Recently, a novel 4D Einstein–Gauss–Bonnet gravity has been proposed by Glavan and Lin (2020 Phys. Rev. Lett. 124 081301) by rescaling the coupling α → α / ( D − 4 ) and taking the limit D → 4 at the level of equations of motion. This prescription, though was shown to bring non-trivial effects for some spacetimes with particular symmetries, remains mysterious and calls for scrutiny. Indeed, there is no continuous way to take the limit D → 4 in the higher D -dimensional equations of motion because the tensor indices depend on the spacetime dimension and behave discretely. On the other hand, if one works with 4D spacetime indices the contribution corresponding to the Gauss–Bonnet term vanishes identically in the equations of motion. A necessary condition (but may not be sufficient) for this procedure to work is that there is an embedding of the 4D spacetime into the higher D -dimensional spacetime so that the equations in the latter can be properly interpreted after taking the limit. In this note, working with 2D Einstein gravity, we show several subtleties when applying the method used in (2020 Phys. Rev. Lett. 124 081301).

We present the calculation of the Feynman path integral in real time for tunneling in quantum mechanics and field theory, including the first quantum corrections. For this purpose, we use the well-known fact that Euclidean saddle points in terms of real fields can be analytically continued to complex saddles of the action in Minkowski space. We also use Picard-Lefschetz theory in order to determine the middle-dimensional steepest- descent surface in the complex field space, constructed from Lefschetz thimbles, on which the path integral is to be performed. As an alternative to extracting the decay rate from the imaginary part of the ground-state energy of the false vacuum, we use the optical theorem in order to derive it from the real-time amplitude for forward scattering. While this amplitude may in principle be obtained by analytic continuation of its Euclidean counterpart, we work out in detail how it can be computed to one-loop order at the level of the path integral, i.e. evaluating the Gaußian integrals of fluctuations about the relevant complex saddle points. To that effect, we show how the eigenvalues and eigenfunctions on a thimble can be obtained by analytic continuation of the Euclidean eigensystem, and we determine the path-integral measure on thimbles. This way, using real-time methods, we recover the one-loop result by Callan and Coleman for the decay rate. We finally demonstrate our real-time methods explicitly, including the construction of the eigensystem of the complex saddle, on the archetypical example of tunneling in a quasi-degenerate quartic potential.

The scalar field theory with potential $V(\phi )=\frac{1}{2}{m}^2{\phi }^2-\frac{1}{4}g{\phi }^4$ ($g>0$) is ill defined as a Hermitian theory but in a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it is well defined, and it has a positive real energy spectrum for the case of spacetime dimension $D=1$. While the methods used in the literature do not easily generalize to quantum field theory, in this paper the path-integral representation of a $\mathcal{P}\mathcal{T}$-symmetric $-g{\phi }^4$ theory is shown to provide a unified formulation for general $D$. A new conjectural relation between the Euclidean partition functions $Z^{\mathcal{P}\mathcal{T}}(g)$ of the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric theory and $Z_{\text{Herm}}(\lambda )$ of the $\lambda {\phi }^4$ ($\lambda >0$) Hermitian theory is proposed: $\log Z^{\mathcal{P}\mathcal{T}}(g)=\frac{1}{2}\log Z_{\text{Herm}}(-g+\mathrm{i}{0}^{+})+\frac{1}{2}\log Z_{\text{Herm}}(-g-\mathrm{i}{0}^{+})$. This relation ensures a real energy spectrum for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric $-g{\phi }^4$ field theory. A closely related relation is rigorously valid in $D=0$. For $D=1$, using a semiclassical evaluation of $Z^{\mathcal{P}\mathcal{T}}(g)$, this relation is verified by comparing the imaginary parts of the ground-state energy $E_0^{\mathcal{P}\mathcal{T}}(g)$ (before cancellation) and $E_{0,\mathrm{Herm}}(-g\pm \mathrm{i}{0}^{+})$.

The Bödeker-Moore thermal friction [1] is usually used to determine whether or not a bubble wall can run away. However, the friction on the wall is not necessarily a monotonous function of the wall velocity and could have a maximum before it reaches the Bödeker-Moore limit. In this paper, we compare the maximal hydrodynamic obstruction, a frictional force that exists in local thermal equilibrium, and the Bödeker-Moore thermal friction. We study the former in a fully analytical way, clarifying its physical origin and providing a simple expression for its corresponding critical phase transition strength above which the driving force cannot be balanced out by the maximal hydrodynamic obstruction. We find that for large parameter space, the maximal hydrodynamic obstruction is larger than the Bödeker-Moore thermal friction, indicating that the conventional criterion for the runaway behavior of the bubble wall may have to be modified. We also explain how to apply efficiently the modified criterion to particle physics models and discuss possible limitations of the analysis carried out in this paper.

Determining the bubble wall velocity in first-order phase transitions is a challenging task, requiring the solution of (coupled) equations of motion for the scalar field and Boltzmann equations for the particles in the plasma. The collision terms appearing in the Boltzmann equation present a prominent source of uncertainty as they are often known only at leading log accuracy. In this paper, we derive upper and lower bounds on the wall velocity, corresponding to the local thermal equilibrium and ballistic limits. These bounds are completely independent of the collision terms. For the ballistic approximation, we argue that the inhomogeneous plasma temperature and velocity distributions across the bubble wall should be taken into account. This way, the hydrodynamic obstruction previously observed in local thermal equilibrium is also present for the ballistic approximation. This is essential for the ballistic approximation to provide a lower bound on the wall velocity. We use a model-independent approach to study the behaviour of the limiting wall velocities as a function of a few generic parameters, and we test our developments in the singlet extended Standard Model.