Lepton number violating signals of a parity symmetric model at $\mu$TRISTAN

This paper investigates a parity-symmetric extension of the Standard Model that naturally explains small neutrino masses while allowing for substantial TeV-scale lepton number violation, demonstrating that a 10 TeV $\mu^+\mu^+$ collider could probe $W'$ boson masses up to 16 TeV through distinctive lepton number violating signals.

The Gist

The Big Picture: Fixing the Universe's "Glitch"

Imagine the Standard Model of particle physics as a perfectly tuned car engine. It runs great, but there's one tiny, annoying glitch: a part of the engine that should be silent is making a weird noise (this is the "Strong CP problem"). Physicists have a proposed fix called Parity Symmetry.

Think of Parity Symmetry like a mirror. If you look in a mirror, your left hand becomes your right hand. This theory suggests that the universe has a hidden "mirror world" where every particle has a twin. In our world, we have "Left-Handed" particles; in the mirror world, they have "Right-Handed" twins. This mirror setup naturally silences that annoying engine noise.

However, to make this mirror work, the theory requires new, heavy particles to exist. The paper asks: If we build a super-powerful microscope (a particle collider), can we see these mirror particles?

The New Machine: The µTRISTAN Collider

The authors are proposing a specific type of particle collider called µTRISTAN.

• The Fuel: Instead of smashing protons (like the Large Hadron Collider does), this machine smashes muons (heavy cousins of electrons).
• The Setup: It smashes two positive muons ($\mu^+$) together.
• The Goal: To reach energies of 10 to 30 TeV (trillion electron volts). This is like hitting a tennis ball with the force of a speeding train to see what tiny fragments fly off.

The Mystery: Lepton Number Violation

In the standard rules of the universe, there is a "conservation law" called Lepton Number. Think of it like a bank account balance. If you start with a certain amount of "lepton money," you must end with the same amount. You can't just create or destroy it.

This new theory suggests that in the mirror world, this rule can be broken. The "Lepton Number" can be violated.

• The Analogy: Imagine a magic trick where a magician pulls two rabbits out of a hat, but the hat was empty to begin with. In the real world, this is impossible. But in this "Mirror World," the universe allows the magician to break the rules.

The paper focuses on a specific "magic trick" reaction:
$$\mu^+ + \mu^+ \rightarrow W^+ + W'^+$$
Two positive muons collide and turn into two positive "W" particles (one standard, one new mirror version). Because we started with two positive particles and ended with two positive particles, but the process involves breaking the "Lepton Number" rule, this is a smoking gun for new physics.

The Cast of Characters

1. The $W'$ Boson: The "Mirror Twin" of the standard W boson. It's heavy and hard to find.
2. The Heavy Neutrinos ($S$): Invisible, heavy ghosts that mix with regular neutrinos. They are the key to why the mirror world exists.
3. The "Off-Shell" Trick: Sometimes, the mirror particle ($W'$) is too heavy to be created directly. But, like a ghost that can briefly appear and disappear, it can exist for a split second as a "virtual" particle to help the reaction happen.

The Detective Work: How to Find Them

The authors calculated what would happen if we built this 10 TeV muon collider.

Scenario A: The Heavy Hitter (Direct Production)
If the mirror particle ($W'$) isn't too heavy (around 10–16 TeV), the collider can create it directly.

• The Signal: The collision creates a burst of energy that turns into jets of particles and charged leptons.
• Why it's special: In the "background noise" of the universe (standard physics), you usually get missing energy because neutrinos escape. In this new signal, nothing escapes. All the energy is accounted for in visible particles. It's like a crime scene where the thief left no footprints, but the safe is wide open and the money is piled up on the floor. It's a "background-free" signal, meaning if we see it, it's definitely new physics.

Scenario B: The Ghostly Touch (Off-Shell Production)
Even if the mirror particle is too heavy to create directly (say, 20 TeV), the collider can still "feel" its presence.

• The Analogy: Imagine trying to lift a heavy boulder. You can't lift it directly, but if you push against a lever connected to it, the boulder moves slightly. The collider pushes against the heavy $W'$ via a "virtual" connection.
• The Reach: By studying these subtle interactions, a 10 TeV collider could actually detect particles up to 16 TeV in mass.

Scenario C: The Lighter Ghosts (Single Production)
If the heavy neutrinos are actually quite light (sub-TeV), the collider can produce them singly.

• The Signal: A muon hits a photon (from the beam), creates a heavy neutrino, and the neutrino decays into a charged lepton and a W boson.
• The Twist: This process flips the "charge" of the lepton (a positive muon turns into a negative electron or tau). It's like a chameleon changing color instantly. This is a very clean signal because nature rarely does this by accident.

The Conflict: The "Double Beta" Detective

There is another detective on the case: Neutrinoless Double Beta Decay. This is a rare radioactive process happening in rocks deep underground.

