Gravitational form factors of pseudoscalar mesons in a contact interaction

Given the unique role played by the gravitational form factors (GFFs) in unraveling the internal mechanics of hadrons, we examine the GFFs of ground state pseudoscalar mesons <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>π</a:mi></a:mrow></a:math>, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>η</c:mi><c:mi>c</c:mi></c:msub></c:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>η</e:mi><e:mi>b</e:mi></e:msub></e:math> and the hypothetical <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>η</g:mi><g:mi>s</g:mi></g:msub><g:mo stretchy="false">(</g:mo><g:mi>s</g:mi><g:mover accent="true"><g:mi>s</g:mi><g:mo stretchy="false">¯</g:mo></g:mover><g:mo stretchy="false">)</g:mo></g:math>. We adopt the coupled framework of Dyson-Schwinger and Bethe-Salpeter equations within a contact interaction and employ a novel approach to the dressed amputated meson-meson scattering amplitude, which makes connection with the energy-momentum tensor and with the GFFs. The resulting GFFs fulfill the anticipated symmetry constraints. The corresponding charge and mass radii and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>D</m:mi></m:math> term are also computed. We show that the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>D</o:mi></o:math> term for the pseudoscalar mesons is bounded within the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mo stretchy="false">(</q:mo><q:mo>−</q:mo><q:mn>1</q:mn><q:mo>,</q:mo><q:mo>−</q:mo><q:mn>1</q:mn><q:mo>/</q:mo><q:mn>3</q:mn><q:mo stretchy="false">)</q:mo></q:math> range; these bounds correspond to the massless (chiral limit) and infinitely massive cases, respectively. Considering the current interest in the GFFs, understanding the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi>D</u:mi></u:mrow></u:math> term of pseudoscalar mesons and their GFFs can provide an important first step for future endeavors in the field.