Chirality loss during brane merging: a universal power law from the Jackiw-Rebbi index
We investigate the rate at which chiral fermion localisation is lost when two domain walls merge in extra-dimensional braneworld scenarios, using the $(1+1)$-dimensional Jackiw-Rebbi framework as a controlled analytical laboratory. As the inter-brane separation $d$ decreases, left- and right-handed zero modes hybridise and chiral asymmetry is progressively lost. We show that the spatial separation between the chiral zero modes follows a universal power law $|\Delta_{\mathrm{abs}}|\propto d^{\gamma}$ in the merging limit $d\to 0^{+}$, with the critical exponent $\gamma$ determined solely by the Jackiw-Rebbi topological index $N_{\mathrm{JR}}$, and independent of the fermionic mass gap, the integrability of the scalar sector, and the detailed shape of the domain wall profile. Comparing the integrable sine-Gordon model with four members of the non-integrable double sine-Gordon family, all sharing $N_{\mathrm{JR}}=1$, we find $\gamma\in[0.930,0.985]$. For the sine-Gordon model we derive the closed-form overlap integral $I(d)=2d/\sinh(2d)$, from which the exact chiral separation follows as a ratio of hyperbolic functions without free parameters. This result identifies $\gamma$ as the crossover plateau of a local effective exponent $\gamma_{\mathrm{eff}}(d)$, explaining the sub-unit value analytically and tracing the universality to the P\"{o}schl-Teller structure of the $N_{\mathrm{JR}}=1$ zero mode. The universality of $\gamma$ implies that the rate of four-dimensional Yukawa coupling collapse during brane merging is a topological invariant, insensitive to the microscopic scalar dynamics generating the walls.