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, 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 |Δ N 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 |Δ N eff | ≲ 10 -6 . We also comment on the impact of deviations from a pure Yukawa potential due to the presence of a thermal width.