We study the contributions for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>K</a:mi><a:msup><a:mi>η</a:mi><a:mo>′</a:mo></a:msup></a:math> pair originating from the scalar intermediate state <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msubsup><c:mi>K</c:mi><c:mn>0</c:mn><c:mo>*</c:mo></c:msubsup><c:mo stretchy="false">(</c:mo><c:mn>1430</c:mn><c:mo stretchy="false">)</c:mo></c:math> in the three-body decays <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>B</g:mi><g:mo stretchy="false">→</g:mo><g:mi>K</g:mi><g:msup><g:mi>η</g:mi><g:mo>′</g:mo></g:msup><g:mi>h</g:mi></g:math> (<j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:mi>h</j:mi><j:mo>=</j:mo><j:mi>π</j:mi></j:mrow></j:math>, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:mi>K</l:mi></l:mrow></l:math>) within the perturbative QCD approach. The contribution of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi>K</n:mi><n:mn>0</n:mn><n:mo>*</n:mo></n:msubsup><n:mo stretchy="false">(</n:mo><n:mn>1430</n:mn><n:mo stretchy="false">)</n:mo><n:mo stretchy="false">→</n:mo><n:mi>K</n:mi><n:msup><n:mi>η</n:mi><n:mo>′</n:mo></n:msup></n:math> is described by the Flatté formula with coupled channels <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>K</s:mi><s:mi>π</s:mi></s:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>K</u:mi><u:mi>η</u:mi></u:math>, and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>K</w:mi><w:msup><w:mi>η</w:mi><w:mo>′</w:mo></w:msup></w:math>. The strong coupling constants <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>g</y:mi><y:mrow><y:msubsup><y:mi>K</y:mi><y:mn>0</y:mn><y:mo>*</y:mo></y:msubsup><y:mi>K</y:mi><y:msup><y:mi>η</y:mi><y:mrow><y:mo stretchy="false">(</y:mo><y:mo>′</y:mo><y:mo stretchy="false">)</y:mo></y:mrow></y:msup></y:mrow></y:msub></y:math> are extracted from <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>g</cb:mi><cb:mrow><cb:msubsup><cb:mi>K</cb:mi><cb:mn>0</cb:mn><cb:mo>*</cb:mo></cb:msubsup><cb:mi>K</cb:mi><cb:mi>π</cb:mi></cb:mrow></cb:msub></cb:math> within flavor SU(3) symmetry. In spite of the strong depression by phase space near the threshold of <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>K</eb:mi><eb:msup><eb:mi>η</eb:mi><eb:mo>′</eb:mo></eb:msup></eb:math>, the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>C</gb:mi><gb:mi>P</gb:mi></gb:math> averaged branching fractions for the <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>B</ib:mi><ib:mo stretchy="false">→</ib:mo><ib:msubsup><ib:mi>K</ib:mi><ib:mn>0</ib:mn><ib:mo>*</ib:mo></ib:msubsup><ib:mo stretchy="false">(</ib:mo><ib:mn>1430</ib:mn><ib:mo stretchy="false">)</ib:mo><ib:mi>h</ib:mi><ib:mo stretchy="false">→</ib:mo><ib:mi>K</ib:mi><ib:msup><ib:mi>η</ib:mi><ib:mo>′</ib:mo></ib:msup><ib:mi>h</ib:mi></ib:math> decays are predicted to be on the order of <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msup><ob:mn>10</ob:mn><ob:mrow><ob:mo>−</ob:mo><ob:mn>8</ob:mn></ob:mrow></ob:msup></ob:math> to <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msup><qb:mn>10</qb:mn><qb:mrow><qb:mo>−</qb:mo><qb:mn>5</qb:mn></qb:mrow></qb:msup></qb:math>, which are non-negligible for the corresponding three-body <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mi>B</sb:mi></sb:math> decays. Since the <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mrow><ub:mi mathvariant="normal">K</ub:mi><ub:mi>η</ub:mi></ub:mrow></ub:math> system is almost decoupled from the even-spin strange mesons under flavor SU(3) symmetry, those quasi-two-body <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:mi>B</xb:mi></xb:math> decays with subprocess <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:msubsup><zb:mi>K</zb:mi><zb:mn>0</zb:mn><zb:mo>*</zb:mo></zb:msubsup><zb:mo stretchy="false">(</zb:mo><zb:mn>1430</zb:mn><zb:mo stretchy="false">)</zb:mo><zb:mo stretchy="false">→</zb:mo><zb:mi>K</zb:mi><zb:mi>η</zb:mi></zb:math> shall have quite small branching ratios and are not taken into account in this work. We also estimate that the branching fraction for <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msubsup><ec:mi>K</ec:mi><ec:mn>0</ec:mn><ec:mo>*</ec:mo></ec:msubsup><ec:mo stretchy="false">(</ec:mo><ec:mn>1430</ec:mn><ec:mo stretchy="false">)</ec:mo><ec:mo stretchy="false">→</ec:mo><ec:mi>K</ec:mi><ec:msup><ec:mi>η</ec:mi><ec:mo>′</ec:mo></ec:msup></ec:math> is about one fifth of that for <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:msubsup><jc:mi>K</jc:mi><jc:mn>0</jc:mn><jc:mo>*</jc:mo></jc:msubsup><jc:mo stretchy="false">(</jc:mo><jc:mn>1430</jc:mn><jc:mo stretchy="false">)</jc:mo><jc:mo stretchy="false">→</jc:mo><jc:mi>K</jc:mi><jc:mi>π</jc:mi></jc:math>. The predictions for the relevant decays are expected to be tested by the LHCb and Belle-II experiments in the future. Published by the American Physical Society 2024
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