Probing lepton number violation at FCC-ee

The paper proposes that the FCC-ee can probe lepton number violation and test neutrino mass ordering through high-multiplicity final-state signatures like $e^+e^-\to N\overline{N}\to \ell^+\ell^+ 4j$, which are enabled by the minimal linear seesaw mechanism to avoid suppression by small neutrino masses and yield over 1,000 expected events with negligible Standard Model background.

The Gist

Imagine the universe is a giant, complex puzzle, and one of the most mysterious pieces is the neutrino. These are tiny, ghost-like particles that zip through everything without leaving a trace. For decades, physicists have known these particles have mass, but they are so incredibly light that it's like trying to weigh a single grain of sand on a scale designed for elephants. The big question is: Why are they so light, and what rules govern them?

This paper proposes a new way to solve this mystery using a massive particle collider called FCC-ee (Future Circular Collider), which is planned to be built in Europe. Here is the story of their proposal, broken down into simple concepts.

1. The "Ghost" Problem: Why We Can't See the Usual Suspects

In the past, scientists looked for heavy versions of neutrinos (let's call them "Heavy Neutrinos") to explain why the regular ones are so light. This is based on a theory called the "Seesaw Mechanism." Think of a seesaw: if one side (the heavy neutrino) is very heavy, the other side (the light neutrino) must be very light.

However, in the old versions of this theory, the heavy neutrinos were so massive and so "hidden" that they were impossible to create in current particle smashers. It was like trying to find a needle in a haystack, but the needle was made of invisible glass.

2. The New Idea: A "Leaky" Seesaw

The authors suggest a specific, slightly different version of the seesaw called the Linear Seesaw.

• The Analogy: Imagine the old theory was a perfectly sealed vault; you couldn't get in. The new theory is like a vault with a small, controlled leak.
• How it works: In this model, the heavy neutrinos can be created much more easily because they don't rely on a tiny, weak connection to the light ones. Instead, they are produced by a strong "Yukawa coupling" (think of this as a strong magnetic pull).
• The Result: At the FCC-ee, we could potentially create thousands of these heavy neutrinos, whereas other models predict we might see zero.

3. The "Magic Trick": Lepton Number Violation (LNV)

The most exciting part of the paper is about a phenomenon called Lepton Number Violation (LNV).

• The Rule: In the Standard Model of physics, there's a rule that says "leptons" (like electrons) must be created in pairs: one positive, one negative. It's like a law of conservation: you can't just create a positive electron out of thin air without a negative one to balance the books.
• The Violation: The authors propose that if these heavy neutrinos are their own antiparticles (called Majorana particles), they can break this rule.
• The Signature: The paper predicts a very specific "smoking gun" event:
  • Two electrons collide.
  • They create two heavy neutrinos.
  • These heavy neutrinos decay into two positively charged leptons (like two positive electrons) and four jets of particles (like a spray of debris).
  • Why it's special: In the standard world, seeing two positive electrons come out of a collision is practically impossible. If we see this, it proves that the "law of conservation" was broken, confirming that neutrinos are their own antiparticles.

4. The "Oscillation" Dance

The paper introduces a fascinating twist involving oscillations.

• The Analogy: Imagine two twins, Alice and Bob, who look almost identical but have a tiny difference in their heartbeat. If they stand still, you can tell them apart. But if they start running and spinning very fast, they blur together.
• The Physics: The heavy neutrinos come in pairs that are almost identical. As they travel through the detector, they can "oscillate" (switch back and forth) between being a particle and an antiparticle.
• The Connection to Mass: The speed of this switching depends on the difference in their mass. Interestingly, this difference is linked to the known mass differences of the light neutrinos we already know about.
• The Twist: By counting how many "two-positive-lepton" events happen, scientists could potentially figure out the ordering of neutrino masses (which is the heaviest and which is the lightest) without needing a separate experiment. It's like solving a puzzle by looking at the shadow it casts.

5. The Prediction: A Crowd of Events

The authors ran the numbers for the FCC-ee collider.

• The Setup: They looked at two energy levels (91 GeV and 240 GeV).
• The Background: In the standard world, the "noise" (background events that look like the signal) is virtually zero. It's a silent room.
• The Result: They predict seeing over 1,000 events (O(10³)) where two same-sign leptons appear.
• Why it matters: Because the background is so low, finding even a few of these events would be a massive discovery. Finding 1,000 would be a slam-dunk confirmation of this new physics.

Summary

In simple terms, this paper says:

1. Stop looking for the needle in the haystack: The old way of finding heavy neutrinos is too hard.
2. Try the new door: The "Linear Seesaw" model opens a door where we can easily create these heavy particles.
3. Watch for the magic trick: If we see two positive electrons appearing together with a spray of debris, it proves that neutrinos are their own antiparticles and that a fundamental rule of the universe is broken.
4. Read the dance: The way these particles switch identities tells us about the mass hierarchy of neutrinos.

The authors believe the FCC-ee is the perfect place to catch this "magic trick" in action, potentially revolutionizing our understanding of why the universe has mass the way it does.

Technical Summary

Technical Summary: Probing Lepton Number Violation at FCC-ee

Problem Statement
The origin of neutrino mass and the nature of lepton number violation (LNV) remain central questions in particle physics. While neutrino oscillations confirm non-zero neutrino masses, the underlying mechanism is unknown. In the classic high-scale seesaw framework, the mediators responsible for neutrino mass generation are super-heavy and inaccessible at current or near-future colliders. Furthermore, conventional low-scale seesaw searches (e.g., $pp \to \ell^+ N \to \ell^+ \ell^+ jj$) are often suppressed by the small light-heavy neutrino mixing required to generate tiny neutrino masses. Additionally, LNV effects in neutrino oscillations are strongly helicity-suppressed, making neutrinoless double beta decay the primary probe for $\Delta L = 2$ processes. There is a need for collider-based probes that can access LNV without being suppressed by small mixing angles or neutrino mass scales, particularly at the proposed Future Circular Collider in electron-positron mode (FCC-ee).

