arXiv subjects

Carolina Figueiredo

Publications and source records attributed to Carolina Figueiredo.

At least 19 recordsLinked to original sources

arXiv2026

The Cosmological $δ$-shift

We consider the extension of the $δ$-shift connecting Tr$(ϕ^3)$ and pion amplitudes to observables in cosmology. We start in flat-space and explain how one can extract the pion wavefunction and correlators from those of Tr$(ϕ^3)$ theory, via a simple kinematic deformation. As opposed to the amplitude, the $δ$-shift for the wavefunction is no longer a linear deformation on the kinematics since the wavefunction depends on magnitudes rather than the squares of momenta. We "linearize" this deformation by using the frequency representation - allowing us to recast the contribution from any graph as a sum over residues. We then show how this construction extends to de Sitter, and find that, remarkably, in the frequency representation, the pion de Sitter wavefunction and correlators can be formulated in precisely the same way as in flat-space. Amongst other things, this unified formulation reveals a simple universal soft-limit for flat-space and de Sitter, where, in the soft-limit, the wavefunction turns into a sum of wavefunctions in a mixed theory of pions and scalars.

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arXiv2026

The Very Nearly Right Theory of Flavor

A striking empirical observation about the CKM matrix is that the angles of the unitarity triangle $(α, β, γ)$ are very close to $(π/2, π/8, 3 π/8) $, simple fractions of $π$ that are suggestive of an underlying theory linking flavor and spontaneous CP violation. However, relating this empirical observation to an underlying theory of flavor is challenging, since the unitarity triangle is a complicated function of the Yukawa matrices. In this letter we present a simple picture for the Yukawas where this direct link is possible. We begin by parametrizing the ten-dimensional space of flavor data via "nine-link textures", full-rank $Y_{u,d}$ matrices with a total of nine non-zero entries, with a single CP violating phase. Fitting the ten parameters of all such textures to the flavor data reveals a wonderful surprise: the CP phases cluster tightly around multiples of $π/8$! This happens because the entries of $Y_{u,d}$ naturally define a "Yukawa triangle", in most cases identical to the unitarity triangle at leading order in small flavor parameters. Most interestingly, these two triangles are not the same beyond leading order, yielding precise predictions for $(α, β, γ)$ with calculable deviation from $(π/2, π/8, 3 π/8)$, which can be decisively excluded or strongly confirmed by the next generation of experimental measurements of the angles. The 9-link textures are sparse and their determinants are naturally real, which taken together with spontaneous CP violation can resolve the strong CP problem.

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arXiv2026

Combinatorics of the Cosmohedron

The cosmohedron was recently proposed as a polytope underlying the cosmological wavefunction for $\text{Tr}(Φ^3)$ theory. Its faces were conjectured to be in bijection with Matryoshkas, which are obtained from a subdivision of a polygon by sequentially wrapping groups of polygons into larger polygons. In this paper we prove the correctness of this construction, and elucidate its combinatorial structure. Cosmohedra generalize to a wider class of $\mathcal{X}$ in $Y$ polytopes, where we chisel a polytope from the family $\mathcal{X}$ at each vertex of a polytope $Y$. We sketch a new application of these chiseled polytopes to the physics of ultraviolet divergences in loop-integrated Feynman amplitudes.

math.CO↗

arXiv2026

Generating the fermion mass hierarchy at the TeV scale

We propose a class of theories to generate quark and lepton mass matrices where the scale of new physics is at the TeV scale, without inducing the large flavor and CP violating processes that are often thought to relegate the origin of flavor to energies above $\sim 100$ TeV. The models have new vector-like leptons and quarks, with mass mixings to each other and Yukawa couplings to light Standard Model fields encoded in "chains" reminiscent of dimensional deconstruction. Locality in the chains both generates the hierarchical Standard Model Yukawa matrices, and ensures that CP and flavor violating effects are small, even with the vector-like particles at the TeV scale. A simple extension also generates neutrino masses, whose tiny size is parametrically related to the square of the electron Yukawa coupling. We outline the essential features of these models, explain how fermion mass hierarchies and mixing angles emerge, and explore their phenomenological implications. This framework can be tested both in the final run of the LHC as well as at possible future colliders operating at the 10 TeV scale, and we identify some of the distinctive experimental signatures associated with the production and decay of the new vector-like fermions.

