34. A framework for the multi-channel quantum S-matrix of the DSG kink–antikink system, with a parity-decoupling theorem

We develop a numerical framework for computing the quantum S-matrix of the double sine--Gordon (DSG) kink--antikink system including meson modes. The approach extends the single-channel formalism of the companion paper by restoring the meson field, expanding it in instantaneous eigenmodes of the two-soliton fluctuation operator, and deriving coupled-channel Schrodinger equations on the half-line. The leading correction to the effective mass from the induced metric---corresponding to Option~B of our plan---is computed using perturbation theory for the $r$-dependent eigenmodes. The resulting channel-dependent effective mass includes classical overlap corrections and quantum zero-point kinetic contributions. A key finding is that the quantum correction does not vanish at large separation for the discrete shape modes: the correction kernel $K_n(r)$ approaches a nonzero constant $K_n^{(\infty)}$, reflecting the fact that each soliton carries its own cloud of virtual mesons. For continuum modes, $K_k(r)$ grows as $k^2$ at large $k$, giving an ultraviolet divergence that is cancelled by renormalization of the kink mass. We derive the asymptotic matching procedure with channel-dependent Bessel functions, construct the Jost matrix, extract the S-matrix, and establish unitarity conditions. We identify which results are exact, which are approximate, and which remain uncertain. A complete numerical algorithm is provided, along with error estimates and a candid discussion of limitations. The two-channel $S$-matrix satisfies $|{\det} S| = 1$, $|S_{01}| = |S_{10}|$, and row-unitarity to the printed precision at every $(\kappa, k)$ in the range studied, and the extracted inelastic amplitudes are stable under variation of the matching radius from $r = 20$ to $r = 60$ to four decimal places. The shape-mode contribution to the induced-metric correction of the reduced mass is small throughout the $\kappa$ range considered ($Q_\infty/(M/2) < 1\%$ at the production truncation $N_{\rm modes}=2$). The continuum contribution grows linearly with the box size and remains UV-divergent; it is not included in the present numerical results. The one-meson production probability $|S_{01}|^2$ above threshold peaks at $4.90\%$ at $\kappa = 0.2$ and decreases monotonically by a factor of $\sim 20$ across the range $\kappa \in [0.2, 2.0]$, with the peak ocated at scattering energies $E_{\rm peak} \approx 6.2$ to $15.3$. Diagonalization of the two-channel Hamiltonian on the full line gives the bound-state spectrum of the coupled system: 18 even-parity states at $\kappa = 0.5$, compared with 10 in the single-channel approximation, the additional eight being meson-dressed kink--antikink molecules that exist only when the shape-mode channel is retained. The cubic-vertex mode sum is logarithmically ultraviolet-divergent; it is evaluated with the physical cutoff $k_{\rm cut} = m$ (the meson mass) on the internal meson-mode sum, above which real meson production opens and the two-channel truncation is no longer complete. The two-channel results are exact below the shape-mode threshold $E < \omega_1^{(\infty)} = 1.188$, complete below the two-meson threshold $E < 2m = 2.83$, and approximate at higher energies where the reported peak inelasticities lie. A Feshbach pole of the elastic $S$-matrix is extracted over $\kappa \in [0.01, 0.60]$ by direct complex-energy search for the zero of the outgoing-wave amplitude on the physical sheet; its position rises monotonically from $E_R = 0.9786$ to $E_R = 1.1436$, and its width exhibits the two-minimum structure of an avoided crossing between two closed-channel bound states. The absolute magnitude of the peak carries a systematic uncertainty of order $20$--$30\%$ from the cutoff choice; the $\kappa$-dependence and the qualitative features are robust. The counterterm subtraction for the off-diagonal coupling, the analytic decomposition of the correction kernel $K_n(r)$ as an alternative to the spectral sum, the sign of the bremsstrahlung vertex, the parity-decoupling theorem that makes the two-channel truncation exact below the continuum threshold within the numerical resolution of the shape-mode search, and the direct extraction of the shape-mode Feshbach resonance are the principal new results.