On the equivalence of unitarization prescriptions for the Sommerfeld enhancement

This paper demonstrates that various unitarization prescriptions for the Sommerfeld enhancement in self-interacting dark matter are effectively equivalent and regulator-independent, leading to a unified, regulator-free formula for multi-state systems expressed in terms of the standard enhancement factor, the hard annihilation amplitude, and the scattering $S$-matrix.

The Gist

The Big Picture: Dark Matter and the "Traffic Jam"

Imagine dark matter particles are like cars driving on a highway. Usually, they just zoom past each other without interacting much. But in some theories, these particles have a "long-range force" (like a magnetic pull) that gets stronger the slower they move.

When two dark matter cars get close, this pull drags them together, creating a "traffic jam" or a crowd. This crowd makes it much more likely for them to crash (annihilate) into other particles. This phenomenon is called the Sommerfeld effect.

However, there's a problem. If the traffic jam gets too perfect—like a perfect resonance where the cars line up exactly right—the math predicts that the crash rate becomes infinite. In physics, an infinite crash rate is impossible; it breaks the rules of the universe (a rule called unitarity, which basically says you can't create more stuff than you started with).

The Problem: Three Different Mechanics, One Broken Car

Physicists realized this "infinite crash" problem and proposed three different ways to fix the math so the crash rate stays finite and realistic. Think of these three methods as three different mechanics trying to fix a car that's revving its engine too high:

1. The PSS24 Mechanic (The "Cut-and-Paste" Approach): This method says, "Let's draw a circle around the crash zone. Inside the circle, we use complex rules for the crash. Outside, we use the simple traffic rules." They match the two at the edge of the circle. The problem? The result seems to depend on exactly where you draw that circle.
2. The W25 Mechanic (The "Renormalization" Approach): This method treats the crash rate like a mathematical series that keeps adding up forever. They use a technique called "Renormalization Group" (like a smart filter) to smooth out the infinite parts and make the math work without needing to draw a specific circle.
3. The FP25 Mechanic (The "Self-Interaction" Approach): This method looks at the car's own internal energy and how it interacts with itself. It uses a complex diagrammatic approach (like a flowchart of every possible interaction) to calculate the crash rate directly, including the "self-correction" that stops the engine from revving too high.

The Paper's Discovery: They Are All Doing the Same Thing

The authors of this paper asked: "Are these three mechanics actually fixing the car in the same way, or are they giving us three different answers?"

They found that, despite looking very different on paper, all three methods are essentially equivalent.

Here is the core of their discovery, explained simply:

1. The "Outgoing Wave" Mystery

In all three methods, there is a specific mathematical term that acts as a "brake" to stop the crash rate from going infinite.

• In the PSS24 method, this brake looks like a complicated number that depends on the "irregular" solution (a weird, messy wave function that blows up at the center).
• In the W25 method, this brake is a simple number related to the "phase shift" (how much the wave is delayed).
• In the FP25 method, it's an integral involving messy wave functions.

The paper proves that near a resonance (when the traffic jam is worst), the messy, complicated "brake" in the PSS24 method is actually just a fancy way of writing the simple "phase shift" brake used in the W25 method.

The Analogy: Imagine you are trying to stop a spinning top.

• Mechanic A says, "I need to measure the exact friction of the table at this specific spot."
• Mechanic B says, "I just need to know how fast the top is wobbling."
• The Paper says: "When the top is wobbling dangerously fast (near resonance), measuring the friction at that specific spot gives you the exact same information as measuring the wobble. You don't need the messy measurement; the simple wobble measurement is enough."

2. The "Circle" Doesn't Matter

The PSS24 method relies on drawing a circle (a "matching radius") to separate the short-range crash physics from the long-range traffic physics. The authors showed that even though the math looks like it depends on where you draw that circle, the final answer does not.

The messy parts of the math that depend on the circle cancel each other out perfectly. This means the result is "regulator-independent"—it's a true physical fact, not an artifact of how you chose to do the math.

3. Extending to Complex Systems (The "Wino" Example)

Dark matter isn't always just one type of particle. Sometimes, it's a mix of different types (like a fleet of different cars: sedans, trucks, and motorcycles) that can turn into one another. This is called a "multi-state system."

The paper takes the insight that "the messy brake is actually just the simple brake" and applies it to these complex, multi-particle fleets. They derived a new, simplified formula that works for these complex systems.

They tested this new formula using Wino Dark Matter (a specific, well-known theoretical particle). They compared their new, simplified "brake" against the old, complicated "brake" used in the PSS24 method.

• The Result: The new, simple formula matched the old, complicated one perfectly, even near the most dangerous resonances.

Summary of the Conclusion

The paper concludes that:

1. Equivalence: The three different ways physicists have been trying to fix the "infinite crash" problem are actually saying the same thing.
2. Simplification: You don't need to worry about the messy, "irregular" wave functions or the specific size of the "circle" you draw. You can use a much simpler formula based on the standard "regular" wave functions and the scattering phase shift.
3. Universality: This simplified formula works not just for simple particles, but for complex fleets of interacting dark matter particles (multi-state systems).

In everyday terms: The paper tells us that the three different maps we were using to navigate a dangerous storm are actually pointing to the same safe harbor. We can now throw away the complicated, confusing maps and use a single, simple, and reliable compass that works for everyone.

