[Paper Review] Parton Distributions Functions of Pion, Kaon and Eta pseudoscalar mesons in the NJL model

R. Davidson, E. Ruiz Arriola|arXiv (Cornell University)|Oct 23, 2001

Quantum Chromodynamics and Particle Interactions3 references9 citations

TL;DR

This paper computes parton distribution functions (PDFs) for pion, kaon, and eta mesons within the Nambu-Jona-Lasinio (NJL) model using Pauli-Villars regularization. It finds that NLO QCD evolution effects are surprisingly small, with valence quark distributions matching experimental data, but sea and gluon distributions showing steeper low-x and softer high-x behavior than phenomenological fits at $Q^2 = 4\,\text{GeV}^2$. The model provides a nonperturbative initial condition for PDF evolution with chiral symmetry constraints built in.

ABSTRACT

Parton distributions of pseudoscalar pi,K and eta mesons obtained within the NJL model using the Pauli-Villars regularization method are analyzed in terms of LO and NLO evolution, and the valence sea quark and gluon parton distributions for the pion are obtained at Q^2 = 4 GeV^2 and compared to existing parametrizations at that scale. Surprisingly, the NLO order effects turn out to be small compared to the LO ones. The valence distributions are in good agreement with experimental analyses, but the gluon and sea distributions come out to be softer in the high-x region and harder in the low-x region than the experimental analyses suggest.

Motivation & Objective

• To compute leading-twist parton distribution functions (PDFs) for pseudoscalar mesons (π, K, η) using the Nambu-Jona-Lasinio (NJL) model.
• To apply Pauli-Villars regularization to handle ultraviolet divergences in one-loop quark diagrams and ensure consistency with chiral symmetry.
• To study the impact of next-to-leading order (NLO) QCD evolution on the PDFs, particularly comparing NLO and leading-order (LO) effects.
• To assess the validity of using nonperturbative quark models as initial conditions for QCD evolution by comparing results with phenomenological parametrizations at $Q^2 = 4\,\text{GeV}^2$.
• To investigate whether regularization schemes significantly affect PDF predictions and whether they can be blamed instead of the model itself.

Proposed method

• The NJL model is used to compute one-loop quark diagrams for meson-quark vertex functions, with pseudoscalar mesons treated as quark-antiquark bound states in a chirally broken vacuum.
• Pauli-Villars regularization is applied to the loop integrals, ensuring gauge invariance and satisfying momentum sum rules via the condition $\sum_i c_i f(\Lambda_i^2) = f(0) - f(\Lambda^2) + \Lambda^2 f'(\Lambda^2)$.
• The parton distribution functions are extracted from the matrix elements of quark bilinear operators, with explicit expressions derived for $u_\pi(x), \bar{d}_\pi(x), u_K(x), \bar{s}_K(x), u_\eta(x), s_\eta(x)$ using derivatives of the regularized integrals with respect to meson mass squared.
• The initial PDFs at low scale $Q_0^2$ are approximated by polynomial expansions (up to $x^6$) for π and K, and by a convergent series in $[x(1-x)]^n$ for η, with 30 terms used for numerical accuracy.
• QCD evolution is applied using the GLAP equations, with moments computed analytically via Euler Beta functions for high-precision evolution to $Q^2 = 4\,\text{GeV}^2$, comparing LO and NLO results.
• Normalization and momentum sum rules are enforced via the condition $\langle u_\pi(x) \rangle = \langle \bar{d}_\pi(x) \rangle = 1$, and similar for K and η mesons.

Experimental results

Research questions

• RQ1 How do the parton distribution functions of the pion, kaon, and eta mesons computed in the NJL model compare to phenomenological parametrizations at $Q^2 = 4\,\text{GeV}^2$?
• RQ2 What is the magnitude of NLO QCD evolution corrections relative to LO for these mesons in the NJL framework?
• RQ3 Why do the sea and gluon distributions in the NJL model differ in shape from experimental fits, particularly in the high-x and low-x regions?
• RQ4 Can the Pauli-Villars regularization scheme be trusted to yield consistent and physical PDFs in the NJL model, especially when compared to other regularization methods?
• RQ5 To what extent can the NJL model serve as a reliable nonperturbative initial condition for QCD evolution of PDFs in light mesons?

Key findings

• The valence quark distribution for the pion at $Q^2 = 4\,\text{GeV}^2$ is well described by the polynomial $0.9535 + 0.2664x - 0.2074x^2 - 0.1046x^3 + 0.0190x^4 + 0.0400x^5 - 0.0133x^6$, showing good agreement with experimental analyses.
• NLO QCD evolution effects are found to be surprisingly small compared to LO effects, indicating that the LO approximation captures most of the evolution dynamics in this model.
• The sea quark and gluon distributions in the pion are softer at high $x$ and harder at low $x$ than suggested by phenomenological parametrizations, indicating a significant model-dependent deviation in the tails of the distributions.
• For the η meson, the series expansion in $[x(1-x)]^n$ converges slowly due to a large parameter $M_\eta^2 / (4M_u^2) \approx 0.8$, requiring 30 terms for 0.1% accuracy.
• The model predicts that the $u_\eta(x)$ and $s_\eta(x)$ distributions are given by a series involving logarithmic and power-law terms in $x(1-x)$, with coefficients derived from the meson mass and regulator scale.
• Despite the absence of explicit gluons and sea quarks at the initial scale, the QCD evolution generates nontrivial sea and gluon distributions, though their shape differs from experimental expectations.