Spin Identification of Dark Sector Mediators through Angular Distributions

This paper proposes a method to distinguish between vector and scalar dark sector mediators by analyzing the angular distributions of their decay products, demonstrating that experiments like DUNE, SHiP, and FASER2 can determine the mediator's spin in significant regions of unconstrained parameter space.

The Gist

Imagine the universe is filled with a hidden "dark sector," a shadowy realm of particles that we can't see directly but suspect exists because of how gravity and other forces behave. Scientists believe this dark sector might contain a "messenger" particle that acts as a bridge, connecting our visible world to this dark one.

The big question is: What kind of messenger is it? Is it a spinning top (a vector particle, like a photon) or a smooth, non-spinning ball (a scalar particle)?

This paper proposes a clever way to answer that question by looking at how these messengers decay (break apart) into pairs of electrons and positrons. Here is the breakdown in simple terms:

The Detective's Dilemma

When a dark messenger particle is created in a high-energy collision (like in a particle accelerator), it flies off, travels a short distance, and then decays into an electron and a positron. Scientists can catch these two particles in a detector.

• The Easy Part: By measuring the energy and speed of these two particles, scientists can easily figure out the mass of the messenger and how strongly it interacts with our world.
• The Hard Part: Figuring out the spin (whether it's a vector or a scalar) is much trickier. Usually, you need to know exactly how the messenger was moving the moment it was created to tell the difference. But in these experiments, the messenger is created in a chaotic "mess" of particles, so we can't see its birth. We only see the "crime scene" (the decay) later on.

The "Magic Angle" Solution

The authors of this paper found a "magic angle" that acts like a fingerprint for the particle's spin.

Think of the messenger particle as a spinning arrow (if it's a vector) or a rolling ball (if it's a scalar).

• If it's a Scalar (Ball): When it breaks apart, the electron and positron fly out in all directions equally, like popcorn popping randomly in a pot. The distribution is isotropic (the same everywhere).
• If it's a Vector (Arrow): Because the arrow was spinning, the electron and positron prefer to fly out in specific directions relative to how the arrow was pointing. The distribution is anisotropic (it has a pattern).

The Catch: To see this pattern, you usually need to know exactly how the messenger was spinning when it was born. Since we can't see that, the authors realized they could use a different reference point: the laboratory itself.

They identified an angle that can be calculated using only the information we can measure in the lab (the speed and direction of the electron and positron).

• If the messenger is a Scalar, this angle looks completely random.
• If the messenger is a Vector, this angle shows a distinct, predictable pattern (like a lopsided cloud of popcorn).

The Experiment Plan

The paper checks if major upcoming and current experiments can actually spot this pattern. They looked at four specific "hunting grounds":

1. NA62: A current experiment.
2. FASER2: A new detector at the Large Hadron Collider (LHC).
3. DUNE: A massive neutrino experiment in the US.
4. SHiP: A proposed experiment at CERN.

The Results:

• NA62 likely won't catch enough "crime scenes" (events) to tell the difference between the random popcorn and the patterned popcorn.
• FASER2, DUNE, and SHiP are expected to be powerful enough. Specifically, SHiP is predicted to be the best at this, capable of identifying the spin in large areas of the "unknown" territory where we haven't found dark particles yet.

The Technical Requirement

To pull this off, the detectors need to be very sharp-eyed.

• Imagine trying to see the direction of two tiny sparks flying apart from a distance. If your camera is blurry (low resolution), the sparks look like they are flying randomly even if they aren't.
• The paper calculates that the detectors need a specific level of precision (about the width of a human hair over a distance of 10 meters) to clearly distinguish the "spinning arrow" pattern from the "rolling ball" randomness.

The Bottom Line

If we discover a new dark messenger particle in the next decade, we won't just know that it exists; we will be able to tell what kind of particle it is. By simply measuring the angles at which its decay products fly, experiments like SHiP and DUNE can determine if the dark sector is populated by spinning vectors or smooth scalars, unlocking a deeper understanding of the universe's hidden architecture.

Technical Summary

Technical Summary: Spin Identification of Dark Sector Mediators through Angular Distributions

Problem Statement
A variety of current and planned experiments (e.g., SHiP, NA62, DUNE, FASER2) are designed to detect displaced decays of light, long-lived dark sector particles, such as dark photons ($A'$) or dark scalars ($\phi$). While the discovery of such a state would allow for the determination of its mass (via invariant mass reconstruction) and coupling strength (via decay length measurements), a critical gap remains in the ability to determine the mediator's spin. Distinguishing between a vector boson (dark photon) and a scalar/pseudoscalar mediator is essential for deconstructing the underlying physics model. Previous methods for spin determination often relied on observables requiring full event kinematics (e.g., the angle $\theta^*$ between the electron momentum in the dark photon rest frame and the dark photon momentum in the pion rest frame), which are generally inaccessible in long-lived particle searches where only the decay products are observed.

