Completeness from Gravitational Scattering

Abstract: We prove that symmetry in the presence of gravity implies a version of the completeness hypothesis. For a broad class of theories, we demonstrate that the existence of finitely many charged particles logically necessitates the existence of infinitely many charged particles populating the entire charge lattice. Our conclusions follow from the consistency of perturbative gravitational scattering and require the following ingredients: 1) a weakly coupled ultraviolet completion of gravity, 2) a nonabelian symmetry $G$, gauged or global, whose Cartan subgroup generates the abelian charge lattice, and 3) a spectrum containing some finite set of charged representations, in the simplest cases taken to be a single particle in the fundamental. Under these conditions, the abelian charge lattice is completely filled by single-particle states for $G=SO(N)$ with $N\geq 5$ and $G=SU(N)$ with $N\geq 3$, which in turn implies completeness for other symmetry groups such as $Spin(N)$, $Sp(N)$, and $E_8$. Curiously, a corollary of our results is that the $SU(5)$ and $SO(10)$ grand unified theories have precisely the minimal field content needed to derive completeness using our methodology.

• The paper establishes that gravitational scattering mandates every allowed charge in a nonabelian theory to appear as a physical single-particle state.
• It exploits a twice-subtracted dispersion relation to connect low-energy Wilson coefficients with ultraviolet spectral data.
• A recursive group-theoretic algorithm shows that minimal GUT representations are sufficient to ensure charge completeness under gravitational consistency.

Completeness of the Charge Spectrum from Gravitational Scattering

Introduction and Context

The paper "Completeness from Gravitational Scattering" (2512.11955) establishes, through rigorous bottom-up arguments, that the spectrum of charges in a theory with gravity and a sufficiently large nonabelian symmetry group is necessarily complete. This means every allowed charge in the Cartan lattice of the symmetry group corresponds to a physical single-particle state. The argument is rooted in the requirements of perturbative gravitational scattering and leverages analytic properties of the amplitude, fundamentally invoking dispersion relations and the universality of the gravitational interaction as imposed by the equivalence principle.

The work takes a highly constructive and iterative approach, solidly within the on-shell bootstrap paradigm, and highlights the interplay between gravitational consistency and internal symmetry constraints on the physical particle spectrum.

Dispersion Relations and Gravitational Consistency

The core of the analysis is the twice-subtracted analytic dispersion relation for four-point amplitudes at fixed momentum transfer $t$. For the low-energy theory with gravity, the amplitude features a characteristic $s^2/t$ pole from tree-level graviton exchange. Analyticity and crossing symmetry relate the Wilson coefficients of higher-dimension operators in the effective theory to spectral data—the presence or absence of exchanged states—in the ultraviolet completion. Given the vanishing of the boundary term $b_n(t)$ for $t<0$ (established from Regge bounds and eikonalization), one must account for the non-vanishing Wilson coefficient $c_2(t) \sim 1/t$ through pole contributions in the $s$ or $u$ channel, i.e., through the existence of exchanged particle states.

This logic is robustly realized in weakly coupled theories with tree-level unitarization (as in string theory), and crucially leverages the assumption that all possible pairs of particles interact gravitationally. The approach is manifestly insensitive to whether the internal symmetry is global or gauged.

Charge Completeness Algorithm

The paper develops a systematic, recursive procedure to generate the full charge lattice, beginning with a minimal set of charged states (typically the fundamental representation). For a Lie group $G$ (with Cartan part $H$), charges in the external spectrum span a set $Q \subset \Lambda$, where $\Lambda$ is the weight lattice of $H$. Particle pairs from $Q$ are scattered; unless one of the allowed $s$ or $u$-channel states is already in $Q$, the dispersion relation forces the existence of a new single-particle state with the relevant charge. Subsequent iterations include group-theoretic orbit operations—applying Weyl reflections and outer automorphisms—to populate the symmetry-related charge sectors.

This algorithm is constructive, ensuring that if an exchange is forced by consistency, the corresponding representation (or set of weights) appears in the spectrum. The procedure naturally distinguishes between theories depending on the nature of $G$.

Results for Specific Symmetry Groups

Abelian Groups

For $G = U(1)^k$, the method cannot establish completeness. The freedom for scattering processes to occur in the $u$-channel (with existing states) precludes any conclusive argument for the necessity of new charges. This is in contrast to various swampland heuristics motivated by the absence of global symmetries.

