Dark photon and U(1)$_{B-L}$ gauge boson from dark Higgs boson decays at FASER and SHiP

This paper investigates the sensitivity of the FASER and SHiP experiments to dark photons and U(1)$_{B-L}$ gauge bosons produced via dark Higgs boson decays, incorporating new single-boson production channels, deriving constraints from current FASER data, and projecting future discovery potential for these models including those with freeze-in sterile neutrino dark matter.

The Gist

Imagine the universe as a giant, bustling city. We know a lot about the "Main Street" of this city—the Standard Model of particle physics—which includes the familiar citizens like electrons, protons, and photons. But physicists suspect there's a hidden "Dark Sector" neighborhood nearby, a place where mysterious residents like Dark Matter live. We can't see them directly, but we think they might interact with our world through secret backdoors.

This paper is like a detective report from a team of physicists investigating two specific types of secret backdoors: the Dark Photon and the $U(1)_{B-L}$ Gauge Boson. They are asking: How can we catch these hidden particles using our most advanced particle detectors, FASER and SHiP?

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

1. The Cast of Characters

• The Dark Higgs Boson ($\phi$): Think of this as a "messenger" or a "courier." It's a heavy particle that lives in the Dark Sector. Its job is to travel from the dark neighborhood into our visible world.
• The Dark Photon ($A'$) & The $Z'$ Boson: These are the "packages" the courier is carrying. They are the actual particles that might explain Dark Matter.
• The Experiments (FASER, FASER2, SHiP): These are giant, high-tech "mailboxes" or "traps" located far away from the particle collision point. They are designed to catch particles that are long-lived (they don't decay immediately) and travel in a straight line.

2. The Old Way vs. The New Discovery

For a long time, scientists thought the Dark Higgs courier had only one way to deliver its packages:

• The Old Method (Pair Production): The courier splits in half, dropping two Dark Photons at once. It's like a delivery truck breaking down and dropping two packages simultaneously. This was the only process they knew how to look for.

The Big New Idea in this Paper:
The authors realized the courier has a second, sneaky way to deliver packages:

• The New Method (Single Production): The courier doesn't have to split in half. Instead, it can drop one Dark Photon while keeping a piece of itself (or interacting with a standard particle) to keep moving.
  • Analogy: Imagine a courier walking down the street. Instead of splitting into two people, they hand a package to a passerby (a standard particle) and keep walking. This allows the courier to travel much further before disappearing.

Why does this matter?
If the courier travels further, it can reach detectors that are farther away. This opens up a whole new "neighborhood" of possibilities. It allows scientists to look for Dark Photons that are heavier or interact more strongly than previously thought possible. It's like realizing the courier can walk through walls, meaning we can search for them in places we previously thought were too far to reach.

3. The Investigation at FASER and SHiP

The team ran simulations to see how this new "Single Production" method would look in two major experiments:

• FASER (The Current Detective): Located at the Large Hadron Collider (LHC) in Switzerland. The team used the latest data from FASER to set "exclusion limits."

  • Result: They found that if the Dark Higgs is heavy enough and the interaction is strong enough, the current FASER data already rules out (excludes) certain types of Dark Photons. It's like saying, "We checked this specific alleyway, and the thief isn't hiding there."

• FASER2 and SHiP (The Future Super-Detectives): These are the next-generation experiments.

  • FASER2 will be a larger, more sensitive detector.
  • SHiP (Search for Hidden Particles) is a massive experiment planned for CERN, designed specifically to catch these elusive particles.
  • Result: By including this new "Single Production" method, the sensitivity of these future experiments increases dramatically. They can now hunt for Dark Photons in regions of "parameter space" (a fancy way of saying "combinations of mass and interaction strength") that were previously invisible.

4. The "Freeze-In" Connection

The paper also touches on a specific theory called "Freeze-in Dark Matter."

• Analogy: Imagine a room where the temperature (energy) is slowly dropping. Particles are "freezing" out of the soup of energy and becoming solid matter.
• In this scenario, the Dark Photon or $Z'$ boson acts as the mechanism that helps create the Dark Matter (specifically, "sterile neutrinos").
• The paper shows that if this "Freeze-in" theory is true, the FASER2 and SHiP experiments are perfectly positioned to catch the evidence. It's like having a camera set up exactly where the ice is forming.

