Constraining bulk-to-boundary correlators in the theories with Poincaré symmetry

TL;DR

The paper demonstrates that bulk-to-boundary correlators in Poincaré-symmetric theories are largely fixed by fall-off conditions at null infinity, with a scalar branch fixed up to normalization and a fermionic branch that includes a $\slashed n$ piece and a shifted exponent $\Delta{+}1$. It shows that fermionic Carrollian amplitudes map to momentum-space amplitudes with an extra $\sqrt{\omega}$ factor for each fermionic operator, and it derives explicit four-point Carrollian amplitudes in Yukawa theory and massless QED. By extrapolating bulk fields to the boundary, the authors classify boundary-to-boundary correlators according to the fall-off index $\Delta$, revealing a magnetic branch for scalars when $0\le \Delta<1$ and potential electric-divergent contributions for $\Delta\ge 1$, while fermionic branches lack a magnetic component. Canonical quantization and transformation laws for fermionic boundary fields are developed, enabling a consistent Carrollian amplitude program and providing a bridge to momentum-space scattering data via appropriate integral transforms. The work also discusses methodological contrasts with other extrapolation schemes and outlines future directions, including tensor fields, non-Poincaré contexts, and regularization schemes for infrared divergences in Carrollian theories.

Abstract

It is well known that a general two-point function cannot be uniquely determined in a theory with Poincaré symmetry. In this paper, we show that bulk-to-boundary correlators are highly constrained after imposing suitable fall-off conditions near future/past null infinity. More precisely, scalar bulk-to-boundary correlators are fixed to a unique form up to a normalization constant, whereas fermionic bulk-to-boundary correlators are fixed to a linear superposition of scalar and fermionic branches. This is established by asymptotically expanding the Ward identities, where upon the leading terms decouple from the subleading ones. In the fermionic branch, the power-law exponent of the bulk-to-boundary correlator is greater by one than the fall-off index. Consequently, we revisit the relation between Carrollian correlators and momentum space scattering amplitudes for fermionic operators. In this context, we find that the Fourier transform bridging the two acquires an extra factor of $\sqrtω$ for each fermionic operator. Furthermore, we reduce the bulk-to-boundary correlator to the boundary-to-boundary correlator and identify a critical fall-off index $Δ=1$. For $0 \le Δ< 1$, only a magnetic branch exists for scalars. For $Δ> 1$, the electric branch is always divergent for both scalar and fermionic branches and thus requires regularization.

Figures (2)

• Figure 1: The bulk-to-boundary correlator $D$. The pole of the bulk-to-boundary correlator should be located on the light ray that connects a bulk field located at $x$ and a boundary operator located at $(u,\Omega)\in\mathcal{I}^+$. In general, this is a function of the translation invariant variable $\widehat{u}=u+n\cdot x$.
• Figure 3: The leading the subleading contributions to the two-point correlation function of the composite operator $\Sigma^2$.