New effect in wave-packet scattering of quantum fields

We report calculations of a wave-packet amplitude of the two-body scattering $\varphi \varphi \to \mathrm{\Phi }\to \varphi \varphi$, which leads to the measured probability in realistic experiments. We elucidate the decay amplitude of $\mathrm{\Phi }\to \varphi \varphi$ from this. In such an amplitude of wave packets, there are in and out time boundaries for the initial $\mathrm{\Phi }$ and final $\varphi \varphi$ configurations, respectively. In this paper, we prove that the effect of the in time boundary of $\mathrm{\Phi }\to \varphi \varphi$ emerges from $\varphi \varphi \to \mathrm{\Phi }\to \varphi \varphi$ without assuming any time boundary a priori. This effect has been overlooked in the standard plane-wave formulation and can exhibit distinct phenomena in wide areas of science. We confirm the result in different integration orders. The result is also interpreted as a Stokes phenomenon in the Lefschetz-thimble decomposition.

Figure 1

Schematic figure in the position space $z=(z^0,\mathit{z})$ for the intersection times ${\mathfrak{T}}_{\mathrm{in}\text{−}\mathrm{int}}$, ${\mathfrak{T}}_{\mathrm{out}\text{−}\mathrm{int}}$ and for the center of wave packet ${\mathbf{\Xi }}_{\mathit{a}}$ ($a=1,\dots ,4$) at the arbitrarily chosen reference time $t=0$. The solid and dashed lines denote the trajectories of the wave packets of $\varphi$. The intermediate state of $\mathrm{\Phi }$ is a plane wave that spreads infinitely. The integral over the in- and out-interaction points $x=(x^0,\mathit{x})$ and $y=(y^0,\mathit{y})$ have largest support around the black dots at $z^0={\mathfrak{T}}_{\mathrm{in}\text{−}\mathrm{int}}$ and ${\mathfrak{T}}_{\mathrm{out}\text{−}\mathrm{int}}$, respectively.

Figure 2

Differential probability ${|\mathcal{S}|}^2/\frac{{\kappa }^4}{64\pi M^6}\frac{{\sigma }_1}{{\sigma }_3^2}$ from Eq. (46) as a function of the final wave-packet positions $x_3≔|{\mathit{X}}_3|$ and $x_4≔|{\mathit{X}}_4|$ at time $X_3^0=X_4^0≕T$ ($=100$ in units $M=1$). The upper and lower panels are for momenta on and off the resonance, respectively. The left panels show the bulk contribution only, whereas the right ones show the sum of all the bulk and boundary ones. The difference between the right and left panels is the boundary contribution. This figure verifies our theoretical prediction that the boundary effect is sizable both on and off the resonance.

Figure 3

Shaded region represents convergent directions for $|p^0|\to \infty$ for the integrand in $I(\mathit{p})$. The points $\pm {\mathsf{E}}_{\mathit{p}}$ denote the poles of the propagator of $\mathrm{\Phi }$. The orange, purple, and blue lines represent the integral path for $\mathfrak{I}{p}^0>\mathfrak{I}(-{\mathsf{E}}_{\mathit{p}})$, $\mathfrak{I}{\mathsf{E}}_{\mathit{p}}<\mathfrak{I}{p}^0<\mathfrak{I}(-{\mathsf{E}}_{\mathit{p}})$, and $\mathfrak{I}{p}^0<\mathfrak{I}{\mathsf{E}}_{\mathit{p}}$, respectively.

Figure 4

We plot $e^{\mathfrak{R}\mathcal{F}}$ with sample parameters ${\varsigma }_{+}=10$, ${\omega }_{\varsigma }({\mathit{p}}_{*})$ (=$\mathfrak{R}{p}_{*}^0$) $=0.5$, and $\epsilon =0.3$ in the $E_{\mathit{p}}=1$ units. This corresponds to forward propagation of $\mathrm{\Phi }$ with $\frac{\delta \mathfrak{T}}{{\varsigma }_{+}}$ ($=-\mathfrak{I}{p}_{*}^0$) $=0.5$.

Figure 5

We show contours for $\mathfrak{I}\mathcal{F}=0$ (mod $2\pi$), on a density plot of $\mathfrak{R}\mathcal{F}$ in $[-10,20]$. We have chosen sample parameters ${\varsigma }_{+}=10$, ${\omega }_{\varsigma }({\mathit{p}}_{*})$ (=$\mathfrak{R}{p}_{*}^0$) $=0.5$, and $\epsilon =0.3$ in the $E_{\mathit{p}}=1$ units. The first panel corresponds to forward propagation of $\mathrm{\Phi }$ with $\frac{\delta \mathfrak{T}}{{\varsigma }_{+}}$ ($=-\mathfrak{I}{p}_{*}^0$) $=0.5$, while the second and third to equal-time and backward with $\frac{\delta \mathfrak{T}}{{\varsigma }_{+}}=0$ and $-0.5$, respectively. The red-solid, yellow-dashed, and green-dotted contours are $\mathfrak{I}\mathcal{F}=\mathfrak{I}{\mathcal{F}}_{(*)}$, $\mathfrak{I}{\mathcal{F}}_{(+)}$, and $\mathfrak{I}{\mathcal{F}}_{(-)}$ respectively. The white dots denote the poles at $p^0=\pm {\mathsf{E}}_{\mathit{p}}$ and the blue single circles nearby them are correspondingly the saddle points $p_{(\pm )}^0$. The blue double circle denotes the saddle point $p_{(*)}^0$.

Figure 6

Magnified plot for $\mathfrak{I}\mathcal{F}$ (mod $2\pi$) of the second panel in Fig. 5 near the saddle point $p_{(-)}^0$ (blue circle) and the pole $-{\mathsf{E}}_{\mathit{p}}$ (white dot). We see that three contours and an artifact line terminate in $-{\mathsf{E}}_{\mathit{p}}$: The three contours are $\mathfrak{I}\mathcal{F}=\mathfrak{I}{\mathcal{F}}_{(*)}$ (red-solid), $\mathfrak{I}\mathcal{F}=\mathfrak{I}{\mathcal{F}}_{(+)}$ (yellow dashed), and the antithimble ${\mathcal{K}}_{(-)}$ (green dotted) from $p_{(-)}^0$. The extra artifact line, seemingly consisting of three degenerate contours, is from the branch cut of the logarithmic function in the numerical computation, along which the contributions from all the other Riemann surfaces appear. This way, in each pole in Fig. 5, there always terminate three contours and an artifact line.