A Resonance in Elastic Kink-Meson Scattering

Abstract

We analytically sum the leading bubble diagrams that contribute to the elastic scattering amplitude of a kink and a meson in the $φ^4$ double-well model. We find a single peak, corresponding to the unstable kink state in which the shape mode is excited twice. The peak has the usual Breit-Wigner form, and its imaginary part agrees with the shape mode decay rate found by Manton and Merabet.

Figures (7)

• Figure 1: The diagrams that compute the components $A$, $B$, $C$ and $D$ of the elastic scattering amplitude $R$Bilguun2025elastickinkmesonscatteringphi4. Time runs from right to left. The $\pm k_0$ represent the incoming and outgoing meson momenta, $S$ represents a shape mode, $\phi_0$ is the zero mode and $\pi_0^2$ insertions arise from center of mass kinetic term in $H^\prime_2$. The tadpole represents the tadpole term in Eq. (\ref{['H3']}).
• Figure 2: The three channels of contributions to $D$ are shown. The intermediate states have an unexcited kink and two mesons (left), a shape mode excited kink and one meson (center) and a kink with an unstable twice-excited shape mode (right). The state diagrams are on top, and the corresponding Feynman diagrams on the bottom. The kink is not shown in the state diagrams. In the Feynman diagrams, the mesons are drawn as dashed lines while the kink curves are solid and no momentum is given for them, as we work in the center of mass frame. The kink curve is drawn with more pronounced wiggles when more shape modes are excited. Note that all three channels correspond to the same state diagram, but the Feynman diagrams have different numbers of loops. Time always flows to the left.
• Figure 3: The vertices representing the various interactions $H_3^n$ in state diagrams, where $n$ is the meson number. Note that for each $n$, the diagrams are the same up to the choice of which leg is a shape mode or a meson. The corresponding Feynman diagrams would depend on how excited the kink is at the beginning. Recall that we have restricted our attention to interactions with two shape modes and a meson because only these will contribute to the leading order pole.
• Figure 4: The two channels contributing to $|k_0\rangle_1^{02}$ (left) and $|k_0\rangle_1^{04}$ (right) respectively. The state diagram is shown on top and the Feynman diagram on the bottom. Note that $|k_0\rangle_1^{04}$ arises from virtual shape modes and a virtual meson which are also present in the dressed kink. $|k_0\rangle_1^{02}$ arises from a meson that is absorbed by a kink, twice exciting its shape mode.
• Figure 5: The state (top) and Feynman (bottom) diagrams describing the contributions to $|k_0\rangle_3$ in Eq. (\ref{['k32']}). The intermediate kink plus meson configuration corresponds to a loop in the Feynman diagram, which leads to the $\omega_{k_{}}-\omega_{k_{0}}$ pole. However it corresponds to a line in the state diagram, as the kink is not drawn. The loop in the state diagram leads to the $\omega_{k_{0}}-2\omega_S$ pole. More generally, the contribution to $|k_0\rangle_{2i+1}$ would consist of $i$ loops separated by internal lines, repeating the pattern seen here.