Updated fundamental constant constraints from Planck 2018 data and
  possible relations to the Hubble tension
  We present updated constraints on the variation of the fine structure
constant, $\alpha_{\rm EM}$, and effective electron rest mass, $m_{\rm e}$,
during the cosmological recombination era. These two fundamental constants
directly affect the ionization history at redshift $z\simeq 1100$ and thus
modify the temperature and polarisation anisotropies of the cosmic microwave
background (CMB) measured precisely with {\it Planck }. The constraints on
$\alpha_{\rm EM}$ tighten slightly due to improved {\it Planck} 2018
polarisation data but otherwise remain similar to previous CMB analysis.
However, a comparison with the 2015 constraints reveals a mildly discordant
behaviour for $m_{\rm e}$, which from CMB data alone is found below its local
value. Adding baryon acoustic oscillation data brings $m_{\rm e}$ back to the
fiducial value, $m_{\rm e}=(1.0078\pm0.0067) m_{\rm e,0}$, and also drives the
Hubble parameter to $H_0=69.1\pm 1.2$ [in units of ${\rm km \, s^{-1} \,
Mpc^{-1} }$]. Further adding supernova data yields $m_{\rm e}=(1.0190\pm0.0055)
m_{\rm e,0}$ with $H_0=71.24\pm0.96$. We perform several comparative analyses
using the latest cosmological recombination calculations to further understand
the various effects. Our results indicate that a single-parameter extension
allowing for a slightly increased value of $m_{\rm e}$ ($\simeq 3.5\sigma$
above $m_{\rm e,0}$) could play a role in the Hubble tension.
