Probing $\Delta L=2$ lepton number violating SMEFT operators at the same-sign muon collider

This paper investigates the sensitivity of a 2 TeV same-sign muon collider (μTRISTAN) to eight distinct $\Delta L=2$ dimension-seven SMEFT operators via $\mu^+ \mu^+ \rightarrow W^+W^+/W^+qq'$ processes, demonstrating its potential to probe lepton number violation beyond existing LHC constraints and future FCC projections.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: Hunting for "Ghostly" Violations

Imagine the universe is a giant, well-organized library. In this library, there is a strict rule called Lepton Number Conservation. It's like a librarian who ensures that for every "lepton" (a type of particle like an electron or a muon) that enters the room, one must eventually leave. The total number of lemons (leptons) in the room never changes.

For decades, physicists believed this rule was absolute. But then, they discovered that neutrinos (ghostly, tiny particles) have mass. This discovery hinted that the rule might actually be broken. If the rule is broken, it means lepton number can change by 2 units (∆L = 2). This is called Lepton Number Violation (LNV).

Why do we care? Because if this rule is broken, it could explain:

1. Why neutrinos have mass.
2. Why the universe is made of matter instead of being a mix of matter and antimatter that canceled each other out.

The Problem: The Library is Too Noisy

Scientists want to catch this rule-breaking in action, but it's incredibly rare. It's like trying to hear a whisper in a rock concert.

• The LHC (Large Hadron Collider): This is the current "rock concert." It smashes protons together with massive energy. But because protons are made of messy chunks (quarks), the collisions create a huge amount of "noise" (background events) that drowns out the quiet whisper of the new physics.
• The µTRISTAN (The Proposed Solution): The authors of this paper are proposing a new experiment using a Same-Sign Muon Collider. Imagine instead of smashing messy rocks, we are smashing two identical, clean, blue marbles (muons) together.
  • The Trick: If you smash two positive muons ($\mu^+$) together, the starting "lepton count" is +2.
  • The Goal: If the rule is broken, the final result could be two W bosons (particles that decay into jets of energy) with zero leptons left.
  • The Signal: Starting with +2 and ending with 0 is a smoking gun. It's a clear, undeniable sign that the rule was broken. Because the initial state is so clean, there is very little "noise" to hide the signal.

The Detective Work: The "Fat Jet" Clue

The paper focuses on a specific process: $\mu^+ \mu^+ \to W^+ W^+$ (or similar combinations). When these W particles decay, they don't leave simple tracks; they explode into sprays of particles.

• The Analogy: Imagine the W particles are like firecrackers that explode into a dense, heavy cloud of debris. The scientists call these dense clouds "Fat Jets."
• The Strategy: The researchers are looking for an event where two muons collide, and the only thing left behind are two heavy Fat Jets and nothing else (no missing particles like neutrinos).
• The Filter: They use a mathematical sieve (kinematic cuts) to filter out the background noise. For example, they check the "missing energy." In the standard background noise, invisible neutrinos often escape, taking energy with them. In their "rule-breaking" signal, the energy balance is different. By cutting away the events with too much missing energy, they isolate the rare, interesting events.

The Toolkit: The "SMEFT" Recipe Book

The scientists aren't just looking for any new physics; they are looking for specific "recipes" written in a book called SMEFT (Standard Model Effective Field Theory).

• The Analogy: Think of the Standard Model as a standard cookbook. SMEFT is a book of "secret recipes" that might exist beyond the standard one. These recipes are written as operators (mathematical instructions).
• The Study: The authors looked at 8 specific "recipes" (operators) that involve muons. They calculated how often these recipes would produce the "Two Fat Jets" signal at the µTRISTAN collider.
• The Result: They found that the µTRISTAN collider is incredibly sensitive. It can detect these "secret recipes" much better than the current LHC, and in some cases, even better than the proposed future "super-collider" (FCC-hh), despite having less energy.

The "UV" Connection: Peeking Behind the Curtain

In Section VII, the authors show how these "recipes" (operators) might actually come from a real, physical machine.

• The Analogy: Imagine you see a magic trick (the operator). You want to know how the magician does it. They propose a specific "machine" involving Leptoquarks (heavy particles that act like a bridge between quarks and leptons).
• The Finding: If these heavy Leptoquarks exist, they would create the "secret recipes" the scientists are looking for. The study shows that if the µTRISTAN finds these recipes, it indirectly proves the existence of these heavy Leptoquarks, even if the collider isn't powerful enough to create the Leptoquarks directly. It's like deducing the existence of a giant elephant in the room by seeing the footprints it left behind.

