Flavorful Lepton Number Violation at the EIC

This paper investigates the potential of the Electron-Ion Collider to detect flavorful lepton number violation via resonant heavy neutral lepton production within the $\nu$SMEFT framework, demonstrating that with $100~\mathrm{fb}^{-1}$ of integrated luminosity and improved muon detection capabilities, the EIC could achieve sensitivities comparable to current LHC constraints.

The Gist

Imagine the universe is a giant, complex puzzle. For decades, physicists have been trying to figure out why neutrinos (tiny, ghost-like particles) have mass. One leading theory suggests that these neutrinos are actually "Majorana" particles, meaning they are their own antiparticles. If this is true, it would break a fundamental rule of physics called "Lepton Number Conservation."

This paper is a proposal for how to find the "smoking gun" evidence of this rule-breaking at a new, massive machine called the Electron-Ion Collider (EIC).

Here is the breakdown of their idea, using some everyday analogies:

1. The Mystery Guest: The Heavy Neutral Lepton (HNL)

Think of the Standard Model (our current rulebook of physics) as a party with only three types of guests: electrons, muons, and taus (all different flavors of "leptons").

The authors are looking for a secret, heavy guest called a Heavy Neutral Lepton (HNL). This guest is invisible to the naked eye (neutral) and very heavy (10 to 100 times heavier than a proton).

• The Trick: This guest doesn't just walk in; it "mixes" with the regular guests. It's like a celebrity (the HNL) wearing a disguise to mingle with the crowd (the light neutrinos). Because of this disguise, the HNL can briefly appear in collisions, do something weird, and then vanish.

2. The Crime Scene: The Electron-Ion Collider (EIC)

The EIC is like a high-speed billiard table where they smash electrons into protons.

• The Plan: The authors propose smashing an electron into a proton to create this heavy guest (the HNL) in a "resonant" way. Think of it like pushing a swing at just the right moment to make it go super high.
• The Crime: Once created, this heavy guest immediately decays (explodes) into a positively charged particle (like a positron or a positive muon) and a spray of other particles (jets).
• Why it's a crime: In the normal world, if you start with a negative electron, you should end with a negative electron (or a neutral neutrino). If you end up with a positive particle, the "Lepton Number" has been violated. It's like starting a game of pool with only red balls and suddenly having a blue ball appear on the table.

3. The Challenge: Finding a Needle in a Haystack

The problem is that the "haystack" (background noise) is huge.

• The Haystack: When you smash particles together, nature produces millions of "normal" events that look almost exactly like the crime. For example, a regular collision might accidentally produce a positive particle due to a glitch in the detector or a rare natural process.
• The Needle: The signal they are looking for is very specific: a heavy guest that appears, decays, and leaves a specific pattern of energy and particles.

The authors ran massive computer simulations (like a video game of physics) to figure out how to separate the needle from the haystack. They looked at:

• How fast the particles are moving (Momentum).
• How much energy is missing (Missing Transverse Energy).
• How many "sprays" of particles (jets) are created.

4. The Detective Work: The "Cuts"

To find the signal, they designed a set of strict filters, which they call "cuts."

• The Electron Channel: This is the hardest to solve because the "haystack" is full of look-alikes. It's like trying to find a specific person in a crowd where everyone is wearing the same uniform. They have to rely on the detector not making mistakes (like misidentifying a negative electron as a positive one).
• The Muon Channel (The Golden Ticket): This is the most promising. Muons are like "ghosts" that pass through walls. If the EIC has a good detector for muons, the background noise is much quieter. It's like trying to hear a whisper in a library (Muon channel) versus a rock concert (Electron channel). The authors argue that if the EIC builds a dedicated "muon detector," they could find this signal much more easily.
• The Tau Channel: This involves a heavy particle that decays quickly. It's tricky because the signal is "softer" (less energetic), but if they can reconstruct the decay perfectly, it's another strong path.

5. The Verdict: Can They Do It?

The authors compared the EIC's potential to other giant machines like the LHC (Large Hadron Collider) in Europe.

• The Result: They found that the EIC could be just as good as the LHC at finding these heavy guests, especially if the guests are in the 10–100 GeV mass range.
• The Twist: In a more complex version of the theory (where there are extra "hidden" rules called SMEFT operators), the EIC might actually be better than the LHC at distinguishing between different types of physics. It's like having two different flashlights; one might be brighter, but the other shines a different color that reveals details the first one misses.

