Vector-channel scattering of dark particles in a Sp(4) gauge theory

This paper presents preliminary lattice results on the vector-channel scattering of two pseudoscalar states in an $Sp(4)$ gauge theory with two fundamental Dirac fermions, a candidate model for strongly interacting massive particle dark matter, utilizing Lüscher's formalism to analyze finite-volume energy levels across various fermion masses.

The Gist

Imagine the universe is a giant, invisible ocean. We know there's a lot of "stuff" in this ocean that we can't see, feel, or touch, but we know it's there because it pulls on stars and galaxies with its gravity. We call this invisible stuff Dark Matter.

For decades, scientists have been trying to figure out what this dark matter is made of. The most popular idea was that it's made of heavy, slow-moving particles (like invisible bowling balls) that barely interact with anything else. But recent observations suggest this might not be the whole story.

This paper explores a different, more exotic idea: Dark Matter made of "Strongly Interacting Massive Particles" (SIMPs).

Here is the breakdown of what the researchers did, using simple analogies:

1. The Playground: A New Kind of Universe

The scientists didn't just guess; they built a virtual universe inside a supercomputer.

• The Theory: They created a mathematical model based on a specific type of force called an $Sp(4)$ gauge theory. Think of this as a new set of rules for how particles behave, different from the rules we see in our everyday world (like electromagnetism or gravity).
• The Players: In this virtual world, they introduced two types of fundamental particles (fermions). When these particles interact, they form "bound states," similar to how quarks stick together to form protons in our world.
• The Result: These bound states create light, ghostly particles called pNGBs (pseudo-Nambu-Goldstone Bosons). In this theory, these pNGBs are the Dark Matter.

2. The Problem: How Do They Bounce?

In the SIMP scenario, dark matter doesn't just float around; it crashes into itself.

• The 3-to-2 Dance: Usually, when particles collide, two might turn into two ($2 \to 2$). But in this dark sector, three particles can collide and turn into two ($3 \to 2$). This is like a dance where three partners join hands, and two spin off, leaving the third one behind. This process is crucial for explaining why the universe looks the way it does today.
• The Bounce-Off: However, for this to work without breaking the laws of physics (like making dark matter clump too tightly), the particles need to bounce off each other in a very specific way. This is called scattering.

3. The Experiment: The Lattice Simulation

The researchers wanted to see exactly how these dark particles bounce off each other.

• The Grid: They placed their virtual universe on a 4D grid (like a giant 3D chessboard made of time and space).
• The Test: They fired two dark particles at each other and watched what happened. They were specifically looking at the "Vector Channel."
  • Analogy: Imagine two spinning tops colliding. The "Vector Channel" is a specific way they can spin and bounce. The researchers were looking for a specific "spin-1" collision, which is like checking if the tops wobble in a particular pattern when they hit.

4. The Discovery: The "Heavy" and the "Light"

They ran the simulation with three different settings, which they called Heavy, Medium, and Light. These refer to how heavy the particles are in the simulation.

• The Heavy Case: When the particles were heavy, they bounced off each other gently. The data looked smooth, like a ball rolling over a flat field. The scientists could describe this bounce using a simple formula (Effective Range Expansion).
• The Light Case: When the particles were lighter, something exciting happened. The data showed a resonance.
  • Analogy: Imagine pushing a child on a swing. If you push at just the right rhythm, the swing goes super high. In the "Light" case, the dark particles found a "sweet spot" where they resonated. This suggests the existence of a heavy, short-lived particle (a vector meson, or $\rho'_D$) that acts like a bridge between the colliding particles. It's like a hidden trampoline that briefly catches the particles before they bounce apart.

5. Why Does This Matter?

The ultimate goal is to see if this theory matches what we see in the real universe.

• The Bullet Cluster: Astronomers have watched galaxy clusters smash into each other (like the "Bullet Cluster"). The dark matter in these collisions seems to pass right through, but not too easily. It has a little bit of friction.
• The Check: The scientists calculated how much "friction" (cross-section) their dark particles would have.
  • At the low energies found in our galaxy's halo (where dark matter lives), the "Vector Channel" (the spin-1 bounce) is very quiet. The "friction" is low.
  • However, at higher energies, that "resonance" (the trampoline effect) creates a huge spike in interaction.

