Twist-3 generalized parton distributions of sea quarks at zero skewness in the light-cone quark model

We present a systematic study of twist-3 generalized parton distributions (GPDs) for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mover accent="true"><a:mi>u</a:mi><a:mo stretchy="false">¯</a:mo></a:mover></a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mover accent="true"><e:mi>d</e:mi><e:mo stretchy="false">¯</e:mo></e:mover></e:math> sea quarks in the proton at zero skewness (<i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>ξ</i:mi><i:mo>=</i:mo><i:mn>0</i:mn></i:math>) using the light-cone formalism with overlap representation. The proton wave functions are derived from a meson-baryon fluctuation model that incorporates <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo stretchy="false">|</k:mo><k:mi>q</k:mi><k:mover accent="true"><k:mi>q</k:mi><k:mo stretchy="false">¯</k:mo></k:mover><k:mi>B</k:mi><k:mo stretchy="false">⟩</k:mo></k:math> Fock states, providing a natural framework for investigating sea quark contributions. Within this approach, we compute the complete set of twist-3 GPDs and present the numerical results, including both chiral-odd (<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>H</q:mi><q:mn>2</q:mn></q:msub></q:math>, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>E</s:mi><s:mn>2</s:mn></s:msub></s:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mover accent="true"><u:mi>E</u:mi><u:mo stretchy="false">˜</u:mo></u:mover><u:mn>2</u:mn></u:msub></u:math>, and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msubsup><y:mover accent="true"><y:mi>H</y:mi><y:mo stretchy="false">˜</y:mo></y:mover><y:mn>2</y:mn><y:mo>′</y:mo></y:msubsup></y:math>) and chiral-even (<cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mover accent="true"><cb:mi>E</cb:mi><cb:mo stretchy="false">˜</cb:mo></cb:mover><cb:mrow><cb:mn>2</cb:mn><cb:mi>T</cb:mi></cb:mrow></cb:msub></cb:math>, <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msubsup><gb:mi>H</gb:mi><gb:mrow><gb:mn>2</gb:mn><gb:mi>T</gb:mi></gb:mrow><gb:mo>′</gb:mo></gb:msubsup></gb:math>, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msubsup><ib:mi>E</ib:mi><ib:mrow><ib:mn>2</ib:mn><ib:mi>T</ib:mi></ib:mrow><ib:mo>′</ib:mo></ib:msubsup></ib:math>, and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msubsup><kb:mover accent="true"><kb:mi>H</kb:mi><kb:mo stretchy="false">˜</kb:mo></kb:mover><kb:mrow><kb:mn>2</kb:mn><kb:mi>T</kb:mi></kb:mrow><kb:mo>′</kb:mo></kb:msubsup></kb:math>) distributions for <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mover accent="true"><ob:mi>u</ob:mi><ob:mo stretchy="false">¯</ob:mo></ob:mover></ob:math> and <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mover accent="true"><sb:mi>d</sb:mi><sb:mo stretchy="false">¯</sb:mo></sb:mover></sb:math> quarks at zero skewness. By taking the forward limit, we also calculate the corresponding twist-3 parton distribution functions <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi>e</wb:mi><wb:mo stretchy="false">(</wb:mo><wb:mi>x</wb:mi><wb:mo stretchy="false">)</wb:mo></wb:math> of <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mover accent="true"><ac:mi>u</ac:mi><ac:mo stretchy="false">¯</ac:mo></ac:mover></ac:math> and <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:mover accent="true"><ec:mi>d</ec:mi><ec:mo stretchy="false">¯</ec:mo></ec:mover></ec:math> quarks. The kinetic orbital angular momenta of sea quarks deduced from the twist-3 GPDs are studied and compared to those from the twist-2 GPDs.