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arXiv2025

How gluon leading singularities discover curves on surfaces

We study the leading singularities for pure gluon amplitudes obtained by on-shell gluing of three-particle amplitudes for an arbitrary graph in any number of dimensions. By encoding the polarization vector contractions in a graphical way, on-shell gluing "discovers" curves on surfaces, and we find that the leading singularity is determined by a simple combinatorial question: what are all ways of covering the graph with non-overlapping curves such that each edge is covered exactly once? This precisely matches the formula from the surfaceology formulation of gluons, where the leading singularities are given by maximal residues, with the combinatorial problem arising from the linearized form of the $u$ variables. At loop-level we describe how the novelties associated with spin sums (related with the need for ghosts when working off-shell using Lagrangians) can be easily encoded in this combinatorial picture. Matching the leading singularities also lets us settle an open question in the surface formulation of gluons, determining the exponents of the closed curves at any loop order.

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arXiv2025

Correlator Polytopes

Recently, "cosmohedra" have been introduced as polytopes underlying the cosmological wavefunction for conformally coupled Tr($Φ^3$) theory in FRW cosmologies, generalizing associahedra for flat space scattering amplitudes. In this letter we show that correlation functions are also directly captured by a new polytope - the "Correlatron". The combinatorics of correlation functions is an interesting blend of flat space scattering amplitudes and wavefunctions. This is reflected in the correlatron geometry, which is a one-higher dimensional polytope sandwiched between cosmohedron and associahedron facets. We provide an explicit embedding for the correlatron, which is a natural extension of the "shaving" picture for cosmohedra to one higher dimension. As a byproduct, we also define "graph correlahedra" as polytopes for the contribution to correlators from any fixed graph. We show how the canonical form of these polytopes directly computes the graph correlator, without the power of two weights seen in previous geometric formulations. Finally, we give a prescription for extracting the full correlator from the canonical form of the correlatron.

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arXiv2025

Cuts and Contours

The traditional formulation of string amplitudes via worldsheet integrals provides a parametrization of the moduli space that fails to expose the complete singularity structure of the amplitudes. This problem is solved by the positive parametrization of string amplitudes given by surfaceology. In this work, we use this formalism to study a number of properties of string amplitudes at tree-level and one-loop. We introduce several global prescriptions for an integration contour for which the integrals are finite everywhere in kinematic space. At tree-level, this is done in two ways: one directly implements the Feynman $i\varepsilon$ to analytically continue from Euclidean to Lorentzian worldsheets; the other is a generalization of the closed Pochhammer contour to arbitrary number of points. At loop-level, we present a systematic way of extracting cuts directly from the worldsheet integrand. This provides a powerful set of unitarity constraints, which we use to test the consistency of different "stringy" UV regularizations of field theory amplitudes. In addition, we identify the massive threshold expansion of the integrand, which allows us to reduce the problem to a finite set of Feynman integrals in Schwinger parametrization and provide a straightforward contour prescription reminiscent of its field-theory version.