Technical Summary

Technical Summary: On the equivalence of unitarization prescriptions for the Sommerfeld enhancement

Problem Statement
The Sommerfeld effect describes the enhancement (or suppression) of particle annihilation cross sections due to long-range attractive (or repulsive) potentials, a phenomenon particularly relevant for self-interacting dark matter (DM) at low velocities. In the presence of long-range forces, the standard calculation of the Sommerfeld enhancement, which treats the wavefunction using only the long-range potential, can lead to partial-wave cross sections that violate unitarity bounds at low velocities, particularly near zero-energy or finite-energy resonances.

Recent literature has proposed three distinct approaches to regulate this behavior and restore unitarity:

1. PSS24 (Parikh, Sato, & Slatyer): Separates short- and long-distance physics by matching wavefunctions at a radius $a$, introducing a matching radius-dependent regulator.
2. W25 (Watanabe): Utilizes renormalization group (RG)-improved perturbation theory to resum secular terms in the scattering amplitude.
3. FP25 (Flores & Petraki): Computes the unitarized S-matrix via a self-energy kernel (Bethe-Salpeter equation), incorporating both Hermitian and anti-Hermitian potentials non-perturbatively.

While all three methods yield unitary cross sections, their equivalence was not established. Specifically, the PSS24 approach nominally depends on an ultraviolet (UV) cutoff (the matching radius $a$), whereas the W25 and FP25 approaches are formulated to be cutoff-independent. Furthermore, only PSS24 explicitly covers multi-state systems (relevant for non-Abelian dark sectors), while the others were primarily developed for single-channel scenarios. The central question is whether these prescriptions are consistent and if the cutoff dependence in PSS24 is physical or an artifact of the matching procedure.

Methodology
The authors analyze the single-channel, single-state case first, where the assumptions of all three methods overlap, to establish a "dictionary" between their formalisms. They then generalize the findings to multi-state systems.

1. Single-Channel Analysis:

   • PSS24 vs. W25: The authors demonstrate that the "outgoing-wave regulator" term in PSS24 ($\bar{Z}_\ell$), which depends on the irregular solution of the Schrödinger equation and the matching radius $a$, can be approximated near resonances by a term depending solely on the regular solution and the elastic scattering phase shift ($\delta_\ell$). Specifically, they show that $\bar{Z}_\ell \approx p C_\ell^2 \cot \delta_\ell$, which is the regulator form used in W25. This approximation relies on the behavior of regular and irregular solutions near resonance, where the irregular solution becomes proportional to the regular solution.
   • PSS24 vs. FP25: The authors relate the FP25 self-energy kernel to the PSS24 matching conditions. They show that the FP25 regulator involves a term ($W_\ell$) that can be split into a UV-divergent part (absorbed into the short-range amplitude) and a finite, long-range part. By matching the short-range amplitudes ($\bar{f}_{s,\ell}$ in PSS24 and the DWBA amplitude in FP25), they prove that the regulated S-matrices coincide exactly when the short-range physics is correctly identified.

2. Multi-State Generalization:

   • Extending the single-state insights, the authors derive a regulator-independent prescription for multi-state systems (e.g., wino dark matter). They show that near a resonance, the matrix-valued outgoing-wave regulator $\bar{Z}_\ell$ in the PSS24 formalism can be expressed entirely in terms of the long-range scattering K-matrix ($K_\ell$) and the Sommerfeld enhancement matrix ($\Sigma_{0,\ell}$), removing explicit dependence on the matching radius $a$.

Key Contributions and Results

• Equivalence of Regulators: The paper establishes that the nominally cutoff-dependent PSS24 regulator $\bar{Z}_\ell$ is, to a good approximation, independent of the UV regulator $a$ when the unitarity-preserving corrections are large (i.e., near resonances). In this regime, $\bar{Z}_\ell$ coincides with the W25 regulator ($p C_\ell^2 \cot \delta_\ell$) and the finite part of the FP25 regulator.
• Regulator-Independent Prescription: The authors provide a new, manifestly cutoff-independent formula for the unitarized cross section in multi-state systems. The corrected Sommerfeld factor is written solely in terms of the standard enhancement factor, the hard annihilation amplitude, and the S-matrix (or K-matrix) for scattering in the long-range potential.
• Numerical Validation: Using the wino dark matter model as a benchmark, the authors perform numerical comparisons. They show that the approximate, regulator-independent expression derived from the single-state equivalence matches the exact PSS24 results (which include the full $a$-dependence) to visual accuracy near resonances for both s-wave and p-wave channels.
• Clarification of UV/IR Separation: The work clarifies how UV-divergent terms in the FP25 approach (arising from the irregular solution at the origin) are absorbed into the definition of the short-range scattering amplitude in the PSS24 framework, separating them from the infrared (IR) enhancements driven by the long-range potential.

Significance and Claims
The authors claim that their work demonstrates the consistency of the three major unitarization approaches under the assumption of a hierarchy between the short-range annihilation scale and the long-range potential scale. By proving that the PSS24 regulator is effectively independent of the matching radius near resonances, they resolve the apparent tension between cutoff-dependent and cutoff-independent methods.

The primary significance lies in providing a simplified, regulator-independent prescription for calculating unitarized Sommerfeld-enhanced cross sections in multi-state systems. This allows for the application of the more general multi-state formalism (previously restricted to PSS24) without the computational complexity or ambiguity of choosing a matching radius. The authors note that this equivalence holds when the absorptive physics is short-range and a scale hierarchy exists; in cases where no such hierarchy exists (e.g., capture into bound states with radii comparable to the potential range), the full FP25 formalism may still be required. The paper does not propose new experimental tests but aims to unify the theoretical tools used for dark matter phenomenology.