Multi-H Diagrams in Physics & Topology

• Multi-H diagrams are analytic constructs representing leading-log contributions in gravitational scattering and combinatorial invariants in manifold decompositions.
• They employ on-shell graviton cuts and rapidity renormalization techniques to recursively generate logarithmic corrections at successive post-Minkowskian orders.
• In topology, multi-H diagrams encode handlebody decompositions of closed manifolds, providing a diagrammatic invariant for classifying high-dimensional structures.

A Multi-H diagram is an analytic construct that arises independently in several domains of mathematical physics and topology, most prominently in the analytic structure of high-energy gravitational scattering amplitudes and in the combinatorial representation of multisections of higher-dimensional manifolds. The term has become central to the classification of leading logarithmic contributions in post-Minkowskian expansions in quantum gravity and to the diagrammatic encoding of smooth manifold decompositions in high-dimensional topology.

1. Multi-H Diagrams in High-Energy Gravitational Scattering

In the Regge regime ($s\gg|t|$) of ultra-relativistic two-body gravitational scattering, the elastic $2\to2$ amplitude is eikonal-dominated by ladder (Glauber) graviton exchanges in the $t$-channel. The first nontrivial unitarity cut beyond the eikonal ladder is the "H diagram" at two loops (3PM), consisting of one on-shell soft graviton (Glauber cut) connecting two distinct graviton-exchange rungs. Multi-H diagrams generalize this by inserting $N$ on-shell soft graviton cuts, each connecting a new pair among $M=N+1$ ladder rungs, with no repetition of pairs (the classical limit excludes rung loops). These contributions first arise at $(2N+3)$PM order and generate leading-logarithmic behavior of the form $(G_N^2\,s\,\log s)^N$ (Alessio et al., 14 Nov 2025).

The multi-H tower provides the dominant contribution to the amplitude’s leading logarithms at each post-Minkowskian order beyond the eikonal approximation.

2. Loop Expansion, Factorization, and the H-Tower

The systematic emergence of the multi-H tower is derived via two parallel techniques: effective field theory (EFT) employing rapidity renormalization group equations (RRGE), and the explicit multi-Regge kinematics construction. The EFT factorization writes the amplitude as a convolution $$\mathcal{M}_{2\to2}(s,t) = i\sum_{M} J_{(M)} \otimes S_{(M)} \otimes \bar{J}_{(M)}$$, where $S_{(M)}$ satisfies a RRGE with anomalous dimension $\gamma_{(M)}$. At leading power in $\lambda \equiv t/s$, the classical part of $\gamma_{(M+1)}$ is generated solely by a single graviton connecting any pair of Glauber lines, achieving recursive insertion of $\gamma^{\mathrm{Cl}}$ and thus of $\log s$ at each order. The $M$th iteration resides at $(2M+1)$PM and produces $\log^{M-1}(s)$, corresponding exactly to the class of multi-H diagrams at that order (Alessio et al., 14 Nov 2025).

3. Explicit Computations: Four-Loop “Double-H” Diagram and Multi-Regge Correspondence

At four loops (5PM), the “double-H” diagram (two on-shell soft-graviton cuts) provides the leading double logarithm. Its EFT computation begins with the tree-level $S^{(0)}_{(3)}$ soft function, to which a single $\gamma_{(3)}^{\mathrm{Cl}}$ RRGE insertion is applied, yielding a contribution

$$2\,\operatorname{Re}\,\mathcal{M}^{(4)}(s, q^2) \simeq - (8\pi G_N)^5 s^4 \frac{1}{64\pi^2} \log^2 s\, H_2(q_\perp^2)$$

in momentum space, and a corresponding $b$-space representation featuring explicit double-pole and transcendental constants.

Parallel direct computation via the multi-Regge expansion shows that all $6$ distinct four-rung topologies contribute equally, each with appropriate rapidity ordering factors. Central massless topologies are integrals of products $J(q_2, \ell_1) q_2^2 J(q_2, \ell_2)$. The reduction of these products yields bubble and “kite” integrals, with the latter evaluated using Gegenbauer or Mellin–Barnes techniques. The sum precisely reproduces the EFT result (Alessio et al., 14 Nov 2025).

