Asymptotic normalization coefficients for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mmultiscripts><mml:mi mathvariant="normal">C</mml:mi><mml:mprescripts/><mml:none/><mml:mn>12</mml:mn></mml:mmultiscripts></mml:mrow></mml:math> synthesis and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> factor for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">C</mml:mi><mml:mprescripts/><mml:none/><mml:mn>12</mml:mn></mml:mmultiscripts><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mo>)</mml:mo><mml:mmultiscripts><mml:mi mathvariant="normal">O</mml:mi><mml:mprescripts/><mml:none/><mml:mn>16</mml:mn></mml:mmultiscripts></mml:mrow></mml:math> radiative capture

: The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mmultiscripts><a:mi mathvariant="normal">C</a:mi><a:mprescripts/><a:none/><a:mn>12</a:mn></a:mmultiscripts><a:mo>(</a:mo><a:mi>α</a:mi><a:mo>,</a:mo><a:mi>γ</a:mi><a:mo>)</a:mo><a:mmultiscripts><a:mi mathvariant="normal">O</a:mi><a:mprescripts/><a:none/><a:mn>16</a:mn></a:mmultiscripts></a:mrow></a:math> reaction, determining the survival of carbon in red giants, is of interest for nuclear reaction theory and nuclear astrophysics. A specific feature of the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mmultiscripts><d:mi mathvariant="normal">O</d:mi><d:mprescripts/><d:none/><d:mn>16</d:mn></d:mmultiscripts></d:math> nuclear structure is the presence of two subthreshold bound states, (6.92 MeV, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msup><f:mn>2</f:mn><f:mo>+</f:mo></f:msup></f:math>) and (7.12 MeV, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msup><g:mn>1</g:mn><g:mo>−</g:mo></g:msup></g:math>), that dominate the behavior of the low-energy <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>S</h:mi></h:math> factor. The strength of these subthreshold states is determined by their asymptotic normalization coefficients (ANCs), which need to be known with high accuracy. : The objective of this research is to examine how the subthreshold and ground-state ANCs impact the low-energy <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>S</i:mi></i:math> factor, especially at the key astrophysical energy of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mn>300</j:mn><j:mspace width="0.16em"/><j:mi>keV</j:mi></j:mrow></j:math>. The <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>S</l:mi></l:math> factors are calculated within the framework of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>R</m:mi></m:math>-matrix method using the code. Our total <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mi>S</n:mi></n:math> factor takes into account the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:mi>E</o:mi><o:mn>1</o:mn></o:mrow></o:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mrow><p:mi>E</p:mi><p:mn>2</p:mn></p:mrow></p:math> transitions to the ground state of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mmultiscripts><q:mi mathvariant="normal">O</q:mi><q:mprescripts/><q:none/><q:mn>16</q:mn></q:mmultiscripts></q:math> including the interference of the subthreshold and higher resonances, which also interfere with the corresponding direct captures, and cascade radiative captures to the ground state of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:mmultiscripts><s:mi mathvariant="normal">O</s:mi><s:mprescripts/><s:none/><s:mn>16</s:mn></s:mmultiscripts></s:math> through four subthreshold states: <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"><u:mrow><u:msubsup><u:mn>0</u:mn><u:mn>2</u:mn><u:mo>+</u:mo></u:msubsup><u:mo>,</u:mo><u:mspace width="0.16em"/><u:msup><u:mn>3</u:mn><u:mo>−</u:mo></u:msup><u:mo>,</u:mo><u:mspace width="0.16em"/><u:msup><u:mn>2</u:mn><u:mo>+</u:mo></u:msup></u:mrow></u:math>, and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"><x:msup><x:mn>1</x:mn><x:mo>−</x:mo></x:msup></x:math>. To evaluate the impact of subthreshold ANCs on the low-energy <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"><y:mi>S</y:mi></y:math> factor, we employ two sets of the ANCs. The first selection, which offers higher ANC values, is attained through the extrapolation process [Blokhintsev , ]. The set with low ANC values was employed by deBoer []. A detailed comparison of the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"><z:mi>S</z:mi></z:math> factors at the most effective astrophysical energy of 300 keV is provided, along with an investigation into how the ground-state ANC