Chirality loss during brane merging: a universal power law from the Jackiw-Rebbi index

Abstract: 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 $|Δ{\mathrm{abs}}|\propto d^γ$ in the merging limit $d\to 0^{+}$, with the critical exponent $γ$ 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 $γ\in[0.930,0.985]$. 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 $γ$ as the crossover plateau of a local effective exponent $γ{\mathrm{eff}}(d)$, explaining the sub-unit value analytically and tracing the universality to the Pöschl-Teller structure of the $N{\mathrm{JR}}=1$ zero mode. The universality of $γ$ 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.

• The paper establishes that chiral fermion localization loss follows a universal power law governed by the Jackiw-Rebbi index, independent of integrability or mass gap.
• It uses (1+1)-dimensional models, including sine-Gordon and double sine-Gordon, to analytically and numerically confirm a scaling exponent (γ) close to unity.
• The findings highlight topological invariance in brane merging, imposing robust constraints on effective Yukawa couplings and braneworld phenomenology.

Universal Power Law in Chirality Loss during Brane Merging

Overview

The paper "Chirality loss during brane merging: a universal power law from the Jackiw-Rebbi index" (2605.24739) establishes that during brane merging in extra-dimensional theories, the loss of chiral fermion localization follows a universal power law. This power law is governed by a critical exponent $\gamma$ determined solely by the Jackiw-Rebbi index $N_{\mathrm{JR}}$, which counts the number of topologically protected zero modes. The exponent is shown to be independent of integrability, mass gap, and the detailed scalar field dynamics generating the domain walls. The analysis is carried out using $(1+1)$-dimensional prototypical models (sine-Gordon, double sine-Gordon) as analogues for brane scenarios with two domain walls.

Jackiw-Rebbi Framework and Braneworld Correspondence

The authors employ the Jackiw-Rebbi mechanism, where a Dirac fermion couples to a scalar kink background, leading to localized zero modes at the domain wall. In the $(1+1)$-dimensional model, the zero-mode count $N_{\mathrm{JR}}$ is a topological invariant depending on the asymptotic signs of the scalar field. This framework maps directly to $(4+1)$-dimensional braneworld models, where localization of chiral fermions on branes is crucial for replicating the four-dimensional Standard Model spectrum. The rate at which left- and right-handed zero modes lose spatial separation as two branes merge determines the collapse of four-dimensional Yukawa couplings, and thus the mass hierarchy.

Power-Law Scaling in Chiral Separation

The central observable is the spatial separation $|\Delta_{\mathrm{abs}}|$ of chiral zero modes. As the inter-brane separation $d$ approaches zero, this observable follows the power law $|\Delta_{\mathrm{abs}}| \propto d^\gamma$. The authors numerically extract $\gamma$ by fitting this scaling in models with different scalar potentials (integrable sine-Gordon, non-integrable double sine-Gordon, and $N_{\mathrm{JR}}$0). For the sine-Gordon model ($N_{\mathrm{JR}}$1), they derive an exact analytical expression for the overlap integral,

$N_{\mathrm{JR}}$2

which explains the observed scaling behavior. Numerical fits yield exponents $N_{\mathrm{JR}}$3 across all $N_{\mathrm{JR}}$4 models, with only minor variations attributable to subleading effects such as kink width.

Universality and Topological Invariance

Three robust findings establish the universality of $N_{\mathrm{JR}}$5:

1. Independence from Integrability: The exponent $N_{\mathrm{JR}}$6 remains consistent across integrable and non-integrable models, indicating it is not controlled by the existence of conserved quantities or soliton scattering behavior.
2. Mass Gap Irrelevance: The exponent does not depend on the mass gap of the fermionic sector; both sine-Gordon and double sine-Gordon models (with mass gaps $N_{\mathrm{JR}}$7 and $N_{\mathrm{JR}}$8) yield similar $N_{\mathrm{JR}}$9.
3. Shape Invariance: Detailed features of the scalar field potential and kink shape do not materially affect $(1+1)$0, except for a weak monotonic trend from kink-width corrections.

Thus, $(1+1)$1 is a topological invariant dictated by $(1+1)$2. Empirically, in the Pöschl-Teller series of models, the scaling exponent tracks the topological index: $(1+1)$3, $(1+1)$4, $(1+1)$5 (Pinheiro et al., 15 Apr 2026).

Analytical Origin of Universality

The analytical derivation stems from the algebraic form of the zero-mode wave function and the scaling of the overlap integral in the merging limit. For $(1+1)$6, the relevant zero mode is $(1+1)$7, making the overlap integral $(1+1)$8. The critical exponent $(1+1)$9 appears as the plateau of the effective exponent across the region where the zero-mode hybridization transitions from exponential to power-law suppression. Subleading deviations from universal behavior arise due to corrections in the kink profile but do not affect the leading-order scaling.

Braneworld Implications

The universality of $(1+1)$0 has several important consequences for extra-dimensional model building:

• Model-Independence: The saturation rate of Yukawa couplings during brane merging is independent of microphysical scalar dynamics. All two-brane scenarios with the same $(1+1)$1 exhibit identical chiral collapse rates.
• Constraints on Moduli Dynamics: If inter-brane separation is a dynamical modulus, its evolution drives power-law changes in effective couplings, robust against model details.
• Topological Classification: The exponent $(1+1)$2 can be used as a practical invariant for classifying brane configurations, with direct phenomenological consequences for the four-dimensional spectrum.

Connections to Critical Phenomena

The scaling structure underlying $(1+1)$3 is closely analogous to universality in critical phenomena, where exponents depend only on symmetry and dimensionality. Here, $(1+1)$4 serves as the universality class, $(1+1)$5 plays the role of deviation from criticality, and $(1+1)$6 is analogous to a correlation length.

Future Directions

The paper identifies several avenues for further research:

• Analytical Calculation of $(1+1)$7: Employing instanton calculus or WKB methods in the moduli space to explicitly derive the dependence of $(1+1)$8 on $(1+1)$9.
• Kink Width Corrections: Perturbative analysis of model-specific corrections to the scaling law.
• Curved Spacetime Generalization: Extending universality claims to warped geometries (Randall-Sundrum models) to assess phenomenological applicability.
• Higher Dimensions and Experimental Realization: Studying the scaling in $N_{\mathrm{JR}}$0 and $N_{\mathrm{JR}}$1 dimensional domain walls and possible measurement in bilayer graphene, where the Jackiw-Rebbi mechanism is physically realized.

Conclusion

This work rigorously demonstrates that the power-law rate of chiral localization loss during brane merging is a universal, topological property characterized by the Jackiw-Rebbi index. The critical exponent $N_{\mathrm{JR}}$2 is invariant to integrability, mass gap, and scalar field profile, providing a robust geometric quantity for extra-dimensional model classification and phenomenology. Analytical arguments and numerical results corroborate this universality, with significant implications for braneworld constructions and theoretical particle physics.