Gravitational form factors of the pion in the self-consistent light-front quark model

We present a self-consistent light-front quark model (LFQM) analysis of the pion’s gravitational form factors (GFFs), incorporating the Bakamjian-Thomas (BT) construction consistently throughout the framework. By uniformly applying the BT formalism to both hadronic matrix elements and their associated Lorentz structures, we achieve a current-component-independent extraction of the pion GFFs <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>A</a:mi><a:mi>π</a:mi></a:msub><a:mo stretchy="false">(</a:mo><a:mi>t</a:mi><a:mo stretchy="false">)</a:mo></a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>D</e:mi><e:mi>π</e:mi></e:msub><e:mo stretchy="false">(</e:mo><e:mi>t</e:mi><e:mo stretchy="false">)</e:mo></e:math>, thereby eliminating the light-front zero-mode ambiguities that typically hinder conventional LFQM approaches. By tuning the model parameters, we identify an optimal set that successfully reproduces the decay constant and electromagnetic form factor of the pion, while yielding a <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>D</i:mi></i:math>-term value <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>D</k:mi><k:mi>π</k:mi></k:msub><k:mo stretchy="false">(</k:mo><k:mn>0</k:mn><k:mo stretchy="false">)</k:mo><k:mo>≈</k:mo><k:mo>−</k:mo><k:mn>1</k:mn></k:math>, consistent with predictions from chiral perturbation theory. The <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>D</o:mi></o:math>-term emerges as a sensitive probe of the pion’s internal dynamics, governing its mechanical radius—the largest among the charge, mass, and mechanical radii. We further examine the pion’s spatial structure via its associated two-dimensional light-front densities, including the momentum density, transverse pressure, and shear stress, all of which satisfy the required normalization and von Laue stability conditions. Our results reveal a detailed mechanical landscape: a centrally peaked momentum density that decreases monotonically; a repulsive pressure near the center (up to <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>x</q:mi><q:mo>⊥</q:mo></q:msub><q:mo>=</q:mo><q:mn>0.33</q:mn><q:mtext> </q:mtext><q:mtext> </q:mtext><q:mi>fm</q:mi></q:math>) that transitions to attraction in the outer region; and a shear stress profile peaking at an intermediate distance (<s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>x</s:mi><s:mo>⊥</s:mo></s:msub><s:mo>≈</s:mo><s:mn>0.2</s:mn><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>fm</s:mi></s:math>).