Towards a precision calculation of N eff in the Standard Model IV: Estimating the impact of positronium formation

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, NeffSM. 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 NeffSM of at most |ΔNeff| ∼ 10-4, comparable to other uncertainties in the current benchmark value for NeffSM. 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 NeffSM then closely matches the 𝒪(e2) perturbative result derived in previous works and the 𝒪(e2) contribution is capped at |ΔNeff| ≲ 10-6. We also comment on the impact of deviations from a pure Yukawa potential due to the presence of a thermal width.