Prospects of five-dimensional $L_\mu-L_\tau$ gauge interactions in the light of elastic neutrino-electron scatterings: The scope of the DUNE near detector

This paper investigates the potential of the DUNE near detector to probe a five-dimensional $U(1)_{L_\mu - L_\tau}$ gauge extension of the Standard Model, which addresses the muon $(g-2)$ anomaly, by analyzing elastic neutrino-electron scattering data to explore MeV-scale parameter spaces and interference effects between multiple massive gauge bosons.

The Gist

Imagine the universe as a giant, multi-story building. In our everyday experience, we live on the ground floor (3 dimensions of space + 1 of time). But what if there are hidden floors above us that we can't see? This paper explores a theory where there is a fifth dimension—a tiny, hidden "attic" space that only certain particles can enter.

Here is the breakdown of the paper's story, using simple analogies:

1. The Mystery: The "Wobbly" Muon

Scientists have a particle called the muon (a heavy cousin of the electron). If you spin a muon in a magnetic field, it wobbles. We can predict exactly how much it should wobble based on our current rules of physics (the Standard Model).

However, experiments show the muon is wobbling more than predicted. It's like a spinning top that seems to have a secret weight inside it that we haven't found yet. This is the "Muon Anomalous Magnetic Moment" problem.

2. The Suspect: A Hidden Force

To fix this, physicists proposed a new force called $L_\mu - L_\tau$. Think of this as a secret handshake that only the 2nd and 3rd generation of particles (muons and taus) know how to do. Electrons don't know the handshake.

In this paper, the authors imagine this secret force lives in that 5th dimension. Because it lives in a higher dimension, it doesn't just appear as one particle; it appears as an infinite tower of particles (called Kaluza-Klein or "KK" particles), like a stack of identical-looking boxes, each slightly heavier than the one below it.

3. The Detective Work: The DUNE Experiment

How do we find these invisible boxes? We need a super-sensitive detector. The paper focuses on the DUNE experiment (Deep Underground Neutrino Experiment), specifically its "Near Detector."

• The Setup: Imagine a stream of neutrinos (ghostly particles that pass through walls) shooting past a giant tank of liquid argon.
• The Target: Inside the tank, there are electrons.
• The Game: Usually, neutrinos just bounce off electrons like billiard balls. But if our "secret force" exists, the neutrinos might hit the electron, bounce off a KK particle from the 5th dimension, and then hit the electron again.
• The Result: This extra bounce changes how the electron flies away. It's like hitting a billiard ball with a cue stick that has a hidden, bouncy spring on the end. The ball flies off at a weird angle or speed.

4. The Twist: The "Interference" Dance

The most exciting part of this paper is the concept of interference.

Imagine you are trying to hear a whisper in a noisy room.

• The Noise: The Standard Model (the usual physics).
• The Whisper: The new 5D particles.

Usually, you'd expect the whisper to just add to the noise, making it louder. But in this quantum world, the whisper can sometimes cancel out the noise, or sometimes amplify it, depending on the timing.

The authors found that because there are so many KK particles (the infinite tower), they create a complex dance. Sometimes, the effects of the new particles cancel each other out perfectly, creating a "Blind Spot" or a "Vacant Zone." In these zones, even if the new physics is there, the DUNE detector sees nothing because the signals cancel out. It's like two people shouting the same word at the exact same time, but one is upside down, so the sound waves cancel and you hear silence.

5. The Verdict: What DUNE Can Find

The paper runs simulations to see what DUNE can find over 1, 3, 5, or 7 years of data collection.

• The Good News: DUNE is incredibly powerful. With 5 to 7 years of data, it can scan a huge area of possibilities. It can find these hidden particles even if they are very weak or very heavy, covering the "MeV scale" (a specific size range of energy).
• The Challenge: Because of the "Blind Spots" mentioned above, there are some specific combinations of particle masses and strengths that DUNE might miss. If the new physics hides in a blind spot, DUNE won't see it.
• The Solution: To find the blind spots, we need other experiments (like looking at muon pairs in other ways) to cross-check.

Summary in a Nutshell

This paper is a treasure map.

• The Treasure: A solution to why the muon is wobbling too much.
• The Map: A theory involving a 5th dimension with a tower of hidden particles.
• The Explorer: The DUNE experiment.
• The Obstacle: A magical fog (interference) that sometimes hides the treasure in "blind spots."

The authors conclude that DUNE is likely to find the treasure within the next decade, but we need to be careful because the treasure might be hiding in a spot where the map's signal cancels itself out. If DUNE finds nothing, it doesn't mean the treasure isn't there; it might just mean we are standing in a blind spot and need a different pair of glasses to see it!

Technical Summary

1. Problem Statement

The paper addresses two primary issues in particle physics:

1. The Muon Anomalous Magnetic Moment ($g-2$) Discrepancy: There is a persistent $\sim 5.2\sigma$ tension between the experimental measurement of the muon's anomalous magnetic moment ($a_\mu$) and the Standard Model (SM) theoretical prediction.
2. The Limitations of 4D $U(1)_{L_\mu - L_\tau}$ Models: The standard 4D extension of the SM with a $U(1)_{L_\mu - L_\tau}$ gauge symmetry is a well-motivated solution to the $(g-2)_\mu$ anomaly. However, it typically relies on a single massive gauge boson ($Z'$). The authors propose extending this scenario to five dimensions (5D) to explore a richer phenomenology involving a Kaluza-Klein (KK) tower of massive gauge bosons.

The specific challenge is to determine if the upcoming DUNE (Deep Underground Neutrino Experiment) Near Detector (ND) can probe this 5D scenario, particularly in the MeV mass scale region, which has been largely uncharted by previous experiments like CHARM-II and Borexino.

