arXiv subjects

He-Chen Weng

Publications and source records attributed to He-Chen Weng.

11 recordsLinked to original sources

arXiv2026

de Sitter Wavefunction from Quadrangular Polylogarithms: Chain Graphs

We present an explicit formula for the $n$-site chain graph contribution to the cosmological wavefunction for conformally coupled $\phi^3$ theory in de Sitter space. Our result relies on the recent finding that the symbol of this function satisfies total compatibility with respect to the $A_{2n-2}$ cluster algebra, and that Rudenko's quadrangular polylogarithms provide, by construction, a complete basis for such functions. We prove our formula by directly relating a recursive set of differential equations satisfied by these wavefunction coefficients to a recursive coproduct formula for quadrangular polylogarithms.

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arXiv2026

Landau Analysis of One-Cycle Negative Geometries

We use geometric Landau analysis to determine the singularity structure of four-point, one-cycle negative geometries in $\mathcal{N}=4$ super-Yang-Mills theory, which represent certain contributions to the logarithm of the four-point amplitude or equivalently the normalized quadrangular Wilson loop with a Lagrangian insertion. By analyzing the relevant Landau diagrams recursively, we prove that this quantity has singularities only at $z=-1,0$ and $\infty$ to all loop orders. This represents a first step towards obtaining a non-perturbative resummation for this quantity at next-to-leading order in the expansion over cycles.

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arXiv2026

Cluster Bootstrap for Cosmological Correlators

We show that cosmological wavefunction coefficients associated with $n$-site chain and loop graphs for a cubic scalar theory in de Sitter spacetime have symbol alphabets given by subsets of $A_{2n{-}2}$ and $B_{2n{-}1}$ cluster variables, respectively, and satisfy the associated cluster adjacency properties. The key step in proving this is identifying a precise connection between graph "tubings" that appear in the kinematic flow equation and polygon "triangulations" that encode the combinatorics of cluster compatibility. Our results imply that cosmological wavefunction coefficients in a general power-law FRW cosmology satisfy cluster adjacency to all orders in the $\epsilon$ expansion. We use this information as bootstrap input to show that de Sitter symbols for $n \leq 4$ are uniquely determined by simple physical constraints.

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arXiv2025

Split-Helicity Tree Amplitudes and Flag Cluster Algebras

Recent work has uncovered a connection between the symbol letters of general massless scattering and (permutations of) cluster variables of partial flag varieties. In this paper we explore the cluster adjacency of tree-level gluon amplitudes, specifically focusing on split-helicity amplitudes which can be written in closed form in terms of zigzag diagrams. We check in several cases, and conjecture in general, that the poles in each term satisfy cluster adjacency under a set of permutations that is built from arc permutations of the corresponding zigzag.

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arXiv2025

Symbol Alphabets in QCD and Flag Cluster Algebras

The full 245-letter symbol alphabet for all planar massless two-loop six-point Feynman integrals was recently determined in arXiv:2412.19884 and arXiv:2501.01847. In a parallel mathematical development, it was shown in arXiv:2408.14956 that there is an embedding of the cluster algebra associated to the partial flag variety $Fl_{2,n-2;n}$, which describes the kinematics of $n$ massless particles, into that of the Grassmannian Gr$(n{-}2,2n{-}4)$. In this paper we connect these developments by showing that most of the rational symbol letters can be expressed in terms of flag cluster variables, and that all of the algebraic symbol letters arise from infinite mutation sequences.

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arXiv2024

On Unitarity of Bespoke Amplitudes

We use partial wave unitarity to constrain various bespoke four-point amplitudes. We start by constructing bespoke generalizations of the type I superstring amplitude, which we show satisfy dual resonance and have suitable high-energy limits. By analyzing the behavior of partial wave coefficients for highly massive states, we strictly rule out all bespoke amplitudes with asymptotically non-linear Regge trajectories and place constraints on the first few non-trivial parameters in asymptotically linear cases. Finally, we argue that while a large class of unitary bespoke amplitudes fails to satisfy Regge Sum Rules, there exists a smaller sub-class with a vanishing mass gap that is superpolynomially bounded.

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arXiv2024

The Geometry of the Modular Bootstrap

We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on $R^+{\otimes}\mathbb{Z}$, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as constant computation time with increasing central charge. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above $Δ^*_{gap}$, where $\frac{c{-}1}{12}<Δ^*_{gap}< \frac{c}{12}$, all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.

