Sommerfeld enhancement from unstable final-state particles in dark matter annihilation

This paper investigates the Sommerfeld enhancement in dark matter annihilation into unstable, non-relativistic final-state particles by incorporating their decay widths into the Schrödinger equation, revealing that narrow-width bound states induce resonant enhancements that significantly impact dark matter relic abundance predictions.

The Gist

Here is an explanation of the paper "Sommerfeld enhancement from unstable final-state particles in dark matter annihilation," translated into everyday language with creative analogies.

The Big Picture: The Dark Matter Mystery

Imagine the universe is filled with invisible "Dark Matter" (DM). We know it's there because it holds galaxies together, but we don't know what it's made of. One popular theory is that Dark Matter particles are like shy ghosts that occasionally bump into each other and vanish (annihilate), turning into energy and heavier, unstable particles.

To figure out how much Dark Matter exists today, scientists need to know how often these ghosts bump into each other and vanish. If they vanish too fast, the universe would be empty of Dark Matter. If they vanish too slow, there would be too much.

The Problem: The "Forbidden" Door

Usually, for two particles to turn into two heavier particles, they need a lot of speed (kinetic energy) to pay the "energy toll" to create the heavy stuff.

However, in the early universe, things were cold. The Dark Matter ghosts were moving very slowly. They didn't have enough energy to create the heavy particles. This is called a "forbidden channel." It's like trying to buy a luxury car when you only have enough money for a bicycle. The transaction shouldn't happen.

But, quantum physics is weird. Even if you don't have enough money, you might still get the car if you borrow a little bit from the future (a concept called being "off-shell").

The Twist: The "Long-Range" Friendship

The paper focuses on a specific scenario: What happens if the heavy particles created do have a long-range attraction between them?

Imagine the two heavy particles are created. Even though they are heavy and moving slowly, they have a strong magnetic-like pull between them.

• The Old Way of Thinking: Scientists used to say, "If these heavy particles decay (die) too quickly, they don't have time to feel that pull. So, we just ignore the pull." They put a "speed limit" on the calculation. If the particles were too slow, they assumed the interaction was zero.
• The New Way (This Paper): The authors say, "Wait a minute! Even if they are dying quickly, they still feel the pull for a split second. And if they form a temporary 'dance couple' (a bound state) before dying, it changes everything."

The Core Concept: The Sommerfeld Enhancement

The "Sommerfeld Enhancement" is a fancy name for quantum interference.

The Analogy: The Echo in a Canyon
Imagine you are shouting in a canyon.

• Without the effect: You shout, and the sound travels away.
• With the effect: The canyon walls reflect the sound back to you. The reflected sound mixes with your new shout, making the total sound much louder.

In physics, the "canyon walls" are the long-range forces between the particles. The "sound" is the probability of them annihilating. If the particles are moving slowly, they spend more time near each other, and the "echo" (the wave function) builds up, making the annihilation much more likely than expected.

The Innovation: The "Unstable" Dance

The unique contribution of this paper is handling the fact that the heavy particles are unstable (they decay quickly).

The Analogy: The Flashlight in the Rain
Imagine two people trying to hold hands in the rain.

• The Old Method: If the rain is heavy (the particles decay fast), the old method says, "They can't hold hands; the rain washes them apart too fast." It assumes they never touch.
• The New Method: The authors say, "Even in heavy rain, they can hold hands for a split second. And if they hold hands just right, they form a stable 'couple' (a bound state) that glows brighter before dissolving."

The authors solved a complex math equation (the Schrödinger equation) that includes a "decay timer." This allows them to calculate exactly how much the "echo" (enhancement) boosts the annihilation rate, even when the particles are dying quickly.

The Key Findings

1. Resonance is Real: Sometimes, the particles form a perfect "dance couple" (a bound state) right before they die. This is called a resonance. It's like pushing a swing at exactly the right moment; the swing goes incredibly high. This causes a massive spike in the annihilation rate.
2. The "Off-Shell" Effect: Even when the particles don't have enough energy to be created normally, the math shows they can still appear briefly as "virtual" particles. The old method ignored this; the new method counts it.
3. Changing the Universe's History: Because this enhancement makes Dark Matter vanish faster than we thought, it changes our prediction of how much Dark Matter is left today.
   • The Result: If you use the old method, you might think Dark Matter needs to be a certain heavy weight to match what we see. With this new method, the "sweet spot" for the weight of Dark Matter shifts. It's like recalibrating a scale; the weight that balances the universe is different than we previously calculated.

