Lepton Double Charge Exchange Reactions as Probes for Lepton Number Violation

This paper proposes and analyzes lepton double charge exchange reactions ($A(e^-,e^+)$X) at multi-GeV accelerator facilities as a feasible and powerful probe for lepton number violation and physics beyond the Standard Model, predicting sizable cross sections driven by left-right mixing and heavy neutral leptons that are free from the constraints of traditional decay or capture experiments.

The Gist

Imagine the universe as a giant, complex machine built according to a specific rulebook called the Standard Model. For decades, physicists have been trying to find a "glitch" in this rulebook—a sign that the machine is actually running on a secret, more advanced operating system. One of the biggest clues they are looking for is something called Lepton Number Violation (LNV).

In simple terms, "Lepton Number" is like a cosmic accounting rule: if you start with a certain number of "electron-like" particles, you should end up with the same number. Violation means breaking this rule. If we can prove this rule can be broken, it opens the door to understanding why the universe has more matter than antimatter and what dark matter might be.

Here is what this paper proposes, explained through everyday analogies:

1. The Problem: The "Too Small" vs. "Too Big" Gap

Currently, scientists are hunting for these rule-breaking events in two very different places:

• The Microscopic Lab (Nuclear Decay): They look at heavy atoms slowly decaying over billions of years. It's like waiting for a single grain of sand to fall off a mountain. It's slow, and the "noise" from the atom's internal structure makes it hard to see the signal.
• The Giant Collider (LHC): They smash particles together at near-light speed. This is like crashing two freight trains to see what parts fly off. It's powerful, but it requires massive, expensive machines and produces so much "debris" that finding the specific rule-breaking event is like finding a needle in a haystack.

The Gap: There is a huge empty space between these two approaches. We need a way to bridge the gap between the slow, quiet nuclear world and the chaotic, high-energy crash zone.

2. The Solution: The "Double Charge Exchange" (LDCE)

The authors propose a new experiment using electron beams (like those at the Jefferson Lab in the US) hitting heavy metal targets (like Lead or Oxygen).

Think of this reaction as a Cosmic Swap Meet:

• You shoot an electron (negative charge) at a nucleus.
• Instead of just bouncing off, the nucleus does something weird: it swaps two of its positive charges for the electron's negative charge.
• The result? The electron turns into a positron (a positive anti-electron), and the nucleus loses two protons.

Why is this special?
In our normal universe, you can't just turn an electron into a positron without creating a matching pair of particles to balance the books. If this reaction happens, it proves that neutrinos (ghostly, invisible particles) are their own antiparticles. This is the "smoking gun" for the new physics the authors are looking for.

3. The "Black Box" Connection

The paper uses a famous idea called the Black Box Theorem. Imagine a black box with a mystery inside.

• If you see a specific event happen on the left side of the box (nuclear decay), it proves the mystery exists inside.
• If you see it on the right side (high-energy collisions), it also proves the mystery exists.
• The Paper's Insight: This new experiment (LDCE) is a third side of the box. If we see the reaction here, it mathematically guarantees that the mystery exists, regardless of whether we find it in the slow nuclear decay or the high-energy crashes. It connects all three worlds.

4. How It Works: The "Heavy Neutrino" Bridge

Usually, these reactions are incredibly rare because they rely on "light" neutrinos, which are very shy. However, the authors suggest that at high energies (multi-GeV), the reaction might be driven by Heavy Neutral Leptons (HNLs).

The Analogy:
Imagine trying to push a heavy boulder (the reaction) across a field.

• Light Neutrinos: Are like a tiny pebble. You can push it, but it doesn't go far.
• Heavy Neutrinos: Are like a massive boulder. If they exist, they act as a giant lever. At high energies, this lever makes the reaction happen much more often, creating a signal strong enough to detect.

The paper calculates that if these heavy particles exist, the reaction rate could be high enough to be seen with current technology, especially if we use heavy targets (like Lead-208) and high-energy electron beams.

