Double-Scaling Limit of a Broken Symmetry Quantum Field Theory

Abstract

The Ising limit of a conventional Hermitian parity-symmetric scalar quantum field theory is a correlated limit in which two bare Lagrangian parameters, the coupling constant $g$ and the {\it negative} mass squared $-m^2$, both approach infinity with the ratio $-m^2/g=\alpha >0$ held fixed. In this limit the renormalized mass of the asymptotic theory is finite. Moreover, the limiting theory exhibits universal properties. For a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Lagrangian lacking parity symmetry, whose interaction term has the form $-g(i\varphi {)}^N/N$, the renormalized mass diverges in this correlated limit. Nevertheless, the asymptotic theory still has interesting properties. For example, the one-point Green's function approaches the value $-i{\alpha }^{1/(N-2)}$ independently of the space-time dimension $D$ for $D<2$. Moreover, while the Ising limit of a parity-symmetric quantum field theory is dominated by a dilute instanton gas, the corresponding correlated limit of a $\mathcal{P}\mathcal{T}$-symmetric quantum field theory without parity symmetry is dominated by a constant-field configuration with corrections determined by a weak-coupling expansion in which the expansion parameter (the amplitude of the vertices of the graphs in this expansion) is proportional to an inverse power of $g$. We thus observe a weak-coupling/strong-coupling duality in that while the Ising limit is a strong-coupling limit of the quantum field theory, the expansion about this limit takes the form of a conventional weak-coupling expansion. A possible generalization of the Ising limit to dimensions $D<4$ is briefly discussed.

27 pages, 2 figures

References in corpus (7)

• PT -symmetric harmonic oscillators
• A new PT symmetric complex Hamiltonian with a real spectra
• Solution of Schwinger-Dyson Equations for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theory
• On the relation between Stokes multipliers and the T-Q systems of conformal field theory
• Variational Ansatz for PT-Symmetric Quantum Mechanics
• Complex Square Well --- A New Exactly Solvable Quantum Mechanical Model
• The energy spectrum of complex periodic potentials of the Kronig-Penney type

Cited by in corpus (4)

• Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics
• PT/Non-PT Symmetric and Non-Hermitian Poschl-Teller-Like Solvable Potentials via Nikiforov-Uvarov Method
• Families of particles with different masses in PT-symmetric quantum field theory
• $T\bar{T}$ deformation on non-Hermitian two coupled SYK model