Enhanced effect of quark mass variation in ${}^{229}\mathrm{Th}$ and limits from Oklo data

The effects of the variation of the dimensionless strong interaction parameter $X_q=m_q/{\Lambda }_{\mathrm{QCD}}\hspace{0.5em}$ ($m_q$ is the quark mass, ${\Lambda }_{\mathrm{QCD}}$ is the QCD scale) are enhanced about $1.5\times {10}^5$ times in the 7.6 eV “nuclear clock” transition between the ground and first excited states in the ${}^{229}\mathrm{Th}$ nucleus and about $1\times {10}^8$ times in the relative shift of the 0.1 eV compound resonance in ${}^{150}\mathrm{Sm}$. The best terrestrial limit on the temporal variation of the fundamental constants, $|\delta X_q/X_q|<4\times {10}^{-9}$ at 1.8 billion years ago ($|{\stackrel{·}{X}}_q/X_q|<2.2\times {10}^{-18}{\mathrm{y}}^{-1}$), is obtained from the shift of this Sm resonance derived from the Oklo natural nuclear reactor data. The results for ${}^{229}\mathrm{Th}$ and ${}^{150}\mathrm{Sm}$ are obtained by extrapolation from light nuclei where the many-body calculations can be performed more accurately. The errors produced by such extrapolation may be smaller than the errors of direct calculations in heavy nuclei. The extrapolation results are compared with the “direct” estimates obtained using the Walecka model. A number of numerical relations needed for the calculations of the variation effects in nuclear physics and atomic spectroscopy have been obtained: for the nuclear binding energy $\delta E/E\approx -1.45\hspace{0.5em}\delta m_q/m_q$, for the spin-orbit intervals $\delta E_{\mathit{so}}/E_{\mathit{so}}\approx -0.22\hspace{0.5em}\delta m_q/m_q$, for the nuclear radius $\delta r/r\approx 0.3\hspace{0.5em}\delta m_q/m_q$ (in units of ${\Lambda }_{\mathrm{QCD}}$); for the shifts of nuclear resonances and weakly bound energy levels $\delta E_r\approx 10\hspace{0.5em}\delta X_q/X_q$ MeV.