Reflection coefficient of a reflectionless kink

Classically, reflectionless kinks transmit all incident radiation. Recently, we have used an analyticity argument together with a solution of the Lippmann-Schwinger equation to write down the leading quantum correction to the reflection probability. The argument was fast, but rather indirect. In the present paper, we calculate the reflection coefficient and probability by methodically grinding through the Schrödinger picture time evolution. We find the same answer. This answer contains contributions not considered in the traditional calculation of meson-kink scattering in 1991. However, as a result of these contributions, our total result is zero in the case of the Sine-Gordon model, and so it is consistent with integrability.

Figure 1

Time runs to the left. This is a schematic drawing of the following process. Meson 1 travels. Then it emits and absorbs the same virtual particle and in the process becomes meson 2. This interaction is proportional to $\mathcal{I}(x)$, which falls exponentially in $mx$ far from the kink, thus the interaction necessarily happens close to the kink. However, the kink is not drawn.

Figure 2

Right: while meson 1 approaches, a virtual particle pair comes in and out of existence, leaving behind it a virtual particle. We will say that the virtual particle is created by a tadpole, although one might rightly note that it is emitted by the kink that is never drawn. This virtual particle merges with meson 1, leaving meson 2. Left: meson 1 emits a virtual particle, becoming meson 2. This emitted virtual particle decays via a virtual particle pair tadpole process.

Figure 3

Left: we see meson 1 splitting into two virtual mesons that recombine into meson 2. Right: two virtual mesons and meson 2 are created together, and later the two virtual mesons annihilate themselves together with meson 1.

Figure 4

Meson 1 is absorbed by the kink, leaving two zero modes. The kink also emits meson 2, together with two more zero modes.

Figure 5

Meson 1 is destroyed by a tadpole (right and bottom) or converted into two zero modes (left) and meson 2 is created by a tadpole (left and bottom) or together with two zero modes (right).

Figure 6

Meson 1 converts into meson 2, emitting two zero modes in the process.

Figure 7

Right: a virtual meson is created together with two zero modes. The virtual meson strikes meson 1 and turns it into meson 2. Left: meson 1 nucleates a virtual meson, which decays into two zero modes.

Figure 8

Meson 1 turns into a virtual meson, emitting a zero mode. The virtual meson emits yet another zero mode, converting into meson 2.

Figure 9

This process is as in Fig. 4. However, two of the four zero modes are annihilated by the ${\pi }_0^2/2$ term in the free Hamiltonian $H_2^{\prime }$.

Figure 10

This interaction is as in Fig. 6. However, the two zero modes are absorbed by the ${\pi }_0^2/2$ kinetic term for the kink center of mass.

Figure 11

This interaction is as in Fig. 7. However, the two zero modes are annihilated by the ${\pi }_0^2/2$ term in the free evolution.

Figure 12

Most of this paper is about the computation of this term, which is the only contribution to the scattering amplitude resulting from zero modes. The process is as in Fig. 8, except that the two zero modes are annihilated by the ${\pi }_0^2/2$ in the free evolution operator.

Figure 13

This is as in Fig. 4 except that all four zero modes are annihilated by the kink center of mass kinetic term in the free evolution operator.