Quantum graviton scattering with definite helicities in the null surface formulation

Abstract: We develop the second-order quantum perturbation theory of gravity in the Null Surface Formulation (NSF) of asymptotically flat spacetimes. In this framework all dynamical degrees of freedom are radiative data defined at null infinity; no bulk fields or off-shell propagators enter the construction. Working directly at null infinity, we derive the helicity-resolved Bondi shear and the corresponding out-operators governing nonlinear graviton processes. The formalism naturally generates a gravitational tail amplitude requiring opposite incoming helicities, and a graviton scattering amplitude that factorizes into two tail vertices connected by an on-shell intermediate graviton. Imposing the Poincare limit reproduces the s-channel contribution of the Weinberg tree-level amplitude, while the crossed channels are shown to arise at higher perturbative order. The theory is perturbatively finite for two independent reasons: all intermediate gravitons are strictly on-shell, so no loop integrals over virtual bulk momenta are generated; and the perturbative regime requires Gaussian-smeared graviton states (small Bondi mass relative to the Planck scale), whose suppression propagates recursively through the hierarchy, rendering all energy integrals absolutely convergent at every order without renormalization or counterterms. This finiteness is structurally distinct from the ultraviolet problem of covariant perturbative gravity, where divergences originate in off-shell bulk propagators and asymptotic states are defined only indirectly via an iε bulk prescription. The natural observables of the NSF are spectral-angular distributions on the celestial sphere, which encode BMS supermomentum flux rather than ordinary Poincare momentum conservation. Gravitational memory, MHV helicity selection rules, and the coherent-state classical limit arise naturally within the same framework.

• The paper presents a novel approach that uses the null surface formulation to derive on‐shell, ultraviolet-finite quantum graviton scattering amplitudes.
• It resolves graviton helicities at the operator level, matching Bondi shear to spin-weighted angular kernels on the celestial sphere.
• The study connects BMS symmetry, gravitational memory, and classical limits, setting a foundation for higher-order NSF extensions.

Quantum Graviton Scattering and Ultraviolet Finiteness in the Null Surface Formulation

Overview of NSF and Radiative Degrees of Freedom

The Null Surface Formulation (NSF) establishes quantum gravity perturbation theory exclusively at null infinity, with all dynamical content encoded in the Bondi shear and radiative data on $\mathscr{I}^\pm$. Unlike conventional approaches reliant on bulk fields and off-shell propagators, NSF operates without reference to bulk spacetime, instead organizing the gravitational dynamics through spin-weighted variables on the celestial sphere. Asymptotic quantization, following Ashtekar's construction, provides a Fock space on $\mathscr{I}^+$ with creation and annihilation operators labelled by definite helicities, manifesting the radiative degrees of freedom associated with massless spin-2 excitations.

Helicity-Resolved Perturbative Structure

This work systematically extends the second-order NSF equations to resolve graviton helicities at the operator level, yielding an explicit decomposition of the Bondi shear $\sigma_2^+$ into four sectors classified by frequency transfer and operator structure. The identification of all relevant spin weights for each angular kernel and operator pair enables a rigorous matching of physical processes to their kinematic and angular profiles on the sphere. Each operator group in $\sigma_2^+$ couples to a distinct spin-weighted factor, ensuring that the integrand for quantum amplitudes possesses zero total spin weight for all integration variables, in agreement with the structure of spherical harmonics and eth operators.

Factorization and On-Shell Scattering Amplitudes

A principal result is the direct derivation of quantum graviton $2\to2$ scattering amplitudes using only on-shell radiative data. The amplitude factorizes into two tail vertices joined by a strictly on-shell intermediate graviton, with all energy integrals rendered finite by NSF's structure. Imposing Poincaré conservation, one recovers the $s$-channel contribution of Weinberg's classic tree-level result, $\mathcal{M}=16\pi G\,s^3/(tu)$. The t- and u-channel contributions, absent at second order, are expected to emerge at higher orders from the sum-frequency operator groups in $\sigma_2^+$.

