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Existence of Nambu-Goldstone Modes in a Lattice Nambu-Jona-Lasinio Model with Multi-Flavor Symmetries


Core Concepts
The authors prove the existence of long-range order and Nambu-Goldstone modes associated with the spontaneous breakdown of SU(3) and SU(2) flavor symmetries in a lattice Nambu-Jona-Lasinio model.
Abstract

The paper studies a lattice Nambu-Jona-Lasinio (NJL) model with SU(2) and SU(3) flavor symmetries of staggered fermions in the Kogut-Susskind Hamiltonian formalism. The key findings are:

  1. For the spatial dimension ν ≥ 5, the SU(3) model shows a long-range order at sufficiently low temperatures. For the SU(2) symmetry, there appears a long-range order in the spatial dimension ν ≥ 3 at sufficiently low temperatures. These results also hold in the ground states.

  2. If a long-range order emerges in this type of models, then there appear gapless excitations above the sector of the infinite-volume ground states, which are the Nambu-Goldstone modes associated with the spontaneous breakdown of the global rotational symmetry of flavors.

  3. The number of the linearly independent Nambu-Goldstone modes is equal to the number of the broken symmetry generators on the Hilbert space constructed from a certain symmetry-breaking infinite-volume ground state.

The authors use the method of reflection positivity and Gaussian domination to prove the existence of long-range order. They also establish the existence of Nambu-Goldstone modes and show that the number of these modes equals the number of broken symmetry generators.

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Deeper Inquiries

How can the results of this paper be extended to lower spatial dimensions or other types of lattice models?

The results of this paper, which demonstrate the existence of long-range order and Nambu-Goldstone modes in a lattice Nambu-Jona-Lasinio (NJL) model with SU(2) and SU(3) flavor symmetries, can potentially be extended to lower spatial dimensions by employing different mathematical techniques or modifying the model parameters. For instance, while the current results hold for spatial dimensions ν ≥ 5 for SU(3) and ν ≥ 3 for SU(2), one could explore the use of renormalization group techniques to analyze the behavior of the system as the dimensionality is reduced. Additionally, one might consider the introduction of different types of interactions or coupling constants that could stabilize long-range order in lower dimensions. Moreover, the findings could be applied to other lattice models, such as those involving different gauge groups or fermionic representations. By adapting the methods of reflection positivity and Gaussian domination used in this paper, researchers could investigate similar phenomena in models with different symmetry groups or in the presence of gauge fields. This could lead to a broader understanding of symmetry breaking and the emergence of Nambu-Goldstone modes across various lattice frameworks.

What are the implications of the spontaneous breakdown of flavor symmetry for the physical interpretation of the NJL model as an effective theory of QCD?

The spontaneous breakdown of flavor symmetry in the NJL model has significant implications for its interpretation as an effective theory of quantum chromodynamics (QCD). In QCD, the strong interactions among quarks and gluons can lead to phenomena such as confinement and mass generation for hadrons. The NJL model, being a low-energy effective theory, captures essential features of these dynamics through its four-fermion interactions, which mimic the effects of gluon exchanges in QCD. The results indicating that the SU(3) flavor symmetry is spontaneously broken suggest that the NJL model can effectively describe the mass generation of quarks and the formation of chiral condensates, which are crucial for understanding the emergence of hadronic masses. This symmetry breaking leads to the existence of Nambu-Goldstone modes, which correspond to the low-energy excitations associated with the broken symmetries. These modes are essential for understanding the dynamics of pions and other light mesons in the context of chiral perturbation theory, thereby linking the NJL model to the physical properties of QCD. Furthermore, the findings highlight the importance of lattice formulations in studying non-perturbative aspects of QCD, as they provide a rigorous mathematical framework to analyze symmetry breaking and its consequences. This enhances our understanding of the phase structure of QCD and the conditions under which flavor symmetries may be broken in realistic scenarios.

Are there any connections between the Nambu-Goldstone modes found in this lattice model and the low-energy excitations in continuum gauge theories like QCD?

Yes, there are significant connections between the Nambu-Goldstone modes identified in the lattice NJL model and the low-energy excitations observed in continuum gauge theories such as QCD. In both frameworks, the emergence of Nambu-Goldstone modes is a direct consequence of spontaneous symmetry breaking. In the context of the NJL model, the breaking of flavor symmetry leads to the appearance of massless excitations, which correspond to the Nambu-Goldstone bosons associated with the broken symmetry generators. In continuum gauge theories like QCD, similar low-energy excitations manifest as pions and other light mesons, which are understood as the Goldstone bosons resulting from the spontaneous breaking of chiral symmetry. The relationship between these modes in the NJL model and the physical pions in QCD is particularly relevant, as the NJL model serves as a simplified effective theory that captures the essential dynamics of chiral symmetry breaking in QCD. Moreover, the presence of Nambu-Goldstone modes in both the lattice NJL model and continuum QCD indicates a universal behavior in the low-energy spectrum of systems exhibiting spontaneous symmetry breaking. This connection allows for the application of insights gained from lattice studies to better understand the low-energy dynamics of QCD, including the role of pions in mediating interactions between nucleons and the implications for hadron physics. Thus, the study of Nambu-Goldstone modes in lattice models provides valuable information that can enhance our understanding of the non-perturbative aspects of continuum gauge theories like QCD.
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