Kaon T-even transverse-momentum-dependent distributions and form factors in a self-consistent light-front quark model

TL;DR

This work develops a self-consistent Bakamjian–Thomas–based light-front quark model (fBT-LFQM) for the kaon, enforcing four-momentum conservation at the meson–quark vertex by replacing $M$ with the invariant mass $M_0$ in both hadronic matrix elements and Lorentz prefactors. The approach yields current-component–independent predictions for electromagnetic and scalar form factors, as well as the full set of unpolarized T-even TMDs and their collinear PDFs, effectively resolving light-front zero-mode ambiguities. Using a Gaussian light-front wave function with Kaon-specific parameters, the model produces a Gaussian $f_1^q(x,\bm k_\perp)$ and reveals twist- and flavor-dependent hierarchies in higher-twist TMDs ($f^{\perp q}$, $e^q$, $f_4^q$), including an elliptic deformation in $f_4^q$ from $\gamma^-$ kinematics. The framework connects the nonperturbative scale $\mu_0=0.6$ GeV to NNLO DGLAP evolution, yielding Mellin moments for kaon PDFs at $\mu^2=16$ and $27~\text{GeV}^2$ and highlighting SU(3) flavor breaking and the redistribution of momentum to gluons with evolving scale, thereby providing a coherent bridge from LFWF dynamics to evolved PDFs and facilitating comparison with lattice and phenomenology.

Abstract

We present a self-consistent light-front quark model (LFQM) for the kaon based on the Bakamjian--Thomas (BT) construction and apply it to the electromagnetic and scalar form factors, as well as the full set of unpolarized T-even transverse-momentum-dependent distributions (TMDs) and their collinear parton distribution functions (PDFs). A uniform use of the invariant mass $M_0$ in both hadronic matrix elements and Lorentz prefactors enforces four-momentum conservation at the meson--quark vertex and yields current-component--independent observables, removing light-front zero-mode ambiguities. Properly normalized electromagnetic form factor $F_{K^+}(Q^2)$ is exemplified as a current-component--independent observable by verifying its uniqueness with the explicit computation from all the available components ($γ^+$, $γ^\perp$, and $γ^-$) of the current. In the scalar channel, we compare the direct $f_S(Q^2)$ and mass-factored $F_S(Q^2)$ definitions and show they are not interchangeable within the BT-based LFQM because $M\to M_0(x,{\bf k}_\perp)$ must be implemented inside the integral. Using a Gaussian form for the light-front wave function, the twist-2 TMD $f_1^q$ becomes exactly Gaussian in ${\bf k}_\perp$, while higher-twist TMDs ($f^{\perp q}$, $e^q$, $f_4^q$) display systematic twist/flavor hierarchies, including an elliptic deformation of $f_4^q$ tied to $γ^-$ component. The BT-based LFQM satisfies the forward-limit sum rule for $f_4^q(x)$. We further analyze the perturbative QCD evolution of the valence PDFs for the pion and kaon and report their Mellin moments at representative scales, enabling direct comparison with phenomenology. Overall, the BT-based LFQM provides a unified, current-independent description of meson structure and a consistent bridge from nonperturbative light-front dynamics to evolved collinear PDFs.

Figures (9)

• Figure 1: Charged–kaon EMFF $F_{K^+}(Q^2)$. Top: fBT-LFQM total (solid) with $u$ (dashed) and $s$ (dotdashed) contributions. Bottom: exact, component-independent benchmark (solid) vs. pBT-LFQM $\gamma^-$ extraction (dashed), showing the missing LF zero mode.
• Figure 2: The $u$-quark contribution to the mass-factored scalar form factor $F_S^{u/\pi}(Q^2)$ of the $\pi^+$ obtained in the fBT-LFQM (solid) and pBT-LFQM (dashed).
• Figure 3: The $u$-quark (top) and $s$-quark (bottom) contributions to the mass–factored scalar form factor $F_S^{q/K}(Q^2)$ of the $K^+$, obtained in the fBT-LFQM (solid) and pBT-LFQM (dashed).
• Figure 4: Comparison of $u$-quark PDFs in $K^+$: $e^{u/K}(x)$ (left) and $f_4^{u/K}(x)$ (right), shown for the fBT-LFQM (solid) and the pBT-LFQM (dashed).
• Figure 5: Contour maps of the unpolarized T-even TMDs $(f_1^q,\,f^{\perp q},\,e^q,\,f_4^q)$ for $u$ in $\pi^+$ (top), $u$ in $K^+$ (middle), and $s$ in $K^+$ (bottom), shown in the $(x,k_\perp)$ plane (color: red = high, violet = low). For $f_4^q$ we fix $\theta=\pi/2$ ($k_x=0$) to display the maximal profile.