The Conclusion: Why This Matters

The paper concludes that a Same-Sign Muon Collider is a "super-powered microscope" for this specific type of physics.

• Cleanliness: Because it starts with two identical muons, the background noise is minimal.
• Sensitivity: It can spot the "whisper" of Lepton Number Violation that the noisy LHC misses.
• Impact: Finding this violation would be a Nobel Prize-level discovery. It would confirm that the universe has a hidden mechanism for breaking symmetry, potentially explaining why we exist at all.

In short: The authors are saying, "Stop smashing messy rocks (protons) and start smashing clean blue marbles (muons). If we do that, we can finally hear the whisper of the universe's deepest secrets."

Technical Summary

Here is a detailed technical summary of the paper "Probing $\Delta L = 2$ lepton number violating SMEFT operators at the same-sign muon collider."

1. Problem Statement

The Standard Model (SM) conserves lepton number ($L$), but experimental evidence of neutrino oscillations implies neutrinos have mass, necessitating physics beyond the SM (BSM). While the dimension-5 Weinberg operator can generate neutrino masses, its direct probe at colliders is difficult due to the extremely high energy scale required ($\Lambda \sim 10^{12}$ TeV for electron flavor).

Lepton Number Violation (LNV) can also occur via higher-dimensional operators. Specifically, dimension-7 operators with $\Delta L = 2$ offer a window to new physics at the TeV scale. However, constraints on these operators are currently weak, particularly for muon-flavored interactions, as existing limits from neutrinoless double-beta decay ($0\nu\beta\beta$) primarily constrain electron-flavor operators.

The central problem addressed is: How effectively can a future same-sign muon collider ($\mu^+\mu^+$) probe these $\Delta L = 2$ dimension-7 SMEFT operators compared to current LHC limits and future hadron collider projections (FCC-hh)?

2. Methodology

A. Theoretical Framework

• SMEFT Approach: The authors analyze the Standard Model Effective Field Theory (SMEFT) Lagrangian extended by dimension-7 operators.
• Operator Selection: They focus on the 12 independent $\Delta L = 2$ dimension-7 operators. Of these, eight contribute at tree level to the processes $\mu^+\mu^+ \to W^+W^+$ and $\mu^+\mu^+ \to W^+qq'$.
• Flavor Structure: The analysis restricts operators to muon-flavor ($\mu\mu$ and $\mu\mu qq'$) to avoid constraints from $0\nu\beta\beta$ and lepton flavor violation. Quark indices are restricted to the first two generations.
• Classification: Operators are categorized by Lorentz structure: $\Psi^2H^2D^2$, $\Psi^2H^3D$, $\Psi^4D$, and $\Psi^4H$.

B. Collider Setup and Simulation

• Facility: The study assumes the $\mu$TRISTAN collider, a proposed same-sign muon collider operating at $\sqrt{s} = 2$ TeV with an integrated luminosity of $1 \text{ ab}^{-1}$.
• Signal Processes:
  • $\mu^+\mu^+ \to W^+W^+$ (via contact interactions or $W$-boson fusion).
  • $\mu^+\mu^+ \to W^+qq'$ (via contact interactions).
• Final State Signature: The analysis focuses on events with two fat jets ($2J$).
  • For $W^+W^+$: Both $W$ bosons decay hadronically.
  • For $W^+qq'$: The $W$ and the quarks form collimated fat jets.
• Backgrounds: The dominant SM background is $W^+W^+\nu\nu$ (where neutrinos escape detection). Subdominant backgrounds include $\mu^+\nu W^+$, $\mu^+\mu^+Z$, and $W^+Z\mu^+\nu$.
• Simulation Chain:
  • Matrix Elements: Generated using FeynRules (for SMEFT implementation) and MG5_aMC.
  • Parton Shower/Hadronization: Pythia8.
  • Detector Simulation: Delphes3 (using default cards).
• Event Selection & Cuts:
  • Object Selection: Two fat jets ($N_J=2$), no isolated leptons or photons.
  • Kinematic Cuts:
    1. $M_{J2} > 10$ GeV (suppresses soft radiation backgrounds).
    2. Missing Energy ($\not{E}_T$) $< 400$ GeV (crucial for rejecting SM backgrounds with neutrinos).
    3. Invariant mass of jet pair ($M_{JJ}$) $> 1600$ GeV (signal peaks near $\sqrt{s}$, background is softer).

