

This paper proposes a new way to find those heavy, hidden particles (called Heavy Neutral Leptons, or N) using a future machine called a Muon Collider.

Here is a breakdown of their ideas using simple analogies:

1. The Machine: A Muon Collider as a "Boson Factory"

Usually, particle colliders smash two particles together head-on, like two cars crashing. But the authors suggest that at very high energies, a Muon Collider acts differently. Because muons are unstable and emit energy easily, they act like a factory that shoots out a stream of "force carriers" (particles called Z' bosons) before they even collide.

Think of it like this: Instead of two people throwing rocks at each other, imagine they are standing on a windy cliff. The wind (the initial state radiation) blows so hard that it creates a storm of invisible "wind particles" (the Z' bosons). These wind particles then crash into each other to create new things. This is called Vector Boson Fusion.

2. The New Players: The "Z'" and the "Heavy Higgs"

The paper studies a specific theory where there is a new force (like a hidden version of electromagnetism) carried by a new particle called Z'. This theory also predicts a new, heavy version of the famous Higgs boson, called H.

The Z' Boson: A new messenger particle that only talks to muons and tau particles (not electrons or protons), making it hard to find with current machines.
The Heavy Higgs (H): A heavy cousin of the standard Higgs boson.

3. The Two Ways to Find the Hidden Particle (N)

The authors propose two different "paths" to create the heavy neutral lepton (N) using these wind particles (Z' bosons):

Path A: The "Resonance" Route (With the Heavy Higgs)
Imagine two Z' particles collide and briefly merge to form a heavy Higgs particle (H), which then instantly splits into two heavy neutral leptons (N).

The Analogy: Two people (Z') throw a ball at a trampoline (H). The trampoline bounces the ball and splits it into two new balls (N).
Why it's special: Usually, this process is very weak because the Higgs bosons don't like to mix. But in this specific theory, the new Z' and the new Higgs are best friends; they interact strongly without needing to "mix" awkwardly. This makes the process much more likely to happen.

Path B: The "Direct" Route (Without the Heavy Higgs)
What if the Heavy Higgs is too heavy to be created? The authors say we can still find the N particles. The two Z' particles can swap a heavy neutral lepton back and forth (like a game of catch) to create a pair of N particles directly.

The Analogy: Even if the trampoline is too heavy to jump on, the two people can still throw a ball back and forth so hard that it creates two new balls out of thin air.

4. The "Smoking Gun" Signature

How do we know we found these invisible N particles? They decay (break apart) into other particles.

The N particles turn into a muon (a heavy electron) and a pair of jets (sprays of particles from a W boson).
Because the N particle is its own antiparticle (a Majorana particle), it can decay into a muon with a positive charge or a negative charge with equal probability.
The Signature: If we create two N particles, there is a chance they both decay into muons with the same charge (e.g., two positive muons).
The Analogy: Imagine a magic factory that usually makes red and blue balls. If you see two red balls come out at the same time, you know something strange happened, because the factory shouldn't make two reds together. This "same-sign" muon signal is a clear sign of "Lepton Number Violation," proving new physics is at work.

5. The Results: What the Simulations Show

The authors ran computer simulations for three different sizes of Muon Colliders: 3 TeV, 10 TeV, and 30 TeV.

The "Fat-Jet" Trick: The two particles from the N decay are often so energetic that they smash together and look like one big blob of energy, called a "fat-jet." The researchers treat this blob as a single object to make counting easier.
Counting Muons: The number of muons they can actually see in the detector depends on how heavy the particles are and how fast the collider is running.
- Lighter particles/Slower colliders: You see more muons (4 muons).
- Heavier particles/Faster colliders: The muons fly off so fast they miss the detector, or they merge, leaving you with fewer visible muons (2 or 3 muons).
The Verdict:
- With the Heavy Higgs: The signal is very strong. Even at the lower energy (3 TeV), they can find these particles. At the highest energy (30 TeV), they could find particles that are incredibly heavy (up to 8.4 TeV) and the new force carrier (Z') could be very heavy (up to 23 TeV).
- Without the Heavy Higgs: The signal is weaker (like trying to hear a whisper in a storm). It's harder to find, but the 10 TeV and 30 TeV colliders could still do it if they run long enough.

Summary

This paper argues that a future Muon Collider is the perfect place to hunt for these heavy, ghostly particles. By using the "wind" of Z' bosons to smash together, we can create heavy particles that reveal themselves by breaking apart into pairs of same-colored muons. The authors show that this method works better than traditional ways, especially if the new particles are very heavy, offering a clear roadmap for how to solve the mystery of neutrino mass.

Technical Summary: New Avenues of Heavy Neutral Lepton at Muon Collider

Problem Statement
The origin of sub-eV neutrino masses remains an open question in particle physics. While the Type-I seesaw mechanism introduces heavy neutral leptons (HNLs, denoted as $N$ ) to explain this, the natural mixing between light and heavy neutrinos ( $V_{\ell N} \sim \sqrt{m_\nu/m_N}$ ) is typically too small for detection at current colliders. Furthermore, standard production channels for HNLs, such as those mediated by the Standard Model (SM) Higgs or vector boson fusion ( $WW/ZZ \to H \to NN$ ), are often suppressed by small mixing angles or kinematic thresholds. The paper investigates the potential of multi-TeV muon colliders to probe HNLs within a $U(1)_{L_\mu-L_\tau}$ gauge extension of the Type-I seesaw model. In this framework, the muon collider acts effectively as an electroweak gauge boson collider due to initial state radiation, potentially enhancing production cross-sections via new vector boson fusion channels involving a new gauge boson $Z'$ and a heavy Higgs $H$ .

