Self-consistent computation of pair production from non-relativistic effective field theories in the Keldysh-Schwinger formalism

This paper employs non-relativistic effective field theory and the Keldysh-Schwinger formalism to self-consistently compute pair production and Sommerfeld-enhanced annihilation, demonstrating that this approach unitarizes the cross-section in a temperature-dependent manner while confirming vacuum results and revealing that bound states remain on-shell during out-of-equilibrium decay despite having Breit-Wigner spectral functions.

The Gist

Imagine you are trying to predict how often two strangers in a crowded room will bump into each other and shake hands (annihilate). In the world of physics, these "strangers" are particles of Dark Matter, and the "room" is the early universe.

This paper tackles a very specific, tricky problem: What happens when these particles are so attracted to each other that they almost stick together, but then also have a chance to be created from thin air by the heat of the room?

Here is the breakdown using everyday analogies:

1. The Problem: The "Crowded Dance Floor" Effect

In the past, physicists knew that if two particles have an attractive force (like a magnet), they get pulled closer together. This makes them much more likely to collide and annihilate. This is called the Sommerfeld Enhancement.

Think of it like a dance floor. If two people really like each other, they drift toward the center. The more they drift, the more likely they are to bump into someone else.

• The Issue: In some scenarios, this attraction is so strong that the math predicts they will collide more often than physically possible. It's like predicting that a single dance floor can hold 10,000 people when it only fits 100. The math breaks the "rules of the universe" (called unitarity).

2. The Old Fix: The "Self-Consistent" Solution

Previous scientists fixed this by realizing that the particles aren't just drifting; they are also annihilating as they get close.

• The Analogy: Imagine the dance floor has a "self-destruct" button. As the dancers get too close, they vanish. The old method calculated the dance floor size including the fact that people vanish as they crowd together. This fixed the math for particles floating in a cold, empty room (vacuum).

3. The New Twist: The "Hot Room" and "Reverse Magic"

This paper asks: What if the room is hot and crowded with other energy?
In the early universe, the heat is so intense that it doesn't just destroy particles; it can also create them out of thin air (pair production).

• The Analogy: Now, imagine the dance floor is not just a place where people vanish, but also a place where the heat of the room occasionally spawns new dancers.
• The Challenge: You have to calculate the dance floor size while accounting for:
  1. People being pulled together (attraction).
  2. People vanishing when they touch (annihilation).
  3. New people appearing out of the heat (creation).
  4. Crucially: Doing all this at the same time without breaking the math.

4. The Big Discovery: The "Ghostly" Bound States

The most surprising finding of this paper concerns "Bound States."

• What are they? Sometimes, two particles get so attracted they get stuck together, like a couple holding hands in the middle of the dance floor. They form a temporary "pair."
• The Paradox: Because these pairs are unstable (they eventually vanish), standard physics says they should be "fuzzy" or "blurred" (like a ghost with a wide shape).
• The Result: The authors found that even though these pairs are "fuzzy" in their energy, they act like solid, sharp objects when they decay.
  • The Metaphor: Imagine a ghost that is so blurry you can't see its edges. But the moment it disappears, it vanishes from a single, precise point rather than fading out slowly.
  • Why it matters: This confirms that even in a chaotic, hot environment, these temporary pairs behave in a very predictable, "on-the-dot" way when they break apart.

5. The Method: The "Two-Path" Approach

To solve this, the authors used a sophisticated mathematical toolkit called Non-Relativistic Effective Field Theory (NREFT) combined with the Keldysh-Schwinger formalism.

• Simple Translation: They used two different maps to navigate the same territory.
  1. Map A (NREFT): A general map that looks at the whole crowd.
  2. Map B (pNREFT): A zoomed-in map that focuses specifically on the pairs holding hands.
• The Victory: Both maps led to the exact same destination. This gives them high confidence that their answer is correct.

Summary

This paper is a masterclass in fixing the math of the early universe. It confirms that:

1. The "Crowded Dance Floor" math works even when you add the heat of the early universe.
2. New particles can be created from the heat, and the math handles this smoothly.
3. Temporary particle pairs are weird: they look blurry (fuzzy energy) but act sharp (precise location) when they die.

Why should you care?
Dark Matter makes up most of the matter in the universe. To understand how much Dark Matter exists today, we need to know exactly how it behaved in the first few seconds after the Big Bang. This paper ensures our calculations for that era are accurate, preventing us from making mistakes about the invisible stuff that holds galaxies together.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of calculating Sommerfeld-enhanced annihilation cross sections for non-relativistic dark matter (DM) candidates, particularly in regimes where nearly zero-energy bound states exist.

• The Unitarity Violation: In the presence of attractive long-range potentials (e.g., Yukawa), the scattering wave function at the origin, $|\psi(0)|^2$, can be strongly enhanced. For small velocities, this leads to an annihilation cross section $(\sigma v)$ that scales as $1/v^2$, potentially violating the partial-wave unitarity bound: $(\sigma v) \leq 4\pi/(m^2 v)$.
• Limitations of Previous Approaches: Previous literature resolved this for vacuum scattering states by self-consistently including the short-distance annihilation potential in the Schrödinger equation (Lippmann-Schwinger procedure). However, these approaches:
  1. Focused primarily on scattering states in vacuum.
  2. Neglected the reverse process: pair creation from the thermal environment, which is crucial for chemical equilibration during the early stages of DM freeze-out.
  3. Did not rigorously determine the nature of bound states in a non-equilibrium thermal environment (i.e., whether they are on-shell or acquire an off-shell Breit-Wigner shape due to finite decay widths).