• The Constraint: If the mirror particles are too light or interact too strongly, this underground process would happen too often. Since we haven't seen it happening that often, it puts a limit on how light the mirror particles can be.
• The Loophole: The authors found a clever way around this. If the three types of heavy neutrinos have slightly different masses (they aren't identical twins), the underground detector might not see the signal, but the collider will. It's like tuning a radio: the underground detector is tuned to one frequency and hears static, while the collider is tuned to a different frequency and hears a clear song.

The Bottom Line

This paper is a blueprint for a future experiment. It says:

1. Build a 10 TeV Muon Collider.
2. Smash positive muons together.
3. Look for a specific pattern of particles where energy is perfectly conserved and no neutrinos escape.

If we see this, we will have proven that:

• The universe has a "mirror world" (Parity Symmetry).
• We have solved the mystery of the Strong CP problem.
• We have discovered why neutrinos have mass.

It's a high-stakes game of hide-and-seek where the "hider" is a new symmetry of nature, and the "seeker" is a machine capable of reaching energies we've never touched before. If the machine is built, the hider might finally be caught.

Technical Summary

1. Problem Statement

The paper addresses two major open questions in particle physics:

1. The Strong CP Problem: Why the QCD $\theta$-term is vanishingly small. The authors utilize a Parity Symmetric (Left-Right Symmetric) extension of the Standard Model (SM) where parity is a fundamental symmetry, naturally forbidding the $\theta$-term at high energies.
2. Neutrino Mass Origin: How to generate small neutrino masses while allowing for large Lepton Number Violation (LNV) at the TeV scale.
   • The Tension: In many models (like the Inverse Seesaw), LNV is suppressed by the same small parameter that generates small neutrino masses, rendering LNV signals unobservable at colliders.
   • The Goal: To demonstrate a framework where neutrino masses are small (via radiative corrections) but LNV processes are unsuppressed and observable at future high-energy muon colliders ($\mu^+\mu^-$).

2. Methodology and Model Framework

Theoretical Model

The authors propose a minimal Parity Symmetric model with the gauge group $SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_X$.

• Higgs Sector: Contains the SM Higgs doublet $H_L$ and its parity partner $H_R$. $H_R$ acquires a vacuum expectation value (VEV) $v_R$ (breaking $SU(2)_R$), while $H_L$ has VEV $v_L$ (breaking $SU(2)_L$).
• Fermion Sector: Includes three generations of singlet Weyl fermions $S$.
  • $S$ mixes with left-handed neutrinos ($\nu_L$) and right-handed antineutrinos ($\bar{\nu}_R$) via Yukawa couplings $x$.
  • This mixing creates pseudo-Dirac fermions ($N$ and $S'$) with mass $m_N \approx x v_R$.
• Neutrino Mass Generation:
  • At tree level, active neutrinos are massless because the determinant of the neutral lepton mass matrix vanishes.
  • Small Majorana masses for $S$ ($M_S$) are introduced.
  • Radiative Mechanism: Finite neutrino masses are generated only at the one-loop level (via $Z$ and Higgs boson exchanges).
  • Key Feature: The LNV is driven by the Yukawa coupling $x$ (which is $\mathcal{O}(1)$), not by the small Majorana mass $M_S$. Thus, LNV signals are decoupled from the smallness of neutrino masses.

Collider Strategy

The study focuses on a $\mu^+\mu^+$ collider (a specific mode of a muon collider) with center-of-mass energies ($\sqrt{s}$) of 10 TeV and 30 TeV.

• Signal Process: $\mu^+\mu^+ \to W^+ W'^+$.
  • This process violates lepton number by two units ($\Delta L = 2$).
  • $W'$ is the heavy gauge boson of $SU(2)_R$.
  • The process proceeds via the exchange of heavy neutral leptons ($N/S$).
• Background Suppression: The signal is essentially background-free. SM backgrounds (e.g., $\mu^+\mu^+ \to W^+ W^+ \nu \nu$) involve escaping neutrinos, whereas the signal deposits the full collider energy into visible hadronic jets and charged leptons.

3. Key Contributions and Analysis

A. On-Shell $W'$ Production ($\sqrt{s} > m_{W'}$)

• Process: Direct production of an on-shell $W'$ boson.
• Cross Section: The cross section $\sigma(\mu^+\mu^+ \to W^+ W'^+)$ is calculated analytically. It scales with $|\lambda_{\mu\mu}|^2$ (where $\lambda = U_{PMNS} U_{PMNS}^T$) and the ratio $(m_N/m_{W'})^4$ in the high-energy limit.
• Result: For degenerate heavy neutrinos ($m_N \approx m_{W'}$), the cross section is substantial ($\mathcal{O}(10-100)$ ab). A 10 TeV collider can probe $W'$ masses up to 10 TeV via on-shell production.