Methodology
The authors analyze the production of heavy neutral lepton pairs ($NN$) at multi-TeV muon colliders ( $\sqrt{s} = 3, 10, 30$ TeV) through two distinct mechanisms:

With Heavy Higgs: The process $Z'Z' \to H \to NN$ , where the heavy Higgs $H$ is produced via $Z'$ -fusion and subsequently decays into an HNL pair.
Without Heavy Higgs: The process $Z'Z' \to NN$ (via $t$ - and $u$ -channel exchange of $N$ ) and the direct s-channel production $\mu^+\mu^- \to Z'^* \to NN$ , applicable when the heavy Higgs mass $m_H$ exceeds the collision energy ( $m_H > \sqrt{s}$ ).

The study focuses on the lepton number violation (LNV) signature arising from the decay $N \to \mu^\pm jj$ , where the two jets from the $W$ boson decay are reconstructed as a single "fat-jet" ( $J$ ). The final state signature is $\mu^+\mu^- \mu^\pm \mu^\pm JJ$ . The analysis considers three specific channels based on the number of muons detected in the main detector ( $|\eta| < 2.5$ ):

Four-muon channel: $3\mu^\pm + \mu^\mp$
Three-muon channel: $3\mu^\pm$ (pure LNV)
Two-muon channel: $2\mu^\pm$

The authors utilize a simulation chain involving FeynRules for model implementation, MadGraph5_aMC@NLO for event generation, Pythia8 for parton showering, and Delphes3 for detector simulation. They apply pre-selection cuts on transverse momentum and pseudorapidity, followed by specific cuts on muon multiplicity, fat-jet multiplicity, and the invariant mass $m_{\mu J}$ to reconstruct the HNL mass. Backgrounds from SM processes ( $\mu^+\mu^- \to \mu^+\mu^- WW$ , $WWZ$, and $\mu^\pm \nu WWW$ ) are estimated, with particular attention to charge misidentification and missing muons.

Key Contributions and Results

Dominance of New Vector Boson Fusion: The paper demonstrates that for multi-TeV muon colliders, the cross-sections for $Z'Z'$ fusion processes increase logarithmically with energy, often surpassing the s-channel heavy Higgs-strahlung ( $\mu^+\mu^- \to Z'H$ ) and direct pair production, particularly when $m_{Z'} \ll \sqrt{s}$ .
Decoupling Limit Viability: Unlike canonical $WW/ZZ \to H \to NN$ processes which are suppressed by the Higgs mixing angle $\alpha$ , the new process $Z'Z' \to H \to NN$ remains viable even in the decoupling limit ( $\alpha \to 0$ ) because the $Z'Z'H$ coupling is independent of the SM-Higgs mixing.
Kinematic Dependence of Signatures: The number of observable muons in the main detector is highly sensitive to the masses of $N$ $N$ and $Z'$ $Z^{'}$ .
- At lower energies (3 TeV) or for heavier $Z'$ , the four-muon channel ( $3\mu^\pm + \mu^\mp$ ) is dominant.
- At higher energies (30 TeV) or for lighter $Z'$ , the two-muon channel ( $2\mu^\pm$ ) becomes dominant as forward muons fall outside the main detector acceptance.
Discovery Reach with Heavy Higgs:
- For a benchmark ( $m_N=200$ GeV, $m_H=3m_N$ , $m_{Z'}=500$ GeV, $g'=0.6$ ), the 3 TeV collider can discover the signal with $\sim 1.35$ fb $^{-1}$ via the four-muon channel.
- The 30 TeV collider offers the largest discovery reach, capable of probing $m_N < 8.4$ TeV and $m_{Z'} < 23$ TeV with $g'=0.6$ .
- The 30 TeV stage can probe gauge couplings as low as $g' \gtrsim 0.12$ .
Discovery Reach without Heavy Higgs:
- In the absence of the resonant Higgs enhancement ( $m_H > \sqrt{s}$ ), cross-sections are significantly lower (several orders of magnitude).
- At 3 TeV, only the four-muon channel is promising, requiring $\sim 241$ fb $^{-1}$ .
- At 30 TeV, the three-muon channel ( $3\mu^\pm$ ) becomes the most promising due to lower backgrounds, despite the two-muon channel having a larger cross-section.
- The 30 TeV collider can probe $m_N < 13$ TeV and $m_{Z'} < 5.1$ TeV with $g'=0.6$ , and $g' \gtrsim 0.22$ for fixed $m_{Z'}=500$ GeV.

Significance
The paper claims that these new vector boson fusion channels provide alternative pathways to probe the intrinsic features of heavy neutral leptons, specifically in regions of parameter space where traditional mixing-angle-dependent searches are insensitive. The study highlights that the multi-TeV muon collider is uniquely suited for this investigation due to its nature as a gauge boson collider, allowing for the exploration of $U(1)_{L_\mu-L_\tau}$ extensions even in the decoupling limit of the heavy Higgs. The authors emphasize that the specific signature $\mu^+\mu^- \mu^\pm \mu^\pm JJ$ offers a clean probe of lepton number violation, with the dominant detection channel varying based on the collider energy and the mass spectrum of the new physics. The results suggest that a 30 TeV muon collider could significantly extend the discovery reach for HNLs and the associated $Z'$ boson compared to lower-energy machines.