Abstract
We extend the study of the two-dimensional euclidean ϕ4 theory initiated in ref. [1] to the ℤ2 broken phase. In particular, we compute in perturbation theory up to N4LO in the quartic coupling the vacuum energy, the vacuum expectation value of ϕ 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 ℤ2 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 N8LO in the quartic coupling. In this way we can access the strong coupling region of the ℤ2 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.
Highlights
Ge scalar field theories [1]
We report the numerical results obtained from the truncated perturbative series and from its Borel resummation
We have considered conformal mapping and PadeBorel approximants methods to obtain a numerical estimate of the Borel function
Summary
Ge scalar field theories [1]. These include the 2d φ4 theory with both m2 > 0, known already to be Borel resummable for parametrically small couplings [26], and m2 < 0. In the Z2 broken phase the only description that can be explored by resumming the perturbative series and without encountering phase transitions is the part of the weakly coupled branch with 0 ≤ g ≤ gc(w). This is the region we will mostly focus on, we will explore other sectors of the phase diagram. In ref. [1] we instead focused on the region 0 ≤ g ≤ gc in the black line of figure 1
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