The Bottom Line

This paper is a preliminary report from a massive computer experiment. The researchers successfully built a virtual dark sector and measured how its particles bounce.

They found that:

1. Their model works and produces stable dark matter particles.
2. There is evidence of a resonance (a hidden, heavy particle) that influences how dark matter interacts.
3. While this specific type of bounce is quiet at the low energies of our galaxy, it could be very important in other parts of the universe or in the early universe.

In short: They built a digital sandbox to test a new theory of dark matter. They found that in this theory, dark matter particles don't just bounce; they sometimes hit a "hidden trampoline" that changes how they interact. This helps scientists understand if this specific theory could explain the mysterious dark matter we see in the sky.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the nature of Dark Matter (DM), specifically within the Strongly Interacting Massive Particle (SIMP) paradigm. Unlike Weakly Interacting Massive Particles (WIMPs), SIMPs propose a dark sector where DM consists of pseudo-Nambu-Goldstone Bosons (pNGBs) arising from the dynamical breaking of a global symmetry.

• The SIMP Mechanism: Thermal relic density is established via $3 \to 2$ number-changing annihilations (driven by the Wess-Zumino-Witten anomaly) rather than the standard $2 \to 2$ freeze-out.
• Theoretical Model: The authors focus on an $Sp(4)$ gauge theory coupled to two fundamental Dirac fermions. This model is minimal and compatible with astrophysical constraints. The global symmetry breaking pattern is $SU(4) \to Sp(4)$, yielding 5 pNGBs ($\pi_D$) transforming in the 5-dimensional representation of the unbroken $Sp(4) \sim SO(5)$.
• The Specific Challenge: While $3 \to 2$ processes drive the relic density, $2 \to 2$ self-scattering of pNGBs is crucial for addressing small-scale structure formation issues (e.g., the core-vs-cusp problem). However, these interactions are constrained by Bullet Cluster observations ($\sigma/m_{DM} \lesssim 1 \text{ cm}^2/\text{g}$).
• Gap in Knowledge: Leading-order Effective Field Theory (EFT) is insufficient in the relevant parameter space. Heavy resonances, specifically vector mesons ($\rho_D, \rho'_D$), play a critical role in both $2 \to 2$ scattering and $3 \to 2$ semi-annihilation. The paper aims to compute the spin-1 ($p$-wave) scattering amplitude of two pNGBs to understand the role of these vector resonances.

2. Methodology

The authors employ Lattice Field Theory to perform non-perturbative numerical simulations of the $Sp(4)$ gauge theory.

• Lattice Setup:

  • Action: Standard Wilson plaquette action for the gauge field and Wilson-Dirac formulation for fermions.
  • Parameters: Three distinct sets of lattice parameters $(\beta, am_0)$ were simulated, labeled "heavy" ($6.9, 0.92$), "medium" ($7.05, 0.863$), and "light" ($7.05, 0.867$).
  • Volumes: Multiple lattice volumes were generated for each parameter set to perform finite-volume analysis.
  • Algorithm: Hybrid Monte Carlo (HMC) with Hasenbusch acceleration using the HiRep code.

• Operator Construction:

  • To isolate the spin-1 channel, the authors constructed interpolating operators transforming under the irreducible representations (irreps) of the octahedral group $O_h$ and its little groups (LG) for boosted frames.
  • Two types of operators:
    1. Single-meson (Vector) operators: $O_{V\mu} = \sum_x \bar{Q}\Gamma_\mu Q e^{i\vec{P}\cdot\vec{x}}$.
    2. Two-meson (pNGB-pNGB) operators: Constructed as antisymmetric combinations of two pNGB operators to ensure the correct symmetry properties for the spin-1 channel.
  • Projection: Operators were projected onto specific LG irreps (e.g., $T_1, A_1, E, B_1$) using group theory projection formulas to isolate the $J^P = 1^-$ channel.

