Oscillating Resonances: Imprints of ultralight dark matter at colliders

This paper proposes that ultralight dark matter can be detected at colliders not as a standard isolated resonance, but as a "smeared" oscillating resonance caused by a mediator whose mass varies over time, offering a unique discovery channel that complements existing constraints from atomic clocks.

The Gist

The Big Picture: A Shifting Target

Imagine you are trying to find a specific radio station. Usually, a radio station broadcasts on one fixed frequency (say, 101.5 FM). If you tune your dial to that exact spot, the signal is loud and clear. This is how scientists usually look for new particles at colliders like the LHC or Belle II: they look for a sharp, distinct "peak" in their data, like a clear radio station.

However, this paper suggests that if Ultralight Dark Matter (ULDM) exists, it acts like a giant, invisible ocean wave that the entire universe is sitting on. As this wave passes through our particle detectors, it doesn't just sit there; it gently pushes and pulls on the fundamental rules of physics.

Specifically, it causes the "mass" of a potential new particle (called a mediator) to wobble back and forth. Instead of being a fixed 500 MeV (a unit of mass), the particle might be 490 MeV one second, 510 MeV the next, and back to 500 MeV after a few hours or days.

The Problem: The "Smudged" Peak

If you try to find this particle using standard methods, you are in trouble.

• The Static World: In a normal world, the particle is always 500 MeV. All the data points pile up neatly at 500, creating a tall, sharp mountain (a resonance peak).
• The Oscillating World: Because the mass is constantly changing, the data points don't pile up in one spot. Instead, they get spread out over a range (say, 490 to 510).

The Analogy: Imagine trying to take a photo of a hummingbird's wings. If you use a fast shutter speed, you see a sharp image. If you use a slow shutter speed while the wings are flapping, you get a blurry, smeared-out image.
In the collider, the "shutter speed" is the total time the experiment runs (years). The "wings" are the dark matter oscillating. The result is that the sharp mountain of data gets flattened into a wide, low hill. To a standard computer algorithm looking for a sharp peak, this signal might look like background noise and get ignored.

The Twist: Why This is Good News

The authors argue that this "smearing" isn't a dead end; it's actually a unique fingerprint.

1. Weaker Limits: Because the signal is smeared, current experiments haven't been able to rule out these particles as strictly as they thought. The "rules" for what is allowed are actually much looser than we believed.
2. The "Threshold" Trick: Sometimes, the particle's mass is just below the energy needed to decay into two muons (a type of particle). In a static world, it would never decay. But because the mass wobbles up and down, it occasionally "jumps" over the energy threshold and decays. This allows scientists to see particles that should theoretically be invisible.

How to Find the Signal: Two New Strategies

The paper proposes two clever ways to find this "smudged" signal, which standard searches miss.

Strategy 1: The "Double-Hump" Detective (Mass-Binned Data)

If you look at the smeared data, you won't see one peak in the middle. You will see two smaller peaks at the edges of the range (like a "W" shape or two hills with a valley in the middle).

• The Method: The authors created an algorithm that looks for these two edge peaks. Once it finds them, it calculates the distance between them to figure out how much the mass is wobbling. It then mathematically "unsmears" the data to reconstruct the original, sharp peak.
• The Catch: This works well if the signal is strong, but it can't tell you exactly how many particles were created, only what they looked like.

Strategy 2: The "Time-Travel" Fourier Transform (Time-Stamped Data)

This is the most powerful method. Colliders record the exact time every single particle collision happens.

• The Method: Instead of just looking at the mass, the scientists look at the timing of the events. They use a mathematical tool called a Fast Fourier Transform (FFT) (think of it as a super-advanced music equalizer) to scan the timeline for a repeating rhythm.
• The Result: Even if the signal is buried in noise, if it has a specific rhythm (e.g., it happens more often every 10 hours), the FFT will find that frequency. Once they find the rhythm, they can "fold" the data, aligning all the events to the same point in the cycle. This reconstructs the original sharp peak perfectly, even if the background noise is loud.

The Bottom Line

The paper concludes that if we find a particle at a collider that doesn't sit still but instead "breathes" or oscillates with a specific rhythm, it would be a smoking gun for Ultralight Dark Matter.

While precision experiments (like atomic clocks) are very good at measuring tiny changes in constants, this paper shows that colliders are actually very competitive in finding these specific types of dark matter. By changing how we look for the data—searching for oscillations and rhythms rather than just static peaks—we might finally catch a glimpse of the invisible dark matter that makes up most of our universe.

Technical Summary

Technical Summary: Oscillating Resonances: Imprints of Ultralight Dark Matter at Colliders

Problem Statement
Ultralight dark matter (ULDM), with masses in the range $10^{-22} \text{ eV} \lesssim m_\phi \lesssim 10^{-15} \text{ eV}$, is a compelling candidate for dark matter that can account for the observed relic abundance via the misalignment mechanism. In this regime, ULDM behaves as a coherently oscillating classical background field. While interactions between ULDM and Standard Model (SM) fields are typically constrained by precision measurements of fundamental constants (e.g., atomic clocks, equivalence principle tests), these constraints rely on effective field theory (EFT) operators.