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arXiv2025

Surface Gauge Invariance, Soft Limits and the Transmutation of Gluons into Scalars

Over the past year, the "scalar-scaffolding" formalism has revealed a number of new features of gluon amplitudes. In this paper, we leverage these developments to study two distinct but related questions, linked by the scaffolding statement of gauge invariance. We start by revisiting the soft expansion of gluon amplitudes. The scaffolding picture allows for a precise definition of the soft limit and a canonical way to expand the amplitude. At tree-level, this reproduces the classic Weinberg soft theorem, and at one-loop, using surface kinematics, we derive an extension of this theorem valid at the level of the loop integrand. We then switch gears and describe a new relationship between gluon and scalar amplitudes. The expression of surface gauge invariance naturally suggests a certain differential operator acting on individual external gluons. Remarkably, we find that, both for the tree-level amplitude and the surface one-loop integrand, repeated applications of this operator transmutes gluon amplitudes/integrands into those of Tr$(ϕ^3)$ scalars. This is an interesting counterpart to the $δ$-shift connection that lifts "stringy" Tr$(ϕ^3)$ amplitudes to those of gluons.

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arXiv2025

Splitting Regions and Shrinking Islands from Higher Point Constraints

We study constraints from higher-point amplitudes on $2 \to 2$ scattering in the context of effective field theory (EFT) using the perturbative numerical S-matrix bootstrap. Specifically, we investigate the class of weakly coupled EFTs with amplitudes that obey the hidden zero and split conditions that are known to hold both for Tr($Φ^3$) theory and for certain string tree amplitudes, including at 4-point the beta function. Requiring the splitting condition for the 5-point amplitude not only fixes nearly all its contact terms, but it also imposes non-linear constraints among the 4-point EFT Wilson coefficients. When included in the bootstrap, the resulting allowed region consistent with positivity is no longer convex but is restricted to a smaller non-convex region - which has a sharp corner near the string beta function! Assuming the absence of an infinite spin tower at the mass gap, the allowed region bifurcates into a trivial region (with states only above a chosen cutoff) and an island that continues to shrink around the string as more constraints are included in the bootstrap. The numerics indicate that in the absence of single-mass infinite spin towers the string beta function is the unique 4-point amplitude compatible with hidden zero and the 5-point splitting constraints. The analysis provides a prototype example for how features of higher-point amplitudes constrain the bootstrap of 4-point amplitudes.

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arXiv2025

Soft Factorisation and Exponentiation from Schwinger-Space Geometry

Infrared divergences in Quantum Field Theory govern the low-energy dynamics of many physical theories, and their understanding is a crucial ingredient in predicting the outcomes of collider experiments. We present a novel approach to deriving the structure of these divergences by employing the Schwinger parametrization of Feynman integrals. After using tropical geometry to identify divergent limits, we study the all-orders asymptotic properties of Feynman diagrams via matrix manipulations of graph Laplacians, which allows us to analyse their IR behaviour systematically. We explicitly demonstrate the soft-hard factorization of the integrand for a broad class of diagrams, and reveal that when written in terms of "worldline distances", topologically distinct diagrams asymptote to the same integrand. In particular, for the case of Quantum Electrodynamics, we use this fact to show how ladder-type diagrams combine in Schwinger-parameter space to yield the correct exponentiated soft anomalous dimension. This framework provides a foundation for extending these methods to more complex theories like Quantum Chromodynamics and offers a pathway towards a systematic understanding of infrared divergences in perturbative amplitudes.

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arXiv2025

Scalar-Scaffolded Gluons and the Combinatorial Origins of Yang-Mills Theory

We present a new formulation for Yang-Mills scattering amplitudes in any number of dimensions and at any loop order, based on the same combinatorial and binary-geometric ideas in kinematic space recently used to give an all-order description of Tr $ϕ^3$ theory. We propose that in a precise sense the amplitudes for a suitably "stringy" form of these two theories are identical, up to a simple shift of kinematic variables. This connection is made possible by describing the amplitudes for $n$ gluons via a "scalar scaffolding", arising from the scattering of $2n$ colored scalars coming in $n$ distinct pairs of flavors fusing to produce the gluons. Fundamental properties of the "$u$-variables", describing the "binary geometry" for surfaces appearing in the topological expansion, magically guarantee that the kinematically shifted Tr $ϕ^3$ amplitudes satisfy the physical properties needed to be interpreted as scaffolded gluons. These include multilinearity, gauge invariance, and factorization on tree- and loop- level gluon cuts. Our "stringy" scaffolded gluon amplitudes coincide with amplitudes in the bosonic string for extra-dimensional gluon polarizations at tree-level, but differ (and are simpler) at loop-level. We provide many checks on our proposal, including matching non-trivial leading singularities through two loops. The simple counting problem underlying the $u$ variables autonomously "knows" about everything needed to convert colored scalar to gluon amplitudes, exposing a striking "discovery" of Yang-Mills amplitudes from elementary combinatorial ideas in kinematic space.