5. The Bottom Line

This paper is a roadmap for the future of particle physics.

1. We found a new path: We realized Dark Higgs bosons can produce single Dark Photons, not just pairs.
2. We updated the map: This new path means we can search for these particles in places we thought were off-limits.
3. We have a plan: The current FASER experiment is already constraining the possibilities, and the future FASER2 and SHiP experiments are expected to be powerful enough to either find these particles or definitively rule out this specific theory of Dark Matter.

In short, the authors are telling us: "Don't just look for the courier dropping two packages. Look for the courier dropping one package and walking away. If you do, you might finally catch a glimpse of the Dark Sector."

Technical Summary

1. Problem Statement

The paper addresses a gap in the search for feebly interacting dark sector particles, specifically dark photons ($A'$) and U(1)$_{B-L}$ gauge bosons ($Z'$).

• Context: These particles are often produced in experiments like FASER, FASER2, and SHiP via the decay of a Dark Higgs boson ($\phi$).
• Limitation of Previous Studies: Existing literature and experimental analyses primarily focused on the pair production of vector bosons from Dark Higgs decays ($\phi \to A'A'$ or $\phi \to Z'Z'$).
• The Gap: This production channel is kinematically forbidden if the vector boson mass ($m_{V}$) is greater than half the Dark Higgs mass ($m_\phi/2$). Furthermore, even when allowed, the pair production channel might not dominate if the kinetic mixing or gauge coupling is small.
• Objective: The authors investigate a previously under-analyzed production channel: the single vector boson production associated with Standard Model (SM) particles via the decay $\phi \to A' + \text{SM}$ (or $Z' + \text{SM}$). They aim to determine if this channel expands the sensitivity of long-lived particle (LLP) searches to new regions of parameter space, particularly for larger kinetic mixing ($\epsilon$) or smaller gauge couplings ($g'$).

2. Methodology

The authors employ a theoretical and phenomenological approach to model the production and detection of these particles.

A. Theoretical Framework

Two models are analyzed:

1. Dark Photon Model: A secluded $U(1)_D$ symmetry with kinetic mixing ($\epsilon$) between the dark photon and the SM hypercharge boson. The Dark Higgs ($\phi$) breaks $U(1)_D$.
2. Gauged U(1)$_{B-L}$ Model: A gauge symmetry based on Baryon minus Lepton number. It requires three right-handed neutrinos to cancel anomalies. The Dark Higgs breaks $U(1)_{B-L}$.

B. Production Mechanisms

The study considers three distinct production processes of the vector boson from Dark Higgs decays:

1. Process (A): On-shell pair production: $\phi \to A'A'$ (or $Z'Z'$).
2. Process (B): New Focus: On-shell single production with SM particles: $\phi \to A' f \bar{f}$ (or $Z' f \bar{f}$). This occurs via an off-shell vector boson in the final state or direct interaction, dominated by the diagram where the Dark Higgs decays into a real vector boson and a virtual one that splits into fermions.
3. Process (C): Off-shell pair production: $\phi^* \to A'A'$ (via meson decays like $B \to K \phi^* \to K A'A'$).

The authors derive the differential decay widths for Process (B), showing that for the Dark Photon model, the amplitude is proportional to $g'\epsilon$ and is unsuppressed by scalar mixing, making it dominant over other single-production diagrams unless $g'$ is extremely small.

C. Experimental Simulation

• Experiments: FASER (Run 2 data), FASER2 (Future HL-LHC), and SHiP (Fixed-target experiment).
• Production Source: Dark Higgs bosons are produced primarily via $B$-meson decays ($B \to X_s \phi$).
• Kinematics: The authors simulate the momentum and angular distributions of $B$ mesons using Pythia 8 (Monash tune) for LHC (FASER/FASER2) and specific target configurations for SHiP.
• Signal Definition: A signal event is defined as the vector boson decaying into a pair of charged particles (e.g., $e^+e^-$) or photons inside the detector volume.
• Calculations: The expected number of signal events ($N_A, N_B, N_C$) is calculated by integrating production cross-sections, branching ratios, and the probability of decay within the detector geometry ($P_{det}$), accounting for the boost factor ($\beta\gamma$) and decay lengths.