2. Methodology

Theoretical Framework

• 5D Setup: The authors consider a 5D spacetime where the SM particles are confined to a 3-brane at position $y = y_{SM}$, while the $U(1)_{L_\mu - L_\tau}$ gauge field propagates in the bulk.
• Boundary Conditions: They impose twisted boundary conditions (Neumann at $y=0$ and Dirichlet at $y=\pi R$). This ensures no massless zero-mode exists, generating a tower of massive KK gauge bosons ($V^{(n)}$) with masses $M_n$.
• Geometries: Both flat and warped (Randall-Sundrum-like) extra-dimensional geometries are analyzed.
• Effective Lagrangian: The 5D theory is reduced to an effective 4D Lagrangian via KK decomposition. Key parameters include:
  • $g'$: The effective 4D gauge coupling.
  • $\epsilon_4$: The effective kinetic mixing parameter between the new gauge bosons and the SM hypercharge.
  • $m_{KK}$: The mass scale of the first KK mode.
  • $f_n$: The wavefunction profile of the $n$-th KK mode at the SM brane.

Phenomenological Analysis

• Process: The study focuses on Elastic Neutrino-Electron Scattering (E$\nu$ES): $\nu_X + e^- \to \nu_X + e^-$.
• Interference Effects: The amplitude includes contributions from SM $W$ and $Z$ bosons, as well as the exchange of the entire KK tower of $V^{(n)}$ bosons. The authors emphasize the importance of interference between these intermediate states, which can be constructive or destructive.
• Experimental Inputs:
  • DUNE ND: Simulated using the expected neutrino flux (FHC and RHC modes), fiducial mass (75 tons of LAr), and running time (1 to 7 years). Backgrounds (CCQE, $\pi^0$) are estimated with systematic uncertainties.
  • Existing Constraints: Current limits from CHARM-II, Borexino, and TEXONO are incorporated.
  • Muon $(g-2)_\mu$: The parameter space is constrained by the requirement to explain the $(g-2)_\mu$ anomaly within a $2\sigma$ range.
• Statistical Analysis: A $\chi^2$ analysis is performed using a bin-by-bin approach on the recoil energy ($T_e$) and the variable $E_e \theta_e^2$ to distinguish signal from background.

3. Key Contributions

1. First Dedicated 5D Analysis for DUNE: This is the first study to systematically evaluate the discovery potential of the DUNE Near Detector for a 5D $U(1)_{L_\mu - L_\tau}$ model via E$\nu$ES.
2. Role of the KK Tower: The paper demonstrates that the contribution of the entire KK tower (not just the lightest mode) is crucial. Even though heavier modes decouple, their cumulative effect and interference with the SM significantly alter the cross-section.
3. Interference "Blind Spots": The authors identify specific regions in the parameter space ($g'$ vs. $m_{KK}$) where destructive interference between the SM amplitudes and the new physics amplitudes cancels out the deviation. These "vacant zones" or "blind spots" are insensitive to DUNE data, even after 7 years of running.
4. Comparison of Geometries: The study provides a direct comparison between flat and warped extra dimensions, showing that while the qualitative features (blind spots) persist, the quantitative sensitivity and the required coupling strengths differ.
5. Kinetic Mixing Sensitivity: The analysis shows that DUNE can probe extremely small kinetic mixing parameters ($\epsilon_4 \sim 10^{-7}$) in the flat case, provided the data accumulation is sufficient, due to the interference effects.

4. Results

• Discovery Potential:
  • With 5 to 7 years of DUNE ND data, the experiment can probe a significant portion of the parameter space that explains the $(g-2)_\mu$ anomaly.
  • The sensitivity extends to very small gauge couplings ($g'$) and kinetic mixing ($\epsilon_4$), covering regions previously inaccessible to CHARM-II and Borexino.
• Blind Spots:
  • In both flat and warped scenarios, distinct "vacant zones" appear in the exclusion plots. In these regions, the new physics contribution is effectively canceled by interference with the SM, making the signal indistinguishable from the background.
  • These zones are a unique signature of the KK tower structure.
• Coupling Strength:
  • The 5D scenario generally requires larger gauge couplings ($g'$) to explain $(g-2)_\mu$ compared to the 4D vanilla scenario. This is attributed to the non-vector-like nature of the effective interactions in the mass eigenbasis (due to kinetic mixing), which introduces negative contributions to $(g-2)_\mu$.
  • However, for very small $\epsilon_4$, the vector contribution dominates, allowing for smaller $g'$ values.
• Convergence: The authors verified that summing up to $n_{KKmax} = 50$ is sufficient for convergence in the MeV mass scale regime.

5. Significance

• Complementarity: The paper highlights that DUNE's Near Detector is a powerful, yet underutilized, tool for probing MeV-scale new physics. It complements other searches like Neutrino Trident Production and beam dump experiments.
• Model Discrimination: The presence of "blind spots" due to interference is a unique prediction of the 5D KK tower scenario. If DUNE observes a deviation consistent with the model but misses specific parameter regions, it would strongly support the multi-boson (KK tower) hypothesis over a single $Z'$ model.
• Future Directions: The authors note that to probe the "blind spots," processes involving muons in the initial or final state (e.g., Neutrino Trident production) are necessary, as the KK bosons couple directly to muons but only indirectly to electrons (via mixing).
• Theoretical Robustness: The paper confirms that the 5D $U(1)_{L_\mu - L_\tau}$ framework remains a viable and testable solution to the muon $(g-2)$ anomaly, with the DUNE experiment poised to either discover it or severely constrain its parameter space.

In conclusion, the paper establishes that the DUNE Near Detector has the potential to revolutionize our understanding of MeV-scale extra dimensions and $L_\mu - L_\tau$ gauge symmetries, provided that the complex interference effects of the KK tower are properly accounted for in the analysis.