C. Statistical Analysis

• Method: Binned $\chi^2$ analysis using the Neyman definition.
• Observable: The angular distribution of the heavier fat jet relative to the beam axis ($\cos \theta_1$), divided into 5 bins.
• Scenarios:
  1. Single Operator: 1-parameter fit (4 degrees of freedom).
  2. Two-Operator: 2-parameter fit (3 degrees of freedom) to study correlations.
• Validity: Ensures the EFT cutoff $\Lambda > \sqrt{s}$ (2 TeV) to maintain perturbative validity.

3. Key Contributions

1. Comprehensive Operator Scan: The first detailed study of all eight relevant $\Delta L = 2$ dimension-7 operators specifically for muon-flavor at a same-sign muon collider.
2. Optimized Signal Strategy: Demonstrated that the "two fat jets" signature combined with strict missing energy cuts effectively isolates LNV signals from the dominant $W^+W^+\nu\nu$ background, which is a significant challenge in hadron colliders.
3. Comparison with FCC-hh: Provided a direct comparison between a lower-energy lepton collider ($\mu$TRISTAN, 2 TeV) and a high-energy hadron collider (FCC-hh, 100 TeV), highlighting the unique advantages of the lepton collider environment.
4. UV Completion Case Study: Constructed a specific New Physics scenario involving two heavy scalar leptoquarks ($S$ and $R$) that generates the operator $O_{LLQdH1}$ at tree level while avoiding the dimension-5 Weinberg operator at leading order. This links collider limits to specific BSM mass scales.

4. Key Results

• Sensitivity to Wilson Coefficients ($C_i/\Lambda^3$):

  • The $\mu$TRISTAN achieves significantly stronger bounds than current LHC limits (13 TeV, 139 fb$^{-1}$) for all operators, typically improving sensitivity by one to three orders of magnitude.
  • Best Constraints: The operator $O_{LLduD}$ (four-fermion contact interaction) yields the tightest bounds due to the contact interaction nature and light final states.
  • $W^+W^+$ Channel: Operators contributing to $W^+W^+$ ($O_{LHD1}, O_{LHD2}, O_{LHDe}$) show massive improvements over LHC. For instance, sensitivity to $O_{LHD2}$ improves by $\mathcal{O}(10^3)$ compared to LHC.

• Comparison with FCC-hh (100 TeV, 30 ab$^{-1}$):

  • Despite operating at 50 times lower energy and 30 times lower luminosity, $\mu$TRISTAN outperforms FCC-hh projections for the $\Psi^2H^2D^2$ and $\Psi^2H^3D$ classes (operators involving $W$-fusion or Higgs interactions).
  • For four-fermion operators ($W^+qq'$), $\mu$TRISTAN achieves comparable sensitivity to FCC-hh, with slight advantages for $O_{LLQdH2}$ and $O_{LeudH}$.
  • Reason: The clean initial state ($\mu^+\mu^+$) and lack of QCD backgrounds allow for precise kinematic reconstruction and superior signal-to-background ratios, compensating for the lower energy.

• Indirect Constraints on Leptoquarks:

  • In the UV completion model, the projected $\mu$TRISTAN limits on $O_{LLQdH1}$ translate to indirect constraints on leptoquark masses ($m_S, m_R$) that are an order of magnitude stronger than current LHC limits.

• Weinberg Operator (Appendix A):

  • The study also projected sensitivity to the muon-flavor diagonal Weinberg operator, finding $\Lambda_{\mu\mu} > 399$ TeV (95% C.L.), surpassing FCC-hh projections by an order of magnitude, though still below direct neutrino mass limits.

5. Significance

• Probing the TeV Scale: The paper establishes that same-sign muon colliders are uniquely powerful tools for probing LNV at the TeV scale, a regime inaccessible to current low-energy experiments and difficult to probe cleanly at hadron colliders due to backgrounds.
• Complementarity: It highlights that lepton colliders are not just "lower energy" alternatives to hadron colliders; for specific BSM signatures (like $\Delta L = 2$ operators), they offer superior discovery potential due to the clean initial state and precise kinematic control.
• Guidance for Future Physics: The results provide concrete targets for the design of future muon colliders and suggest that if LNV exists at the TeV scale, $\mu$TRISTAN could be the facility to discover it, potentially shedding light on the origin of neutrino mass and the matter-antimatter asymmetry of the universe (via leptogenesis).
• Model Independence: By using the SMEFT framework, the results remain model-independent, covering a broad class of New Physics scenarios without committing to a specific UV completion.