Beyond leading twist: ρ meson decay constants and distribution amplitudes in a self-consistent light-front quark model

In this study, we present a comprehensive analysis of decay constants and chiral-even and chiral-odd distribution amplitudes (DAs) up to twist 4 for the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ρ</a:mi></a:math> meson in the standard light-front quark model (LFQM) based on the Bakamjian-Thomas (BT) construction. For the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ρ</c:mi></c:math> meson, which possesses both longitudinal (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>h</e:mi><e:mo>=</e:mo><e:mn>0</e:mn></e:mrow></e:math>) and transverse <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">(</g:mo><g:mi>h</g:mi><g:mo>=</g:mo><g:mo>±</g:mo><g:mn>1</g:mn><g:mo stretchy="false">)</g:mo></g:math> polarizations, two types of decay constants, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msubsup><k:mi>f</k:mi><k:mi>ρ</k:mi><k:mo stretchy="false">∥</k:mo></k:msubsup></k:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi>f</n:mi><n:mi>ρ</n:mi><n:mo>⊥</n:mo></n:msubsup></n:math>, arises accordingly. We demonstrate that these decay constants can be self-consistently extracted from both local (<p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msup><p:mi>z</p:mi><p:mi>μ</p:mi></p:msup><p:mo>=</p:mo><p:mn>0</p:mn></p:math>) and nonlocal (<r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msup><r:mi>z</r:mi><r:mi>μ</r:mi></r:msup><r:mo>≠</r:mo><r:mn>0</r:mn></r:math>) matrix elements <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mo stretchy="false">⟨</t:mo><t:mn>0</t:mn><t:mo stretchy="false">|</t:mo><t:mover accent="true"><t:mi>q</t:mi><t:mo stretchy="false">¯</t:mo></t:mover><t:mo stretchy="false">(</t:mo><t:mi>z</t:mi><t:mo stretchy="false">)</t:mo><t:mi mathvariant="normal">Γ</t:mi><t:mi>q</t:mi><t:mo stretchy="false">(</t:mo><t:mo>−</t:mo><t:mi>z</t:mi><t:mo stretchy="false">)</t:mo><t:mo stretchy="false">|</t:mo><t:mi>ρ</t:mi><t:mo stretchy="false">(</t:mo><t:mi>P</t:mi><t:mo>,</t:mo><t:mi>h</t:mi><t:mo stretchy="false">)</t:mo><t:mo stretchy="false">⟩</t:mo></t:math>, with <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mrow><ib:mi mathvariant="normal">Γ</ib:mi><ib:mo>=</ib:mo><ib:mo stretchy="false">(</ib:mo><ib:msup><ib:mrow><ib:mi>γ</ib:mi></ib:mrow><ib:mrow><ib:mi>μ</ib:mi></ib:mrow></ib:msup><ib:mo>,</ib:mo><ib:msup><ib:mrow><ib:mi>σ</ib:mi></ib:mrow><ib:mrow><ib:mi>μ</ib:mi><ib:mi>ν</ib:mi></ib:mrow></ib:msup><ib:mo>,</ib:mo><ib:msup><ib:mrow><ib:mi>γ</ib:mi></ib:mrow><ib:mrow><ib:mi>μ</ib:mi></ib:mrow></ib:msup><ib:msub><ib:mrow><ib:mi>γ</ib:mi></ib:mrow><ib:mn>5</ib:mn></ib:msub><ib:mo>,</ib:mo><ib:mn mathvariant="bold">1</ib:mn><ib:mo stretchy="false">)</ib:mo></ib:mrow></ib:math>, in a manner independent of current components, polarizations, and reference frames. In particular, we emphasize the role of nonlocal matrix elements involving axial-vector and scalar currents, where mixing between <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msubsup><ob:mi>f</ob:mi><ob:mi>ρ</ob:mi><ob:mo stretchy="false">∥</ob:mo></ob:msubsup></ob:math> and <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"><rb:msubsup><rb:mi>f</rb:mi><rb:mi>ρ</rb:mi><rb:mo>⊥</rb:mo></rb:msubsup></rb:math> occurs. We show that this mixing is consistently resolved through the BT construction, ensuring the proper extraction of these decay constants. Additionally, we investigate the structure of chiral-even DAs (<tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:msubsup><tb:mi>ϕ</tb:mi><tb:mrow><tb:mn>2</tb:mn><tb:mo>;</tb:mo><tb:mi mathvariant="normal">V</tb:mi></tb:mrow><tb:mo stretchy="false">∥</tb:mo></tb:msubsup><tb:mo>,</tb:mo><tb:msubsup><tb:mi>ϕ</tb:mi><tb:mrow><tb:mn>3</tb:mn><tb:mo>;</tb:mo><tb:mi mathvariant="normal">V</tb:mi></tb:mrow><tb:mo stretchy="false">∥</tb:mo></tb:msubsup><tb:mo>,</tb:mo><tb:msubsup><tb:mi>ψ</tb:mi><tb:mrow><tb:mn>3</tb:mn><tb:mo>;</tb:mo><tb:mi mathvariant="normal">A</tb:mi></tb:mrow><tb:mo>⊥</tb:mo></tb:msubsup><tb:mo>,</tb:mo><tb:msubsup><tb:mi>ϕ</tb:mi><tb:mrow><tb:mn>4</tb:mn><tb:mo>;</tb:mo><tb:mi mathvariant="normal">V</tb:mi></tb:mrow><tb:mo stretchy="false">∥</tb:mo></tb:msubsup></tb:math>) and chiral-odd DAs (<cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:msubsup><cc:mi>ϕ</cc:mi><cc:mrow><cc:mn>2</cc:mn><cc:mo>;</cc:mo><cc:mi mathvariant="normal">T</cc:mi></cc:mrow><cc:mo>⊥</cc:mo></cc:msubsup><cc:mo>,</cc:mo><cc:msubsup><cc:mi>ϕ</cc:mi><cc:mrow><cc:mn>3</cc:mn><cc:mo>;</cc:mo><cc:mi mathvariant="normal">T</cc:mi></cc:mrow><cc:mo>⊥</cc:mo></cc:msubsup><cc:mo>,</cc:mo><cc:msubsup><cc:mi>ψ</cc:mi><cc:mrow><cc:mn>3</cc:mn><cc:mo>;</cc:mo><cc:mi mathvariant="normal">S</cc:mi></cc:mrow><cc:mo stretchy="false">∥</cc:mo></cc:msubsup><cc:mo>,</cc:mo><cc:msubsup><cc:mi>ϕ</cc:mi><cc:mrow><cc:mn>4</cc:mn><cc:mo>;</cc:mo><cc:mi mathvariant="normal">T</cc:mi></cc:mrow><cc:mo>⊥</cc:mo></cc:msubsup></cc:math>) beyond leading twists, and present their corresponding <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:mi>ξ</jc:mi></jc:math>-moments and Gegenbauer moments. These results provide deeper insight into the nonperturbative structure of vector mesons and demonstrate the robustness and self-consistency of the LFQM based on the BT framework.