2. Methodology

The authors employ Non-Relativistic Effective Field Theory (NREFT) and Potential Non-Relativistic EFT (pNREFT) within the Keldysh-Schwinger (Closed-Time-Path, CTP) formalism. This framework allows for the treatment of non-equilibrium quantum field theory and thermal effects.

• Formalism:
  • The system is described by a static potential and a local annihilation/creation operator $\Gamma$ on the CTP contour.
  • The evolution of the particle number density $n_\eta$ is derived from the NREFT action, resulting in a transport equation dependent on a four-point correlation function $G_{\eta\xi}$.
• Two Computational Approaches:
  1. NREFT Approach: The Martin-Schwinger hierarchy of correlation functions is truncated at the four-point level. This yields a closed set of integral equations (Bethe-Salpeter type) solved self-consistently.
  2. pNREFT Approach: The problem is projected onto the two-particle subspace (center-of-mass and relative coordinates). This provides closed differential equations for the four-point function, which are solved analytically.
• Key Technical Steps:
  • Derivation of the number density equation including both annihilation and pair creation.
  • Calculation of the self-consistent spectral function and four-point correlators, accounting for the imaginary part of the potential (annihilation width).
  • Analysis of the gradient expansion (leading order) versus exact analytic solutions for initial value problems to capture time-dependent dynamics and coherence effects.

3. Key Contributions and Results

A. Unitarization of the Sommerfeld Effect (Scattering States)

The authors confirm that the self-consistent computation of the four-point function restores partial-wave unitarity for scattering states.

• Result: The self-consistent annihilation cross section respects the unitarity bound $(\sigma v) \leq 4\pi/(m^2 v)$.
• Temperature Dependence: When pair creation is included, the unitarization becomes temperature-dependent. The effective matching coefficient $\Gamma_R$ acquires thermal corrections (Bose-enhancement factors).
• Consistency: In the non-relativistic, dilute limit, the results match previous vacuum-based Schrödinger equation calculations, validating the use of those methods for standard DM freeze-out predictions.

B. Bound State Dynamics (The Core Novelty)

The paper provides the first self-consistent analysis of bound-state decay via annihilation in a thermal environment, addressing whether bound states are on-shell or off-shell.

• The Paradox: Spectral functions for unstable particles typically exhibit a Breit-Wigner distribution (off-shell) due to finite decay widths. One might expect bound states to behave similarly.
• The Finding: The self-consistent computation reveals that bound states remain strictly on-shell during their out-of-equilibrium decay.
  • Mathematically, the statistical correlator $G^{+-}$ for bound states contains a delta function $\delta(E - E_n)$ rather than a Breit-Wigner peak.
  • The decay manifests as an exponential time evolution of the number density ($\dot{n}_n = -\Gamma_{dec} n_n$), not as a broadening of the energy distribution.
• Mechanism: This result arises from the specific structure of the Kadanoff-Baym equations when solved for the initial value problem. The "diamond operator" (gradients) converts the exponential series into a geometric series, leading to a first-order differential equation in time for the density, while the energy constraint remains sharp.
• Equilibrium Limit: In the long-time limit with pair creation included, the system reaches thermal equilibrium where the spectral function does take a Breit-Wigner form (consistent with the fluctuation-dissipation theorem). However, in the collision term governing number density evolution, the bound states effectively behave as on-shell degrees of freedom.

C. Comparison of NREFT and pNREFT

The authors demonstrate that both the NREFT (hierarchy truncation) and pNREFT (two-particle subspace) approaches yield identical results for scattering states and consistent results for bound states. This validates the use of pNREFT as a computationally simpler alternative for these problems.

4. Significance

1. Validation of DM Predictions: The work confirms that standard vacuum-based Schrödinger equation approaches for calculating Sommerfeld-enhanced annihilation are robust for dark matter freeze-out, even when considering thermal corrections, as these corrections are exponentially suppressed in the non-relativistic regime.
2. Resolution of the Bound-State Paradox: It clarifies a subtle theoretical point: while finite decay widths imply a continuous spectral function (Breit-Wigner), the chemical equilibration of bound states in a non-equilibrium setting proceeds via on-shell dynamics. This aligns with exact solutions of Kadanoff-Baym equations for decaying elementary particles.
3. Framework for Future Studies: The derivation of the number density equation including pair creation provides a rigorous foundation for studying chemical equilibration in the early universe, particularly for scenarios involving resonant annihilation or bound-state formation (e.g., Wino, Higgsino, or hidden sector DM).
4. Methodological Advance: The paper successfully integrates non-relativistic EFTs with the Keldysh-Schwinger formalism to handle non-equilibrium processes involving both scattering and bound states, offering a template for future calculations in thermal field theory.

Conclusion

Binder and Wang successfully extend the self-consistent treatment of Sommerfeld enhancement to include thermal pair production and bound-state dynamics. They prove that while spectral functions may broaden, the dynamical evolution of bound-state number densities in the early universe follows standard on-shell Boltzmann equations. This reconciles the apparent contradiction between finite decay widths and on-shell behavior, providing a solid theoretical basis for precision calculations in dark matter phenomenology.