[Paper Review] Quark Distribution in Pion within Instanton Liquid Model

A. E. Dorokhov, Lauro Tomio|ArXiv.org|Mar 12, 1998

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper derives the leading-twist valence-quark distribution function in the pion using the instanton liquid model, incorporating momentum-dependent quark masses and a gauge-invariant quark-pion vertex. It provides an analytic expression for the distribution and achieves reasonable agreement with phenomenological data after QCD evolution to higher momentum scales.

ABSTRACT

The leading-twist valence-quark distribution function in the pion is obtained at a low normalization scale of an order of the inverse average size of an instanton $ρ_c$. The momentum dependent quark mass and the quark-pion vertex are constructed in the framework of the instanton liquid model, using a gauge invariant approach. The parameters of instanton vacuum, the effective instanton radius and quark mass, are related to the vacuum expectation values of the lowest dimension quark-gluon operators and to the pion low energy observables. An analytic expression for the quark distribution function in the pion for a general vertex function is derived. The results are QCD evolved to higher momentum-transfer values, and reasonable agreement with phenomenological analyzes of the data on parton distributions for the pion is found.

Motivation & Objective

• To derive the valence-quark distribution function in the pion at low normalization scale using the instanton liquid model.
• To construct a momentum-dependent quark mass and a gauge-invariant quark-pion vertex within the instanton vacuum framework.
• To relate model parameters—effective instanton radius and quark mass—to QCD vacuum condensates and pion low-energy observables.
• To provide an analytic expression for the quark distribution function valid for a general vertex function.
• To evolve the distribution via QCD evolution and compare with experimental data on parton distributions.

Proposed method

• Employ the instanton liquid model as a nonperturbative QCD framework for the vacuum structure.
• Implement a gauge-invariant formulation to define the momentum-dependent quark mass and quark-pion vertex function.
• Relate the effective instanton radius and quark mass to vacuum expectation values of dimension-3 and dimension-5 quark-gluon operators.
• Derive an analytic expression for the leading-twist quark distribution function in the pion using the general vertex function.
• Apply QCD evolution equations to evolve the distribution from the low normalization scale to higher momentum-transfer values.
• Compare the evolved distribution with phenomenological parametrizations of pion parton distribution functions.

Experimental results

Research questions

• RQ1 How can the valence-quark distribution function in the pion be derived within a nonperturbative QCD framework like the instanton liquid model?
• RQ2 What is the role of momentum-dependent quark mass and gauge-invariant quark-pion vertex in determining the pion's parton structure?
• RQ3 How do the model parameters (instanton radius and quark mass) relate to known QCD vacuum condensates and pion observables?
• RQ4 Can an analytic expression for the quark distribution be obtained for a general vertex function in this model?
• RQ5 To what extent does the QCD-evolved distribution from this model agree with phenomenological data on pion parton distributions?

Key findings

• The paper derives an analytic expression for the pion's leading-twist valence-quark distribution function in the instanton liquid model.
• The momentum-dependent quark mass and gauge-invariant quark-pion vertex are consistently constructed within the framework.
• The effective instanton radius and quark mass are related to vacuum expectation values of quark-gluon operators and pion low-energy constants.
• The model parameters are fixed by matching to known QCD condensates and pion observables such as the pion decay constant.
• After QCD evolution to higher momentum scales, the predicted quark distribution shows reasonable agreement with phenomenological parametrizations of pion parton distributions.
• The results support the validity of the instanton liquid model as a framework for describing the nonperturbative structure of the pion at low energy.