Summary

This paper is a blueprint for a treasure hunt.

1. The Treasure: A heavy, invisible particle that breaks the rules of physics.
2. The Map: A new collider (EIC) smashing electrons into protons.
3. The Strategy: Use strict filters to ignore the noise and look for a specific "positive particle" signature.
4. The Key: Building a better detector for muons (and understanding tau particles) is crucial. If they do this, the EIC could solve one of the biggest mysteries in physics: Why do neutrinos have mass, and are they their own antiparticles?

If they succeed, it wouldn't just be a new discovery; it would rewrite the rulebook of the universe.

Technical Summary

1. Problem Statement

The paper investigates the potential of the Electron-Ion Collider (EIC) to detect Lepton Number Violation (LNV) mediated by Heavy Neutral Leptons (HNLs), also known as sterile neutrinos.

• Context: While neutrinoless double beta decay ($0\nu\beta\beta$) is the primary probe for $\Delta L = 2$ processes, it is limited to the electron flavor sector. High-energy colliders offer a complementary avenue, particularly for HNLs in the 10–100 GeV mass range.
• Gap: Previous studies at HERA, LHeC, and early EIC proposals focused primarily on the electron channel ($e^- p \to e^+ + X$) or lepton-number conserving (LNC) processes. There is a lack of detailed analysis regarding flavor-violating LNV signals (specifically $\mu$ and $\tau$ final states) at the EIC, including realistic detector effects, hadronization, and background suppression strategies.
• Goal: To quantify the EIC's sensitivity to HNLs in the $\nu$SMEFT framework (Standard Model Effective Field Theory extended with sterile neutrinos), focusing on resonant production and decay into different lepton flavors ($e, \mu, \tau$).

2. Theoretical Framework

The authors employ the $\nu$SMEFT framework, which extends the Standard Model with $n$ singlet HNLs ($N_R$).

• Mass Scale: They assume $n$ HNLs with masses near or below the electroweak scale, while heavier states are integrated out. The analysis focuses on a single kinematically accessible HNL ($N_4$) with mass $m_4 \in [10, 100]$ GeV.
• Lagrangian: The effective Lagrangian includes dimension-4 Yukawa interactions and mass terms, as well as higher-dimensional operators (dim-5 to dim-7) generated by integrating out heavy physics.
• Mixing Mechanism: The primary production and decay mechanism considered is the mixing between active neutrinos and the heavy $N_4$. The relevant charged-current vertex is proportional to the mixing angle $U_{\alpha 4}$ (where $\alpha \in \{e, \mu, \tau\}$).
• Process: The signal process is the resonant production of $N_4$ followed by its decay:
  $$e^- p \to N_4 j + X \to \ell^+_\alpha W^{*-} j + X \to \ell^+_\alpha + k \text{ jets} + X$$
  where $\ell^+_\alpha$ is a positively charged lepton (violating lepton number) and $k$ is the number of jets (2 or 3).

3. Methodology

The study utilizes a full simulation chain to assess EIC sensitivity:

• Event Generation:
  • Matrix Element: MadGraph5_aMC@NLO (v3.6.2) was used to generate signal and background events.
  • Parton Shower & Hadronization: Pythia8 (v8.315) handled parton showering and hadronization.
  • Matching: The MLM matching scheme was employed to avoid double-counting between matrix-element jets and shower emissions.
• Detector Simulation:
  • Tool: Delphes3 (v3.5.0) with a customized card based on the EIC Yellow Report.
  • Parameters: $\sqrt{s} = 141$ GeV ($18 \times 275$ GeV), integrated luminosity $\mathcal{L} = 100$ fb$^{-1}$, and electron polarization $P_e = 70\%$.
  • Crucial Assumption: Since the EIC Yellow Report does not specify muon capabilities, the authors modified the Delphes card to include a hypothetical muon detector with efficiency and angular response identical to the electron detector.
• Signal and Background Modeling:
  • Signal: $e^- p \to \ell^+_\alpha + k j + X$ ($\alpha = e, \mu, \tau$).
  • Backgrounds:
    1. Deep Inelastic Scattering (DIS): Both Neutral Current (NC-DIS) and Charged Current (CC-DIS). NC-DIS is reducible for the electron channel via charge misidentification ($P_{\text{misID}}$) but irreducible for $\mu/\tau$ via heavy-flavor decays.
    2. Heavy Quark Production: Boson-gluon fusion ($e^- p \to e^- Q\bar{Q} + X$) where heavy quarks decay to leptons.
    3. W Boson Production: Negligible compared to DIS and heavy quarks.
• Analysis Strategy:
  • A cut-based analysis was performed (though the authors note AI/ML could improve this).
  • Kinematic Variables: Transverse momentum ($p_T$), missing transverse energy ($E_T^{\text{miss}}$), scalar sum of jet $p_T$ ($H_T$), and invariant mass ($M_{\ell jj}$).
  • Selection Cuts: Specific cuts were defined for each channel to maximize signal-to-background ratios (detailed in Boxes I–IV of the paper).

