Muon Beam Dump Experiments explicate five-dimensional nature of $U(1)_{L_{\mu}-L_{\tau}}$

This paper investigates how present and future muon beam dump experiments can probe the five-dimensional nature of $U(1)_{L_\mu-L_\tau}$ interactions by leveraging the enhanced signals from multiple Kaluza-Klein gauge bosons and their distinct decay channels to provide direct evidence for extra dimensions while constraining parameters via the muon $(g-2)$ anomaly.

The Gist

Imagine our universe as a giant, multi-story building. For decades, physicists have been convinced that all the "residents" (particles like electrons and quarks) live on the ground floor, which we call 4D space-time (three dimensions of space plus time).

But what if some residents are actually living on the 5th floor, a dimension we can't see or touch? This paper is a detective story about how we might catch a glimpse of these "5th-floor" residents using a very specific type of particle: the Muon.

Here is the story of the paper, broken down into simple concepts.

1. The Mystery: A Ghostly Force

Physicists are looking for a new, invisible force carrier called a $Z'$ boson. Think of this particle as a "ghost messenger."

• The 4D Theory: In the standard view, there is only one type of this ghost messenger. It's like having a single key that opens a specific lock.
• The 5D Theory: The authors propose that this force actually lives in a 5th dimension. In this scenario, the "ghost messenger" isn't just one particle; it's an infinite tower of clones.
  • The Analogy: Imagine a guitar string. When you pluck it, it doesn't just make one note; it makes a fundamental note plus many higher harmonics (overtones). In the 5D world, the $Z'$ boson is like that string. We see the "fundamental note" (the first particle), but there are also "higher notes" (called Kaluza-Klein or KK modes) that are heavier and heavier versions of the same particle.

2. The Detectives: Muon Beam Dump Experiments

To find these ghosts, the paper looks at four different "detective teams" (experiments) currently running or planned: NA64µ, M3, MuSIC, and a Future Muon Dump.

They all use the same basic trick: The Beam Dump.

• The Setup: They fire a high-speed beam of Muons (heavy cousins of electrons) at a thick wall of material (like lead, tungsten, or water).
• The Goal: Most muons crash into the wall and stop. But occasionally, a muon might "sneak" a ghost messenger ($Z'$) out of the wall before it stops.
• The Two Types of Clues:
  1. The "Invisible" Clue (NA64µ & M3): The ghost messenger flies away and disappears into thin air (decaying into invisible neutrinos). The detector sees a muon that suddenly lost energy, but nothing else. It's like seeing a magician make a ball vanish.
  2. The "Visible" Clue (MuSIC & Future Dump): The ghost messenger flies a bit, then decays into a pair of new muons. The detector sees two new particles appearing out of nowhere. This is like the magician making two rabbits appear from a hat.

3. The Big Discovery: Why 5D is Different

The paper's main point is that if the universe is 5D, the "Infinite Tower of Clones" changes the game in two exciting ways:

A. More Ghosts = More Signals
In a 4D world, you only have one ghost messenger to look for. In a 5D world, you have a whole tower of them. Even if the first one is hard to find, the second, third, or fourth might be easier to spot depending on the energy of the beam.

• The Result: The experiments become much more sensitive. They can find these particles in places where a 4D theory says they shouldn't exist.

B. The "Fingerprint" of Extra Dimensions
This is the coolest part. If the "Visible Clue" experiments (MuSIC) see the ghost messenger decay into two muons, they can measure the mass of the messenger.

• In 4D: You would see just one specific mass.
• In 5D: You would see multiple distinct masses (the fundamental note and the overtones).
• The Analogy: If you hear a single piano key, it's just a note. If you hear a chord with specific, distinct notes played together, you know it's a complex instrument. Finding multiple distinct masses would be the "smoking gun" proof that the force lives in a 5th dimension.

4. The "Mixing" Complication

The paper also discusses a tricky variable called Kinetic Mixing.

