$\lambda {\varphi }^4$ Theory II: The Broken Phase Beyond NNNN(NNNN)LO

Abstract

We extend the study of the two-dimensional euclidean ${\varphi }^4$ theory initiated in ref. [1] to the ${\mathbb{Z}}_2$ broken phase. In particular, we compute in perturbation theory up to N${}^4$LO in the quartic coupling the vacuum energy, the vacuum expectation value of $\varphi$ and the mass gap of the theory. We determine the large order behavior of the perturbative series by finding the leading order finite action complex instanton configuration in the ${\mathbb{Z}}_2$ broken phase. Using an appropriate conformal mapping, we then Borel resum the perturbative series. Interestingly enough, the truncated perturbative series for the vacuum energy and the vacuum expectation value of the field is reliable up to the critical coupling where a second order phase transition occurs, and breaks down around the transition for the mass gap. We compute the vacuum energy using also an alternative perturbative series, dubbed exact perturbation theory, that allows us to effectively reach N${}^8$LO in the quartic coupling. In this way we can access the strong coupling region of the ${\mathbb{Z}}_2$ broken phase and test Chang duality by comparing the vacuum energies computed in three different descriptions of the same physical system. This result can also be considered as a confirmation of the Borel summability of the theory. Our results are in very good agreement (and with comparable or better precision) with those obtained by Hamiltonian truncation methods. We also discuss some subtleties related to the physical interpretation of the mass gap and provide evidence that the kink mass can be obtained by analytic continuation from the unbroken to the broken phase.

arXiv commentv1: 36 pages, 11 figures, 8 tables; v2: 37 pages, 12 figures, 8 tables. Added figure 12, minor improvements, typos fixed, matches JHEP version

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Cited by in corpus (23)

• Uniformization and Constructive Analytic Continuation of Taylor Series
• Exploring Hamiltonian Truncation in $\mathbf{d}=2+1$
• Resurgence in the O(4) sigma model
• Conformal and Uniformizing Maps in Borel Analysis
• Analytic resurgence in the O(4) model
• Uniformizing Lee-Yang Singularities
• Schwinger pair production from Padé-Borel reconstruction
• The Two-Loop ${\varphi }^4$ Kink Mass
• A new renormalon in two dimensions
• Renormalization Scheme Dependence, RG Flow and Borel Summability in ${\varphi }^4$ Theories in $d<4$
• Real time lattice correlation functions from differential equations
• Self-Dualities and Renormalization Dependence of the Phase Diagram in 3d $O(N)$ Vector Models
• Transitioning from equal-time to light-front quantization in ${\varphi }_2^4$ theory
• The semiclassical energy density of kinks and solitons
• Cut-Off Kinks
• Critical behavior of the 2d scalar theory: resumming the ${\mathrm{N}}^8\mathrm{LO}$ perturbative mass gap
• The Breakdown of Resummed Perturbation Theory at High Energies
• Tadpoles and vacuum bubbles in light-front quantization
• Quantum phase transitions in a bidimensional $O(N)\times {\mathbb{Z}}_2$ scalar field model
• Perturbative versus Non-Perturbative Renormalization
• Universality, Lee-Yang singularities and series expansions
• Trans-series from condensates in the non-linear sigma model
• Nonperturbative dynamics of (2+1)d ${\varphi }^4$-theory from Hamiltonian truncation