Revisiting radiative transitions of charmonium states in a covariant confined quark model

We calculate the amplitudes and branching fractions of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>χ</a:mi> <a:mrow> <a:mi>c</a:mi> <a:mi>J</a:mi> </a:mrow> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mn>1</a:mn> <a:mi>P</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> , <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>h</e:mi> <e:mi>c</e:mi> </e:msub> <e:mo stretchy="false">(</e:mo> <e:mn>1</e:mn> <e:mi>P</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> , <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>J</i:mi> <i:mo>/</i:mo> <i:mi>ψ</i:mi> <i:mo stretchy="false">(</i:mo> <i:mn>1</i:mn> <i:mi>S</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>ψ</m:mi> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:math> radiative decays in the framework of the covariant confined quark model. First, we explicitly show that all calculated amplitudes are gauge invariant. Next, we use experimental data for three decay channels <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>J</q:mi> <q:mo>/</q:mo> <q:mi>ψ</q:mi> <q:mo stretchy="false">→</q:mo> <q:msub> <q:mi>η</q:mi> <q:mi>c</q:mi> </q:msub> <q:mi>γ</q:mi> </q:math> , <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msub> <t:mi>χ</t:mi> <t:mrow> <t:mi>c</t:mi> <t:mn>0</t:mn> </t:mrow> </t:msub> <t:mo stretchy="false">→</t:mo> <t:mi>J</t:mi> <t:mo>/</t:mo> <t:mi>ψ</t:mi> <t:mi>γ</t:mi> </t:math> , and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>ψ</w:mi> <w:mo stretchy="false">(</w:mo> <w:mn>2</w:mn> <w:mi>S</w:mi> <w:mo stretchy="false">)</w:mo> <w:mo stretchy="false">→</w:mo> <w:msub> <w:mi>χ</w:mi> <w:mrow> <w:mi>c</w:mi> <w:mn>0</w:mn> </w:mrow> </w:msub> <w:mi>γ</w:mi> </w:math> to fit the model parameters including the charm quark mass and the meson size parameter. Finally, we use the obtained parameters to predict the branching fractions of the decays <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:msub> <bb:mi>h</bb:mi> <bb:mi>c</bb:mi> </bb:msub> <bb:mo stretchy="false">→</bb:mo> <bb:msub> <bb:mi>η</bb:mi> <bb:mi>c</bb:mi> </bb:msub> <bb:mi>γ</bb:mi> </bb:math> , <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:msub> <eb:mi>χ</eb:mi> <eb:mrow> <eb:mi>c</eb:mi> <eb:mn>1</eb:mn> <eb:mo stretchy="false">(</eb:mo> <eb:mi>c</eb:mi> <eb:mn>2</eb:mn> <eb:mo stretchy="false">)</eb:mo> </eb:mrow> </eb:msub> <eb:mo stretchy="false">→</eb:mo> <eb:mi>J</eb:mi> <eb:mo>/</eb:mo> <eb:mi>ψ</eb:mi> <eb:mi>γ</eb:mi> </eb:math> , and <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>ψ</jb:mi> <jb:mo stretchy="false">(</jb:mo> <jb:mn>2</jb:mn> <jb:mi>S</jb:mi> <jb:mo stretchy="false">)</jb:mo> <jb:mo stretchy="false">→</jb:mo> <jb:msub> <jb:mi>χ</jb:mi> <jb:mrow> <jb:mi>c</jb:mi> <jb:mn>1</jb:mn> <jb:mo stretchy="false">(</jb:mo> <jb:mi>c</jb:mi> <jb:mn>2</jb:mn> <jb:mo stretchy="false">)</jb:mo> </jb:mrow> </jb:msub> <jb:mi>γ</jb:mi> </jb:math> . Our predictions agree well with experimental data.