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

Comments: 24 pages, 5 figures, 2 tables, v2: Added a new section discussing an alternative production scenario for positronium v3: Minor improvements, matches published version in JCAP