Determination of $Q_{\mathrm{EC}}$ values of $T=1/2$ mirror nuclei at LEBIT

The ongoing search for evidence of physics beyond the standard model is one of the driving forces in fundamental physics research. Particularly crucial for testing the validity of the electroweak model is the unitarity of the Cabibbo-Kobayashi-Maskawa matrix. The most stringent test is the verification of the first row condition $V^2_{ud}+V^2_{us}+V^2_{ub}=1$. Therefore, the dominating element, $V_{ud}$, has to be determined with high precision. Complementary methods are measurements of the free neutron lifetime, pion decay rates, superallowed decay properties of $T=1$ nuclei and the newer approach of decay property measurements of $T=1/2$ isospin doublets. In the latter two approaches $V_{ud}$ is determined via $\mathcal{F}$t values to which the transition energy $Q_{\mathrm{EC}}$ contributes in the fifth power. We report the first high-precision measurement of $Q_{\mathrm{EC}}$ of the mixed Fermi-Gamow-Teller decays of the three T=1/2 nuclei $^{11}$C, $^{21}$Na and $^{29}$P. The uncertainties of all values were reduced significantly and their contribution to $V_{ud}$ thereby reduced the same order as the theoretical uncertainties.