# Gap and Bethe-Salpeter equations in Hamiltonian lattice QCD with Wilson fermions

## Abstract

The dynamical breaking of symmetries in strong-coupling large-N/sub c/ Hamiltonian lattice QCD with Dirac fermions, thoroughly studied by Smit, is reconsidered and clarified here in another language. We use the Bogoliubov-Valatin variational method to formulate the gap equation and obtain the condensate, the dynamical mass, and the shift in energy density between the invariant and the broken vacua. We also solve the Bethe-Salpeter equation and get the quantum numbers, dispersion laws, and Dirac wave functions of the meson spectra in the whole Brillouin zone. The eightfold fermion multiplicity reflects in the meson spectrum through 8N/sub f/ /sup 2/ Goldstone bosons by the breaking U(4N/sub f/)->U(2N/sub f/) x U(2N/sub f/ ) (N/sub f/ = number of flavors). We repeat the program in the presence of a current mass and a Wilson term; we study the stability of the vacuum and see how the high degeneracy of Goldstone bosons is lifted. Although a massless pseudoscalar can be obtained by adjusting the current mass and the Wilson coupling, we find that the vacuum is not chiral degenerate and therefore this massless pseudoscalar is not a Goldstone boson. Although not chiral degenerate, the vacuum satisfies nevertheless the weaker condition of being a flat minimum:more »

- Authors:

- Publication Date:

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

- OSTI Identifier:
- 5764230

- Resource Type:
- Journal Article

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

- Additional Journal Information:
- Journal Volume: 33:10

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; LATTICE FIELD THEORY; BETHE-SALPETER EQUATION; FERMIONS; QUANTUM CHROMODYNAMICS; DIRAC EQUATION; ENERGY DENSITY; GOLDSTONE BOSONS; HAMILTONIANS; MULTIPLICITY; QUANTUM NUMBERS; QUARK MODEL; STRONG-COUPLING MODEL; SYMMETRY BREAKING; VARIATIONAL METHODS; WAVE FUNCTIONS; BOSONS; COMPOSITE MODELS; DIFFERENTIAL EQUATIONS; ELEMENTARY PARTICLES; EQUATIONS; FIELD THEORIES; FUNCTIONS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; PARTIAL DIFFERENTIAL EQUATIONS; PARTICLE MODELS; POSTULATED PARTICLES; QUANTUM FIELD THEORY; QUANTUM OPERATORS; WAVE EQUATIONS; 645400* - High Energy Physics- Field Theory; 645204 - High Energy Physics- Particle Interactions & Properties-Theoretical- Strong Interactions & Properties

### Citation Formats

```
Le Yaouanc, A, Oliver, L, Pene, O, and Raynal, J h.
```*Gap and Bethe-Salpeter equations in Hamiltonian lattice QCD with Wilson fermions*. United States: N. p., 1986.
Web. doi:10.1103/PhysRevD.33.3098.

```
Le Yaouanc, A, Oliver, L, Pene, O, & Raynal, J h.
```*Gap and Bethe-Salpeter equations in Hamiltonian lattice QCD with Wilson fermions*. United States. doi:10.1103/PhysRevD.33.3098.

```
Le Yaouanc, A, Oliver, L, Pene, O, and Raynal, J h. Thu .
"Gap and Bethe-Salpeter equations in Hamiltonian lattice QCD with Wilson fermions". United States. doi:10.1103/PhysRevD.33.3098.
```

```
@article{osti_5764230,
```

title = {Gap and Bethe-Salpeter equations in Hamiltonian lattice QCD with Wilson fermions},

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

abstractNote = {The dynamical breaking of symmetries in strong-coupling large-N/sub c/ Hamiltonian lattice QCD with Dirac fermions, thoroughly studied by Smit, is reconsidered and clarified here in another language. We use the Bogoliubov-Valatin variational method to formulate the gap equation and obtain the condensate, the dynamical mass, and the shift in energy density between the invariant and the broken vacua. We also solve the Bethe-Salpeter equation and get the quantum numbers, dispersion laws, and Dirac wave functions of the meson spectra in the whole Brillouin zone. The eightfold fermion multiplicity reflects in the meson spectrum through 8N/sub f/ /sup 2/ Goldstone bosons by the breaking U(4N/sub f/)->U(2N/sub f/) x U(2N/sub f/ ) (N/sub f/ = number of flavors). We repeat the program in the presence of a current mass and a Wilson term; we study the stability of the vacuum and see how the high degeneracy of Goldstone bosons is lifted. Although a massless pseudoscalar can be obtained by adjusting the current mass and the Wilson coupling, we find that the vacuum is not chiral degenerate and therefore this massless pseudoscalar is not a Goldstone boson. Although not chiral degenerate, the vacuum satisfies nevertheless the weaker condition of being a flat minimum: a finite number of derivatives vanish in the chiral direction. We conjecture that higher derivatives will converge to zero as the coupling decreases, chiral degeneracy being recovered only in the continuum weak-coupling limit. For any value of the Wilson term, the mass of a local baryon is just equal to N/sub c/M/sub dyn/ where M/sub dyn/ is the solution of the gap equation.},

doi = {10.1103/PhysRevD.33.3098},

journal = {Phys. Rev. D; (United States)},

number = ,

volume = 33:10,

place = {United States},

year = {1986},

month = {5}

}