5. Why This Matters Now

• Feasibility: Unlike the massive colliders, this experiment could be done with existing equipment (like the Jefferson Lab) by simply changing the target and the detectors.
• The "Sweet Spot": It operates in an energy range (10 GeV) that hasn't been explored for this specific type of physics. It's a "terra incognita" (unknown territory) that is ripe for discovery.
• The Stakes:
  • If they find it: It's a Nobel Prize-level breakthrough, proving neutrinos are their own antiparticles and revealing new laws of physics.
  • If they don't find it: It's still a huge win. It sets strict limits on where these new particles can't be, forcing physicists to rethink their theories and guiding future experiments.

Summary

This paper is a proposal to build a new bridge between the quiet world of nuclear physics and the violent world of particle colliders. By shooting high-speed electrons at heavy atoms and watching for a specific "double swap" of electric charge, they hope to catch a glimpse of a hidden particle that could rewrite our understanding of the universe. It's a low-cost, high-reward gamble that could finally crack the code of why the universe exists.

Technical Summary

1. Problem Statement

The search for physics Beyond the Standard Model (BSM), specifically Lepton Number Violation (LNV), faces a significant challenge: the extreme energy gap between low-energy nuclear processes and high-energy collider events.

• Current Approaches:
  • Nuclear Scale: Neutrinoless double beta decay ($0\nu\beta\beta$) operates at MeV scales but is constrained by complex nuclear structure uncertainties.
  • Collider Scale: High-energy processes like same-charge lepton pairs plus di-jets ($lljj$) at the LHC operate at TeV scales but require massive energies to bridge the gap to nuclear phenomena.
• The Gap: There is a lack of experimental methods capable of bridging the MeV–TeV gap to probe LNV dynamics in a controlled environment.
• Proposed Solution: The authors propose investigating Lepton Double Charge Exchange (LDCE) reactions, specifically $A(e^-, e^+)X$, on nuclear targets at multi-GeV accelerator facilities. This approach aims to bypass the structural constraints of $0\nu\beta\beta$ while remaining feasible with existing technology.

2. Methodology

The paper employs a theoretical framework combining the Left-Right Symmetric Model (LRSM) with a phenomenological approach to estimate cross-sections.

• Theoretical Framework (LRSM):

  • The study utilizes the LRSM, which treats left-handed (LH) and right-handed (RH) lepton currents on equal footing.
  • The effective low-energy Lagrangian includes terms for LR-mixing induced by RH currents and weak gauge boson mixing.
  • The reaction is modeled as a second-order double charged current (DCC) process. Unlike $0\nu\beta\beta$ (a $t$-channel exchange), LDCE is an $s$-channel process, allowing direct experimental access to the intermediate states.
  • The amplitude is expressed as a sum over helicity-projected lepton states involving Majorana neutrinos (both light and heavy).

• Black Box Theorem (BBT) Application:

  • The authors invoke the generalized Black Box Theorem, which posits that if any LNV process exists (e.g., $0\nu\beta\beta$, $lljj$, or LDCE), then neutrinos must be Majorana fermions. Thus, observing LDCE would confirm the Majorana nature of neutrinos.

• Phenomenological Estimation:

  • Since full-scale calculations for multi-GeV energies are complex, the authors use a semi-phenomenological approach.
  • They parameterize available $\nu_\mu$ and $\bar{\nu}_\mu$ charged current (CC) cross-section data using Lorentzian and Gaussian distributions.
  • Assuming lepton universality, these empirical matrix elements are adapted to calculate the $(e^-, \nu_e)$ and $(\bar{\nu}_e, e^+)$ amplitudes required for the LDCE process.
  • The calculation accounts for three distinct nuclear response regions: Quasi-Elastic (QE), Resonance Excitation (RE), and Deep-Inelastic (DI) scattering.
  • A "no-recoil" approximation is used for intermediate states, assuming the intermediate nuclear mass $M_C \gg m_i$ (neutrino mass).