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arXiv2025

Surface Kinematics and "The" Yang-Mills Integrand

It has been a long-standing challenge to define a canonical loop integrand for non-supersymmetric gluon scattering amplitudes in the planar limit. Naive integrands are inflicted with $1/0$ ambiguities associated with tadpoles and massless external bubbles, which destroy integrand-level gauge invariance as well as consistent on-shell factorization on single loop-cuts. In this letter, we show that this essentially kinematical obstruction to defining "the" integrand for Yang-Mills theory has a structural solution, handed to us by the formulation of gluon amplitudes in terms of curves on surfaces. This defines "surface kinematics" generalizing momenta, making it possible to define "the" integrand satisfying both a (surface generalized) notion of gauge-invariance and consistent loop-cuts. The integrand also vanishes at infinity in appropriate directions, allowing it to be recursively computed for non-supersymmetric Yang-Mills theory in any number of dimensions. We illustrate these ideas through one loop for all multiplicity, and for the simplest two-loop integrand.

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arXiv2025

Open String Amplitudes: Singularities, Asymptotics, and New Representations

Open string amplitudes at tree level have been studied for over fifty years, but there is no known analytic form for general $n$-point amplitudes, and their conventional representation in terms of worldsheet integrals does not make many of their most basic physical properties manifest. Recently, a formulation of these amplitudes exposing the underlying "binary geometry" via the use of "$u$" variables has given us many insights into their basic features. In this paper, we initiate a systematic exploration of fundamental aspects of open string amplitudes from this new point of view. We begin by giving explicit expressions for the factorization of amplitudes at general massive levels. We then study the asymptotic behavior when subsets of kinematic variables become large, delineating regimes with exponential (generalized hard scattering) and power-law (generalized Regge) behavior. We also give precise expressions for the asymptotics, which reveal another example of the recently observed property of factorization away from poles. We finally derive new recursion relations and infinite series representations for the amplitude, and for five points, we present a new closed-form expression for the amplitude that for the first time gives its analytic continuation to all of kinematic space.

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arXiv2024

Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons

Recent years have seen the emergence of a new understanding of scattering amplitudes in the simplest theory of colored scalar particles - the Tr$(ϕ^3)$ theory - based on combinatorial and geometric ideas in the kinematic space of scattering data. In this paper we report a surprise: far from the toy model it appears to be, the ''stringy'' Tr$(ϕ^3)$ amplitudes secretly contain the scattering amplitudes for pions, as well as non-supersymmetric gluons, in any number of dimensions. The amplitudes for the different theories are given by one and the same function, related by a simple shift of the kinematics. This discovery was spurred by another fundamental observation: the tree-level Tr$(ϕ^3)$ field theory amplitudes have a hidden pattern of zeros when a special set of non-planar Mandelstam invariants is set to zero. Furthermore, near these zeros, the amplitudes simplify, by factoring into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for pions and gluons are observed to also vanish in the same kinematical locus. These properties further generalize to the ''stringy'' Tr$(ϕ^3)$ amplitudes. There is a unique shift of the kinematic data that preserves the zeros, and this shift is precisely the one that unifies colored scalars, pions, and gluons into a single object. We will focus in this paper on explaining the hidden zeros and factorization properties and the connection between all the colored theories, working for simplicity at tree-level. Subsequent works will describe this new formulation for the Non-linear Sigma Model and non-supersymmetric Yang-Mills theory, at all loop orders.