4. Key Contributions

1. Flavor-Dependent Analysis: Unlike previous works focusing on the electron channel, this paper provides a comprehensive study of $\mu$ and $\tau$ channels. It highlights that the muon channel offers a significantly cleaner signature due to the absence of charge misidentification issues and lower SM backgrounds.
2. Realistic Simulation: The inclusion of parton showering, hadronization, and detector effects (via Delphes) distinguishes this work from purely parton-level studies (like Ref. [18]). This reveals that resonant peaks in invariant mass distributions are significantly broadened, rendering simple mass-window cuts less effective than $H_T$ and $p_T$ cuts.
3. Comprehensive Background Handling: The study explicitly models charge misidentification for electrons and secondary lepton production from heavy quark decays for all channels, providing a robust background estimation.
4. $\nu$SMEFT Context: The analysis is framed within the general $\nu$SMEFT up to dimension seven, acknowledging that higher-dimensional operators could interfere with mixing-induced signals, though the primary focus remains on the mixing mechanism for benchmarking.

5. Results

• Sensitivity Projections:
  • The EIC with $100$ fb$^{-1}$ luminosity can probe mixing angles $|U_{\alpha 4} U_{e 4}| / U_4^2$ down to $\sim 10^{-5} - 10^{-4}$ for $m_4 \in [10, 100]$ GeV.
  • Muon Channel ($e^- p \to \mu^+ + jj$): This channel shows the best sensitivity, comparable to or exceeding current LHC constraints (CMS/ATLAS) in the 20–60 GeV mass range, provided a dedicated muon detector exists.
  • Tau Channel: Sensitivity is currently limited by the assumption of reconstructing $\tau$ only via its leptonic decay ($\tau \to \mu \nu \nu$). The authors note that including hadronic $\tau$ reconstruction would significantly improve sensitivity.
  • Electron Channel: Sensitivity is degraded by charge misidentification backgrounds ($P_{\text{misID}} \approx 10^{-3}$), reducing reach by an order of magnitude compared to the optimistic scenario.
• Comparison with Other Probes:
  • General $\nu$SMEFT Scenario: In the presence of higher-dimensional operators that can cancel indirect constraints, the EIC offers competitive sensitivity to the LHC.
  • Restricted 3+1 Scenario: If no higher-dimensional operators exist, indirect constraints (from $0\nu\beta\beta$, LEP, LHC, and meson decays) are generally stronger (by 1–2 orders of magnitude) than the projected EIC sensitivity. However, the EIC remains competitive with $\mu \to e$ conversion searches in the 20–60 GeV range.
• Cut Performance: The proposed kinematic cuts ($H_T > 50$ GeV, $p_T^\ell > 5$ GeV) effectively suppress DIS and heavy quark backgrounds while retaining signal efficiency.

6. Significance and Outlook

• Motivation for Muon Detection: The study strongly argues that the EIC's potential to discover BSM physics is critically dependent on the inclusion of a high-efficiency muon detector. Without it, the cleanest LNV channel ($\mu^+$) cannot be exploited.
• Complementarity: The EIC provides a unique kinematic regime (ep collisions) that complements pp colliders (LHC) and low-energy experiments. It is particularly sensitive to resonant production mechanisms that might be obscured by different backgrounds at the LHC.
• Future Work: The authors call for:
  1. Detailed experimental assessment of muon detection and hadronic $\tau$ reconstruction at the EIC.
  2. Theoretical extensions to include simultaneous contributions from mixing and higher-dimensional operators (dim-6, dim-7) to fully disentangle the origin of any potential signal.
  3. Application of multivariate analysis (AI/ML) to further improve sensitivity.

In conclusion, this paper establishes that the EIC is a powerful, yet under-explored, facility for probing Lepton Number Violation, specifically in the flavor-violating sectors, provided that appropriate detector capabilities (muon identification) are realized.