Extracting Dark-Matter Mass from Angular Scanning

This paper proposes a novel method to determine the mass scale of ambient dark matter in two-dimensional directional detectors by analyzing how event rates and their angular curvature depend on the orientation of the detection plane relative to the dark-matter flow, a concept validated using a Graphene-Josephson-Junction detector as a proof of principle.

The Gist

The Big Mystery: Invisible Rain

Imagine you are standing in a field during a heavy rainstorm. You can't see the raindrops, but you can feel them hitting your umbrella. If you want to know how heavy the raindrops are (are they tiny mist or heavy hail?), you usually look at how hard they hit you.

In the world of physics, Dark Matter is like that invisible rain. It fills our galaxy, passing through us all the time. Scientists have been trying to catch these "raindrops" for decades. The problem? Most detectors are like simple "on/off" switches. They can tell you if a dark matter particle hit them, but they can't measure how hard it hit. Without knowing the force of the hit, it's very hard to figure out the mass (weight) of the particle.

The New Idea: Spinning the Umbrella

The authors of this paper, Jeong, Kim, and Park, propose a clever new trick. Instead of trying to measure the "force" of the hit, they suggest rotating the detector to see how the "wind" of dark matter changes the number of hits.

Here is the analogy:
Imagine you are holding a flat, 2D sheet of paper (like a piece of graphene) in a wind tunnel.

• Scenario A: You hold the paper flat, facing the wind head-on. The wind hits the edge of the paper. Very little air actually passes through the surface area to push against it.
• Scenario B: You tilt the paper so it catches the wind like a sail. Now, the wind is blowing parallel to the surface, pushing against the whole sheet.

The paper argues that dark matter behaves similarly. Because our Solar System is moving through the galaxy (like a car driving through a swarm of bugs), the dark matter "wind" is coming from a specific direction (toward the constellation Cygnus).

The "Heavy" vs. "Light" Particle Trick

The core discovery is that heavy and light dark matter particles react differently to the angle of this wind.

• Heavy Dark Matter: These are like bowling balls. Even if they are moving slowly, they have enough energy to knock over a pin (trigger the detector). It doesn't matter much if they hit the detector at a weird angle; they will still trigger it. The "hit rate" stays fairly steady no matter how you tilt the detector.
• Light Dark Matter: These are like ping-pong balls. They are so light that they need to be moving very fast to knock over a pin. If they hit the detector at a shallow angle, they might not have enough speed to trigger it. But if they hit it head-on (parallel to the surface), they might trigger it.

The Result: If you rotate your detector and count the hits, the pattern of "hits vs. angle" will look different depending on whether the particles are heavy or light.

• If the curve is flat, the particles are likely heavy.
• If the curve is steep (lots of hits at one angle, almost none at another), the particles are likely very light.

The "Virtual Spin" (No Moving Parts Needed)

You might ask, "Do we have to physically build a giant robot to spin the detector?"
The answer is no.

The Earth is already spinning, and it orbits the Sun, which orbits the center of the Galaxy. This means the angle between our detector and the "dark matter wind" changes automatically throughout the day and year.

The authors created a software tool called DarkWind. Think of it like a weather app, but instead of predicting rain, it predicts the direction of the dark matter wind based on your location and the time of day.

• Real Life: You leave your detector sitting still in a lab.
• Virtual Life: The software calculates, "At 2:00 PM, the wind is hitting your detector at a 45-degree angle. At 8:00 PM, it's hitting at 10 degrees."

By comparing the number of hits recorded at 2:00 PM vs. 8:00 PM, scientists can map out the "angular spectrum" and figure out the mass of the dark matter without ever moving a single screw.

Why This Matters

Most current experiments struggle to find "superlight" dark matter because they rely on measuring energy, which is hard to do with these tiny particles. This new method uses direction as the key.

It's like trying to guess the size of a fish in a dark pond.

• Old way: Try to feel the fish with a net (hard to feel small fish).
• New way: Watch how the ripples change when the wind blows from different directions. The pattern of the ripples tells you exactly how big the fish is.

Summary

1. The Problem: We can't easily weigh invisible dark matter particles because our detectors only count "hits," not "force."
2. The Solution: Use the fact that dark matter comes from a specific direction (the "wind").
3. The Method: Observe how the number of hits changes as the Earth rotates and changes the angle of the detector relative to the wind.
4. The Payoff: The shape of this change reveals the mass of the dark matter. Heavy particles ignore the angle; light particles are very sensitive to it.
5. The Tool: A computer program (DarkWind) does the math, so we don't need to physically spin our detectors.

This paper opens a new door for finding the lightest, most elusive forms of dark matter that have been hiding in plain sight.

Technical Summary

1. Problem Statement

The detection of superlight dark matter (DM) (masses in the keV-to-MeV range) presents a unique challenge for direct detection experiments.

• The "On-Off" Limitation: Many emerging detectors (e.g., Transition Edge Sensors, Graphene Josephson Junctions) operate on an "on-off" principle. They register an event if the energy deposit exceeds a threshold ($E_{th}$) but cannot precisely measure the magnitude of the recoil energy.
• Mass Determination Failure: Conventional DM mass extraction relies on analyzing the differential recoil energy spectrum. Without precise energy resolution, this method fails.
• Threshold Dependency: While varying the detection threshold is one approach, it is often experimentally difficult or impossible to adjust thresholds dynamically with high precision.
• Isotropic Assumption: Standard DM velocity distributions are often assumed to be isotropic, which washes out directional information. However, the motion of the Solar System and Earth creates a directional "DM wind," a feature often underutilized in 2D detectors.

