Three-body resonances of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>α</mml:mi> <mml:mi>α</mml:mi> <mml:mi>M</mml:mi> </mml:math> clusters ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inli

Motivated by the recently obtained Hadrons to Atomic nuclei from Lattice QCD potentials for the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>N</a:mi> <a:mtext>−</a:mtext> <a:mi>ϕ</a:mi> </a:mrow> </a:math> , <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mi>N</c:mi> <c:mtext>−</c:mtext> <c:mi>J</c:mi> <c:mo>/</c:mo> <c:mi>ψ</c:mi> </c:mrow> </c:math> , and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:mi>N</e:mi> <e:mtext>−</e:mtext> <e:msub> <e:mrow> <e:mi>η</e:mi> </e:mrow> <e:mrow> <e:mi>c</e:mi> </e:mrow> </e:msub> </e:mrow> </e:math> interactions, we investigate the structure of the exotic nuclei <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:mmultiscripts> <g:mrow> <g:mi>Be</g:mi> </g:mrow> <g:mprescripts/> <g:mrow> <g:mi>ϕ</g:mi> </g:mrow> <g:mrow> <g:mn>8</g:mn> </g:mrow> </g:mmultiscripts> </g:mrow> </g:math> , <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mmultiscripts> <i:mrow> <i:mi>Be</i:mi> </i:mrow> <i:mprescripts/> <i:mrow> <i:mi>J</i:mi> <i:mo>/</i:mo> <i:mi>ψ</i:mi> </i:mrow> <i:mrow> <i:mn>8</i:mn> </i:mrow> </i:mmultiscripts> </i:math> , and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mmultiscripts> <k:mrow> <k:mi>Be</k:mi> </k:mrow> <k:mprescripts/> <k:msub> <k:mi>η</k:mi> <k:mi>c</k:mi> </k:msub> <k:mrow> <k:mn>8</k:mn> </k:mrow> </k:mmultiscripts> </k:mrow> </k:math> as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>α</m:mi> <m:mo>+</m:mo> <m:mi>α</m:mi> <m:mo>+</m:mo> <m:mi>M</m:mi> </m:math> three-body systems ( <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>M</o:mi> </o:math> denotes the meson). The bound and resonant states are calculated consistently using the Gaussian expansion method, with resonances identified via the complex scaling method. For the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>α</q:mi> <q:mi>ϕ</q:mi> </q:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>α</s:mi> </s:math> -charmonium interactions, a folding potential is constructed based on the Hadrons to Atomic nuclei from Lattice (HAL) QCD potentials and fitted to a Woods-Saxon form. We find that the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>ϕ</u:mi> </u:math> meson exhibits a strong “gluelike” effect, binding the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:msubsup> <w:mn>0</w:mn> <w:mn>1</w:mn> <w:mo>+</w:mo> </w:msubsup> </w:math> , <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msubsup> <y:mn>2</y:mn> <y:mn>1</y:mn> <y:mo>+</y:mo> </y:msubsup> </y:math> , and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msubsup> <ab:mn>4</ab:mn> <ab:mn>1</ab:mn> <ab:mo>+</ab:mo> </ab:msubsup> </ab:math> resonant states of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mrow> <cb:mmultiscripts> <cb:mrow> <cb:mi>Be</cb:mi> </cb:mrow> <cb:mprescripts/> <cb:none/> <cb:mrow> <cb:mn>8</cb:mn> </cb:mrow> </cb:mmultiscripts> </cb:mrow> </cb:math> into stable states and significantly reducing the <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mrow> <eb:mi>α</eb:mi> <eb:mtext>−</eb:mtext> <eb:mi>α</eb:mi> </eb:mrow> </eb:math> distance. In contrast, the interactions of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>J</gb:mi> <gb:mo>/</gb:mo> <gb:mi>ψ</gb:mi> </gb:math> and <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:msub> <ib:mi>η</ib:mi> <ib:mi>c</ib:mi> </ib:msub> </ib:math> with the nucleus are weaker, forming only shallow bound states with the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:msubsup> <kb:mn>0</kb:mn> <kb:mn>1</kb:mn> <kb:mo>+</kb:mo> </kb:msubsup> </kb:math> state of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"> <mb:mrow> <mb:mmultiscripts> <mb:mrow> <mb:mi>Be</mb:mi> </mb:mrow> <mb:mprescripts/> <mb:none/> <mb:mrow> <mb:mn>8</mb:mn> </mb:mrow> </mb:mmultiscripts> </mb:mrow> </mb:math> and even increasing the <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mrow> <ob:mi>α</ob:mi> <ob:mtext>−</ob:mtext> <ob:mi>α</ob:mi> </ob:mrow> </ob:math> separation. Notably, our analysis predicts weakly bound <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"> <qb:mrow> <qb:mi>α</qb:mi> <qb:mtext>−</qb:mtext> <qb:mi>J</qb:mi> <qb:mo>/</qb:mo> <qb:mi>ψ</qb:mi> </qb:mrow> </qb:math> states in the <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"> <sb:msub> <sb:mrow> <sb:mmultiscripts> <sb:mrow> <sb:mi>S</sb:mi> </sb:mrow> <sb:mprescripts/>