• The Analogy: Imagine the ghost messenger is trying to talk to the muons, but it's wearing a disguise (mixing with the photon/electromagnetic force).
• Sometimes, the disguise helps the ghost hide better (making it harder to find).
• Other times, the disguise actually helps the ghost interact more strongly, making it easier to find.
• The authors show that even if this mixing is tiny, it changes the "search map" significantly, opening up new areas where we might find these particles.

5. The "G-2" Update

For a long time, scientists thought these particles could explain a weird measurement called the Muon (g-2) (a tiny wobble in how muons spin).

• The Plot Twist: Recent measurements show the "wobble" actually matches the Standard Model perfectly. The "anomaly" is gone.
• The Good News: This doesn't kill the theory. It just means we can't use the "wobble" to find the particles anymore. Instead, we have to rely on the Beam Dump experiments described above. The paper shows that even without the "wobble" mystery, these experiments are powerful enough to hunt down the 5D particles.

Summary: What's the Takeaway?

This paper is a roadmap for the next decade of particle physics. It tells us:

1. Don't just look for one particle. If extra dimensions exist, we are looking for a whole family of them.
2. Muon beams are the best flashlights. They are the most effective way to shine a light on these hidden dimensions.
3. Mass reconstruction is key. If we can catch the "ghost messenger" decaying into muons and measure its mass, and we see multiple different masses, we will have proven that our universe has a hidden 5th dimension.

It's like searching for a specific type of bird. In a 4D world, you look for one species. In a 5D world, you are looking for a whole flock of related species, and finding even one of the rare ones proves the forest is bigger than we thought.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the search for light, feebly interacting particles (FIPs) that couple to muons, specifically within the framework of a five-dimensional (5D) $U(1)_{L_\mu - L_\tau}$ gauge extension of the Standard Model (SM).

• Context: While the 4D version of this model is well-studied (often motivated by the muon $(g-2)$ anomaly, though recent data suggests the anomaly may be resolved), the 5D scenario introduces a tower of Kaluza-Klein (KK) excitations of the gauge boson ($Z'$).
• The Challenge: Distinguishing between a 4D model with a single $Z'$ and a 5D model with multiple $Z'$ states is difficult. Standard constraints often treat the new physics as a single effective particle.
• Goal: The authors aim to determine if current and future muon beam dump experiments (NA64$\mu$, M3, MuSIC, and a future high-energy setup) can probe the 5D nature of this interaction, specifically looking for signatures of multiple KK modes and the effects of kinetic mixing.

2. Theoretical Framework (Methodology)

The authors construct a 5D model where:

• Geometry: The SM fields are confined to a 3-brane at $y = y_{SM}$, while the $U(1)_{L_\mu - L_\tau}$ gauge field propagates in the 5D bulk ($y \in [0, \pi R]$).
• Boundary Conditions: Twisted boundary conditions (Neumann at $y=0$, Dirichlet at $y=\pi R$) are applied, ensuring no massless mode exists in the spectrum.
• Mass Spectrum: The gauge bosons appear as a tower of massive KK modes ($Z'^{(n)}$) with masses $M_n = (2n-1)m_{KK}$, where $m_{KK} = 1/(2R)$.
• Kinetic Mixing: The model includes a kinetic mixing parameter $\epsilon_4$ between the $Z'$ and the SM hypercharge boson, which induces couplings to electrons and modifies the vector/axial-vector couplings to muons.
• Effective Lagrangian: The authors derive the effective 4D Lagrangian, explicitly calculating the mixing between the $Z$ boson and the KK tower, and determining the vector ($C_V$) and axial-vector ($C_A$) couplings for each mode.

3. Methodology: Signal Estimation

The study analyzes four specific muon beam dump experiments, categorized by their detection channels:

A. Invisible Decay Channels ($Z' \to \text{invisible}$)

• Experiments: NA64$\mu$ (CERN, 160 GeV beam) and M3 (Fermilab, 15 GeV beam).
• Process: Muon Bremsstrahlung ($\mu A \to \mu A Z'$) followed by invisible decay ($Z' \to \nu\bar{\nu}$).
• Method:
  • Calculated the differential cross-section using the Weizsäcker-Williams (WW) approximation.
  • Summed the signal events over the first $n_{KK}^{max}$ modes.
  • Applied 90% CL exclusion limits based on the number of signal events ($N_{signal} > 2.3$).