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arXiv2024

NLSM $\subset$ Tr$(ϕ^3)$

Scattering amplitudes for the simplest theory of colored scalar particles - the Tr($Φ^3$) theory - have recently been the subject of active investigations. In this letter we describe an unanticipated wider implication of this work: the Tr($Φ^3$) theory secretly contains Non-linear Sigma Model (NLSM) amplitudes to all loop orders. The NLSM amplitudes are obtained from Tr$(Φ^3)$ amplitudes by a unique shift of kinematic variables. We show that this shifted kinematics produces amplitudes for a cubic theory with a linear term in potential, with extrema spontaneously breaking $U(N) \to U(N-k) \times U(k)$. The Goldstone amplitudes for this theory coincide with those of pions in the $U(N) \times U(N) \to U(N)$ chiral Lagrangian to all orders in the planar limit. We also give a purely on-shell understanding of this correspondence, showing integrands defined by the kinematic shifts have the correct residues on poles and appropriately produce the Adler zero. Finally, we discuss how similar kinematic shifts produce certain infinite classes of mixed amplitudes of pions and Tr($Φ^3$) scalars, most of which are not interpretable from the Lagrangian description.

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arXiv2024

Cosmohedra

It has been a long-standing challenge to find a geometric object underlying the cosmological wavefunction for Tr($ϕ^3$) theory, generalizing associahedra and surfacehedra for scattering amplitudes. In this note we describe a new class of polytopes -- "cosmohedra" -- that provide a natural solution to this problem. Cosmohedra are intimately related to associahedra, obtained by "blowing up" faces of the associahedron in a simple way, and we provide an explicit realization in terms of facet inequalities that further "shave" the facet inequalities of the associahedron. We also discuss a novel way for computing the wavefunction from cosmohedron geometry that extends the usual connection with polytope canonical forms. We illustrate cosmohedra with examples at tree-level and one loop; the close connection to surfacehedra suggests the generalization to all loop orders. We also briefly describe "cosmological correlahedra" for full correlators. We speculate on how the existence of cosmohedra might suggest a "stringy" formulation for the cosmological wavefunction/correlators, generalizing the way in which the Minkowski sum decomposition of associahedra naturally extend particle to string amplitudes.

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arXiv2024

All-order splits and multi-soft limits for particle and string amplitudes

The most important aspects of scattering amplitudes have long been thought to be associated with their poles. But recently a very different sort of "split" factorizations for a wide range of particle and string tree amplitudes have been discovered away from poles. In this paper, we give a simple, conceptual origin for these splits arising from natural properties of the binary geometry of the curve integral formulation for scattering amplitudes for Tr$(Φ^3)$ theory. The most natural way of "joining" smaller surfaces to build larger ones directly produces a choice of kinematics for which higher amplitudes factor into lower ones. This gives a generalization of splits to all orders in the topological expansion. These splits allow us to access and compute loop-integrated multi-soft limits for particle and string amplitudes, at all loop orders. This includes split factorizations and multi-soft limits for pion and gluon amplitudes, that are related to Tr$(Φ^3)$ theory by a simple kinematical shift.

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arXiv2024

Multiparticle Factorization and the Rigidity of String Theory

Is string theory uniquely determined by self-consistency? Causality and unitarity seemingly permit a multitude of putative deformations, at least at the level of two-to-two scattering. Motivated by this question, we initiate a systematic exploration of the constraints on scattering from higher-point factorization, which imposes extraordinarily restrictive sum rules on the residues and spectra defined by a given amplitude. These bounds handily exclude several proposed deformations of the string: the simplest "bespoke" amplitudes with tunable masses and a family of modified string integrands from "binary geometry." While the string itself passes all tests, our formalism directly extracts the three-point amplitudes for the low-lying string modes without the aid of worldsheet vertex operators.

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