Why It Matters

This paper is like upgrading the software on a GPS.

• Old GPS: "You are here. The road ahead is blocked. Stop."
• New GPS: "You are here. The road ahead looks blocked, but there's a secret tunnel (resonance) and a shortcut (off-shell particles) you can take. Keep going!"

By accounting for the fact that the "passengers" (annihilation products) are unstable but still interact, the authors provide a more accurate map of the early universe. This helps physicists narrow down exactly what Dark Matter might be, ruling out some theories and highlighting others.

In a nutshell: The authors found that even when Dark Matter creates heavy, short-lived particles, those particles still "feel" each other's pull. This pull creates a quantum echo that makes Dark Matter vanish faster than we thought, which changes our calculations of how much Dark Matter exists in the universe today.

Technical Summary

Here is a detailed technical summary of the paper "Sommerfeld enhancement from unstable final-state particles in dark matter annihilation" by Abe, Sato, and Yamanaka.

1. Problem Statement

The paper addresses a specific gap in the calculation of Dark Matter (DM) relic abundance, specifically in "forbidden" annihilation channels. In these scenarios, DM particles ($\chi_1$) annihilate into heavier unstable particles ($\chi_2$).

• Kinematic Threshold: Because $m_2 > m_1$, the annihilation products are produced with very low kinetic energy (non-relativistic) near the kinematic threshold.
• Final-State Interactions: If $\chi_2$ and its antiparticle $\bar{\chi}_2$ experience long-range interactions (e.g., via a light mediator), their wavefunctions are distorted, leading to a Sommerfeld Enhancement (SE) of the annihilation cross-section.
• The Instability Issue: Unlike stable particles, $\chi_2$ decays with a width $\Gamma$. Previous studies (e.g., Ref [12]) treated this instability by introducing an arbitrary velocity cutoff ($v_{cut} \approx \sqrt{\Gamma/m_2}$). Below this velocity, the SE was assumed to be negligible.
• The Flaw: This cutoff method fails to account for:
  1. Resonant Bound States: Near the threshold, bound states of $\chi_2\bar{\chi}_2$ can form. Even if the particles are unstable, these bound states can significantly enhance the cross-section via resonances.
  2. Off-Shell Contributions: Annihilation processes where the final state particles are off-shell (virtual) are kinematically allowed below the threshold but are artificially set to zero in the cutoff method.
  3. Threshold Behavior: Analogous to top-quark pair production ($t\bar{t}$) in collider physics, the cross-section near the threshold is non-zero and resonant even when the energy is below the nominal mass threshold.

2. Methodology

The authors develop a rigorous quantum mechanical framework to calculate the final-state SE, treating the decay width $\Gamma$ consistently within the Schrödinger equation.

• Formalism: They model the process $\chi_1 \bar{\chi}_1 \to \chi_2 \bar{\chi}_2$ using a coupled-channel approach.
  • Initial State ($\chi_1$): Treated as a plane wave (or with initial state SE if applicable, though the paper focuses on final state).
  • Final State ($\chi_2$): Described by a Schrödinger equation with a complex energy term to account for decay:
    $$\left[ -\frac{1}{2\mu_2}\nabla^2 + V(r) - (E_2 + i\Gamma) \right] \Psi_2(r) = u^* \delta^3(r) \Psi_1(0)$$
    Here, $E_2 = E - 2\delta m$ is the kinetic energy relative to the threshold, and $i\Gamma$ introduces the decay width.
• Green's Function Approach: The solution is obtained using the Green's function $G_2(r; E_2 + i\Gamma)$ of the final state. The annihilation cross-section is proportional to the imaginary part of the Green's function at the origin:
  $$\sigma v_{rel} \propto \text{Im} \, G_2(0; E_2 + i\Gamma)$$
• Comparison: They compare this "full" formulation against the "cutoff method" (Ref [12]), where the SE factor is set to 1 if the velocity is below $v_{cut}$.
• Potentials Tested: The formalism is applied to two specific long-range potentials:
  1. Attractive Coulomb Potential: $V(r) = -\alpha/r$.
  2. Attractive Hulthén Potential: $V(r) = -\frac{\alpha m^* e^{-m^* r}}{1 - e^{-m^* r}}$ (approximating a screened Yukawa potential).

