Direct-detection constraints on inelastic dark matter with a scalar mediator

This paper calculates direct detection constraints on inelastic Dirac dark matter with a scalar mediator and leptophilic couplings, demonstrating that the p-wave velocity suppression of annihilation opens viable MeV-GeV mass parameter space and that liquid-xenon experiments like XENON1T, PandaX-4T, and LZ can constrain sub-MeV mass splittings via spin-independent dark matter-electron scattering.

The Gist

The Big Picture: Hunting for the "Invisible"

Imagine the universe is a giant, dark room filled with furniture. We know the furniture is there because we can see its shadow and feel the air moving around it (this is Dark Matter). But we've never actually touched the furniture or seen what it looks like.

Scientists have built super-sensitive "motion detectors" (like the XENON1T, PandaX-4T, and LZ experiments) filled with liquid xenon. These detectors are waiting for a dark matter particle to bump into an atom in the liquid, creating a tiny flash of light or an electric signal.

For a long time, these detectors were looking for heavy, slow-moving dark matter. But they haven't found anything. This paper asks: "What if the dark matter is actually light and fast, but it's playing a tricky game of hide-and-seek?"

The Main Character: "Inelastic" Dark Matter

Usually, scientists imagine dark matter as a billiard ball. When it hits an electron, it bounces off, losing a little energy, but stays the same ball.

This paper proposes a different idea: Inelastic Dark Matter.
Think of this dark matter not as a billiard ball, but as a transformer toy (like a robot that turns into a car).

• State 1 (The Robot): The light, stable version we usually see.
• State 2 (The Car): A slightly heavier, excited version.

The "Inelastic" part means that when the dark matter interacts, it must change its form. It can't just bounce; it has to switch states.

The Two Scenarios: The "Energy Bill"

The paper looks at two ways this "transformer" can interact with the electrons in the detector, depending on the "energy bill" (the mass difference between the two states).

1. The Endothermic Scenario (The "Up-Switch")

• The Analogy: Imagine you are trying to push a heavy box up a small hill. You need to spend extra energy just to get it to the top.
• What happens: The dark matter (the Robot) hits an electron and tries to turn into the heavier version (the Car). To do this, it has to steal energy from the electron.
• The Result: Because the dark matter has to pay an "energy tax" to switch forms, it leaves less energy behind for the detector to see. It's like a thief who spends so much of the stolen money on bribes that there's almost nothing left for the police to find.
• The Finding: The paper finds that if the "energy tax" is too high, the detectors can't see it at all. But if the tax is tiny, the detectors can still catch it, though it's harder than catching normal dark matter.

2. The Exothermic Scenario (The "Down-Switch")

• The Analogy: Imagine you are holding a compressed spring. When you let go, it snaps open and releases a burst of extra energy.
• What happens: The dark matter starts as the heavy version (the Car) and switches to the light version (the Robot). Because it's shedding weight, it releases extra energy into the electron.
• The Result: This is a "bonus" event. The electron gets hit with a double whammy: the impact of the collision plus the energy released by the dark matter changing form.
• The Finding: This is the exciting part! The paper shows that if the universe is full of these heavy "Cars" ready to switch to "Robots," our detectors could see a much brighter signal than expected. It opens up a "sweet spot" where dark matter with a mass between 100 MeV and 500 MeV (very light for a particle) could finally be detected.

The Messenger: The "Scalar Portal"

How does the dark matter talk to the electrons? It needs a messenger.

• The Analogy: Imagine the dark matter and the electron are in two different rooms. They can't talk directly. They need a messenger to run between the rooms and pass a note.
• The Paper's Messenger: This paper uses a Scalar Mediator (a specific type of particle).
• The Twist: This messenger is "Leptophilic," which means it only likes to talk to electrons (and other light particles), not to heavy nuclei. It's like a messenger who refuses to talk to the heavy furniture and only whispers to the light dust motes. This explains why previous experiments looking for hits on heavy nuclei might have missed it.

What Did They Actually Do?

The authors took the data from three massive underground experiments (XENON1T, PandaX-4T, and LZ) and ran a new simulation. They asked:
"If dark matter is this 'transformer' type, and it uses this specific 'messenger,' would these experiments have seen it by now?"

The Verdict:

1. For the "Up-Switch" (Endothermic): The experiments are still looking, but the signal is weak. If the mass difference is too big, the signal disappears completely.
2. For the "Down-Switch" (Exothermic): This is the gold mine. The experiments are sensitive enough to rule out a huge chunk of possibilities. If this type of dark matter exists in the "sweet spot" mass range, these detectors should have seen it. Since they haven't (yet), the authors have drawn a "No Trespassing" sign on a large area of the map where this dark matter cannot exist.