affects this <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"><ab:mi>S</ab:mi></ab:math> factor. : The contribution to the total <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML"><bb:mrow><bb:mi>E</bb:mi><bb:mn>1</bb:mn></bb:mrow></bb:math> and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML"><cb:mrow><cb:mi>E</cb:mi><cb:mn>2</cb:mn><cb:mspace width="4pt"/><cb:mspace width="0.16em"/><cb:mi>S</cb:mi></cb:mrow></cb:math> factors from the corresponding subthreshold resonances at <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML"><fb:mrow><fb:mn>300</fb:mn><fb:mspace width="0.16em"/><fb:mi>keV</fb:mi></fb:mrow></fb:math> are <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML"><hb:mrow><hb:mspace width="0.16em"/><hb:mo>(</hb:mo><hb:mn>71</hb:mn><hb:mo>–</hb:mo><hb:mn>74</hb:mn><hb:mo>)</hb:mo><hb:mo>%</hb:mo></hb:mrow></hb:math> and <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML"><jb:mrow><jb:mo>(</jb:mo><jb:mn>102</jb:mn><jb:mo>–</jb:mo><jb:mn>103</jb:mn><jb:mo>)</jb:mo><jb:mo>%</jb:mo></jb:mrow></jb:math>, respectively. The correlation of the uncertainties of the subthreshold ANCs with the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML"><kb:mrow><kb:mi>E</kb:mi><kb:mn>1</kb:mn></kb:mrow></kb:math> and <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML"><lb:mrow><lb:mi>E</lb:mi><lb:mn>2</lb:mn><lb:mspace width="4pt"/><lb:mspace width="0.16em"/><lb:mi>S</lb:mi><lb:mo>(</lb:mo><lb:mn>300</lb:mn><lb:mspace width="0.16em"/><lb:mi>keV</lb:mi><lb:mo>)</lb:mo></lb:mrow></lb:math> factors is found. The <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML"><pb:mrow><pb:mi>E</pb:mi><pb:mn>1</pb:mn></pb:mrow></pb:math> transition of the subthreshold resonance <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML"><qb:msup><qb:mn>1</qb:mn><qb:mo>−</qb:mo></qb:msup></qb:math> does not depend on the ground-state ANC but interferes constructively with a broad <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML"><rb:mrow><rb:mo>(</rb:mo><rb:mn>9.585</rb:mn><rb:mspace width="0.16em"/><rb:mi>MeV</rb:mi><rb:mo>;</rb:mo><rb:mspace width="0.16em"/><rb:msup><rb:mn>1</rb:mn><rb:mo>−</rb:mo></rb:msup><rb:mo>)</rb:mo></rb:mrow></rb:math> resonance giving (for the present subthreshold ANC) an additional <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML"><ub:mrow><ub:mn>26</ub:mn><ub:mo>%</ub:mo></ub:mrow></ub:math> contribution to the total <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML"><vb:mrow><vb:mi>E</vb:mi><vb:mn>1</vb:mn><vb:mspace width="4pt"/><vb:mspace width="0.16em"/><vb:mi>S</vb:mi><vb:mo>(</vb:mo><vb:mn>300</vb:mn><vb:mspace width="0.16em"/><vb:mi>keV</vb:mi><vb:mo>)</vb:mo></vb:mrow></vb:math> factor. Interference of the <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML"><zb:mrow><zb:mi>E</zb:mi><zb:mn>2</zb:mn></zb:mrow></zb:math> transition through the subthreshold resonance <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML"><ac:msup><ac:mn>2</ac:mn><ac:mo>+</ac:mo></ac:msup></ac:math> with direct capture is almost negligible for small ground-state ANC of <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML"><bc:mrow><bc:mn>58</bc:mn><bc:mspace width="4pt"/><bc:msup><bc:mrow><bc:mi>fm</bc:mi></bc:mrow><bc:mrow><bc:mo>−</bc:mo><bc:mn>1</bc:mn><bc:mo>/</bc:mo><bc:mn>2</bc:mn></bc:mrow></bc:msup></bc:mrow></bc:math>. However, its interference with direct capture for higher ground-state ANC of <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML"><dc:mrow><dc:mn>337</dc:mn><dc:mspace width="4pt"/><dc:msup><dc:mrow><dc:mi>fm</dc:mi></dc:mrow><dc:mrow><dc:mo>−</dc:mo><dc:mn>1</dc:mn><dc:mo>/</dc:mo><dc:mn>2</dc:mn></dc:mrow></dc:msup></dc:mrow></dc:math> is significant and destructive, contributing <fc:math xmlns:fc="http://www.w3.org/1998/Math/MathML"><fc:mrow><fc:mo>−</fc:mo><fc:mn>27</fc:mn><fc:mo>%</fc:mo></fc:mrow></fc:math>. The low-energy <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML"><gc:mrow><gc:msub><gc:mi>S</gc:mi><gc:mrow><gc:mi>E</gc:mi><gc:mn>2</gc:mn></gc:mrow></gc:msub><gc:mrow><gc:mo>(</gc:mo><gc:mn>300</gc:mn><gc:mspace width="0.16em"/><gc:mi>keV</gc:mi><gc:mo>)</gc:mo></gc:mrow></gc:mrow></gc:math> factor experiences a smaller increase when both subthfreshold and the ground-state ANCs rise together due to their anticorrelation, compared to when only the subthreshold ANCs increase. Published by the American Physical Society 2024

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