Can LLP detectors probe the reheating temperature? A case study of vector dark matter

This paper investigates a dark vector dark matter model with a long-lived scalar mediator, demonstrating that long-lived particle searches at the LHC and FCC-hh can probe otherwise inaccessible parameter spaces and place novel constraints on the reheating temperature through the interplay of cosmological freeze-in production and collider signatures.

The Gist

The Big Picture: A Cosmic Cold Case

Imagine the universe is a giant, bustling city. We know that most of the "citizens" in this city are invisible Dark Matter, but we have no idea what they look like or how they were born.

This paper investigates a specific theory about how these invisible citizens (Dark Matter) were created. The authors propose a scenario involving a "messenger" particle that is very shy and takes a long time to show up. They ask a crucial question: Can we catch this messenger in our particle colliders (like the LHC) and, by doing so, figure out how hot the universe was right after the Big Bang?

The Cast of Characters

To understand the story, we need to meet the three main characters in this "Dark Sector":

1. The Dark Matter (The Villain/Protagonist): This is a heavy, invisible particle called a Vector ($V$). It is the stable Dark Matter we are looking for. It's like a ghost that never leaves the party.
2. The Messenger (The Long-Lived Particle): This is a heavy particle called a Scalar ($\phi$). It is unstable and wants to decay, but it is very slow to do so. Think of it as a messenger who gets stuck in traffic for hours before delivering a letter. Because it lives so long, it travels far away from the crash site before it disappears.
3. The Standard Model (The Visible World): This is everything we can see and touch (atoms, light, etc.). The Dark Sector and the Visible World don't talk to each other often; they only interact through a very weak "Higgs Portal" (a secret door).

The Story: How the Universe Was Born

The paper explores two ways the Dark Matter was made:

• The "Freeze-In" Method: Imagine a very cold room where people (particles) are trying to enter. Because the door is so small and the key is so hard to find, only a few people manage to slip in slowly over time. This is how the Dark Matter was created. It didn't happen in a big explosion; it happened through tiny, rare interactions.
• The Reheating Temperature: This is the "temperature" of the universe right after the Big Bang. The paper argues that if the universe wasn't super hot (a "low reheating temperature"), it actually helps create the exact amount of Dark Matter we see today.

The Twist: In this scenario, the Messenger ($\phi$) is created, but it doesn't decay immediately. It travels a long distance before turning into the Dark Matter ($V$) and a visible particle (like a Z-boson or a photon). Because it travels so far, it is called a Long-Lived Particle (LLP).

The Detective Work: Catching the Messenger

The authors are trying to figure out if we can find this Messenger in our giant particle smashers (colliders).

1. The Main Detectors (ATLAS and CMS): These are like the main security cameras in the center of the city. They look for "displaced vertices"—places where a particle decays inside the detector but not right where the collision happened. It's like seeing a car crash, but the car keeps driving for 100 meters before exploding.

   • The Problem: If the Messenger lives too long, it flies right past the main detectors before decaying. If it lives too short, it decays too early to be noticed.

2. The Far Detectors (MATHUSLA, ANUBIS, DELIGHT, FOREHUNT): These are the paper's "secret weapons." Imagine building a giant, empty warehouse 100 meters away from the main security cameras. If the Messenger is slow, it will fly past the main cameras and finally decay inside this distant warehouse.

   • The paper shows that these far detectors are perfect for catching Messengers that live just long enough to escape the main detector but not so long that they fly off into space.

The Big Discovery: Connecting the Dots

The most exciting part of the paper is the connection between the Messenger's speed and the Universe's temperature.

• The Analogy: Imagine you find a frozen ice cube in a room. By measuring how big the ice cube is, you can guess how cold the room was when it formed.
• The Paper's Claim: By measuring how far the Messenger travels (its "lifetime") in our detectors, we can calculate exactly how hot the universe was when it was born (the Reheating Temperature).

Usually, scientists think we can't measure the temperature of the early universe directly. But this paper says: "Yes we can! If we see these specific long-lived particles at the LHC or the future FCC-hh collider, we can work backward to tell you the temperature of the universe."