Methodology
The authors propose a method to identify the mediator spin using an angular observable that is reconstructible solely from the four-momenta of the decay products in the laboratory frame.

1. Theoretical Framework: The study focuses on dark photons produced in pseudoscalar meson decays ($\pi^0, \eta, \eta' \to \gamma A'$) followed by $A' \to e^+e^-$. The authors analyze the angular distribution of the electron in the mediator's rest frame.
2. Spin Correlations: The analysis explicitly includes spin correlations between the production and decay processes. The authors note that ignoring these correlations would incorrectly render the dark photon distribution isotropic.
3. Reconstructible Observable: They identify a specific angle, $\theta$, defined as the angle between the electron momentum in the dark photon rest frame and the dark photon momentum in the laboratory frame. This angle can be calculated using only laboratory-frame kinematic variables ($E_\pm, \vec{p}_\pm$).
4. Distribution Analysis:
   • For a scalar mediator, the angular distribution is isotropic.
   • For a vector mediator (dark photon), the distribution is anisotropic. The probability density function $\rho(\cos \theta)$ depends on the dark photon mass ($m$), the meson mass ($M$), and the fermion mass ($m_f$).
   • The authors derive an analytical expression for $\rho(\cos \theta)$ (Eq. 4) which shows that the anisotropy is a dynamical effect arising from the transverse polarization degrees of freedom of the dark photon.
5. Experimental Sensitivity: Using a log-likelihood analysis, the authors calculate the 95% Confidence Level (CL) sensitivity for distinguishing the anisotropic vector hypothesis from the isotropic scalar hypothesis. They utilize event statistics from existing and future facilities (NA62, FASER2, DUNE, SHiP) based on light meson spectra from EPOSLHC and decay simulations from FORESEE.
6. Resolution Requirements: The study assesses the impact of experimental angular and energy resolution. It determines the maximum angular smearing allowed before the anisotropic and isotropic distributions become indistinguishable.

Key Contributions

• Identification of a Lab-Frame Observable: The paper provides a specific angular observable ($\theta$) that allows for spin determination without needing to reconstruct the full event kinematics (specifically the initial meson momentum), which is often impossible in fixed-target or forward-detector experiments.
• Analytical Distribution Formula: The authors derive a closed-form expression for the electron angular distribution in terms of measurable laboratory variables, explicitly accounting for spin correlations and mass dependencies.
• Facility-Specific Sensitivity Maps: The study presents 95% CL sensitivity contours for NA62, FASER2, DUNE, and SHiP, mapping the regions of the dark photon mass ($m_{A'}$) and kinetic mixing parameter ($\epsilon$) where spin identification is feasible.
• Experimental Requirements: The paper quantifies the necessary detector performance, specifically angular resolution, to successfully measure the anisotropy.

Results

• NA62: The analysis indicates that NA62 does not possess a sufficient event rate to determine the dark photon spin in the studied parameter space.
• FASER2 and DUNE: These experiments are projected to identify the mediator spin in currently unconstrained regions of parameter space, specifically around $\epsilon \sim 10^{-4}$ for FASER2 and $\epsilon \sim 10^{-7}$ for DUNE.
• SHiP: Due to its high collision rate and large fiducial volume, SHiP demonstrates the best expected sensitivity, capable of measuring angular anisotropies over the largest portion of the unconstrained parameter space.
• Resolution Constraints: To distinguish the spin hypotheses, the experiments require an angular resolution of approximately 0.1 mrad (for 10–100 GeV pion energies) to 50 $\mu$rad (for 1 TeV energies). The study notes that this implies a spatial resolution requirement of roughly $\delta x \sim 0.1$ mm for a detector length of $\sim 10$ m.
• Mass Dependence: The anisotropy is most pronounced for lighter dark photons ($m \ll M$). As the dark photon mass approaches the meson mass ($m \to M$), the angular dependence diminishes, and the distribution approaches isotropy.

Significance and Claims
The paper claims that the proposed method offers a viable pathway to "deconstruct" the underlying physics model of a discovered dark sector mediator. By demonstrating that the mediator spin can be determined using only the decay products' four-momenta, the authors provide a practical tool for future long-lived particle experiments. The work establishes that while mass and coupling can be inferred from standard kinematic measurements, the spin determination requires specific angular distribution measurements that are now shown to be experimentally feasible for major upcoming facilities like SHiP, FASER2, and DUNE. The authors emphasize that these results represent "best-case sensitivities," assuming background-free environments and no other exotic production mechanisms.