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arxiv: 2604.11553 · v1 · submitted 2026-04-13 · ✦ hep-ph

Recognition: unknown

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

Tobias Binder , Edward Wang
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:38 UTC · model grok-4.3

classification ✦ hep-ph
keywords Sommerfeld enhancementbound statespair productionKeldysh-Schwinger formalismnon-relativistic effective field theoryunitarityBreit-Wigner spectral function
0
0 comments X

The pith

Self-consistent Keldysh-Schwinger computation shows bound states remain on-shell during out-of-equilibrium decay in non-relativistic EFTs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a method for incorporating pair-creation effects self-consistently into the calculation of four-point correlation functions using non-relativistic effective field theory and the Keldysh-Schwinger formalism. The approach addresses the unitarization of Sommerfeld-enhanced annihilation cross sections, making it temperature dependent when pair production is included. It confirms earlier vacuum results for scattering states and demonstrates that bound states stay on-shell in their decay processes even with finite decay widths that produce Breit-Wigner spectral functions. Readers would care because this provides a consistent framework for handling large annihilation rates near bound-state thresholds, relevant for accurate predictions in particle physics and cosmology.

Core claim

By self-consistently including pair-creation effects in the computation of four-point correlation functions within non-relativistic effective field theories using the Keldysh-Schwinger formalism, the unitarization of the Sommerfeld effect becomes temperature dependent. For bound states, the computation shows they remain on-shell in out-of-equilibrium decay despite Breit-Wigner spectral functions from finite decay widths, aligning with expectations from Kadanoff-Baym equations.

What carries the argument

The self-consistent four-point correlation function computed in the Keldysh-Schwinger formalism applied to non-relativistic effective field theory, incorporating the short-distance annihilation potential.

If this is right

  • Unitarization of Sommerfeld-enhanced cross sections becomes temperature dependent due to pair-creation effects.
  • Results for scattering states in vacuum are recovered up to small thermal corrections in the non-relativistic dilute regime.
  • Bound states are analyzed beyond leading annihilation decay, remaining on-shell in decay.
  • Consistency with analytic solutions of Kadanoff-Baym equations for decaying particles in thermal baths is maintained.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This framework could be applied to calculate relic abundances in dark matter models involving bound-state formation at finite temperatures.
  • Similar methods might resolve off-shell issues in other out-of-equilibrium quantum field theory processes.
  • Extensions to include more thermal effects could test the small-correction assumption in denser regimes.

Load-bearing premise

The results hold up to small thermal corrections in the non-relativistic and dilute regime of the pairs.

What would settle it

A calculation of the four-point correlation function that does not keep bound states on-shell during decay while using Breit-Wigner spectral functions would contradict the central claim.

read the original abstract

Sommerfeld-enhanced annihilation cross sections in the presence of nearly zero-energy bound states can become so large that perturbative partial-wave unitarity appears to be violated. Previous literature incorporated the short-distance annihilation potential self-consistently into the computation of the Schr\"odinger wave function at the origin, leading to the unitarization of the Sommerfeld effect in vacuum. We employ non-relativistic effective field theory methods and the Keldysh-Schwinger formalism to additionally include pair-creation effects in the self-consistent computation of four-point correlation functions, which renders the unitarization temperature dependent. Up to small thermal corrections in the non-relativistic and dilute regime of the pairs, we confirm the previous results based on the Schr\"odinger equation approach for scattering states in vacuum. For the first time, we analyze bound-state contributions beyond their leading decay via annihilation. Interestingly, our self-consistent computation of the four-point correlation function shows that bound states remain on-shell in their out-of-equilibrium decay, even though their spectral functions take the form of Breit-Wigner distributions due to finite decay widths. While this may appear paradoxical, it aligns with expectations from earlier results based on exact analytic solutions of the Kadanoff-Baym equations for a decaying elementary particle in a thermal environment.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper develops a non-relativistic effective field theory (NREFT) framework combined with the Keldysh-Schwinger real-time formalism to compute four-point correlation functions for pair production and annihilation self-consistently. It incorporates thermal pair-creation effects, rendering the unitarization of Sommerfeld-enhanced annihilation temperature-dependent. The work confirms prior vacuum Schrödinger-equation results for scattering states (up to small thermal corrections in the NR dilute regime) and extends the analysis to bound states, demonstrating that they remain on-shell in out-of-equilibrium decay despite acquiring Breit-Wigner spectral functions from finite decay widths; this is shown to align with exact Kadanoff-Baym solutions for decaying particles.

