Exploring high multiplicity amplitudes: The quantum mechanics analogue of the spontaneously broken case

Calculations of high multiplicity Higgs amplitudes exhibit a rapid growth that may signal an end of perturbative behavior or even the need for new physics phenomena. As a step toward this problem we consider the quantum mechanical equivalent of $1\to n$ scattering amplitudes in a spontaneously broken ${\varphi }^4$-theory by extending our previous results on the quartic oscillator with a single minimum [Phys. Rev. D 98, 096007 (2018)] to transitions $⟨n|\widehat{x}|0⟩$ in the symmetric double-well potential with quartic coupling $\lambda$. Using recursive techniques to high order in perturbation theory, we argue that these transitions are of exponential form $⟨n|\widehat{x}|0⟩\sim \exp (F(\lambda n)/\lambda )$ in the limit of large $n$ and $\lambda n$ fixed. We apply the methods of “exact perturbation theory” put forward by Serone et al. in [Phys. Rev. D 96, 021701 (2017); J. High Energy Phys. 05 (2017) 056] to obtain the exponent $F$ and investigate its structure in the regime where tree-level perturbation theory violates unitarity constraints. We find that the resummed exponent is in agreement with unitarity and rigorous bounds derived by Bachas [Nucl. Phys. B377, 622 (1992)].

Figure 1

Tree-level contribution to the holy grail function $F$ in the double scaling limit $n\to \infty$, $\epsilon =\lambda n=\text{const}$. It exhibits a minimum at $\epsilon =4\sqrt{2}$ and a root at $\epsilon =4\sqrt{2}e$.

Figure 2

Leading order holy grail function $F=F^{\mathrm{tree}}+F_0$ in the double scaling limit $n\to \infty$ with $\epsilon =\lambda n$ fixed. The label denotes the highest coefficient in $\epsilon$ that is included in the series representation of $F_0$.

Figure 3

Diagonal Borel-Padé approximants of the holy grail function $F$ in the double scaling limit $n\to \infty ,\epsilon =\lambda n=\text{const}$. Higher order corrections in the $1/n$-expansion are neglected. $F$ is obtained with the ansatz of exact perturbation theory using the auxiliary potential (3.10).

Figure 4

Value of the different approximants of the holy grail function $F$ in the double scaling limit $n\to \infty ,\epsilon =\lambda n=\text{const}$ at the minimum, $\epsilon =4\sqrt{2}$, (left panel) and at the root, $\epsilon =4\sqrt{2}e$, (right panel) of the tree-level holy grail function. The $k$ denotes the number of coefficients of the power series of $F$ taken into account in the corresponding approximation scheme. The different approximation schemes shown are Padé, Borel-Padé (BP), Meijer G (MG) and Shafer (Sh), all of which are discussed in Appendix pp2. The label $\alpha$-Exp corresponds to the scheme proposed in Appendix pp3.

Figure 5

Different approximants (to highest available order) of the holy grail function $F$ in the double scaling limit $n\to \infty ,\epsilon =\lambda n=\text{const}$ (as in Fig. 3) compared to WKB estimates [58] and a rigorous bound derived by Bachas [59], labeled WKB and B, respectively. The other labels are as in Fig. 4.

Figure 6

Sign of each coefficient of ${\widehat{F}}_0(\epsilon ,{\epsilon }_0)=\sum _k{\widehat{F}}_{0,k}({\epsilon }_0){\epsilon }^k$ for different values of ${\epsilon }_0$ shown on the vertical axes. Blue dots denote a positive while red dots denote a negative sign, respectively. The left panel corresponds to $v_2=1$ whereas the right panel is for $v_2=1/2$. All input parameters are normalized to the mass, $m^2=1$. Note the different scales for ${\epsilon }_0$.

Figure 7

Diagonal Borel-Padé approximants of the holy grail function $F$ with an estimated large order behavior as discussed in this Appendix using Eq. (c3). We can see that the higher approximants are already nicely converged. Note, however, that this does not represent the full error. There is an additional systematic uncertainty at small $\epsilon$ due to the use of the approximate formula (c4), as well as a general uncertainty due to our guessing of the large order behavior.