Towards a precision calculation of N $_{eff}$ in the Standard Model IV: Estimating the impact of positronium formation
(2025) Journal of Cosmology and Astroparticle Physics — Vol. 10 (2025)
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- We present a first assessment of how the previously unexplored effect of positronium formation can impact on the value of the effective number of neutrino species in the Standard Model, N$_{eff}$$^{SM}$.Adopting a Yukawa form for the electrostatic potential, we discuss two possible scenarios that differ primarily in their assumptions about entropy evolution. The first, out-of-equilibrium scenario assumes that thermal corrections to the potential such as Debye screening prevent positronium from appearing until the temperature drops below a threshold. Once the threshold is reached, entropy generated in the QED sector from the equilibration process, if instantaneous, leads to a variation in N$_{eff}$$^{SM}$ of at most |ΔN$_{eff}$| ∼ 10$^{-4}$,comparable to other uncertainties in the current benchmark value for N$_{eff}$$^{SM}$. A more gradual formation could however yield a larger change.The second, equilibrium scenario assumes the QED sector to stay in equilibrium at all times. In this case, we show that cancellations between the first, s-wave bound- and scattering-states contributions ensure that it is possible to evolve the system across the bound-state formation threshold without generating entropy in the QED sector. The corresponding change in N$_{eff}$$^{SM}$ then closely matches the 𝒪(e$^{2}$) perturbative result derived in previous works and the 𝒪(e$^{2}$) contribution is capped at |ΔN$_{eff}$| ≲ 10$^{-6}$.We also comment on the impact of deviations from a pure Yukawa potential due to the presence of a thermal width.
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Drewes, M., Georis, Y., & et al. (2025). Towards a precision calculation of N $_{eff}$ in the Standard Model IV: Estimating the impact of positronium formation. Journal of Cosmology and Astroparticle Physics, 10. https://doi.org/10.1088/1475-7516/2025/10/069 (Original work published 2025)