The Results

• LHC (Current Collider): The current Large Hadron Collider can catch these particles if the universe wasn't too hot. It can probe temperatures roughly between 10 and 1,000 degrees (in energy units).
• FCC-hh (Future Super-Collider): The proposed Future Circular Collider is much bigger and more powerful. It can catch these particles even if the universe was incredibly hot (up to 100,000 degrees).
• Complementary: The main detectors and the far detectors are like two different types of fishing nets. One catches small fish close to the boat; the other catches big fish far away. Together, they cover almost all possibilities.

Conclusion

This paper proposes a clever detective story. If we build these new "far detectors" and catch a specific type of slow-moving, long-lived particle, we won't just find Dark Matter. We will also solve a mystery about the very first moments of the universe, telling us exactly how hot it was when the game began.

In short: Catching a slow, shy particle in a distant detector could tell us the temperature of the Big Bang.

Technical Summary

1. Problem Statement

The nature of Dark Matter (DM) remains an open question, with minimal Standard Model (SM) extensions increasingly constrained by direct detection experiments. The Freeze-In (FI) mechanism offers a solution where DM is produced via extremely feeble interactions, evading direct detection bounds. However, these small couplings typically render such models unobservable at colliders.

Recent interest has focused on low reheating temperatures ($T_{RH}$) in the early universe. In scenarios with two new fields (a DM candidate and a heavier mediator), low $T_{RH}$ can enhance the production of DM while simultaneously making the mediator Long-Lived (LLP). The specific problem addressed is whether the decay signatures of such LLPs at colliders (specifically the LHC and future FCC-hh) can not only detect the model but also constrain or probe the reheating temperature, a cosmological parameter usually inaccessible to particle physics experiments.

2. Methodology

Theoretical Model

The authors propose an extension of the singlet-scalar Higgs portal featuring:

• A massive dark vector boson $V_\mu$ (the DM candidate).
• A real scalar singlet $\phi$ (a long-lived mediator).
• Both are odd under a new $Z_2$ symmetry.
• Interactions:
  • Higgs portal: $\lambda_{HS} \phi^2 |H|^2$.
  • Dark-axion portal (dimension-5 operator): $\frac{c_5}{\Lambda} \phi V_{\mu\nu} \tilde{B}^{\mu\nu}$, where $V_{\mu\nu}$ and $B_{\mu\nu}$ are field strength tensors for the dark $U(1)_V$ and SM hypercharge $U(1)_Y$.
• Key Decay Channels: The mediator $\phi$ decays via $\phi \to \gamma V$ and, crucially for this study, $\phi \to Z V$ (with an on-shell $Z$ boson). The $Z$ subsequently decays to visible fermions ($Z \to f\bar{f}$), creating displaced vertex signatures.

Cosmological Framework

• Production Mechanisms: The authors solve Boltzmann equations to calculate the relic abundance of $V$. They consider:
  1. Freeze-In (FI): Production from the thermal bath and late decays of $\phi$.
  2. Super-WIMP (SW): Production from the decay of $\phi$ after it has chemically decoupled.
• Thermalization: They determine the conditions under which $\phi$ reaches thermal equilibrium (dependent on $\lambda_{HS}$ and $m_\phi$) and calculate the required $T_{RH}$ to satisfy the observed DM relic density ($\Omega_{DM} h^2$).
• Constraints: They verify that the model satisfies Big-Bang Nucleosynthesis (BBN) bounds regarding extra radiation ($\Delta N_{eff}$) and that the DM remains "cold" enough.

Collider Phenomenology

• Process: $pp \to \phi \to Z V \to (f\bar{f}) V$. The $V$ escapes detection (Missing Transverse Energy, MET), while the $Z$ decay products form a displaced vertex.
• Detectors Analyzed:
  • LHC: Main detectors (ATLAS/CMS) using Displaced Vertex + MET (DV+MET) searches; Far detectors (MATHUSLA, ANUBIS).
  • FCC-hh: Main detectors (Central and Forward Trackers) and proposed far detectors (DELIGHT, FOREHUNT).
• Tools: Simulations were performed using MadGraph5_aMC@NLO for event generation, Pythia8 for showering/hadronization, and micrOMEGAs for relic density calculations. The Displaced Decay Counter (DDC) tool was used to estimate far detector acceptances.

