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

TL;DR

This work quantifies how a transient population of positronium in the early-universe QED plasma could affect the Standard Model prediction for the effective number of neutrinos, $N_{ m eff}^{ m SM}$. By modeling the mediator with a Yukawa potential and considering two limiting formation scenarios—out-of-equilibrium (potentially entropy-adding) and equilibrium (entropy-conserving)—the authors derive bounds on $ riangle N_{ m eff}$ using entropy conservation, Boltzmann-like rates, and non-perturbative many-body tools. In the out-of-equilibrium case, instantaneous formation yields $| riangle N_{ m eff}| o O(10^{-4})$, with larger shifts possible for slower formation; in the equilibrium case, non-ideal gas corrections computed via the Beth–Uhlenbeck framework give $| riangle N_{ m eff}| aisebox{0.2ex}{$ leq$} 10^{-6}$. Collectively, the results suggest positronium could at most induce a sub-permille change to $N_{ m eff}^{ m SM}$, well below current and near-future experimental sensitivities, though robust control of bound-state spectral properties remains a key challenge for future work.

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 $|ΔN_{\rm eff}| \sim 10^{-4}$, 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 $|ΔN_{\rm eff}| \lesssim 10^{-6}$. We also comment on the impact of deviations from a pure Yukawa potential due to the presence of a thermal width.

Figures (5)

• Figure 1: We display the (negative) change in $N_\mathrm{eff}^{\rm SM}$, $-\Delta N_\mathrm{eff}$, on a logarithmic scale as a function of the temperature $T_{\mathrm{Ps}}^{\mathrm{eq}}$ at which both $n=1$ positronium states reach equilibrium. We use two different approximations to compute $\Delta N_\mathrm{eff}$: The blue line represents estimates from the entropy argument of section \ref{['sec:entropy']} assuming instantaneous positronium equilibration. The green and orange lines denote $\Delta N_\mathrm{eff}$ from non-instantaneous production modelled with the parameterisation \ref{['eq:parampositroniumformation']} for the choices of $\Delta T/T^{\rm eq}_{\rm Ps} \in \{0.1,1\}$, respectively. Positronium formation is prevented by Debye screening in the grey-shaded region. In the hatched region, we expect scatterings with the plasma to prevent the formation of stable bound states, but more detailed computations are required to quantify the exact impact. The three horizontal lines represent, from top to bottom, the current sensitivity of Planck at 95$\%$ C.L. Planck:2018vyg, the planned sensitivity of CMB-S4 CMB-S4:2016ple at 95$\%$ C.L., and the last but one significant digit of the SM benchmark value $N_\mathrm{eff}^{\rm SM}$Gariazzo:2019gyiAkita:2020szlFroustey:2020mcqBennett:2019ewmBennett:2020zkvCielo:2023bqpJackson:2023zklDrewes:2024wbw.
• Figure 2: Debye screening length as a function of the temperature computed from the 1PI-resummed photon propagator (blue) and the HTL approximation (orange). Observe that while the two estimates coincide at $T \gtrsim m_e$, the former is significantly larger at $T \lesssim m_e$. We compare $a_D$ to the Bohr radius $a_0$ of the positronium state in vacuum (black dashed horizontal line). The vertical line represents $T_{D}$, the temperature at which $a_0 = a_D$ and Debye screening starts to be effective at $T>T_{D}$.
• Figure 3: Feynman diagrammatic representation of equation \ref{['eq:p2nr']}. In a symmetric plasma, the "dumbbells" diagrams cancel, leaving us with the "oyster" diagram to the right of the equal sign.
• Figure 4: Breakdown of the various contributions to the coefficient $b_{\ell=0}$ in equation \ref{['eq:BU2']} as a function of temperature.
• Figure 5: Change to the entropy, $\delta s$, as a function of temperature $T$. Left: Breakdown of the contributions from various components. Right: The sum of all contributions as derived from a full non-perturbative computation (black solid) and from the Born approximation (red dashed). The difference between the non-perturbative and Born results (blue solid) remains substantially below the individual result, suggesting that perturbation theory is under control.