Why Does This Matter?

This paper is like a detective narrowing down the suspect list.

• Before, we thought dark matter might be heavy and slow.
• Then we thought it might be light and fast.
• Now, we are testing the idea that it's a shape-shifter.

By proving that certain "shape-shifting" scenarios are impossible (or highly unlikely) based on current data, the authors help us focus our search on the remaining, viable possibilities. They are essentially telling us: "If the dark matter is a transformer, it has to be very specific about how it changes, or it's not there at all."

Summary in One Sentence

This paper uses data from giant underground water tanks to prove that if dark matter is a "shape-shifting" particle that interacts only with electrons, it can't be just any shape-shifter—it has to fit a very specific set of rules, or our detectors would have already found it.

Technical Summary

1. Problem Statement

The paper addresses the search for light thermal dark matter (DM) in the mass range of 1 MeV to 1 GeV. Standard direct-detection experiments (like XENON1T, PandaX-4T, and LZ) have historically focused on heavy WIMPs interacting via nuclear recoil. However, for sub-GeV DM, nuclear recoil energies fall below experimental thresholds. Consequently, electron-recoil scattering has become the primary probe for light DM.

The specific challenge addressed is the inelastic dark matter (iDM) paradigm, where the DM particle exists in two states ($\chi_1$ and $\chi_2$) with a small mass splitting ($\delta = m_{\chi_2} - m_{\chi_1}$).

• Endothermic scattering ($\delta > 0$): Requires kinetic energy to up-scatter $\chi_1 \to \chi_2$, suppressing the signal.
• Exothermic scattering ($\delta < 0$): Releases energy when down-scattering $\chi_2 \to \chi_1$, potentially enhancing the signal.

The authors investigate how these inelastic effects, mediated by a leptophilic scalar (coupling preferentially to electrons), alter the constraints derived from liquid-xenon experiments. They specifically aim to determine if the p-wave velocity suppression of the annihilation cross-section (characteristic of scalar-mediated iDM) opens viable parameter space that is otherwise excluded by thermal relic abundance requirements or collider limits.

2. Methodology

A. Theoretical Framework

• Model: A simplified benchmark model involving a Dirac fermion DM sector ($\chi_1, \chi_2$) coupled to the Standard Model (SM) via a scalar mediator $\varphi$.
• Couplings: The mediator is leptophilic, with flavor-dependent couplings proportional to lepton masses ($c_{\ell\ell} \propto m_\ell$). The interaction is off-diagonal ($\chi_1 \chi_2 \varphi$), driving the inelastic transitions.
• Relic Abundance: The authors assume thermal freeze-out. They note that for small mass splittings ($|\Delta| \ll 1/20$), the thermal relic density curves for inelastic DM coincide with the elastic case. The p-wave suppression of the annihilation cross-section ($\langle \sigma v \rangle \propto v^2$) allows for viable parameter space in the MeV–GeV range that might be excluded in s-wave scenarios.
• Scenarios: Two distinct scenarios are analyzed based on the relative abundance of the excited state ($f_{\chi_2}$):
  1. Endothermic (Up-scattering): $f_{\chi_2} \approx 0$ (only $\chi_1$ exists). Constraints come from $\chi_1 e^- \to \chi_2 e^-$.
  2. Exothermic (Down-scattering): $f_{\chi_2} \approx f_{\chi_1} \approx 1/2$. Constraints come from $\chi_2 e^- \to \chi_1 e^-$.

B. Experimental Data & Analysis

• Experiments: The study utilizes public ionization-electron (S2-only) data from:
  • XENON1T: 258.2 days exposure.
  • PandaX-4T: 95 days commissioning run.
  • LZ: 4.5 tonne-years exposure.
• Kinematics: The authors calculate the minimum DM velocity ($v_{min}$) required to deposit energy $E_d$ given the mass splitting $\delta$.
  • For endothermic scattering, $v_{min}$ increases with $\delta$, suppressing the event rate.
  • For exothermic scattering, $v_{min}$ can approach zero, enhancing the rate for specific mass ranges.
• Cross-Section Calculation:
  • They compute the differential event rate $dR/dE_d$ using non-relativistic effective field theory.
  • They incorporate the atomic ionization factor and the dark matter form factor ($F_{DM}$) to account for the bound-state nature of electrons in xenon.
  • The squared matrix element is derived for the scalar mediator, showing a dependence on the momentum transfer $q$ and mediator mass $m_\varphi$.
• Statistical Approach:
  • Binned Likelihood: A profile-likelihood ratio test is used to derive 90% confidence level (CL) upper limits on the effective cross-section $\bar{\sigma}_e$.
  • Unbinned Bayesian: A conservative estimate is also provided using a Bayesian approach (Eq. 20) to cross-check the sensitivity.

