# Spontaneous breaking of chiral symmetry for confining potentials

## Abstract

Using the Bogoliubov-Valatin variational method, we show that the chiral-invariant vacuum is unstable for a color, fourth-component vector powerlike potential rsup..cap alpha.. (0<..cap alpha..<3) independently of the strength of the coupling constant. The fermion self-energy is negative and dominates over the positive potential energy, destabilizing the vacuum by psi-barpsi pair condensation. This self-energy is finite but infrared singular, reflecting the behavior of the potential at large distances. We give an analytical proof of the fact that the energy of the unbroken vacuum is not minimum. The proof extends to logarithmic potentials as ..cap alpha -->..0, but breaks down for ..cap alpha..> or =3 (number of spatial dimensions) due to severe infrared singularities. If the confining potential possesses a spin-spin piece, there are critical values of its strength, depending on the power ..cap alpha.., beyond which the stability of the chiral-invariant vacuum is restored. In the case of the harmonic oscillator ..cap alpha.. = 2, the gap equation reduces to a nonlinear second-order differential equation. We find (besides the usual chiral degeneracy) an infinite number of solutions breaking chiral symmetry, higher in energy as the number of their nodes increases. We compute the expectation value of psi-barpsi and the mass gapmore »

- Authors:

- Publication Date:

- Research Org.:
- Laboratoire de Physique Theorique et Hautes Energies, Universite de Paris-Sud, Btiment 211, 91405 Orsay, France

- OSTI Identifier:
- 6875230

- Resource Type:
- Journal Article

- Journal Name:
- Phys. Rev. D; (United States)

- Additional Journal Information:
- Journal Volume: 29:6

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; FERMIONS; SELF-ENERGY; QUANTUM CHROMODYNAMICS; CHIRAL SYMMETRY; SYMMETRY BREAKING; COLOR MODEL; COUPLING CONSTANTS; GLUONS; INFRARED DIVERGENCES; POTENTIALS; QUARK-QUARK INTERACTIONS; SPIN; VACUUM STATES; VARIATIONAL METHODS; ANGULAR MOMENTUM; COMPOSITE MODELS; ELEMENTARY PARTICLES; ENERGY; FIELD THEORIES; INTERACTIONS; MATHEMATICAL MODELS; PARTICLE INTERACTIONS; PARTICLE MODELS; PARTICLE PROPERTIES; POSTULATED PARTICLES; QUANTUM FIELD THEORY; QUARK MODEL; SYMMETRY; 645400* - High Energy Physics- Field Theory

### Citation Formats

```
Le Yaouanc, A, Oliver, L, Pene, O, and Raynal, J.
```*Spontaneous breaking of chiral symmetry for confining potentials*. United States: N. p., 1984.
Web. doi:10.1103/PhysRevD.29.1233.

```
Le Yaouanc, A, Oliver, L, Pene, O, & Raynal, J.
```*Spontaneous breaking of chiral symmetry for confining potentials*. United States. https://doi.org/10.1103/PhysRevD.29.1233

```
Le Yaouanc, A, Oliver, L, Pene, O, and Raynal, J. Thu .
"Spontaneous breaking of chiral symmetry for confining potentials". United States. https://doi.org/10.1103/PhysRevD.29.1233.
```

```
@article{osti_6875230,
```

title = {Spontaneous breaking of chiral symmetry for confining potentials},

author = {Le Yaouanc, A and Oliver, L and Pene, O and Raynal, J},

abstractNote = {Using the Bogoliubov-Valatin variational method, we show that the chiral-invariant vacuum is unstable for a color, fourth-component vector powerlike potential rsup..cap alpha.. (0<..cap alpha..<3) independently of the strength of the coupling constant. The fermion self-energy is negative and dominates over the positive potential energy, destabilizing the vacuum by psi-barpsi pair condensation. This self-energy is finite but infrared singular, reflecting the behavior of the potential at large distances. We give an analytical proof of the fact that the energy of the unbroken vacuum is not minimum. The proof extends to logarithmic potentials as ..cap alpha -->..0, but breaks down for ..cap alpha..> or =3 (number of spatial dimensions) due to severe infrared singularities. If the confining potential possesses a spin-spin piece, there are critical values of its strength, depending on the power ..cap alpha.., beyond which the stability of the chiral-invariant vacuum is restored. In the case of the harmonic oscillator ..cap alpha.. = 2, the gap equation reduces to a nonlinear second-order differential equation. We find (besides the usual chiral degeneracy) an infinite number of solutions breaking chiral symmetry, higher in energy as the number of their nodes increases. We compute the expectation value of psi-barpsi and the mass gap for the new vacuum, the lowest solution in energy. The infrared singularity of the massless fermion self-energy is removed for the stable broken solution.},

doi = {10.1103/PhysRevD.29.1233},

url = {https://www.osti.gov/biblio/6875230},
journal = {Phys. Rev. D; (United States)},

number = ,

volume = 29:6,

place = {United States},

year = {1984},

month = {3}

}