2. Methodology

The authors propose a novel method to determine the DM mass scale by exploiting the angular dependence of event rates in effectively two-dimensional (2D) detectors.

Core Physical Mechanism

• Directional Preference: Due to the Solar System's motion toward the Cygnus constellation (the "DM wind"), the DM flux is not isotropic in the Earth's frame.
• 2D Detector Geometry: In 2D detectors (like graphene sheets), the event rate is dominated by the velocity component parallel to the detection plane ($v_{\parallel}$). The normal component contributes negligibly.
• Angular Scanning: By rotating the detector plane relative to the DM wind (angle $\Theta$), the effective parallel velocity distribution changes.
  • When the wind blows parallel to the plane ($\Theta = 90^\circ$), the velocity distribution is "harder" (more high-velocity particles).
  • When the wind blows perpendicular to the plane ($\Theta = 0^\circ$), the distribution is "softer."
• Mass Dependence: The minimum velocity required to overcome the energy threshold ($E_{th}$) depends on the DM mass ($m_\chi$).
  • Heavy DM: Low velocities suffice to trigger a signal; the angular dependence is weak.
  • Light DM: Only high velocities can trigger a signal; the angular dependence is strong because the "hard" tail of the velocity distribution is critical.
• The Observable: The curvature of the event rate distribution as a function of the angle $\Theta$ encodes the DM mass.

Theoretical Framework

• Velocity Profile: The authors derive a modified Maxwell-Boltzmann distribution projected onto the 2D plane ($\tilde{F}(V_{\chi\parallel}; \Theta)$), accounting for the Solar System's velocity ($\vec{v}_\odot$) and the Earth's orbital/rotational motion.
• Event Rate Calculation: The event rate $n_{ev}(\Theta)$ is calculated by integrating the projected velocity profile over the phase space, weighted by the scattering cross-section and the structure functions of the target material.
• Virtual Rotation: Instead of physically rotating the detector (which introduces noise), the authors propose inferring $\Theta$ from the time of detection and the geographical location of the experiment, utilizing the Earth's rotation and orbit to naturally scan different angles.

3. Key Contributions

1. Novel Mass Extraction Technique: Introduced a method to determine DM mass using only the angular distribution of event rates in 2D detectors, bypassing the need for differential energy spectra.
2. Formalism for 2D Detectors: Developed a rigorous mathematical framework to project the 3D DM velocity distribution onto a 2D plane, explicitly deriving the dependence on the angle $\Theta$ between the DM wind and the detector normal.
3. Application to Graphene Josephson Junctions (GJJ): Applied the formalism to the recently proposed GJJ-based superlight DM detector, demonstrating its feasibility.
4. DarkWind Software Package: Developed and released a Python package (DarkWind) that calculates the DM wind direction and the angle $\Theta$ for any detector location and time, accounting for Galactic rotation, Solar peculiar velocity, and Earth's orbital/rotational dynamics.

4. Results

• Numerical Validation: Using the GJJ detector model with a heavy-mediator scenario ($m_\phi = 100$ keV) and a threshold of $E_{th} = 1$ meV, the authors simulated event rates for various DM masses ($1$ keV to $10$ keV).
• Angular Dependence:
  • $m_\chi = 1$ keV: Showed the strongest angular dependence. The event rate varied significantly as $\Theta$ changed, as light particles require high parallel velocities to exceed the threshold.
  • $m_\chi = 10$ keV: Showed the weakest angular dependence. Heavier particles trigger signals even at lower velocities, making the rate less sensitive to the wind's angle.
• Uncertainty Analysis: The DarkWind code estimates the uncertainty in determining $\Theta$ to be approximately $0.33^\circ$, arising from uncertainties in Galactic rotation and the Sun's peculiar velocity. This precision is sufficient to distinguish between different mass hypotheses.
• Cross-Check Potential: For detectors capable of measuring recoil energy (e.g., NEWSdm, DIAMOND), the mass derived from angular scanning can be cross-verified with the mass derived from the energy spectrum.

5. Significance

• Overcoming Experimental Limitations: This method provides a solution for "on-off" detectors that cannot measure recoil energy, a common limitation in next-generation superlight DM searches.
• Broad Applicability: While demonstrated on GJJ detectors, the method is applicable to any effectively 2D detector, including DNA detectors, diamond detectors, and other low-threshold technologies.
• New Discovery Channel: It opens a new avenue for probing the superlight DM sector (keV–MeV), a region that is increasingly difficult to access with traditional WIMP-search techniques.
• Practical Implementation: By utilizing the Earth's natural motion ("virtual rotation") rather than mechanical rotation, the method avoids introducing experimental noise and calibration issues, making it highly feasible for current and future experimental setups.

In conclusion, the paper establishes that the angular correlation of event rates is a powerful observable for extracting the mass of superlight dark matter, transforming the directional nature of the DM wind from a background nuisance into a primary tool for mass determination.