Methodology and Theoretical Framework
The authors propose a search strategy based on the minimal linear seesaw model, a low-scale realization where lepton number symmetry is broken explicitly but softly. Unlike the inverse seesaw or Type-I seesaw, this framework allows for heavy neutrino production that is not suppressed by the small light-heavy neutrino mixing.

• Model Setup: The model introduces singlet fermions $\nu_R$ and $S_R$, and a scalar doublet $\chi$. The Lagrangian includes Yukawa couplings $Y_\nu$ and $Y_S$. Crucially, the lepton-number-violating term $M_L$ is small, allowing the heavy mass scale $M_R$ to remain relatively low while maintaining consistency with observed neutrino masses.
• Production Mechanism: Heavy neutral leptons (NHLs), denoted as $N$ and $\bar{N}$, are produced via the exchange of a charged, leptophilic Higgs boson ($H^\pm$) in the $t$-channel ($e^+e^- \to N\bar{N}$). This process is governed by the Yukawa coupling $Y_S$, which can be sizable ($|Y_S| \sim 1$), bypassing the suppression typical of mixing-driven production.
• Oscillation Dynamics: The two heavy mass eigenstates ($N_4, N_5$) form a quasi-Dirac pair with a mass splitting $\Delta M$. This splitting is directly linked to the neutrino mass ordering: $\Delta M = \Delta m_{31}^2$ for Normal Ordering (NO) and $\Delta M = \Delta m_{21}^2$ for Inverted Ordering (IO).
  • If the heavy neutrinos oscillate between $N$ and $\bar{N}$ before decaying, they can produce Lepton Number Violating (LNV) final states ($e^+e^- \to N\bar{N} \to \ell^\pm \ell^\pm 4j$).
  • Without oscillation, only Lepton Number Conserving (LNC) states ($\ell^\pm \ell^\mp 4j$) are produced.
• Simulation Parameters: The study calculates event yields at FCC-ee center-of-mass energies of $\sqrt{s} = 91$ GeV and $240$ GeV, with integrated luminosities of $205 \text{ ab}^{-1}$ and $10 \text{ ab}^{-1}$, respectively. The analysis assumes a vertex displacement range of $1 \text{ mm} \le x \le 1 \text{ m}$ and fixes parameters such as $|Y_S|=1$, $m_{H^\pm}=1 \text{ TeV}$, and active-sterile mixing $U^2 = 10^{-6}$.

Key Contributions

1. Novel Production Channel: The paper highlights a production mechanism for heavy neutrinos mediated by a charged Higgs boson, which is unsuppressed by the small light-heavy mixing angle. This contrasts with conventional searches where the cross-section is proportional to the mixing squared.
2. Direct LNV Probe via Topology: The authors propose high-multiplicity final states ($e^+e^- \to \ell^\pm \ell^\pm 4j$) as a direct probe of the Majorana nature of neutrinos. This signature features negligible Standard Model background at lepton colliders.
3. Sensitivity to Mass Ordering: The study demonstrates that the ratio of LNV to LNC events depends on the oscillation probability, which is a function of the ratio $\Delta M / \Gamma_N$. Since $\Delta M$ is determined by the neutrino mass ordering (NO vs. IO), the collider event rates can distinguish between these orderings.
4. Quantitative Event Estimates: The paper provides concrete estimates for the number of observable LNV events, showing that the linear seesaw framework can yield $\mathcal{O}(10^3)$ events at FCC-ee under specific parameter choices.

Results

• Event Yields: For the benchmark parameters ($U^2 = 10^{-6}$, $|Y_S|=1$, $m_{H^\pm}=1 \text{ TeV}$), the expected number of LNV events ($e^\pm e^\pm 4j$) is significant, reaching up to $\mathcal{O}(10^3)$ for heavy neutrino masses ($M_N$) where the oscillation condition $\Delta M > \Gamma_N$ is met.
• Mass Dependence: The number of LNV events decreases sharply for high $M_N$ where the decay width $\Gamma_N$ exceeds the mass splitting $\Delta M$ ($\Delta M < \Gamma_N$), rendering oscillations ineffective. This suppression occurs at lower $M_N$ values for Inverted Ordering (IO) compared to Normal Ordering (NO) because $\Delta M_{NO} > \Delta M_{IO}$.
• Low Mass Limit: At very low $M_N$, no events are observed because the heavy neutrino lifetime becomes too long, causing decays to occur outside the detector volume ($x > 1 \text{ m}$).
• Oscillation Regime: When $\Delta M \gtrsim 4\Gamma_N$, the ratio of LNV to LNC events approaches unity ($R \approx 1$), maximizing the LNV signal.

Significance and Claims
The paper claims that the minimal linear seesaw model offers a "novel avenue" to test neutrino physics at high-energy colliders. Specifically:

• It enables a direct probe of the Majorana nature of neutrinos through final-state topology ($\ell^\pm \ell^\pm$) without the suppression factors that plague conventional seesaw searches.
• It provides a method to test the neutrino mass ordering (NO vs. IO) established by oscillation experiments within a collider setting, leveraging the connection between the heavy neutrino mass splitting and light neutrino oscillation parameters.
• The proposed signatures ($e^+e^- \to \ell^\pm \ell^\pm 4j$) are robust against Standard Model backgrounds, making FCC-ee a viable facility for discovering or constraining these low-scale LNV scenarios.

The authors conclude that this framework allows for sizable LNV rates even in regimes where oscillations are not fully resolved, offering a promising target for future FCC-ee operations.