Distribution Functions of Radially Excited Pion using the Light-Front Quark Model

TL;DR

This work investigates the valence-quark structure of the pion across the ground state and the first two radially excited states within a light-front quark model, using a variationally optimized QCD-motivated Hamiltonian and a harmonic-oscillator basis with state mixing. Light-front wavefunctions are built via the Brodsky–Huang–Lepage mapping and Melosh–Wigner rotations to compute the pion DA, PDF, and EMFF, which are then evolved with LO ERBL and NLO DGLAP equations to compare with experimental data and other theoretical approaches. The results show that mixing effects are mild for the $1S$ and $2S$ states but become pronounced for the $3S$ state, significantly impacting the DA, PDF, EMFF, and decay constants, while charge radii increase with radial excitation. The study provides a systematic LFQM analysis of radially excited pions, highlighting mixing as a key degree of freedom and offering predictions that can guide future lattice QCD studies and experimental measurements at facilities like an Electron-Ion Collider.

Abstract

We investigate the internal structure of the ground ($1S$) and the first two radially excited ($2S,3S$) states of the pion within the light-front quark model. The valence Fock sector is described using pure harmonic-oscillator eigenstates and mixed states formed as orthogonal linear combinations of these eigenfunctions. The optimal wavefunction parameters are determined through a variational procedure based on a QCD-motivated effective Hamiltonian. Using the resulting light-front wavefunctions, we study the pion distribution amplitude, parton distribution function, and electromagnetic form factor. After QCD evolution, the ground state distribution amplitude and parton distribution function are found to be in good agreement with available experimental data. At the model scale, the parton distribution functions of the $1S$ and $2S$ states show clear sensitivity to state mixing, while the distribution amplitudes and electromagnetic form factors are weakly sensitive. In contrast, for the $3S$ state, all three observables exhibit a pronounced sensitivity to mixing. The decay constants of the mixed states are also found to decrease sequentially with increasing radial excitation.

Figures (6)

• Figure 1: (Color online) (a) Experimental mass contours of $\pi(3S)$, $\rho(2S)$, and $\rho(3S)$ overlaid on the $\pi(2S)$ base surface; (b) variation of the potential parameter $a$ with the mixing angles; (c) variation of the potential parameter $\alpha_s$ with the mixing angles; and (d) variation of the harmonic parameter $\beta_{\pi}$ with the mixing angles. The best-fit point is indicated by a red dot in all contour plots.
• Figure 2: (Color online) Variation of Decay constant as a function of harmonic parameter $\beta$ for pure states (a) and for the case of mixing (b).
• Figure 3: (Color online) DAs of radially excited pions. (a) DA of the ground-state pion ($\pi (1S)$) at the initial scale $Q_{0}$ and at $Q=2$ GeV for pure and mixed states, compared with the asymptotic form $6x(1-x)$ and E791 data E791:2000xcx; (b) DA of the first excited state ($\pi (2S)$) at $Q_{0}$ for pure and mixed states; (c) DA of the second excited state ($\pi (3S)$) at $Q_{0}$ for pure and mixed states.
• Figure 4: (Color online) PDFs $xf(x)$ of radially excited pions at the model scale $Q^2_{0}=0.20$ GeV$^2$(a) ground state, (b) first excitation, and (c) second excitation, shown for both pure and mixed states.
• Figure 5: (Color online) PDFs of radially excited pions at different $Q^{2}$: (a) ground state $\pi(1S)$ compared with E615 data E615:1989bdaAicher:2010cb at $Q^{2}=16$ GeV$^{2}$, and (b) first excitation $\pi(2S)$ compared with Xu et al. Xu:2025cyj at $Q^{2}=10.24$ GeV$^{2}$. Results are shown for both pure and mixed states.