Tomography of Pions and Kaons in the QCD Vacuum: Transverse Momentum Dependent Parton Distribution Functions

TL;DR

The paper develops a nonperturbative, tomographic description of pion and kaon TMDPDFs by computing low-resolution constituent-quark TMDs within the ILM and attaching a rapidity-dependent soft factor from staple Wilson lines. The approach combines the ILM-based LFWF overlaps for the mesons with a Collins-Soper-Sterman evolution framework that includes both a perturbative CS kernel and a nonperturbative instanton-induced component, connected through a matching scale. At the intrinsic ILM scale, the distributions peak near moderate x and transverse distance b_b ~ ρ; after evolving to higher scales, the TMDs shift toward small x and smaller b_b, reflecting broadened transverse momentum while maintaining nonperturbative input. Comparisons to Drell-Yan data show qualitative agreement within uncertainties, highlighting the need for more precise data and lattice input to constrain the nonperturbative evolution and higher Fock components, and underscoring the ILM's role in connecting QCD vacuum structure to meson TMD phenomenology.

Abstract

We evaluate the pion and kaon transverse momentum dependent parton distribution functions (TMDPDFs) in the instanton liquid model (ILM), a model of the QCD vacuum at low resolution. The relevant TMDs are factored into a constituent quark distribution times a rapidity dependent soft factor from staple-shaped Wilson lines, for fixed parton longitudinal momentum and transverse separation. The results are evolved to higher rapidities using the Collins-Soper (CS) kernel and higher resolution using renormalization group evolution. The comparison to existing extractions of pion TMDs from Drell-Yan (DY) data is briefly discussed.

Figures (13)

• Figure 1: Visualization of the vacuum in gluodynamics, before cooling at a resolution of about $\frac{1}{10}\,{\rm fm}$ (top), and after cooling at a resolution of about $\frac{1}{3}\,{\rm fm}$ (bottom) Moran:2008xq, where the pseudoparticles emerge.
• Figure 2: Wilson lines for (a) SIDIS process with the space-like correlation function and (b) Drell-Yan process with the timelike correlation.
• Figure 3: Soft separation for the quark pion TMD \ref{['tmd']}, which is approximated into a constituent quark distribution without rapidity dependence and a rapidity dependent factor generated by the stapled Wilson line.
• Figure 4: Constituent quark TMD distribution in pion (left) \ref{['eq:pion_tmd']} and kaon (right) \ref{['eq:kaon_tmd']} at low resolution $\mu=1/\rho$ with $\rho=0.313$ fm: (a,b) are the density plots, (c,d) the 3D plots, (e,f) the transverse momentum dependent plots for fixed $x$, and (g,h) the longitudinal momentum dependence for fixed $k_\perp$. The pion parameters are $C_\pi=7.240$, $m_\pi=139.0$ MeV, $M=398.17$ MeV. The kaon parameters are $C_K=6.60$, $m_K=458.0$ MeV, $M_u=394.4$ MeV, $M_s=556.5$ MeV.
• Figure 5: The comparison of ILM estimated pion TMD distribution $\tilde{F}_1^{q/\pi}(x,b_\perp;\mu_0,\zeta)$\ref{['pi_tmd']} at scale $\mu_0\sim1/\rho$ (perturbative gluons excluded) between the one with full unsubtracted quark transverse momentum included (blue, $\zeta=\mu_0^2$) and soft transverse momentum subtracted (red, $\zeta=\mu^2=4$ GeV$^2$) at $x=0.3$ in $b_\perp$ space (a) and in $k_\perp$ space (b)