4. Analytic Continuation and Regge Cut Structure

Multi-H diagrams engender an infinite tower of Regge cuts in the complex angular momentum ($j$) plane, corresponding to overlapping multi-graviton unitarity cuts. On the $z_t = 1+2s/t$ variable, which unifies the $s$-, $u$- and $t$- channel cuts, these diagrams produce higher-order branch points at $z_t = \pm 1$ and branch cuts extending to infinity. Mellin representation analysis reveals a correspondence between $(r+1)$-fold poles in the Mellin variable $\omega$ and logarithmic powers, mapping to the amplitude’s discontinuity structure and its higher-order analytic features (Alessio et al., 14 Nov 2025).

In the $t$-plane, the two- and four-loop bubble and kite integrals introduce branch points at $q_\perp^2=0$ and at shifted thresholds, which in impact parameter $b$-space correspond to cuts at $b=0$ (UV singularity) and essential singularities due to log accumulation.

5. Extraction of Imaginary Parts and Dispersion Relations

The eikonal phase $\delta(s,b)$, defined via $$\tilde{\mathcal{M}}(s,b) = (1 + \Delta_Q)\,e^{2i\delta_{\mathrm{Cl}}}$$, admits a twice-subtracted, $s \leftrightarrow u$ crossing-symmetric dispersion relation in $z_t$. At 5PM, the single logarithmic contribution to $\operatorname{Im} \delta$ is determined using known 3PM data and the real part of the four-loop “double-H” amplitude, giving, in dimensional regularization,

$$\operatorname{Im} \delta_{\mathrm{Cl}}^{(4),1}(b) = \frac{16}{\pi} (\pi b^2 \varpi e^\gamma)^{5\epsilon} \left[ -\frac{1}{\epsilon} - \zeta_3 + 6 + O(\epsilon) \right].$$

This closes the leading-logarithmic characterization of the classical amplitude through 5PM order (Alessio et al., 14 Nov 2025).

6. Relation to Multi-H Diagrams in Manifold Theory

In an unrelated but terminologically analogous context, multi-H diagrams also denote combinatorial data associated to multisections of closed orientable manifolds. Here, a multi-H diagram (or $n$-section diagram) is a tuple $(\Sigma_g; \alpha^1, \dots, \alpha^n)$, encoding homologically independent cut systems of curves on a genus-$g$ surface $\Sigma_g$, derived from the decomposition of the manifold into 1-handlebodies whose intersections are themselves 1-handlebodies, globally intersecting in the surface $\Sigma_g$. The diagram determines a (unique up to diffeomorphism, for $n\leq 6$) closed manifold, and its associated handlebody gluings, thus providing a diagrammatic invariant of the manifold structure (Aribi et al., 2023).

The coincidence of terminology is historical and unrelated to the Regge theory context. However, both usages leverage diagrammatic representations to encode highly nontrivial topological or analytic decompositions: in the former, leading-log graviton cuts; in the latter, manifold decomposition data.

7. Broader Analytical and Physical Implications

The tower of multi-H diagrams, as derived in the EFT rapidity-RG framework and through direct multi-Regge expansions, is fundamental to the systematic organization of leading logarithmic corrections in gravitational amplitudes. The explicit evaluation of amplitudes up to the four-loop “double-H” level and the extraction of the corresponding radiative logarithms in the eikonal phase concretize the connection between dispersion-theoretic analytic structure and field-theoretic unitarity cuts. The appearance of infinite Regge cuts, essential singularities, and a complex hierarchy of branch points reflects the intricate analytic underpinnings of classical gravity in the high-energy regime (Alessio et al., 14 Nov 2025).

In summary, multi-H diagrams are essential tools in both the analytic machinery behind high-energy gravitational scattering and in the topological characterization of high-dimensional manifolds, each embodying a distinct but formally parallel approach to encoding the structural complexity of interactions or decompositions.