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The $^{22}$Ne($p,\gamma$)$^{23}$Na reaction, part of the neon-sodium cycle ofhydrogen burning, may explain the observed anticorrelation between sodium andoxygen abundances in globular cluster stars. Its rate is controlled by a numberof low-energy resonances and a slowly varying non-resonant component. Three newresonances at $E_p$ = 156.2, 189.5, and 259.7 keV have recently been observedand confirmed. However, significant uncertainty on the reaction rate remainsdue to the non-resonant process and to two suggested resonances at $E_p$ = 71and 105 keV. Here, new $^{22}$Ne($p,\gamma$)$^{23}$Na data with high statisticsand low background are reported. Stringent upper limits of 6$\times$10$^{-11}$and 7$\times$10$^{-11}$\,eV (90\% confidence level), respectively, are placedon the two suggested resonances. In addition, the off-resonant S-factor hasbeen measured at unprecedented low energy, constraining the contributions froma subthreshold resonance and the direct capture process. As a result, at atemperature of 0.1 GK the error bar of the $^{22}$Ne($p,\gamma$)$^{23}$Na rateis now reduced by three orders of magnitude.

Direct capture cross section and the $E_p$ = 71 and 105 keV resonances in the $^{22}$Ne($p,\gamma$)$^{23}$Na reaction