Reexamination of the astrophysical<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:math>factor for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math>+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi/><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>Li+<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:math>reaction

Recently, a new measurement of the $^{6}\mathrm{Li}$ (150 A MeV)dissociation in the field of $^{208}\mathrm{Pb}$ has been reported [Hammache et al., Phys. Rev. C 82, 065803 (2010)] to study the radiative capture $\ensuremath{\alpha}+d\ensuremath{\rightarrow}{}^{6}\mathrm{Li}+\ensuremath{\gamma}$ process. However, the dominance of the nuclear breakup over the Coulomb one prevented the information about the $\ensuremath{\alpha}+d\ensuremath{\rightarrow}{}^{6}\mathrm{Li}+\ensuremath{\gamma}$ process from being obtained from the breakup data. The astrophysical ${S}_{24}(E)$ factor has been calculated within the $\ensuremath{\alpha}\ensuremath{-}d$ two-body potential model with potentials determined from the fits to the $\ensuremath{\alpha}\ensuremath{-}d$ elastic scattering phase shifts. However, the scattering phase shift, according to the theorem of the inverse scattering problem, does not provide a unique $\ensuremath{\alpha}\ensuremath{-}d$ bound-state potential, which is the most crucial input when calculating the ${S}_{24}(E)$ astrophysical factor at astrophysical energies. In this work, we emphasize the important role of the asymptotic normalization coefficient (ANC) for ${}^{6}\mathrm{Li}\ensuremath{\rightarrow}\ensuremath{\alpha}+d$, which controls the overall normalization of the peripheral $\ensuremath{\alpha}+d\ensuremath{\rightarrow}{}^{6}\mathrm{Li}+\ensuremath{\gamma}$ process and is determined by the adopted $\ensuremath{\alpha}\ensuremath{-}d$ bound-state potential. Since the potential determined from the elastic scattering data fit is not unique, the same is true for the ANC generated by the adopted potential. However, a unique ANC can be found directly from the elastic scattering phase shift, without invoking intermediate potential, by extrapolation the scattering phase shift to the bound-state pole [Blokhintsev et al., Phys. Rev. C 48, 2390 (1993)]. We demonstrate that the ANC previously determined from the $\ensuremath{\alpha}\ensuremath{-}d$ elastic scattering $s$-wave phase shift [Blokhintsev et al., Phys. Rev. C 48, 2390 (1993)], confirmed by ab initio calculations, gives ${S}_{24}(E)$, which at low energies is about $38%$ less than the other one reported [Hammache et al., Phys. Rev. C 82, 065803 (2010)]. We recalculate also the reaction rates, which are lower than those obtained in that same study [Hammache et al., Phys. Rev. C 82, 065803 (2010)].

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