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arXiv2023

Bootstrapping string theory EFT

We study the space of open string effective field theories by combining the constraint of unitarity and monodromy relations for the four-point amplitude. The latter is a reflection of an underlying disk correlator with singularities at the boundary. By assuming maximal susy the resulting bootstrap isolates Wilson coefficients to at least $10^{-4}$ of the Type-I superstring. Furthermore, utilizing our geometric approach, we obtain the critical dimension of 10 from the low energy coefficients alone. Remarkably, relaxing SUSY but requiring the massless states to carry four-dimensional helicities, the Wilson coefficients are again constrained to superstring values within $10^{-4}$. Thus we conclude that type-I string theory is the unique solution to the monodromy bootstrap with either maximal susy or vector external states. We also introduce Tachyons to the bootstrap and demonstrate for the scattering of external vectors, the bosonic and superstring span the allowed region. Allowed regions for closed string effective field theories are obtained by implementing the KLT relations.

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arXiv2023

(Non)-projective bounds on gravitational EFT

In this paper we study both projective and non-projective constraints on four-dimensional gravitational effective fields theories implied from unitarity, causality and crossing, assuming perturbative UV completions in $M_{\rm pl}$. We derive bounds on the Wilson coefficients of $R^3$ and $D^{2n}R^4$ from its dispersive representation, utilizing both numerical semi-definite programming and analytic geometry analysis. From the former, we derive projective bounds on ratios of couplings and observe accumulation point spectrum populating the boundary of the allowed region. For the latter we consider the non-projective geometry of the EFThedron, which we relate to the known $L$-moment problem in the literature. This allows us to move beyond positivity and incorporate the upper bound from unitarity of the imaginary parts of partial waves. This leads to sharp bounds on individual coefficients, which are of order unity when normalized with respect to the UV scale. Finally, the non-projective geometry also allows us to derive optimal bounds reflecting assumptions of low-spin dominance, improving previous results. We complement the analytic analysis with a simple linear programming approach that validates the bounds.

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arXiv2022

De-projecting the EFThedron

The space of Wilson coefficients of EFT that can be UV completed into consistent theories was recently shown to be described analytically by a positive geometry, termed the EFThedron. However, this geometry, as well as complementary numerical methods of semi-definite programming, have so far focused on the positivity of the partial wave expansion, which allows bounding only ratios of couplings. In this paper we describe how the unitarity upper bound of the partial waves can be incorporated. This new problem can be formulated in terms of the well known $L$-moment problem, which we generalize and solve from a geometrical perspective. We find the non-projective generalization of the EFThedron has an infinite number of non-linear facets, which in some cases have remarkably simple descriptions. We use these results to derive bounds on single couplings, finding that the leading derivative operators are bounded by unity, when normalized by the cut-off scale and loop factors. For general operators of mass dimension $2k$ we find the upper bound is heavily suppressed at large $k$, with an $1/k$ fall-off.

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arXiv2022

Into the EFThedron and UV constraints from IR consistency

Recently it was proposed that the theory space of effective field theories with consistent UV completions can be described as a positive geometry, termed the EFThedron. In this paper we demonstrate that at the core, the geometry is given by the convex hull of the product of two moment curves. This makes contact with the well studied bi-variate moment problem, which in various instances has known solutions, generalizing the Hankel matrices of couplings into moment matrices. We are thus able to obtain analytic expressions for bounds, which perfectly match numerical results from semi-definite programing methods. Furthermore, we demonstrate that crossing symmetry in the IR imposes non-trivial constraints on the UV spectrum. In particular, permutation invariance for identical scalar scattering requires that any UV completion beyond the scalar sector must contain arbitrarily high spins, including at least all even spins $\ell\le28$, with the ratio of spinning spectral functions bounded from above, exhibiting large spin suppression. The spinning spectrum must also include at least one state satisfying a bound $m^2_{J}<M^2 \frac{( J^2-12) ( J^4 - 32 J^2 +204)}{8 (150 - 43 J^2 + 2 J^4)}$, where $J^2=\ell(\ell+1)$, and $M^2$ is the mass of the heaviest spin 2 state in the spectrum.

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