Explore the Origin of Spontaneous Symmetry Breaking from Adaptive Perturbation Method

Document Type : Regular article


1 Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, US

2 School of Physics, Sun Yat-Sen University, Guangzhou 510275, Guangdong, China

3 Key Laboratory of Atomic and Subatomic Structure and Quantum Control, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, Guangdong, China


Spontaneous symmetry breaking occurs when the underlying laws of a physical system are symmetric, but the vacuum state chosen by the system is not. The (3+1)d $\phi^4$ theory is relatively simple compared to other more complex theories, making it a good starting point for investigating the origin of non-trivial vacua. The adaptive perturbation method is a technique used to handle strongly coupled systems. The study of strongly correlated systems is useful in testing holography. It has been successful in strongly coupled QM and is being generalized to scalar field theory to analyze the system in the strong-coupling regime. The unperturbed Hamiltonian does not commute with the usual number operator. However, the quantized scalar field admits a plane-wave expansion when acting on the vacuum. While quantizing the scalar field theory, the field can be expanded into plane-wave modes, making the calculations more tractable. However, the Lorentz symmetry, which describes how physical laws remain the same under certain spacetime transformations, might not be manifest in this approach. The proposed elegant resummation of Feynman diagrams aims to restore the Lorentz symmetry in the calculations. The results obtained using this method are compared with numerical solutions for specific values of the coupling constant $\lambda = 1, 2, 4, 8, 16$. Finally, we find evidence for quantum triviality, where self-consistency of the theory in the UV requires $\lambda = 0$. This result implies that the $\phi^4$ theory alone does not experience SSB, and the $\langle \phi\rangle = 0$ phase is protected under the RG-flow by a boundary of Gaussian fixed-points.


Main Subjects

[1]J. Goldstone, “Field Theories with Superconductor Solutions,” Nuovo Cim. 19, 154-164 (1961). DOI: 10.1007/BF02812722

[2]M. Luscher and P. Weisz, “Scaling Laws and Triviality Bounds in the Lattice phi**4 Theory. 1. One Component Model in the Symmetric Phase,” Nucl. Phys. B 290, 25-60 (1987). DOI: 10.1016/0550-3213(87)90177-5

[3]S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of Spontaneous Symmetry Breaking,” Phys. Rev. D 7, 1888-1910 (1973). DOI: 10.1103/PhysRevD.7.1888

[4]M. Aizenman, “Proof of the Triviality of phi**4 in D-Dimensions Field Theory and Some Mean Field Features of Ising Models for D>4,” Phys. Rev. Lett. 47, 1-4 (1981). DOI: 10.1103/PhysRevLett.47.1

[5]M. Weinstein, “Adaptive perturbation theory. I. Quantum mechanics,” [arXiv:hepth/0510159 [hep-th]].

[6]M. Weinstein, “Adaptive perturbation theory: Quantum mechanics and field theory,” Nucl. Phys. B Proc. Suppl. 161, 238-247 (2006). DOI: 10.1016/j.nuclphysbps.2006.08.059 [arXiv:hep-th/0510160 [hep-th]].

[7]C. T. Ma, “Adaptive Perturbation Method in Quantum Mechanics,” IOP SciNotes 2, no.3, 035202 (2021). DOI: 10.1088/2633-1357/ac12ba [arXiv:1911.08211 [quant-ph]].

[8]C. T. Ma, “Second-Order Perturbation in Adaptive Perturbation Method,” JHAP 2, no.2, 37 (2022). DOI: 10.22128/jhap.2022.586.1034 [arXiv:2004.00842 [hep-th]].
[9]C. T. Ma, “Accurate Study from Adaptive Perturbation Method,” Int. J. Mod. Phys. A 36, no.04, 2150029 (2021). DOI: 10.1142/S0217751X21500299 [arXiv:2007.09080 [hepth]].

[10]J. S. Schwinger, “On the Green’s functions of quantized fields. 1.,” Proc. Nat. Acad. Sci. 37, 452-455 (1951). DOI: 10.1073/pnas.37.7.452

[11]M. Gell-Mann and F. E. Low, “Quantum electrodynamics at small distances,” Phys. Rev. 95, 1300-1312 (1954). DOI: 10.1103/PhysRev.95.1300

Articles in Press, Accepted Manuscript
Available Online from 20 January 2024
  • Receive Date: 07 November 2023
  • Revise Date: 17 January 2024
  • Accept Date: 20 January 2024