Significance. If the central results hold, the manuscript provides a valuable extension of vacuum unitarization techniques to thermal environments, relevant for precision calculations in dark matter annihilation or other processes involving near-threshold bound states. The self-consistent inclusion of pair creation and the resolution of the apparent paradox for bound-state poles constitute a technical advance. Explicit alignment with prior Kadanoff-Baym analytics is a strength, as is the focus on the four-point function rather than two-point approximations.

major comments (2)
  1. [Abstract and main results (four-point function computation)] The central claim that bound states remain on-shell (real part of the pole unshifted) despite Breit-Wigner spectral functions is load-bearing for the out-of-equilibrium analysis. The abstract and main results section assert this follows from the self-consistent four-point correlator, but no explicit expression for the retarded four-point function, the self-energy insertion in the bound-state channel, or the pole-position equation is referenced to demonstrate that the real-part shift vanishes while the imaginary width remains finite.
  2. [Introduction and results summary] The confirmation of vacuum results is qualified by 'up to small thermal corrections in the non-relativistic and dilute regime of the pairs.' No quantitative bound, expansion parameter, or numerical estimate of these corrections (e.g., relative size of thermal mass shifts or pair-creation contributions) is supplied, which is required to delimit the validity of the temperature-dependent unitarization claim.
minor comments (3)
  1. [Abstract] The abstract is information-dense; splitting the description of the bound-state result into a separate sentence would improve readability.
  2. [Methods] Notation for the Keldysh components of the four-point function and the precise definition of the self-consistent potential should be introduced with a short equation or diagram early in the methods section to aid readers unfamiliar with the combined NREFT+Keldysh setup.
  3. [Results] A brief comparison table or plot contrasting the vacuum Schrödinger result, the new thermal four-point result, and the Kadanoff-Baym benchmark would help quantify the size of thermal corrections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive comments. Below we respond point by point to the major remarks. We will revise the manuscript to address both issues as described.

read point-by-point responses

  1. Referee: [Abstract and main results (four-point function computation)] The central claim that bound states remain on-shell (real part of the pole unshifted) despite Breit-Wigner spectral functions is load-bearing for the out-of-equilibrium analysis. The abstract and main results section assert this follows from the self-consistent four-point correlator, but no explicit expression for the retarded four-point function, the self-energy insertion in the bound-state channel, or the pole-position equation is referenced to demonstrate that the real-part shift vanishes while the imaginary width remains finite.

    Authors: We agree that the presentation would benefit from explicit references. In the revised manuscript we will add, both in the abstract and in the main-results summary, direct citations to the retarded four-point function (derived in Sec. III from the Keldysh-Schwinger contour) and to the pole-position equation obtained from the self-consistent Dyson resummation in the bound-state channel. These expressions show that the real part of the pole remains unshifted at leading order in the non-relativistic expansion while the imaginary part is set by the finite decay width; the structure of the thermal self-energy insertions in the four-point function preserves the on-shell condition for the bound-state pole. revision: yes

  2. Referee: [Introduction and results summary] The confirmation of vacuum results is qualified by 'up to small thermal corrections in the non-relativistic and dilute regime of the pairs.' No quantitative bound, expansion parameter, or numerical estimate of these corrections (e.g., relative size of thermal mass shifts or pair-creation contributions) is supplied, which is required to delimit the validity of the temperature-dependent unitarization claim.

    Authors: We accept that a quantitative estimate is needed to delimit the regime. In the revision we will insert a short paragraph (new Sec. II.C or an appendix) that supplies the relevant expansion parameters: the thermal mass shift is O(T^{2}/m) and the pair-creation contribution is suppressed by exp(−m/T) in the non-relativistic dilute limit. We will also give a numerical example for representative values of m and T, thereby making the qualifier 'small thermal corrections' concrete. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper deploys non-relativistic EFT plus the Keldysh-Schwinger formalism to compute four-point correlators that incorporate pair creation, thereby extending the vacuum Schrödinger-equation treatment to finite temperature. The central result—that bound states remain on-shell despite Breit-Wigner spectral functions—is obtained directly from this new self-consistent computation and is only noted to align with independent prior Kadanoff-Baym solutions; no equation is shown to reduce by definition or by fitted-parameter renaming to an input quantity, and no load-bearing step collapses to a self-citation chain. The derivation therefore stands on its own explicit construction rather than on circular re-labeling of its premises.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions of non-relativistic effective field theory and the applicability of the Keldysh-Schwinger formalism in the dilute thermal regime; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Non-relativistic approximation for the pairs
    Invoked when stating that thermal corrections remain small.
  • domain assumption Dilute regime for the pairs
    Stated as the regime in which the results hold up to small corrections.

pith-pipeline@v0.9.0 · 5525 in / 1428 out tokens · 62980 ms · 2026-05-10T15:38:32.034984+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the equivalence of unitarization prescriptions for the Sommerfeld enhancement

    hep-ph 2026-05 accept novelty 7.0

    Different unitarization prescriptions for the Sommerfeld enhancement are equivalent to leading order and yield a regulator-independent formula for multi-state systems written only in terms of the standard enhancement ...

  2. Thermal effects on Dark Matter production during cosmic reheating

    hep-ph 2026-04 unverdicted novelty 4.0

    Thermal corrections to reheating and freeze-in DM production rates are generally small in the computable regime but can be large in constructed counter-examples.

Reference graph

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