3. Key Contributions

1. Exploration of the $\phi \to ZV$ Channel: Unlike previous studies focusing on $\phi \to \gamma V$ or off-shell $Z$, this paper investigates the regime $m_\phi > m_Z$, enabling the on-shell decay $\phi \to ZV$. This provides a distinctive signature with a displaced $Z$ boson decaying to charged fermions.
2. Linking LLPs to Reheating Temperature: The paper establishes a direct quantitative link between the collider observables (event rates and LLP lifetimes) and the cosmological reheating temperature ($T_{RH}$). It demonstrates that observing a signal in a specific parameter space fixes the $T_{RH}$ required to generate the correct DM abundance.
3. Complementarity of Detectors: It highlights the distinct roles of main detectors (sensitive to prompt/short-lived decays) and far detectors (sensitive to long-lived decays) in probing different regions of the $(m_\phi, c\tau_\phi)$ plane.
4. Super-WIMP Sub-dominance: The authors rigorously show that in their parameter space, the Super-WIMP contribution to the DM relic density is negligible compared to Freeze-In, justifying the focus on FI production.

4. Results

Cosmological Constraints

• Freeze-In Dominance: For the benchmark parameters ($m_V \in [0.01, 50]$ GeV, $m_\phi \in [m_Z, 500]$ GeV), Freeze-In is the dominant production mechanism.
• Reheating Temperature Bounds:
  • To reproduce the observed relic density, the cutoff scale $\Lambda$ must be tuned based on $T_{RH}$. Lower $T_{RH}$ requires lower $\Lambda$ (stronger effective coupling), which shortens the $\phi$ lifetime.
  • LHC Sensitivity: The combination of main and far detectors at the High-Luminosity LHC (HL-LHC) can probe $T_{RH}$ in the range $10 \text{ GeV} \le T_{RH} \le 10^2 \text{ GeV}$ (extending to $10^3$ GeV with specific MATHUSLA designs).
  • FCC-hh Sensitivity: The Future Circular Collider (FCC-hh) significantly expands this reach, probing $10 \text{ GeV} \le T_{RH} \le 10^5 \text{ GeV}$.
• BBN Safety: Detailed analysis in Appendix A confirms that the injection of dark photons from $\phi$ decays does not violate BBN constraints on extra radiation ($\Delta N_{eff} \ll 0.1$), as the decays occur early enough or the population is sufficiently diluted.

Collider Sensitivity

• LHC:
  • Main Detectors (ATLAS): The DV+MET search is sensitive to lower $c\tau_\phi$ (prompt to intermediate decays). Optimizing the MET trigger cut to 50 GeV (from the standard 200 GeV) is crucial for sensitivity.
  • Far Detectors (MATHUSLA/ANUBIS): These are sensitive to large $c\tau_\phi$ (decays meters away). The study shows that MATHUSLA's sensitivity is highly dependent on its design size; the updated, smaller design loses sensitivity to the high $T_{RH}$ region for this model.
  • Complementarity: There is no overlap in sensitivity between main and far detectors; they probe disjoint regions of the parameter space.
• FCC-hh:
  • The higher energy ($\sqrt{s}=100$ TeV) and luminosity ($20 \text{ ab}^{-1}$) allow probing mediator masses up to the TeV scale and couplings as small as $\lambda_{HS} \approx 0.1$.
  • The Central Tracker of the main FCC-hh detector dominates the sensitivity, covering the forward tracker's reach. Far detectors (DELIGHT, FOREHUNT) provide complementary validation.

5. Significance

This work demonstrates that Long-Lived Particle (LLP) searches are not just a tool for discovering new particles but also a probe of early-universe cosmology.

• Indirect Measurement of $T_{RH}$: By measuring the mass and lifetime of a mediator $\phi$ at a collider, and assuming the Freeze-In mechanism, one can infer the reheating temperature of the universe. This bridges the gap between high-energy collider physics and cosmological history.
• Validation of Freeze-In: It provides a concrete, testable scenario where the "feeble" interactions required for Freeze-In are compensated by the long lifetime of the mediator, making the model accessible to current and future experiments.
• Experimental Guidance: The paper offers specific guidance for experimentalists, suggesting optimized trigger cuts (e.g., lower MET thresholds) and highlighting the critical importance of far detector designs (like MATHUSLA's size) for probing specific cosmological regimes.

In conclusion, the paper establishes that the interplay between cosmological constraints and LLP searches at the LHC and FCC-hh can place novel, model-dependent bounds on the reheating temperature, covering a range of 5 orders of magnitude ($10 - 10^5$ GeV).