Pionic transitions of the spin-2 partner of X(3872) to χcJ

We investigated the pionic transitions between the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>X</a:mi><a:mn>2</a:mn></a:msub></a:math> [spin-2 partner of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>X</c:mi><c:mo stretchy="false">(</c:mo><c:mn>3872</c:mn><c:mo stretchy="false">)</c:mo></c:math>] and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>χ</g:mi><g:mrow><g:mi>c</g:mi><g:mn>1</g:mn><g:mo>,</g:mo><g:mn>2</g:mn></g:mrow></g:msub></g:math> using a nonrelativistic effective field theory. The <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>X</i:mi><i:mn>2</i:mn></i:msub></i:math> is assumed to be a bound state of the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>D</k:mi><k:mo>*</k:mo></k:msup></k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msup><m:mover accent="true"><m:mi>D</m:mi><m:mo stretchy="false">¯</m:mo></m:mover><m:mo>*</m:mo></m:msup></m:math> mesons and to decay through several kinds of loops, including the bubble, triangle, and box loops. Within the present model, the widths for the single-pion decays <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>X</q:mi><q:mn>2</q:mn></q:msub><q:mo stretchy="false">→</q:mo><q:msup><q:mi>π</q:mi><q:mn>0</q:mn></q:msup><q:msub><q:mi>χ</q:mi><q:mrow><q:mi>c</q:mi><q:mi>J</q:mi></q:mrow></q:msub></q:math> are predicted to be about 3–30 keV. For the dipion decays, the widths are a few keVs. These widths yield a branching fraction of <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mrow><t:msup><t:mrow><t:mn>10</t:mn></t:mrow><t:mrow><t:mo>−</t:mo><t:mn>3</t:mn></t:mrow></t:msup><t:mi>–</t:mi><t:msup><t:mrow><t:mn>10</t:mn></t:mrow><t:mrow><t:mo>−</t:mo><t:mn>2</t:mn></t:mrow></t:msup></t:mrow></t:math>. The ratio <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msub><v:mi>R</v:mi><v:mrow><v:mi mathvariant="normal">c</v:mi><v:mn>0</v:mn></v:mrow></v:msub><v:mo>=</v:mo><v:mi mathvariant="normal">Γ</v:mi><v:mo stretchy="false">(</v:mo><v:msub><v:mi>X</v:mi><v:mn>2</v:mn></v:msub><v:mo stretchy="false">→</v:mo><v:msup><v:mi>π</v:mi><v:mo>+</v:mo></v:msup><v:msup><v:mi>π</v:mi><v:mo>−</v:mo></v:msup><v:msub><v:mi>χ</v:mi><v:mrow><v:mi>c</v:mi><v:mi>J</v:mi></v:mrow></v:msub><v:mo stretchy="false">)</v:mo><v:mo>/</v:mo><v:mi mathvariant="normal">Γ</v:mi><v:mo stretchy="false">(</v:mo><v:msub><v:mi>X</v:mi><v:mn>2</v:mn></v:msub><v:mo stretchy="false">→</v:mo><v:msup><v:mi>π</v:mi><v:mn>0</v:mn></v:msup><v:msup><v:mi>π</v:mi><v:mn>0</v:mn></v:msup><v:msub><v:mi>χ</v:mi><v:mrow><v:mi>c</v:mi><v:mi>J</v:mi></v:mrow></v:msub><v:mo stretchy="false">)</v:mo><v:mo>≃</v:mo><v:mn>1.6</v:mn></v:math>, which is a bit smaller than the expected value of 2, and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>R</gb:mi><gb:mn>21</gb:mn></gb:msub><gb:mo>=</gb:mo><gb:mi mathvariant="normal">Γ</gb:mi><gb:mo stretchy="false">(</gb:mo><gb:msub><gb:mi>X</gb:mi><gb:mn>2</gb:mn></gb:msub><gb:mo stretchy="false">→</gb:mo><gb:mi>π</gb:mi><gb:mi>π</gb:mi><gb:msub><gb:mi>χ</gb:mi><gb:mrow><gb:mi>c</gb:mi><gb:mn>2</gb:mn></gb:mrow></gb:msub><gb:mo stretchy="false">)</gb:mo><gb:mo>/</gb:mo><gb:mi mathvariant="normal">Γ</gb:mi><gb:mo stretchy="false">(</gb:mo><gb:msub><gb:mi>X</gb:mi><gb:mn>2</gb:mn></gb:msub><gb:mo stretchy="false">→</gb:mo><gb:mi>π</gb:mi><gb:mi>π</gb:mi><gb:msub><gb:mi>χ</gb:mi><gb:mrow><gb:mi>c</gb:mi><gb:mn>1</gb:mn></gb:mrow></gb:msub><gb:mo stretchy="false">)</gb:mo><gb:mo>≃</gb:mo><gb:mn>0.85</gb:mn></gb:math>. These ratios are nearly independent of the <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msub><qb:mi>X</qb:mi><qb:mn>2</qb:mn></qb:msub></qb:math> mass and the coupling constants, which might be a good quantity for the experiments. Moreover, the invariant mass spectra of the <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msup><sb:mi>π</sb:mi><sb:mn>0</sb:mn></sb:msup><sb:msub><sb:mi>χ</sb:mi><sb:mrow><sb:mi>c</sb:mi><sb:mi>J</sb:mi></sb:mrow></sb:msub></sb:math> final state for the dipion processes are presented, showing a cusp structure at the <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi>D</ub:mi><ub:msup><ub:mover accent="true"><ub:mi>D</ub:mi><ub:mo stretchy="false">¯</ub:mo></ub:mover><ub:mo>*</ub:mo></ub:msup></ub:math> threshold enhanced and narrowed by the nearby triangle singularity.