This paper addresses a gap in the phenomenology of ULDM: the possibility that these effective operators arise from an explicit mediator field (scalar or vector) with a mass accessible to collider experiments. The authors investigate the consequences of such a mediator existing within an oscillating ULDM background. Specifically, they explore how the direct coupling of the mediator to the ULDM field induces time-dependent oscillations in the mediator's mass and/or couplings. Standard "bump-hunt" resonance searches at colliders assume a static mass; the authors argue that an oscillating mediator would not appear as a narrow peak but as a smeared distribution, potentially evading detection or leading to misinterpretation of limits.

Methodology
The authors develop a framework to model "oscillating resonances" by integrating out mediator fields in specific ultraviolet (UV) completions. They consider two primary scenarios:

1. Scalar Mediator ($S$): Coupling directly to dark matter ($\phi$) and SM fermions.
2. Vector Mediator ($X$, Dark Photon): Coupling to SM fermions via kinetic mixing or gauge couplings, with mass generated by a dark Higgs mechanism.

In these models, the mediator mass ($m_{S,X}$) and/or the kinetic mixing parameter ($\epsilon$) become time-dependent due to the oscillating ULDM field $\phi(t) \propto \cos(\omega t)$. The oscillation frequency $\omega$ is determined by the ULDM mass ($m_\phi$), and the amplitude depends on the dark matter energy density and coupling scales ($\Lambda$).

The authors perform the following analyses:

• Loop Calculations: They compute the loop-induced effective couplings of the mediator to photons and electrons, deriving the resulting variations in the fine-structure constant ($\alpha$) and electron mass ($m_e$). These are compared against existing constraints from atomic clocks (Rb/Cs, BACON), spectroscopy, and equivalence principle tests (MICROSCOPE).
• Collider Phenomenology: They reinterpret existing and projected limits from Belle II, LHCb, and SHiP. They model the signal as an invariant mass distribution smeared over a modulation window defined by the oscillation amplitude.
  • Prompt Searches: They analyze the dilution of the resonance peak in the invariant mass spectrum when the modulation amplitude exceeds the detector resolution. They also investigate "threshold effects," where a mediator with a static mass below the kinematic threshold ($2m_\mu$) can periodically decay into muons when the oscillating mass exceeds the threshold.
  • Displaced Vertex Searches: For beam-dump experiments like SHiP, they analyze the impact of oscillating kinetic mixing on the decay length and signal yield, noting that mass oscillations largely cancel out in the total yield for displaced decays, whereas coupling oscillations induce threshold effects.
• Reconstruction Strategies: The authors propose two algorithms to recover the signal from data:
  1. Mass-binned data: A "peak-finding and peak-merging" algorithm to reconstruct the central mass and modulation amplitude from the characteristic double-peak structure (or smeared distribution) in the invariant mass spectrum.
  2. Time-binned data: A method utilizing event timestamps to apply a Fast Fourier Transform (FFT) to the data. This allows for phase folding, coherent amplification of the signal, and reconstruction of the absolute signal normalization, provided the background level is manageable.

Key Contributions and Results

• Weakening of Collider Limits: The authors demonstrate that standard exclusion limits on the kinetic mixing parameter $\epsilon$ are significantly weakened if interpreted in the context of an oscillating resonance. For LHCb Run-2 and projected Run-6 sensitivities, the limits on $\epsilon$ can be relaxed by a factor of 4–5 for optimal modulation amplitudes ($R_m \approx 12\%$). This occurs because the signal is distributed across multiple mass bins, reducing the local significance in any single bin.
• Threshold Extension: Mass modulation allows resonances with central masses slightly below the kinematic threshold (e.g., $m_X < 2m_\mu$) to produce detectable events during the phase of the oscillation where $m_X(t) > 2m_\mu$. This extends the sensitivity of collider searches to lower masses than static analyses would suggest.
• Comparison with Precision Constraints: The paper shows that the parameter space probed by collider searches for oscillating resonances is not excluded by current atomic clock or equivalence principle constraints. In some regions, collider searches are competitive with or even surpass precision sensor limits.
• Reconstruction Algorithms:
  • Using mass-binned data, the authors show that the central mass and modulation amplitude can be reconstructed if the double-peak structure is resolvable. However, this method cannot recover the total event yield (normalization) without external inputs.
  • Using time-stamped data, the authors show that a Fast Fourier Transform can identify the oscillation frequency and phase. By phase-folding the data, they can reconstruct the unmodulated resonance peak and recover the absolute signal normalization. They quantify the robustness of this method, finding that the signal frequency can be reliably extracted for signal-to-background ratios ($S/B$) down to $\approx 1/3$. Below this ratio, the modulation peak is obscured by noise, and the reconstruction of the absolute yield becomes unreliable.

Significance and Claims
The paper claims that the discovery of an "oscillating resonance" at a collider would constitute a "smoking gun" signal for ultralight dark matter. Unlike a standard resonance, which could be a heavy particle, an oscillating resonance with characteristics matching the predicted ULDM oscillation period and amplitude would directly probe the underlying nature of dark matter as a coherent classical field.

The authors emphasize that while standard bump-hunt strategies are suboptimal for this specific signature, the periodicity of the signal offers a unique handle. By incorporating time-domain information or specific reconstruction algorithms, colliders can regain sensitivity to mediators coupled to ULDM. The work suggests that the "oscillating resonance" signature is a generic feature of UV-complete models where mediators couple quadratically to ULDM, making it a robust target for current and future collider experiments (Belle II, LHCb, SHiP) to complement precision measurements of fundamental constants.