• Analysis Framework:

  • Variational Method: A correlation matrix (up to $3 \times 3$) was constructed involving vector and two-pNGB operators. Generalized Eigenvalue Problem (GEVP) analysis was used to extract the low-lying energy eigenvalues ($E_n$).
  • Lüscher's Formalism: Finite-volume energy shifts were mapped to infinite-volume scattering phase shifts ($\delta_1$) using Lüscher's quantization condition. This relates the discrete energy spectrum in a box of size $L$ to the scattering amplitude $t_1(s)$.
  • Kinematics: The study focused on the elastic region ($2m_{\pi_D} < E_{cm} < 4m_{\pi_D}$).

3. Key Contributions

• First Lattice Study of Vector Channel: This work provides the first lattice determination of the $p$-wave scattering phase shift in the vector channel (the 10-dimensional representation of $Sp(4)$) for this specific dark matter model.
• Operator Basis Extension: The authors extended existing operator bases to include specific gamma-matrix structures ($\gamma_0\gamma_\mu$) necessary to properly interpolate the vector states in the lattice formulation.
• Phenomenological Bridge: The study connects non-perturbative lattice data directly to phenomenological constraints on SIMP dark matter, specifically calculating the scattering cross-section per unit mass ($\sigma/m_{DM}$).

4. Results

The analysis yielded distinct behaviors depending on the fermion mass (lattice parameter set):

• Heavy Case ($am_0 = 0.92$):

  • The phase shifts were small, and the energy levels were close to the non-interacting thresholds.
  • Analysis: The data was fitted using the Effective Range Expansion (ERE) at leading order: $p^3 \cot \delta_1 \approx -1/a_1$.
  • Result: The scattering length was determined to be $a_1 m_{\pi_D} = -1.76^{+0.11}_{-0.47}$. The negative sign indicates an attractive interaction in the 10 channel, contrasting with the repulsive nature found in the 14 channel in previous studies.

• Light Case ($am_0 = 0.867$):

  • The energy spectrum showed clear evidence of a resonance (a level shifting significantly from the non-interacting line).
  • Analysis: The data was fitted using a Breit-Wigner form for a resonance: $\cot \delta_1 = \frac{m_{\rho'_D}^2 - s}{\sqrt{s}\Gamma(s)}$.
  • Result: A resonance was identified, and the coupling constant between the vector resonance and two pNGBs was extracted: $g_{\rho'_D \pi_D \pi_D} = 10.3^{+1.6}_{-1.0}$.

• Scattering Cross Section:

  • Using the extracted phase shifts, the authors calculated the $2 \to 2$ scattering cross-section $\sigma$ normalized by the DM mass ($m_{DM} = 100$ MeV).
  • Kinematic Dependence: At low galactic halo energies ($E_{cm} < 1$ MeV), the 14-channel (scalar) dominates, and the 10-channel (vector) is highly suppressed. However, the vector channel exhibits a prominent peak at higher energies, which could be significant in other kinematic regimes.

5. Significance and Future Outlook

• Validation of SIMP Models: The results demonstrate that vector resonances in $Sp(4)$ gauge theories can be successfully characterized via lattice simulations. This is crucial for validating the SIMP scenario, where such resonances mediate the self-interactions required to solve small-scale structure problems.
• Constraint on Parameter Space: The finding that the vector channel is attractive but suppressed at low energies helps refine the parameter space where SIMP models satisfy both the Bullet Cluster constraints and the requirements for structure formation.
• Future Work: The authors note that these are preliminary results. Future work will involve:
  • Extending the analysis to the 14-dimensional channel and $3 \to 2$ processes.
  • Addressing lattice systematics (continuum limit, infinite volume extrapolation).
  • Exploring a broader range of parameters to fully map the phenomenological viability of the model.

In summary, this paper establishes a robust lattice framework for studying vector-channel scattering in $Sp(4)$ dark matter models, providing the first non-perturbative evidence of vector resonances and their coupling strengths, which are essential inputs for realistic SIMP dark matter phenomenology.