Nonabelian Groups: SO(N), SU(N)

SO(N) ($N \geq 5$)

For orthogonal groups, the vector representation (fundamental) and its Weyl orbits suffice to fill the entire integer charge lattice via the recursive algorithm. The discrete center is always covered by the allowed states, and the iteration rapidly exhausts the lattice under the action of reflections and signed permutations.

SU(N) ($N \geq 3$)

For unitary groups, completeness is nontrivial due to the presence of a larger center ($\mathbb{Z}_N$) and the structure of the weight lattice in fundamental weight coordinates. Here, the minimal starting set must contain representatives in each central charge sector ($z=0,1,\ldots,N-1$, as defined by $N$-ality), which in physical terms matches the field content of grand unified theories. For instance, in SU(5) GUTs, the $\mathbf{5},\mathbf{10},\mathbf{24}$ and their conjugates exactly cover all central charge sectors and thus guarantee spectral completeness. This is a necessary and sufficient condition for completeness and establishes that the ordinary GUT matter content is precisely minimal.

The recursive algorithm proceeds by constructing and orbiting multicorner polytopes (triangles, hexagons, etc.) in the weight lattice, filling out the charge spectrum by successive scattering and Weyl group actions.

Other Groups

The arguments generalize to $Spin(N)$, $Sp(N)$, and the exceptional group $E_8$, with suitable initial representations (e.g., vector and spinor for $Spin(N)$, adjoint for $E_8$). For $Spin(N)$, both integer and half-integer weights are generated, covering the full lattice with both vector and spinor starting points. For $E_8$, starting with the adjoint ($\mathbf{248}$) is sufficient to iterate through the entire even self-dual charge lattice.

Strong and Contradictory Claims

The paper establishes that, for nonabelian $G$ with sufficiently large rank and a minimal but complete initial set of representation charges, the presence of gravity and the consistency of four-point scattering amplitudes enforce completeness of the charge spectrum by single-particle states. The claim is simultaneously stronger than the swampland completeness conjecture (which allows for multi-particle states) and weaker in the sense that some initial minimally charged states are assumed.

The analysis explicitly finds that, for both $SU(5)$ and $SO(10)$ grand unified theories, the minimal representation content needed for completeness matches exactly the phenomenological requirements to embed standard model matter, i.e., any strict subset of the standard GUT field content fails to enforce completeness under this methodology.

Theoretical and Practical Implications

The findings have profound structural constraints on ultraviolet completions of quantum gravity and on the allowed low-energy spectra of matter-coupled gravity. Completeness, as established here, is not a phenomenological assumption but emerges from the mathematical consistency of the S-matrix in the presence of gravity. This tightens the link between internal symmetry, charge quantization, and the spectrum required for unitarity and causality of gravitational scattering—directly relating swampland-type conjectures to on-shell constructibility.

This also showcases that standard constructions in GUTs ensure, rather than arbitrarily assume, that all possible charges are realized by single-particle states, an important consistency check for bottom-up model building.

Future Directions

Several directions are suggested. First, the necessity versus sufficiency for representation completeness (i.e., presence of all irreducible representations) is not established for all cases; the analysis for abelian $G$ remains out of reach. Second, whether higher-point amplitudes or unitarity constraints could enforce stricter versions of completeness (including representation instead of just charge completeness) is a topic left open. Relaxations, such as considering loop-level scattering (with multi-particle states), or refinements, such as additional assumptions about symmetry breaking or spacetime symmetry, could further sharpen these constraints.

Moreover, the methodology could be extended to the analysis of more exotic symmetry groups or to broader classes of bottom-up swampland conjectures, including the weak gravity conjecture, where similar analytic tools might enforce additional spectrum or coupling bounds.

Conclusion

The work provides a rigorous, constructive argument that gravitational consistency enforces charge completeness for ample classes of nonabelian symmetry groups, with the minimal spectrum matching the requirements of GUT models. This result strengthens the relation between quantum gravity, internal symmetry, and the structure of the physical spectrum and exemplifies the utility of on-shell bootstrap techniques in quantum field theory and quantum gravity phenomenology.