3. Key Contributions

1. Unified Formalism: The paper provides a formulation that naturally incorporates both bound state resonances (for narrow widths) and off-shell contributions (for wide widths) without ad-hoc cutoffs.
2. Resonant Enhancement: They demonstrate that bound states of the unstable final particles ($\chi_2\bar{\chi}_2$) create sharp resonances in the annihilation cross-section. These resonances occur at energies corresponding to the binding energies of the bound states ($E_n$), even when $E < 2\delta m$ (below the kinematic threshold).
3. Off-Shell Physics: The formalism correctly predicts a non-zero annihilation cross-section below the kinematic threshold due to off-shell intermediate states, a feature completely missed by the cutoff method.
4. Consistency Condition: They derive a condition for the validity of their Born approximation treatment: the partial decay width of the bound state back into the initial DM state ($\Gamma_{2\to1}$) must be much smaller than the total decay width $\Gamma$.

4. Results

The authors present numerical results for a benchmark scenario ($m_1 = 1$ TeV, $m_2 \approx 1.01 m_1$, $\alpha = 0.2$):

• Cross-Section Behavior:

  • Below Threshold: The "full result" shows significant enhancement below the threshold ($E_2 < 0$) due to resonances and off-shell effects. The cutoff method predicts zero cross-section here.
  • Resonances: Distinct peaks appear in $\sigma v_{rel}$ corresponding to $1s, 2s, 3s$bound states. The position and width of these peaks depend on$\Gamma$and$\alpha$.
  • Large Width Limit: As $\Gamma$ increases, the resonant peaks are suppressed (broadened), and the cross-section approaches the "free" result (no SE), but remains non-zero below threshold due to off-shell effects. The cutoff method fails to capture this, reverting to the tree-level cross-section.

• Thermally Averaged Cross-Section ($\langle \sigma v \rangle$):

  • At low temperatures (large $x = m_1/T$), the resonant contributions from bound states significantly boost $\langle \sigma v \rangle$ compared to the cutoff method.
  • The deviation is most pronounced when the mass ratio $m_2/m_1$ allows the thermal distribution to overlap with the bound state resonances.

• Relic Abundance ($\Omega h^2$):

  • The inclusion of final-state SE and bound states leads to a larger annihilation cross-section during freeze-out.
  • Consequently, the predicted relic density is lower than that calculated using the cutoff method.
  • To match the observed DM density, the required mass splitting ($m_2/m_1$) or coupling $\alpha$ must be adjusted. The paper shows that the cutoff method can overestimate the relic abundance by orders of magnitude in specific parameter regions, leading to incorrect constraints on DM models.

5. Significance

• Correcting DM Constraints: This work implies that previous constraints on "forbidden" DM models (which relied on the cutoff method) may be inaccurate. Models that were thought to be ruled out (due to overproduction of DM) might actually be viable if the resonant enhancement is properly calculated.
• Analogy to Collider Physics: The paper successfully bridges the gap between DM phenomenology and collider physics (specifically $t\bar{t}$ production near threshold), applying well-established non-relativistic QCD techniques to the early universe context.
• General Applicability: While focused on a single-component scalar DM, the methodology is applicable to multi-component models (e.g., composite dark matter) and scenarios involving heavy unstable mediators.
• Theoretical Rigor: It resolves the ambiguity of how to treat unstable particles in non-perturbative SE calculations, moving from heuristic cutoffs to a first-principles quantum mechanical derivation.

In conclusion, the paper establishes that ignoring the decay width in the Schrödinger equation (or using a simple velocity cutoff) leads to a significant underestimation of the Sommerfeld enhancement in forbidden DM channels, particularly when bound states are involved. This necessitates a re-evaluation of the parameter space for various Dark Matter models.