3. Key Contributions

1. Scalar Portal Constraints: This is one of the first comprehensive studies deriving direct-detection constraints specifically for scalar-mediated inelastic DM interacting with electrons. Previous works focused largely on vector mediators (dark photons).
2. Kinematic Enhancement in Exothermic Regime: The paper rigorously demonstrates that for exothermic scattering with a specific mass splitting, the sensitivity of liquid-xenon detectors is significantly enhanced. This occurs when the DM mass satisfies the resonance condition:
   $$m_{\chi_1} \simeq \frac{E_d}{|\Delta|}$$
   where $E_d$ is the deposited energy and $\Delta$ is the relative mass splitting.
3. Velocity Suppression Implications: The authors highlight that the p-wave suppression of the annihilation cross-section in scalar-mediated models allows the DM to avoid over-closure constraints in the MeV–GeV range, making the direct-detection limits the primary probe for this specific model.
4. Comparison of Experiments: The study provides a comparative analysis of XENON1T, PandaX-4T, and LZ, showing how their different energy thresholds and exposures affect sensitivity to different DM mass ranges.

4. Results

• Endothermic Scenario ($\Delta > 0$):

  • Constraints weaken as the mass splitting increases.
  • For very small splittings ($\Delta \ll 10^{-7}$), the constraints are nearly identical to the elastic case.
  • For $\Delta \approx 10^{-7}$, deviations appear at higher masses ($m_{\chi_1} > 1$ GeV for LZ; $> 100$ MeV for XENON1T/PandaX-4T).
  • The kinematically forbidden region begins at $\Delta > 10^{-6}$.

• Exothermic Scenario ($\Delta < 0$):

  • Significant Sensitivity Enhancement: When $|\Delta| \approx 10^{-5}$, the experiments show peak sensitivity (lowest upper limits on $\sigma_e$) at specific masses:
    • PandaX-4T: $\sim 10^{-2}$ GeV (10 MeV).
    • XENON1T: $\sim 10^{-1}$ GeV (100 MeV).
    • LZ: $\sim 1$ GeV.
  • In these "resonant" mass regions, the cross-section limits reach down to $\sigma_e \sim 10^{-45} \text{ cm}^2$.
  • The exothermic scenario allows the exclusion of a significant portion of the parameter space in the 100 MeV – 500 MeV mass range, which is difficult to probe with elastic scattering alone.

• Comparison with Other Probes:

  • The derived limits are comparable to or stronger than those expected from the NA64$\mu$ experiment (muon fixed-target) for the specific parameter space considered.
  • The constraints are generally weaker than those for vector mediators due to the p-wave suppression in the annihilation channel, but the direct detection limits remain the most stringent for the electron-scattering channel in this model.

5. Significance

• Opening Viable Parameter Space: The work demonstrates that inelastic DM with a scalar mediator can evade thermal relic constraints (due to p-wave suppression) while remaining testable via direct detection.
• Optimization of Search Strategies: The identification of the "resonant" mass region ($m_{\chi} \approx E_d/|\Delta|$) provides a clear target for future analyses of existing data and the design of future low-threshold detectors. It suggests that looking for specific mass splittings can dramatically improve sensitivity.
• Complementarity: The study reinforces the necessity of combining data from multiple experiments (XENON1T, PandaX-4T, LZ) with varying thresholds to cover the full MeV–GeV mass spectrum for inelastic scenarios.
• Theoretical Consistency: By explicitly modeling the scalar portal and leptophilic couplings, the paper provides a consistent framework for interpreting low-energy electron recoil excesses (such as the XENON1T anomaly) within the context of inelastic dark matter, distinguishing them from elastic or vector-mediated interpretations.

In conclusion, the paper establishes that exothermic inelastic scattering mediated by a scalar particle offers a unique and highly sensitive window for detecting light dark matter in the MeV–GeV range, potentially allowing current liquid-xenon experiments to probe cross-sections orders of magnitude lower than previously thought possible for this specific model class.