Ultraviolet Finiteness and Absence of Bulk Loops

NSF perturbation theory is shown to be perturbatively finite due to two independent mechanisms:

1. Strict On-Shell Intermediate Gravitons: All intermediate states are physical, satisfying $k^2=0$. This precludes loop integrals over virtual momenta and eliminates the source of ultraviolet divergences endemic to bulk Feynman-diagram gravity.
2. Gaussian-Smeared Operator States: The requirement of a genuinely quantum regime ($GM/(c^2\ell_{\mathrm{Pl}})\ll1$) forces the use of Gaussian-smeared graviton operators. This propagates recursively through the perturbative hierarchy, with each order acquiring an additional Gaussian suppression factor. As a result, all energy integrals decay faster than any inverse power and are absolutely convergent, obviating any need for renormalization or counterterms.

This structural finiteness is distinct from the non-renormalizability revealed by Goroff and Sagnotti for bulk gravity. The NSF's finiteness is intrinsic and arises from its foundational insistence on asymptotic, on-shell radiative states and physically motivated smearing, not from auxiliary regulator artifacts.

Infrared Behavior and Collinear Singularities

The analysis reveals that infrared divergences emerge exclusively from collinear configurations in the forward-scattering region, analogous to standard soft theorems in perturbative gravity. These divergences are localized and are not tied to UV physics or bulk loops; a complete treatment would require consideration of degenerate soft states and BMS resummation.

NSF Observables: BMS Supermomentum Flux and Memory

NSF fundamentally alters the observable content of quantum gravity. Rather than differential cross sections derived from four-momentum conservation, the natural observables are spectral-angular distributions on the celestial sphere, directly measuring BMS supermomentum flux. The frequency delta functions in tail amplitudes arise from Bondi-time Fourier transforms, not imposed momentum conservation, thereby reflecting the underlying infinite-dimensional BMS symmetry rather than the finite-dimensional Poincaré group.

The multipole decomposition of tail amplitudes provides direct access to radiated supermomentum charges, connecting quantum graviton emission processes to the full hierarchy of BMS multipoles. The gravitational memory effect, a permanent displacement resulting from nonlinear radiation, appears as the zero-frequency limit of tail processes and is regulated naturally by Gaussian smearing, making memory finite for realistic wave-packet states.

Helicity Selection Rules and MHV Structure

The second-order NSF amplitudes automatically reflect maximally helicity violating (MHV) selection rules: same-helicity channels vanish or are suppressed, with leading contributions arising from opposite-helicity pairs. This structure arises directly from spin-weight matching in null-surface equations and reproduces familiar results from on-shell gravity amplitudes, including their independence from external helicities.

Classical Limit and Coherent-State Reduction

The NSF framework reduces seamlessly to classical nonlinear gravity in the coherent-state limit, with operator expectation values reproducing classical solutions for $\mathscr{I}^+$0. Quantum corrections—such as helicity-dependent angular distributions and nontrivial multipole mixing—can be transparently linked to quantum structure in operator products.

Higher-Order NSF and Future Directions

At higher orders, sum-frequency operator groups are predicted to activate, yielding the full crossed channels of the Weinberg amplitude and additional radiative corrections. The recursive Gaussian suppression mechanism is expected to persist, maintaining ultraviolet finiteness throughout the perturbative hierarchy. Open questions concern the persistence of peeling properties and the realization of fully nonlinear dynamics entirely at null infinity.

Conclusion

The Null Surface Formulation presents a structurally distinct, on-shell quantum gravity framework wherein all physical content is determined at null infinity through radiative data. Graviton scattering, helicity selection rules, BMS symmetry, gravitational memory, and ultraviolet finiteness are naturally encoded in the operator structure and angular kernels, with no reference to bulk fields or off-shell propagation. NSF's adherence to the asymptotic structure of spacetime, its compatibility with the infinite-dimensional BMS group, and the absence of ultraviolet divergences position it as a promising candidate for reformulating quantum gravity beyond conventional approaches. Future research includes third-order computations and detailed exploration of non-Poincaré channels and soft-sector resummation in NSF gravity.