Reaction Studies of Lepton Number Violation

This paper proposes lepton number violation (LNV) reactions on nuclei at accelerators as a promising new approach to quantitatively study lepton number violation, such as in double beta decay, by investigating heavy ion double single charge exchange and direct Majorana double charge exchange reactions alongside isotensor two-body transition densities and pion potentials.

The Gist

Imagine the universe is built on a set of strict rules, like the laws of a giant, cosmic game. One of these rules is "Lepton Number Conservation." Think of this like a bank account where you can't just create money out of thin air; the total amount of "lepton currency" (particles like electrons and neutrinos) must always stay the same.

However, physicists suspect there might be a loophole in this rule—a way to break the bank. If we find it, it proves there is "New Physics" beyond our current understanding of the universe. This paper is about building better tools to find that loophole.

Here is a simple breakdown of the three main ideas in the paper, using everyday analogies:

1. The "Double Swap" Game (Nuclear Reactions)

The author is studying a specific type of nuclear reaction called Double Charge Exchange (DCE).

• The Analogy: Imagine two teams of people (nuclei) standing in a line. Usually, if one person swaps places with another, it's a simple trade. But in this "Double Swap" game, two people from Team A swap with two people from Team B simultaneously.
• The Goal: The scientists want to see how the teams rearrange themselves. This rearrangement is called "isotensor spectroscopy." It's like looking at the internal wiring of a machine to see how it handles a double shock.
• Why it matters: This process mimics what happens in Double Beta Decay, a rare event where a nucleus changes its identity by swapping two particles. If this happens without emitting neutrinos (which carry away the "lepton currency"), it breaks the rules of the universe. By studying the "Double Swap" in the lab, we can understand the hidden mechanics of that rare decay.

2. The Two Ways to Swap (DSCE vs. MDCE)

The paper discusses two different ways this "Double Swap" can happen, like two different delivery methods for a package.

• Method A: The Relay Race (DSCE)
  • How it works: Two heavy ions (like fast-moving bowling balls) collide. They pass a "baton" (a force carrier) back and forth in two steps. First, they swap one thing, then they swap another.
  • The Metaphor: It's like two people passing a ball to each other, then passing it back. It happens in a sequence.
• Method B: The Teleportation (MDCE)
  • How it works: This is more direct. Instead of a relay, the two nuclei interact instantly through a "pion" (a tiny particle that acts like a messenger).
  • The Metaphor: Imagine the two nuclei are connected by a magical, invisible rubber band (the pion). When one moves, the other feels it immediately. This creates a "pion potential," which is like a map showing how strong that invisible connection is.
• The Discovery: The author calculated these "maps" for the first time. They found that this connection is very short-range (like a whisper heard only by someone standing right next to you), unlike the long-range whispers of neutrinos.

3. The "Electron-Positron" Shortcut (Lepton DCE)

This is the most exciting new idea in the paper. Instead of smashing heavy nuclei together, the author suggests using a particle accelerator to fire electrons at a target and watch them turn into anti-electrons (positrons).

• The Analogy: Imagine you have a machine that turns a "good" coin (electron) into a "bad" coin (positron) instantly. In the real world, this is forbidden. But if the universe has a secret loophole (Lepton Number Violation), this machine might work.
• The Experiment: The paper proposes firing high-energy electrons at a heavy target (like Lead). If the "loophole" exists, the electron will vanish and a positron will appear, changing the nucleus in the process.
• Why it's cool:
  • Control: Unlike waiting for a rare natural decay (which might take billions of years), we can control this experiment in a lab.
  • Energy: The paper suggests that using very high-energy beams (like those planned for future colliders) makes this reaction much more likely to happen. It's like turning up the volume on a radio; the louder you turn it up, the easier it is to hear the signal.
  • The Result: The author did the math and estimated that if we use a heavy target like Lead and a powerful beam, we might finally see this "forbidden" reaction.

The Big Picture

Think of this paper as a blueprint for a new detective agency.

1. Old Detective Work: We've been waiting for nature to show us a crime (Double Beta Decay), but it happens so rarely we can barely see it.
2. New Detective Work:
   • Tool 1: We are building better microscopes (Heavy Ion Collisions) to understand the "crime scene" (the nucleus) in extreme detail.
   • Tool 2: We are proposing a new crime scene simulator (Electron-Positron reactions) where we can force the crime to happen in a controlled environment to catch the culprit.

If successful, this research won't just tell us how atoms work; it will tell us why the universe exists at all, potentially revealing physics that is currently hidden from our view.

Technical Summary

Based on the provided paper "Reaction Studies of Lepton Number Violation" by H. Lenske, here is a detailed technical summary covering the problem, methodology, key contributions, results, and significance.

1. Problem Statement

The primary scientific challenge addressed is the quantitative understanding of Lepton Number Violation (LNV), specifically in the context of Neutrinoless Double Beta Decay (MDBD).

• The Gap: While MDBD is a signature of physics beyond the Standard Model (BSM), its theoretical interpretation relies heavily on nuclear matrix elements (NMEs) involving isotensor two-body transition densities. These nuclear configurations are currently poorly understood and rarely studied experimentally.
• The Need: There is a critical need for experimental and theoretical tools to bridge the gap between low-energy nuclear physics (MDBD) and high-energy collisional experiments to better constrain the nuclear physics inputs required for interpreting LNV signals.

2. Methodology

The paper employs a dual approach, combining heavy-ion nuclear reaction theory with a proposed new accelerator-based lepton scattering framework.