3. Key Contributions

1. Identification of a New Dominant Channel: The paper establishes that the single vector boson production ($\phi \to A' + \text{SM}$) is a critical channel, particularly when:
   • The vector boson is heavier than $m_\phi/2$ (where pair production is kinematically forbidden).
   • The kinetic mixing $\epsilon$ is relatively large ($\epsilon > 10^{-5}$), which would normally cause the Dark Higgs to decay too quickly to reach the detector if only pair production were considered. However, in this channel, the Dark Higgs can still be long-lived enough to travel to the detector while producing a short-lived vector boson.
2. Updated Exclusion Limits: Using the latest FASER results (177 fb$^{-1}$), the authors derive new exclusion limits for the Dark Photon model, specifically constraining the parameter space where $g' \sim 0.05 - 0.2$ and $\alpha > 10^{-3}$.
3. Expanded Sensitivity Projections:
   • FASER2: Demonstrates that including Process (B) significantly extends the sensitivity to larger $\epsilon$ values (up to $\sim 10^{-3}$) compared to the "vanilla" dark photon model. It also shows sensitivity to smaller $\epsilon$ regions due to the rectangular detector geometry and the specific kinematics of the single-production channel.
   • SHiP: Shows that SHiP's large decay volume allows it to probe an even wider parameter space than FASER2.
4. Application to Freeze-in Dark Matter: The study applies these results to the U(1)$_{B-L}$ model with sterile neutrino Dark Matter produced via the freeze-in mechanism. They show that FASER2 and SHiP can probe the specific parameter space ($g_{B-L} < 10^{-7}$) required to explain the observed Dark Matter relic abundance, a region inaccessible to standard $Z'$ production searches.

4. Key Results

• FASER Constraints: The latest FASER null results exclude a region of the Dark Photon parameter space for $m_\phi = 0.5 - 3.0$ GeV, specifically for gauge couplings $g' \in [10^{-5}, 0.05]$ and scalar mixing $\alpha \gtrsim 10^{-3}$.
• FASER2 Sensitivity:
  • The inclusion of $\phi \to A' + \text{SM}$ opens up a "vertical" extension in the sensitivity plot (higher $\epsilon$) that was previously unexplored.
  • For $m_\phi = 2.0$ GeV, the single-production channel dominates the sensitivity for $\epsilon > 10^{-5}$.
  • The rectangular shape of the FASER2 detector does not hinder sensitivity; in fact, the new production mechanism compensates for geometric acceptance issues.
• SHiP Sensitivity: SHiP is projected to cover a significantly larger parameter space than FASER2 due to its higher proton-on-target count ($6 \times 10^{20}$) and larger decay volume, reaching down to very small $\epsilon$ and $g_{B-L}$ values.
• U(1)$_{B-L}$ & Dark Matter: The analysis confirms that the freeze-in production of sterile neutrino Dark Matter in the U(1)$_{B-L}$ model is testable. The single-production channel allows FASER2 and SHiP to probe $g_{B-L}$ values as low as $10^{-7}$, which are necessary to generate the correct relic density for DM masses $m_N \gtrsim 10$ GeV.

5. Significance

• Paradigm Shift in LLP Searches: This work challenges the assumption that pair production is the only viable channel for Dark Higgs decays in LLP searches. It demonstrates that single production is essential for a complete understanding of the Dark Higgs phenomenology, especially for heavier vector bosons or larger mixing parameters.
• Optimization of Future Experiments: The results provide crucial guidance for the design and analysis strategies of FASER2 and SHiP. Ignoring the single-production channel would lead to an underestimation of the discovery potential, particularly for models with larger kinetic mixing.
• Connection to Dark Matter: By linking the production of gauge bosons from Dark Higgs decays to the freeze-in mechanism of sterile neutrino Dark Matter, the paper bridges the gap between collider LLP searches and cosmological Dark Matter constraints, offering a concrete path to test the freeze-in scenario.
• Comprehensive Coverage: The study provides a unified framework covering both Dark Photon and U(1)$_{B-L}$ models, incorporating the latest experimental data and future projections, thereby setting a new benchmark for sensitivity studies in the dark sector.