Mass gap of the φ 4 Hamiltonian (4.26) for a range of quartic couplings ¯ λ (x-axis), for a different set of UV scaling dimensions ∆ (different curves, see legend).

Contexts in source publication

Context 1

... contribution to the vacuum energy at second order can then be extracted from eq. (3.14), by truncating the integral in eq. ...

Context 2

... figure 14, we plot precisely these curves ∆E(x, ∆). As always, we subtract the Casimir energy according to (3.1), and we divide the mass gap by ∆ such that we can compare curves of different UV dimensions ∆. ...

Context 3

... plot provides evidence that the curves ∆E(x, ∆) indeed have a finite limit as ∆ → ∞, although we cannot increase ∆ further than ∆ ≈ 10 for computational reasons. Semi-quantitatively, the plot is consistent with a critical coupling of the order of λ/m 2 ≈ 3. Numerically, the convergence rate decreases rapidly for large couplings: this can be seen for example by the large error bars appearing in figure 14. Therefore, analyzing the phase transion in more detail is not feasible in the current set-up, even by going to rather large cutoffs with ∼ 5 · 10 4 states. ...

Context 4

... prescription (3.1) does not always lead to positive energies, contrary to differences of energies of a Hamiltonian, which are positive by construction. The value of e.g. the first excited state E 1 (Λ) − E Ω (Λ − ∆) can indeed be negative, as seems to happen to points with large couplings in figure 14. This would indicate that the vacuum exchanges roles with an excited state (which has different quantum numbers!). ...