A. Heavy Ion Double Charge Exchange (DCE) Reactions

The author analyzes two distinct mechanisms for DCE reactions ($A + A' \to B + B'$), where nuclear charge changes by $\pm 2$ units:

1. Double Single Charge Exchange (DSCE): A sequential, second-order process involving two nucleon-nucleon ($NN$) isovector $T$-matrices ($T_{NN}$).
2. Direct Majorana Double Charge Exchange (MDCE): A first-order ion-ion interaction but second-order within each nucleus, mediated by the pion-nucleon ($T_{\pi N}$) isovector $T$-matrix.

• Theoretical Framework: The study utilizes Feynman diagrams to derive amplitude equations.
  • MDCE Amplitude: Calculated using nuclear two-body isotensor transition form factors and a pion propagator ($D_{\pi\pi}$). It is evaluated in the closure approximation, leading to the derivation of pion potentials ($U_{\pi}$) which mirror the neutrino potentials used in MDBD calculations.
  • DSCE Amplitude: Analyzed via a kernel ($K_{\alpha\beta}$) acting as an effective isotensor interaction, transforming the $t$-channel formulation into an $s$-channel representation to probe intrinsic nuclear responses.
• Specific Application: Theoretical calculations were performed for the reaction $^{76}\text{Se}(^{18}\text{O}, ^{18}\text{Ne})^{76}\text{Ge}$, which is the inverse of the MDBD candidate $^{76}\text{Ge} \to ^{76}\text{Se}$.

B. Lepton Double Charge Exchange (LDCE) Reactions

A novel theoretical proposal is introduced for accelerator experiments: $(e^-, e^+)$ reactions on nuclear targets.

• Framework: Based on the Left-Right Symmetric Model (LRSM).
• Mechanism: The process is treated as a second-order perturbation theory problem. The amplitude involves charged current (CC) amplitudes for $(e^-, \nu)$ and $(\bar{\nu}, e^+)$ transitions.
• Approach: Due to the lack of data at high energies, a semi-phenomenological approach is adopted. CC matrix elements are extracted from existing compilations (up to the deep-inelastic region) and extrapolated to evaluate the cross-sections for the $s$-channel scattering process.

3. Key Contributions

• First-Time Investigation of Isotensor TBTD: The paper presents the first theoretical study of isotensor two-body transition densities (TBTD) in momentum space. It specifically maps the double-Fermi, double-Gamow-Teller, and mixed Fermi-Gamow-Teller components for the $^{76}\text{Se} \to ^{76}\text{Ge}$ transition.
• Derivation of Pion Potentials: The author derives pion potentials ($U_{00}, U_{22}$) for the MDCE mechanism. These potentials define the strength of double-Fermi and double-Gamow-Teller components and serve as a direct analog to the neutrino potentials in MDBD, but with much shorter correlation lengths ($\sim 1$ fm).
• Proposal of LDCE as a New Probe: The paper introduces Lepton DCE (LDCE) reactions as a promising, previously unexplored method to investigate LNV under full laboratory control at accelerators.
• Cross-Section Estimation: It provides the first theoretical estimates for $(e^-, e^+)$ cross-sections on heavy nuclei, scaling with energy and target mass.

4. Results

• Nuclear Structure Insights:
  • The pion potentials ($U_{ij}$) in MDCE are found to be almost independent of external momenta and nuclear masses in collinear kinematics.
  • The correlation lengths for these potentials are short-range ($\sqrt{\langle x^2 \rangle} \simeq 1$ fm), contrasting with the long-range correlations of neutrino potentials in MDBD.
  • The momentum space TBTD for the $^{76}\text{Se} \to ^{76}\text{Ge}$ transition was successfully mapped, showing dependence on the total momentum transfer $q$ and individual momenta $p_{1,2}$ extending to $\pm 400$ MeV/c.
• LDCE Cross-Sections:
  • The total cross-section for $(e^-, e^+)$ reactions on a $^{208}\text{Pb}$ target scales approximately as $\sigma \sim E^3$.
  • At a beam energy of 10 GeV, the estimated cross-section is $\sigma_{e^-e^+} \sim |\Gamma_{BSM}|^2 10^{-38} \text{cm}^2$, where $\Gamma_{BSM}$ is the dimensionless LNV parameter (estimated $\lesssim 10^{-7}$).
  • The results indicate that using heavy target nuclei and high-energy electron beams (e.g., at the planned Electron-Ion Collider or Jefferson Lab) offers the most favorable conditions for detection.

5. Significance

• Bridging Scales: The work successfully bridges the gap between low-energy nuclear spectroscopy (relevant for DBD) and high-energy collision physics.
• Experimental Guidance: By characterizing the isotensor response and pion potentials, the study provides essential nuclear structure inputs needed to interpret future MDBD experiments (like the NUMEN project).
• New Discovery Channel: The proposal of LDCE reactions offers a distinct, controllable experimental pathway to search for LNV, potentially complementing or surpassing the limitations of passive decay experiments.
• Theoretical Foundation: It establishes the theoretical formalism for isotensor two-body transition densities, a crucial but previously neglected component in the calculation of neutrinoless double beta decay rates.

In summary, Lenske's paper advances the field by providing the first detailed spectroscopic analysis of isotensor nuclear responses via heavy-ion reactions and proposing a viable, high-energy accelerator-based alternative for directly probing lepton number violation.