B. Off-Shell $W'$ Production ($\sqrt{s} < m_{W'}$)

• Process: $\mu^+\mu^+ \to W^+ q \bar{q}'$ via an off-shell $W'^*$.
• Methodology: The authors integrate the differential cross section over the invariant mass of the quark pair ($m_{qq'}$), applying a lower cut ($m_{cut}$) to suppress SM backgrounds.
• Result: Even when $m_{W'} > \sqrt{s}$, the process is observable.
  • At $\sqrt{s} = 10$ TeV, the reach extends to $m_{W'} \approx 16$ TeV.
  • At $\sqrt{s} = 30$ TeV, the reach extends even further.
  • The signal is characterized by a high invariant mass of visible particles peaking at $\sqrt{s}$.

C. Single Production of Heavy Neutral Leptons

• Scenario: If $m_N \ll m_{W'}$, the $W'$-mediated processes are suppressed.
• Process: $\mu^+\mu^+ \to \mu^+ W^+ S$ (Single production of $S$).
• Mechanism: Dominated by collinear photon emission (Effective Photon Approximation).
• Result: Cross sections can reach $\mathcal{O}(100-1000)$ ab for sub-TeV $m_N$. While the LNV decay $S \to \ell^- q \bar{q}'$ has a small branching ratio, the LNV-conserving decay $S \to \ell^+ W^-$ (with lepton flavor violation) offers a complementary discovery channel.

D. Relaxing Neutrinoless Double Beta Decay ($0\nu\beta\beta$) Constraints

• Constraint: $0\nu\beta\beta$ experiments (e.g., $^{136}$Xe) constrain the parameter space, typically requiring $m_{W'} \gtrsim 10-17$ TeV if the heavy neutrinos are degenerate and $\lambda_{ee} \approx 1$.
• Solution (Non-Degenerate Case): The authors show that by allowing non-degenerate heavy neutrino masses and specific textures for the Yukawa coupling matrix $x$, the element $\tilde{\lambda}_{ee}$ (relevant for $0\nu\beta\beta$) can be suppressed (e.g., $\tilde{\lambda}_{ee} < 0.125$) while keeping $\tilde{\lambda}_{\mu\mu}$ (relevant for the collider) large ($\approx 1$).
• Impact: This allows $m_{W'}$ to be as low as 8 TeV (comparable to HL-LHC sensitivity) while maintaining a strong signal at the muon collider.

4. Results Summary

Feature	Finding
Collider Reach (On-Shell)	10 TeV $\mu^+\mu^+$ collider can probe $m_{W'} \approx 10$ TeV.
Collider Reach (Off-Shell)	10 TeV $\mu^+\mu^+$ collider can probe $m_{W'} \approx 16$ TeV.
Signal Characteristics	$\mu^+\mu^+ \to W^+ W'^+ \to W^+ + \text{jets/leptons}$. No missing energy; full energy deposition.
Background	Negligible. SM backgrounds are suppressed by missing neutrinos and kinematic cuts.
$0\nu\beta\beta$ vs. Collider	$0\nu\beta\beta$ constrains $m_{W'} \gtrsim 10$ TeV for degenerate cases. Non-degenerate tuning relaxes this to $\sim 8$ TeV, making the collider signal robust.
Heavy Neutrino Mass	If $m_N \ll m_{W'}$, single production $\mu^+\mu^+ \to \mu^+ W^+ S$ becomes the dominant probe with large cross sections.

5. Significance and Conclusion

• Decoupling LNV from Neutrino Mass: The paper provides a concrete theoretical realization where small neutrino masses arise from loop-suppressed radiative corrections, while the underlying LNV interactions are unsuppressed. This challenges the conventional wisdom that small neutrino masses imply unobservable LNV at colliders.
• Muon Collider Potential: It establishes the $\mu^+\mu^+$ collider as a unique and powerful machine for discovering TeV-scale parity symmetry. Unlike hadron colliders (LHC) or electron colliders (FCC/CEPC), the $\mu^+\mu^+$ initial state allows for direct $\Delta L = 2$ processes with negligible background.
• Complementarity: The study highlights the necessity of combining $0\nu\beta\beta$ searches with collider experiments. While $0\nu\beta\beta$ probes the electron sector, the muon collider probes the muon sector, allowing for a full reconstruction of the flavor structure of the parity symmetric model.
• Discovery Prospects: The authors conclude that a 10 TeV muon collider could discover the $W'$ boson and the associated parity symmetry up to masses of 16 TeV, significantly extending the reach of current and planned LHC runs.

In summary, the paper argues that the Parity Symmetric Model offers a compelling solution to the Strong CP problem and neutrino mass hierarchy, with distinctive, background-free signatures at future high-energy muon colliders that are not suppressed by the smallness of neutrino masses.