[Paper Review] Study of two--pion light--cone distribution amplitudes in the resonance region and at low energies

M. V. Polyakov|arXiv (Cornell University)|Sep 22, 1998

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper derives dispersion relations that express two-pion light-cone distribution amplitudes (DA's) across a broad range of two-pion invariant masses, including the resonance region, using ππ scattering phase shifts and low-energy subtraction constants. Using an effective quark-pion chiral Lagrangian from the instanton vacuum, it quantitatively determines these constants, enabling complete determination of both chirally even and odd two-pion DA's, and connects resonance DA's to two-pion DA's near resonance masses, with applications to hard two-pion production and extraction of meson structure from ππ mass spectra.

ABSTRACT

The dispersion relations which allow to express the two-pion light-cone distribution amplitudes (DA's) in a wide range of two-pion invariant masses (including the resonance region) in terms of $\\pi\\pi$ scattering phase shifts and of a few low-energy subtraction constants are derived. The corresponding subtraction constants can be determined in the low energy region--where the effective chiral theory is applicable. In this region we use an effective quark-pion chiral lagrangian derived from the instanton vacuum to make quantitative estimates of the subtraction constant and hence to fix completely two-pion DA's (both chirally even and odd) in a wide range of two-pion invariant masses including the resonance one. We show that the distribution amplitudes of the resonances ($\ ho, f_2, \ ho_3$, etc.) can be expressed in terms of the two-pion DA's at invariant mass of two pions close to the mass of resonance. The quantitative estimates of the resonance DA's (chirally even and odd) at low normalization point are made. Certain soft pion theorems relating the two-pion DA to the pion distribution amplitude are proven. Applications of $2\\pi$DA's for a hard production of two pions in the reaction resonance region are discussed. In particular, we argue that studying the shape of \\pi\\pi mass spectra (not the absolute cross section!) in diffractive pions production experiments one can extract the deviation of the meson (\\pi and non-perturbative information about structure of mesons.

Motivation & Objective

• To establish a framework for computing two-pion light-cone distribution amplitudes (DA's) across a wide range of two-pion invariant masses, including the resonance region.
• To determine low-energy subtraction constants using an effective quark-pion chiral Lagrangian derived from the instanton vacuum, enabling quantitative estimates of DA's in the low-energy domain.
• To relate the distribution amplitudes of resonances (e.g., ρ, f₂, ρ₃) to the two-pion DA's evaluated near their respective masses.
• To explore applications of two-pion DA's in hard exclusive processes, particularly in diffractive two-pion production.
• To demonstrate that the shape of ππ mass spectra—rather than absolute cross sections—can reveal non-perturbative meson structure and deviations from pion DA behavior.

Proposed method

• Derives dispersion relations that express two-pion light-cone DA's in terms of ππ scattering phase shifts and low-energy subtraction constants.
• Applies an effective quark-pion chiral Lagrangian based on the instanton vacuum to compute the subtraction constants quantitatively in the low-energy region.
• Uses the derived subtraction constants to fix both chirally even and odd two-pion DA's across the full range of two-pion invariant masses, including resonances.
• Establishes a mapping between resonance DA's and two-pion DA's evaluated at invariant masses near the resonance mass.
• Proves soft pion theorems that relate two-pion DA's to the pion distribution amplitude.
• Analyzes the implications for hard exclusive two-pion production, emphasizing the sensitivity of ππ mass spectra to non-perturbative meson structure.

Experimental results

Research questions

• RQ1 How can two-pion light-cone distribution amplitudes be consistently described across the resonance region and low-energy domain using dispersion relations and phase shifts?
• RQ2 What is the quantitative value of the low-energy subtraction constants in the dispersion representation of two-pion DA's, and how can they be computed from effective field theory?
• RQ3 To what extent can the distribution amplitudes of resonances like ρ, f₂, and ρ₃ be reconstructed from two-pion DA's near their respective masses?
• RQ4 What constraints do soft pion theorems impose on the relation between two-pion DA's and the pion distribution amplitude?
• RQ5 Can the shape of the ππ mass spectrum in diffractive two-pion production experiments reveal non-perturbative meson structure beyond the pion DA?

Key findings

• The subtraction constants in the dispersion representation of two-pion DA's are quantitatively determined using an effective quark-pion chiral Lagrangian derived from the instanton vacuum.
• Both chirally even and odd two-particle distribution amplitudes are fully fixed across a wide range of two-pion invariant masses, including the resonance region.
• The distribution amplitudes of resonances such as ρ, f₂, and ρ₃ are expressed in terms of two-pion DA's evaluated at invariant masses close to the resonance mass.
• Soft pion theorems are proven that relate the two-pion DA to the pion distribution amplitude, providing consistency checks and constraints.
• The shape of the ππ mass spectrum in diffractive two-pion production experiments is shown to encode non-perturbative information about meson structure, even without measuring absolute cross sections.
• Quantitative estimates of resonance DA's at low normalization points are obtained, enabling phenomenological applications in hard exclusive processes.