Towards a precision calculation of $N_{\rm eff}$ in the Standard Model IV: Impact of positronium formation

Abstract: 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, $N_{\rm eff}^{\rm SM}$. 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 $N_{\rm eff}^{\rm SM}$ of at most $|\Delta N_{\rm eff}| \sim 10^{-4}$, comparable to other uncertainties in the current benchmark value for $N_{\rm eff}^{\rm SM}$. 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 $N_{\rm eff}^{\rm SM}$ then closely matches the $\mathcal{O}(e^2)$ perturbative result derived in previous works and the $\mathcal{O}(e^4)$ contribution is capped at $|\Delta N_{\rm eff}| \lesssim 10^{-6}$. We also comment on the impact of deviations from a pure Yukawa potential due to the presence of a thermal width.