B. Visible Decay Channels ($Z' \to \mu^+\mu^-$)

• Experiments: MuSIC (proposed Muon Synchrotron Ion Collider, 8.9 TeV) and a Future High-Energy Muon Beam Dump (1.5 TeV).
• Process: Muon Bremsstrahlung followed by visible decay into muon pairs.
• Method:
  • Calculated production cross-sections and decay probabilities within a displaced detector geometry.
  • Accounted for the decay length ($\ell_{Z'} = \gamma / \Gamma_{Z'}$) of each KK mode.
  • Key Feature: The ability to reconstruct the invariant mass of the muon pair allows for the identification of distinct mass peaks corresponding to different KK modes ($M_n$).

4. Key Contributions and Results

A. Enhancement via Multiple KK Modes

• Signal Summation: The total signal is the sum of contributions from all kinematically accessible KK modes.
• Result: Moving from a single-mode truncation ($n_{KK}^{max}=1$) to a five-mode truncation ($n_{KK}^{max}=5$) significantly enhances the exclusion sensitivity, particularly for NA64$\mu$ and M3. The presence of multiple modes broadens the parameter space where constraints can be set.

B. Distinguishing 4D vs. 5D via Mass Reconstruction

• Crucial Finding: In experiments with displaced detectors (MuSIC and Future Beam Dump), the decay $Z'^{(n)} \to \mu^+\mu^-$ allows for mass reconstruction.
• Significance: Observing multiple distinct mass peaks (corresponding to $M_1, M_2, \dots$) would provide direct evidence of the 5D nature of the interaction, distinguishing it from a 4D model with a single $Z'$.
• Kinematic Islands: The "island structure" in exclusion plots arises because the detector only accepts particles with specific lifetimes (decay lengths). Higher KK modes have different masses and lifetimes, creating distinct sensitivity regions.

C. Impact of Kinetic Mixing ($\epsilon_4$)

• The authors systematically analyzed non-zero kinetic mixing ($\epsilon_4 = 10^{-4}, 10^{-5}, 10^{-6}$).
• Cancellation Effects: They identified regions in the $(m_{KK}, g')$ parameter space where the contributions from the gauge coupling ($g'$) and kinetic mixing ($\epsilon_4$) cancel each other out, creating "blind spots" where the signal vanishes.
• Enhanced Reach: Conversely, for small $g'$, non-zero $\epsilon_4$ can significantly extend the reach of experiments, making them sensitive to parameter regions that would otherwise be inaccessible.

D. Role of Muon $(g-2)$

• With the latest experimental data showing consistency between the SM prediction and the measured muon $(g-2)$ (resolving the long-standing anomaly), the authors use this null result to exclude specific regions of the parameter space where new particles would have contributed to the anomaly.

5. Significance and Conclusion

• Direct Evidence of Extra Dimensions: The paper demonstrates that muon beam dump experiments are not just sensitive to the existence of new particles but can discriminate between 4D and 5D origins of the $U(1)_{L_\mu - L_\tau}$ interaction. The observation of multiple mass peaks in the $\mu^+\mu^-$ channel is the "smoking gun" for the KK tower.
• Complementarity: The combination of "invisible" searches (NA64$\mu$, M3) and "visible" searches (MuSIC, Future Beam Dump) covers different regions of the parameter space, providing a comprehensive probe.
• Robustness: The study confirms that truncating the KK tower at $n_{KK}^{max}=5$ is sufficient for log-log plot accuracy in exclusion limits, as higher modes are suppressed by cross-section and phase space.
• Future Outlook: Even with the resolution of the $(g-2)$ anomaly, the $U(1)_{L_\mu - L_\tau}$ framework remains a viable candidate for Dark Matter mediation and Hubble tension resolution. The proposed muon experiments offer a unique pathway to test the geometric origin (extra dimensions) of these interactions.

In summary, this work provides a rigorous phenomenological roadmap for using next-generation muon beam dump experiments to test the existence of extra dimensions through the specific signature of a Kaluza-Klein tower of gauge bosons.