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The ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio is a significant indicator ofnucleosynthesis and mixing processes during hydrogen burning in stars. Itsvalue mainly depends on the relative rates of the${}^{12}\mathrm{C}(p,\gamma){}^{13}\mathrm{N}$ and${}^{13}\mathrm{C}(p,\gamma){}^{14}\mathrm{N}$ reactions. Both reactions havebeen studied at the Laboratory for Underground Nuclear Astrophysics (LUNA) inItaly down to the lowest energies to date ($E_\mathrm{c.m.} =60\,\mathrm{keV}$) reaching for the first time the high energy tail of hydrogenburning in the shell of giant stars. Our cross sections, obtained with bothprompt $\gamma$-ray detection and activation measurements, are the most preciseto date with overall systematic uncertainties of $7-8\%$. Compared with most ofthe literature, our results are systematically lower, by $25\%$ for the${}^{12}\mathrm{C}(p,\gamma){}^{13}\mathrm{N}$ reaction and by $30\%$ for${}^{13}\mathrm{C}(p,\gamma){}^{14}\mathrm{N}$. We provide the most precisevalue up to now of $(3.6 \pm 0.4)$ in the $20-140\,\mathrm{MK}$ range for thelowest possible ${}^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio that can beproduced during H burning in giant stars.

New proton-capture rates on carbon isotopes and their impact on the astrophysical $^{12}\mathrm{C}/{}^{13}\mathrm{C}$ ratio