Does the Higgs field give the electron its mass?
In plain words
The electron couples to the Higgs so weakly (strength about 3 in a million) that the coupling has never been detected. Measuring it would show whether the Higgs field is the source of the mass of the particle that sets the size of atoms.
Precise statement
Measure the electron Yukawa coupling $y_e$ (SM value $\sqrt{2}\,m_e/v \sim 2.9e-6$), or $\kappa_e = y_e/y_e^{\mathrm{SM}}$. The best direct bound, CMS $\operatorname{BR}(H \to e+ e-) < 3.0e-4$ at 95 percent CL (Phys. Lett. B 846, 137783, 2023), corresponds to $\left|\kappa_e\right| < \sim 240$ (approximate, using $\operatorname{BR}_{\mathrm{SM}} \sim 5e-9$). An answer is $\kappa_e$ measured with $O(1)$ precision, enough to separate $\kappa_e = 1$ from $\kappa_e = 0$.
What would settle it
Resonant s-channel production $e+ e- \to H$ at $\sqrt{s}=m_{H}$ with a center-of-mass energy spread of about 4 MeV at FCC-ee, projected to reach $1.3\,\sigma$ for the SM coupling (an upper limit of 1.6 times $y_{e}^{\mathrm{SM}}$) per interaction point per year with $10\,\mathrm{ab}^{-1}$ (d'Enterria, Poldaru, Wojcik, Eur. Phys. J. Plus 137, 201, 2022, arXiv:2107.02686).
Status in the literature
Unverified note
Only upper bounds exist as of 2026; FCC-ee feasibility studies reach sensitivity near the SM value.