3. Key Contributions

• Proposal of LDCE as a New Probe: The paper identifies $A(e^-, e^+)X$ reactions as a viable, previously unexplored avenue for LNV searches, distinct from decay experiments.
• Formalism Development: It presents a second-order formalism incorporating LRSM dynamics, explicitly separating helicity-diagonal mass terms from helicity-non-diagonal momentum-dependent mixing terms.
• Energy-Momentum Dependence: The authors demonstrate that LDCE cross-sections are dominated by energy-momentum dependent LR-mixing terms rather than Majorana mass terms (for light neutrinos).
• Feasibility Analysis: The study provides the first numerical estimates for LDCE cross-sections at multi-GeV energies (up to 100 GeV), identifying specific experimental conditions (beam energy, target mass) that maximize sensitivity.

4. Key Results

• Cross-Section Magnitude:
  • At beam energies around 10 GeV, inclusive total LDCE cross-sections are predicted to be approximately $100 \times |\Gamma_{BSM}|^2$ fb (femtobarns), where $|\Gamma_{BSM}|^2$ represents the unknown BSM vertex strength.
  • The cross-section scales strongly with energy, following a power law $\sigma \propto T_{lab}^3$.
  • The cross-section also increases significantly with target mass number $A$, showing a clear preference for heavy nuclei ($A > 100$, e.g., $^{208}\text{Pb}$).
• Role of Neutrino Masses:
  • Light Neutrinos (LNL): For light neutrinos ($m_i \sim 1$ eV), the Majorana mass terms are heavily suppressed (quenched) by factors of $R_i \approx 10^{-13}$ at 10 GeV.
  • Heavy Neutral Leptons (HNL): For heavy neutrinos ($m_i \sim 100$ GeV), the mass terms become comparable to momentum terms ($R_i \sim O(1)$), potentially inducing a significant enhancement in cross-sections even at lower beam energies.
• Kinematics:
  • Prompt positrons ($e^+$) are expected to be emitted at forward angles ($5^\circ - 15^\circ$) with energies close to the beam energy.
  • To distinguish signal from background, the detection of the positron must be in coincidence with other reaction products (hadrons, pions) to reconstruct the final nuclear configuration.
• Facility Potential:
  • Jefferson Lab (JLab): With 12 GeV beams, JLab could potentially reach sensitivities down to $|\Gamma_{BSM}|^2 \approx 10^{-6}$.
  • Electron-Ion Collider (EIC): Operating at center-of-mass energies up to 140 GeV, the EIC could offer a cross-section increase of approximately 4 orders of magnitude, despite lower luminosity.

5. Significance and Outlook

• Bridging the Energy Gap: LDCE reactions offer a unique "middle ground" between nuclear decay experiments and high-energy collider searches, probing LNV in a regime where nuclear structure effects are less dominant than in $0\nu\beta\beta$ but energies are lower than the LHC.
• Model Independence: The approach relies on the general LRSM framework without requiring specific assumptions about BSM mechanisms beyond the existence of LR mixing.
• Experimental Feasibility: The paper argues that LDCE experiments are feasible with existing equipment (e.g., JLab) under full laboratory control.
• Scientific Impact:
  • Positive Observation: Even the detection of a few unambiguous LDCE events would constitute a major breakthrough, proving neutrinos are Majorana fermions and confirming LNV.
  • Null Result: The absence of events would still be highly valuable, establishing upper limits on LNV in a previously unexplored energy region and guiding future second-generation experiments.
• Future Directions: The authors highlight the need to study recoil effects (crucial for HNLs) and the potential impact of higher-dimensional operators, which are currently unknown in the multi-GeV regime.

In conclusion, the paper establishes a strong theoretical and phenomenological basis for using multi-GeV electron beams to search for Lepton Number Violation via double charge exchange, offering a promising, complementary path to current BSM research.