Deletion Does Not Measure Contribution in Coupled-Channel Dynamics

Abstract

In projected descriptions of quantum dynamics, the importance of an eliminated degree of freedom is routinely assessed by deleting it and measuring the system's response. This conflates two effects: the channel's intrinsic contribution and the reorganization of the surviving model space. Here we disentangle them in continuum-discretized coupled-channels (CDCC) scattering, decomposing the Feshbach dynamic polarization potential (DPP) channel by channel while keeping the full Green's function intact, and comparing with conventional bin-deletion from the coupled equations. For $d$+$^{58}$Ni the two approaches reproduce the same elastic $S$-matrix to 0.45\%, yet a channel ranked first by one diagnostic is ranked fifth by the other. A frozen-basis protocol, zeroing couplings without reducing the basis, yields rankings that track the DPP closely ($ρ_{\rm DPP,frozen} = 0.94$) and are uncorrelated with standard deletion ($ρ_{\rm frozen,del} = -0.37$), establishing that the discrepancy is dominated by model-space reorganization. Pairwise analysis reveals quantum anti-synergy: adjacent channels partially cancel through off-diagonal Green's-function coherence, in all 10 tested pairs by the DPP and 8 of 10 by deletion. The asymmetry between excluding a degree of freedom from the effective interaction and deleting it from the model space is algebraic and general; basis-preserving decoupling, implementable in any coupled-channel code, isolates the reorganization component.

Figures (2)

• Figure 1: Single-bin importance $\mathcal{I}_\alpha = |\Delta S_\alpha|/|S_\mathrm{full}|$ from DPP channel-exclusion (blue), frozen-basis CDCC (green, couplings zeroed but basis retained), and standard CDCC bin-deletion (pink) for $d$+$^{58}$Ni at 21.6 MeV, $J = 8$. (a) $l = 2$: DPP and frozen-basis agree closely ($\rho_{\rm DPP,frozen} = 0.94$); standard deletion gives a qualitatively different ranking ($\rho_{\rm frozen,del} = -0.37$). (b) $l = 0$: same pattern ($\rho_\mathrm{DPP,frozen} = 0.77$).
• Figure 2: Quantum anti-synergy in $d$+$^{58}$Ni at $J = 8$. (a) Argand diagram for pair P$_3$ ($l = 2$, [4--6]$\times$[6--8] MeV): $\Delta S_\alpha$ (blue) and $\Delta S_\beta$ (pink) drawn tip-to-tail overshoot $\Delta S_{\alpha\beta}$ (purple dashed); the gap (orange) is the anti-synergy. (b) Interaction term $I$ for all 10 adjacent pairs. DPP (blue) predicts $I < 0$ universally; CDCC (pink) agrees in 8/10 cases. P$_4$ and P$_6$ (anomalous $I > 0$) involve maximum $\mathcal{Q}$-space reorganization.