How to understand "analytical continuation" in the context of instantons?

In the recent years there is a renewed understanding of the role of the analytic continuation in path integrals, please see the following work by Witten. The consequences of this understanding are really exciting. They allow for example to understand the Chern-Simons theory for non-integer levels (again Witten).

The analytic continuation techniques are based on the Picard-Lefschetz theory which basically states that a saddle point integral can be associated with a convergent integral over a cycle in a complexified space. The convergent integration cycles are known by the name Lefshetz thimbles.

The integration over complex trajectories was already encountered in coherent states path integrals; please see for example the following work by Stone, Park and Garg. These integrals are formulated on phase space which can be considered as a complexification of the configuration space.

As for the case of the double-well: Please see the following work by Cherman and Ünsal, where they consider a family of analytic continuations of time: $$t = e^{i\alpha} \tau \quad0 \lt \alpha \le \frac{\pi}{2}$$ (The Euclidean path integral corresponds to the special case of $\alpha = \frac{\pi}{2}$). For each value of $\alpha$ they find a complex instanton solution whose corresponding saddle point action is equal to the correct value. The only problem is that the limit to the Minkowskian value $\alpha = 0$ is singular.

For a more thorough explanation on the Picard- Lefschetz theory and more examples, please see the following thesis by: Yuya Tanizaki

TL;DR: OP's title question (v7) about instantons in the Minkowski signature is physically meaningless. It is an irrelevant mathematical detour run amok. The connection to physics/Nature is established via a Wick rotation of the full Euclidean path integral, not bits and pieces thereof. Within the Euclidean path integral, it is possible to consistently expand over Euclidean instantons, but it meaningless to Wick rotation the instanton picture to Minkowski signature.

In more details, let there be given a double-well potential

$$V(x)~=~\frac{1}{2}(x^2-a^2)^2. \tag{A}$$

The Minkowskian and Euclidean formulations are connected via a Wick rotation

$$ t^E e^{i\epsilon}~=~e^{i\frac{\pi}{2}} t^M e^{-i\epsilon}.\tag{B} $$

We have included Feynman's $i\epsilon$-prescription in order to help convergence and avoid branch cuts and singularities. See also this related Phys.SE post.

I) On one hand, the Euclidean partition function/path integral is

$$\begin{align} Z^E~=~&Z(\Delta t^E e^{i\epsilon})\cr ~=~& \langle x_f | \exp\left[-\frac{H \Delta t^E e^{i\epsilon}}{\hbar}\right] | x_i \rangle \cr ~=~&N \int [dx] \exp\left[-\frac{S^E[x]}{\hbar} \right],\end{align}\tag{C} $$

with Euclidean action

$$\begin{align} S^E[x] ~=~&\int_{t^E_i}^{t^E_f} \! dt^E \left[ \frac{e^{-i\epsilon}}{2} \left(\frac{dx}{dt^E}\right)^2+e^{i\epsilon}V(x)\right]\cr ~=~& \int_{t^E_i}^{t^E_f} \! dt^E \frac{e^{-i\epsilon}}{2} \left(\frac{dx}{dt^E}\mp e^{i\epsilon}\sqrt{2V(x)}\right)^2 \cr &\pm \int_{x_i}^{x_f} \! dx ~\sqrt{2V(x)}.\end{align}\tag{D}$$

and real regular kink/anti-kink solution$^1$

$$\begin{align} \frac{dx}{dt^E}\mp e^{i\epsilon}\sqrt{2V(x)}~\approx~&0 \cr ~\Updownarrow~ & \cr x(t^E) ~\approx~&\pm a\tanh(e^{i\epsilon}\Delta t^E). \end{align}\tag{E}$$

Note that a priori space $x$ and time $t^E$ are real coordinates in the path integral (C). To evaluate the Euclidean path integral (C) via the method of steepest descent, we need not complexify space nor time. We are already integrating in the direction of steepest descent!

II) On the other hand, the corresponding Minkowskian partition function/path integral is

$$\begin{align} Z^M~=~&Z(i \Delta t^M e^{-i\epsilon})\cr ~=~& \langle x_f | \exp\left[-\frac{iH \Delta t^M e^{-i\epsilon}}{\hbar} \right] | x_i \rangle\cr ~=~& N \int [dx] \exp\left[\frac{iS^M[x]}{\hbar} \right],\end{align}\tag{F} $$

with Minkowskian action

$$\begin{align} S^M[x]~=~&\int_{t^M_i}^{t^M_f} \! dt^M \left[ \frac{e^{i\epsilon}}{2} \left(\frac{dx}{dt^M}\right)^2-e^{-i\epsilon}V(x)\right]\cr ~=~& \int_{t^M_i}^{t^M_f} \! dt^M \frac{e^{-i\epsilon}}{2} \left(\frac{dx}{dt^M}\mp i e^{i\epsilon}\sqrt{2V(x)}\right)^2 \cr &\pm i \int_{x_i}^{x_f} \! dx ~\sqrt{2V(x)},\end{align}\tag{G}$$

and imaginary singular kink/anti-kink solution

$$\begin{align} \frac{dx}{dt^M}\mp i e^{i\epsilon}\sqrt{2V(x)}~\approx~&0 \cr ~\Updownarrow~ & \cr x(t^M)~\approx~&\pm i a\tan(e^{-i\epsilon}\Delta t^M)\cr ~=~&\pm a\tanh(i e^{-i\epsilon}\Delta t^M). \end{align}\tag{H}$$

It is reassuring that the $i\epsilon$ regularization ensures that the particle starts and ends at the potential minima:

$$ \lim_{\Delta t^M\to \pm^{\prime}\infty} x(t^M)~=~(\pm a) (\pm^{\prime} 1). \tag{I}$$

Unfortunately, that seems to be just about the only nice thing about the solution (H). Note that a priori space $x$ and time $t^M$ are real coordinates in the path integral (F). We cannot directly apply the method of steepest descent to evaluate the Minkowski path integral. We need to deform the integration contour and/or complexify time and space in a consistent way. This is governed by Picard-Lefschetz theory & the Lefschetz thimble. In particular, the role of the imaginary singular kink/anti-kink solution (H) looses its importance, because we cannot expand perturbatively around it in any meaningful way.

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$^1$ The explicit (hyperbolic) tangent solution (E) is an over-simplified toy solution. It obscures the dependence of finite initial (and final) time $t^E_i$ (and $t^E_f$), moduli parameters, and multi-instantons. We refer to the literature for details.