$$\rho $$ meson transverse momentum-dependent parton distributions

Abstract In this paper, the light-front wave functions (LFWFs) of the $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> meson are evaluated in the framework of the Nambu–Jona-Lasinio model using the proper time regularization scheme. The transverse momentum-dependent parton distributions (TMDs) of the $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> meson are derived from the overlap representations of the LFWFs. We investigate the $$\varvec{k}_{\perp }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mo>⊥</mml:mo> </mml:msub> </mml:math> -weighted moments and the x -dependent average transverse momentum $$\langle k_{\perp }^n(x)\rangle _{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>⊥</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msub> </mml:math> of $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> meson TMDs. The $$\langle k_{\perp }^n(x)\rangle _{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>⊥</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msub> </mml:math> shows the typical transverse momenta of quark TMDs in our model. Our findings indicate that the average transverse momenta of $$\langle k_{\perp }(x)\rangle _{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mo>⊥</mml:mo> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msub> </mml:math> fall within the range of [0.3, 0.45] GeV, while $$\langle k_{\perp }^2(x)\rangle _{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:msubsup> <mml:mi>k</mml:mi> <mml:mrow> <mml:mo>⊥</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msub> </mml:math> are in the region of [0.15, 0.30] $$GeV^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>e</mml:mi> <mml:msup> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . Our TMDs and parton distribution functions (PDFs) exhibit excellent adherence to positive constraints. The investigation of spin densities within the $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> meson in transverse momentum space reveals axially symmetric distributions for quarks and targets polarized in either the longitudinal or transverse direction. Conversely, unpolarized symmetric distributions occur when the quark is longitudinally polarized and the target is transversely polarized, while exhibiting dipolar distortions in the opposite scenario. A comparison between TMDs from LFWFs and those from a covariant approach demonstrates that TMDs from LFWFs more effectively satisfy positive constraints than their counterparts from the covariant approach. Furthermore, the x -moments of $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> meson PDFs indicate that $$\langle x\rangle _{g_L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>x</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:msub> </mml:math> is larger using LFWFs than with the covariant approach. Additionally, analysis of quark spin densities within the $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> meson reveals that $$\rho _{\uparrow \downarrow }\left( k_x,k_y\right) =f-g_L-\frac{1}{3}f_{LL}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>↑</mml:mo> <mml:mo>↓</mml:mo> </mml:mrow> </mml:msub> <mml:mfenced> <mml:msub> <mml:mi>k</mml:mi>