Virtual contributions from K * in the B → K * π h and B → K ρ h decays

Inspired by the significant virtual contributions of the subprocesses <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>D</a:mi> <a:mo>*</a:mo> </a:msup> <a:mo>/</a:mo> <a:msup> <a:mi>B</a:mi> <a:mo>*</a:mo> </a:msup> <a:mo stretchy="false">→</a:mo> <a:mi>D</a:mi> <a:mi>π</a:mi> </a:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>ρ</d:mi> <d:mo stretchy="false">→</d:mo> <d:mi>ω</d:mi> <d:mi>π</d:mi> <d:mo>/</d:mo> <d:mi>K</d:mi> <d:mover accent="true"> <d:mi>K</d:mi> <d:mo stretchy="false">¯</d:mo> </d:mover> </d:math> in the three-body hadronic <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>B</i:mi> </i:math> meson decays, we study the off-shell effects of the resonance <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi>K</k:mi> <k:mo>*</k:mo> </k:msup> </k:math> decaying into the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msup> <m:mi>K</m:mi> <m:mo>*</m:mo> </m:msup> <m:mi>π</m:mi> </m:math> or <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>K</o:mi> <o:mi>ρ</o:mi> </o:math> system in the decays <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>B</q:mi> <q:mo stretchy="false">→</q:mo> <q:msup> <q:mi>K</q:mi> <q:mo>*</q:mo> </q:msup> <q:mi>π</q:mi> <q:mi>h</q:mi> </q:math> and <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>B</t:mi> <t:mo stretchy="false">→</t:mo> <t:mi>K</t:mi> <t:mi>ρ</t:mi> <t:mi>h</t:mi> </t:math> (with <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mrow> <w:mrow> <w:mi>h</w:mi> <w:mo>=</w:mo> <w:mi>π</w:mi> </w:mrow> <w:mo>,</w:mo> <w:mrow> <w:mi>K</w:mi> </w:mrow> </w:mrow> </w:math> ) within the perturbative QCD approach. 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:msup> <y:mi>K</y:mi> <y:mo>*</y:mo> </y:msup> <y:msup> <y:mi>K</y:mi> <y:mo>*</y:mo> </y:msup> <y:mi>π</y:mi> </y:mrow> </y:msub> </y:math> and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msub> <ab:mi>g</ab:mi> <ab:mrow> <ab:msup> <ab:mi>K</ab:mi> <ab:mo>*</ab:mo> </ab:msup> <ab:mi>K</ab:mi> <ab:mi>ρ</ab:mi> </ab:mrow> </ab:msub> </ab:math> involved in this work are derived from the coupling constant <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:msub> <cb:mi>g</cb:mi> <cb:mrow> <cb:mi>ρ</cb:mi> <cb:mi>ω</cb:mi> <cb:mi>π</cb:mi> </cb:mrow> </cb:msub> </cb:math> under the flavor SU(3) symmetry. The <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>C</eb:mi> <eb:mi>P</eb:mi> </eb:math> averaged branching fractions for the quasi-two-body decays <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>B</gb:mi> <gb:mo stretchy="false">→</gb:mo> <gb:msup> <gb:mi>K</gb:mi> <gb:mo>*</gb:mo> </gb:msup> <gb:mo stretchy="false">[</gb:mo> <gb:mo stretchy="false">→</gb:mo> <gb:msup> <gb:mi>K</gb:mi> <gb:mo>*</gb:mo> </gb:msup> <gb:mi>π</gb:mi> <gb:mo stretchy="false">]</gb:mo> <gb:mi>h</gb:mi> </gb:math> and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"> <mb:mi>B</mb:mi> <mb:mo stretchy="false">→</mb:mo> <mb:msup> <mb:mi>K</mb:mi> <mb:mo>*</mb:mo> </mb:msup> <mb:mo stretchy="false">[</mb:mo> <mb:mo stretchy="false">→</mb:mo> <mb:mi>K</mb:mi> <mb:mi>ρ</mb:mi> <mb:mo stretchy="false">]</mb:mo> <mb:mi>h</mb:mi> </mb:math> are predicted to be on the order of <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"> <sb:msup> <sb:mn>10</sb:mn> <sb:mrow> <sb:mo>−</sb:mo> <sb:mn>9</sb:mn> </sb:mrow> </sb:msup> </sb:math> to <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"> <ub:msup> <ub:mn>10</ub:mn> <ub:mrow> <ub:mo>−</ub:mo> <ub:mn>7</ub:mn> </ub:mrow> </ub:msup> </ub:math> in this study. And the branching fractions for the decays <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"> <wb:mi>B</wb:mi> <wb:mo stretchy="false">→</wb:mo> <wb:msup> <wb:mi>K</wb:mi> <wb:mo>*</wb:mo> </wb:msup> <wb:mo stretchy="false">[</wb:mo> <wb:mo stretchy="false">→</wb:mo> <wb:mi>K</wb:mi> <wb:mi>ρ</wb:mi> <wb:mo stretchy="false">]</wb:mo> <wb:mi>h</wb:mi> </wb:math> are found to be around half of the corresponding results for the <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"> <cc:mi>B</cc:mi> <cc:mo stretchy="false">→</cc:mo> <cc:msup> <cc:mi>K</cc:mi> <cc:mo>*</cc:mo> </cc:msup> <cc:mo stretchy="false">[</cc:mo> <cc:mo stretchy="false">→</cc:mo> <cc:msup> <cc:mi>K</cc:mi> <cc:mo>*</cc:mo> </cc:msup> <cc:mi>π</cc:mi> <cc:mo stretchy="false">]</cc:mo> <cc:mi>h</cc:mi> </cc:math> channels, mainly due to the phase space difference between the <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"> <ic:mi>K</ic:mi> <ic:mi>ρ</ic:mi> </ic:math>