Quantum fluctuation energies over a spatially inhomogeneous field background in a chiral soliton model

TL;DR

This work develops and applies a spectral phase-shift method to compute quantum fluctuation energies of quarks on spatially inhomogeneous chiral backgrounds within a linear sigma model. By solving hedgehog-anchored Dirac equations, constructing grand-spin-parity channels, and evaluating scattering phase shifts, the authors obtain a finite one-loop vacuum energy after Born-subtraction renormalization, with explicit Γ_2 and Γ_4 counterterms. The results show that continuum and discrete-quark contributions yield a substantial, predominantly negative vacuum energy that competes with the classical soliton energy, underscoring the significance of quantum fluctuations in solitonic hadron descriptions. The methodology provides a self-consistent framework for exploring hadron structure and the QCD phase diagram in nonuniform vacua, with potential extensions to finite temperature and dense quark matter.

Abstract

Based on chiral soliton models, the quantum fluctuation energies of quarks over a spatially inhomogeneous meson field background have been thoroughly studied. We have used a systematic calculation scheme initiated by Schwinger, in which the loop quantum fluctuation energies are evaluated by a nontrivial level summation over the eigenvalue spectrum of the effective Hamiltonian of the system. The effective Hamiltonian can be constructed by one loop effective action of fluctuations of quarks over a static chiral soliton field background. The corresponding Dirac equation is obtained. In a static and spatially spherical case and by the hedgehog ansatz the radial part and the angular part of the grand spin of the wave function for the Dirac equation can be separated. Due to the soliton background the eigenvalue spectrum are distorted. The scattering phase shift can be determined by solve the radial equations at different momentum. The density of states in momentum space can be derived. The effective Hamiltonian has been diagonalized in a Hilbert space where the eigenfunctions are labeled by the parity, grand spin and energy. The renormalization scheme can be carried out by a Born subtraction of the phase shift and the compensating Feynman diagram renormalization. Finally the finite quantum fluctuation energies over chiral soliton background at different parities and grand spins have been numerically evaluated, compared and discussed.

Figures (8)

• Figure 1: The solutions $f_{1}(r),\,g_{1}(r)$, $\sigma (r)$ and $\pi (r)$ of the chiral soliton equations.
• Figure 2: The phase shift $\delta_{G}$ as a function of momentum $k$ for both parities $\Pi=\pm(-1)^{G}$ and both signs $\omega=\pm$ of the energy.
• Figure 3: The subtracted phase shift function $\bar{\delta}_0(k)$ with respect to momentum $k$ in the case of $\Pi=\pm$ and $\omega=\pm$.
• Figure 4: The subtracted phase shift $\bar{\delta}_G$ as a function of momentum $k$ for $G = 1,\, 2,\, 3,\,4$ in the case of $\Pi=+$ and $\omega=+$.
• Figure 5: The subtracted phase shift $\bar{\delta}_G$ as a function of momentum $k$ for $G = 1,\, 2,\, 3,\,4$ in the case of $\Pi=-$ and $\omega=+$.