A stringent constraint on the fractional change of proton $g$-factor

We report a constraint on the cosmological variation of the proton g-factor, $g_{\rm~p}$. By comparing the measured redshifts between mboxH [1316 1287 1054 1032 1026 1171 1062 1032 1026 817 374 340 333 324]21 cm and OH 18 cm lines observed with the newly commissioned Five-hundred-meter Aperture Spherical radio Telescope (FAST) toward PKS 1413+135 at $z$ = 0.24671, we obtain $\Delta~g_{\rm~p}/g_{\rm~p}=~(-4.3\pm2.5)\times10^{-5}$, which is more than two orders of magnitude more sensitive than previous constraints. In addition, we obtain sensitive constraints of $\Delta~(\mu\alpha^{2})/(\mu\alpha^{2})~=~(2.0\pm1.2)\times10^{-5}$ and $\Delta~(\mu\alpha^{2}g_{\rm~p}^{0.64})/(\mu\alpha^{2}g_{\rm~p}^{0.64})~=~(-4.7\pm1.9)\times10^{-6}$.