arXiv subjects

Qu Cao

Publications and source records attributed to Qu Cao.

At least 19 recordsLinked to original sources

arXiv2026

The Hydrotope in the Water-Wave Action

The Hydrotope gives a geometric representation of tree-level water-wave amplitudes with two negative spatial momenta as the volume of a hyperplane slice of a box. We trace the origin of this geometry directly to the water-wave action. Writing the $n$-point contact interaction as $\mathcal{V}_n=\sum_{i<j}w_iw_jh_{ij}^{(n)}$, we show that when the two marked momenta have the same sign and every spectator has the opposite sign, the corresponding coefficient is $h_{ij}^{(n)}=2H_n$, where $H_n/(n{-}3)!$ is the Hydrotope volume. More generally, every fixed-pair coefficient admits a denominator-free ordered-flag representation as an oriented sum of $(n{-}3)$-dimensional box volumes. We then sum all two-minus trees by cutting each at the unique vertex joining its two minus branches. The coefficients multiplying minus-minus, minus-plus, and plus-plus frequency bilinears reduce, respectively, to $(2^{n{-}1}{-}2)H_n$, $0$, and $2H_n$, and immediately reproduce the known amplitude $2^{n{-}1}w_1w_2H_n$. Thus, the Hydrotope-and a broader class of related box-slice geometries-is already encoded locally in the water-wave action. This points to further hidden simplicity and geometric structure in general water-wave amplitudes.

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arXiv2026

NLSM amplitudes from a quartic two-derivative theory

We revisit the well-known nonlinear sigma model (NLSM), an effective field theory describing the scattering of $\mathrm{SU}(N)$ Goldstone bosons and characterized by an infinite tower of two-derivative interactions. We introduce a local scalar Lagrangian involving two scalar fields $\psi^\pm$ carrying opposite "polarities", whose interacting part consists of a single polynomial quartic two-derivative operator, and prove that it reproduces planar NLSM amplitudes at all loop orders. This quartic two-derivative formulation reveals a previously hidden simplicity of the NLSM: its Feynman rules involve only a single quartic interaction vertex, making the Adler zero and the leading double-soft factor manifest at all loop orders on generalized cuts and significantly improving the efficiency of high-multiplicity computations. We also suggest a possible analogous description of the NLSM +$ \phi^3 $ theory in terms of a finite tower of local interactions.

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arXiv2026

Uniqueness and Analytic Structures of Bosonic String Effective Amplitudes

We revisit the zero-transcendentality sector of bosonic string effective amplitudes with spin-1 external states, conjectured to correspond to a mass-deformed $(DF)^2$ theory, known as the $(DF)^2{+}\text{YM}$ theory. Imposing gauge invariance, locality, and cyclicity under minimal assumptions uniquely fixes a set of dimension-raising operators and leads to a recursive construction of amplitudes from Yang-Mills amplitudes in the $\alpha'{\to}0$ limit. At finite $\alpha'$, certain derivative operators dressed with gauge invariant and $\alpha'$-dependent factors, what we call $\textit{inverse operators}$, reconstruct the full bosonic string effective amplitudes, yielding compact expressions that universally factorize into tachyon-pole coefficients times Yang-Mills-Scalar amplitudes. This structure holds at arbitrary multiplicity and also extends to the amplitudes of the pure $(DF)^2$, $(DF)^2{+}\phi^{3}$ and $(DF)^2{+}\text{YM}{+}\phi^{3}$ theories.

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arXiv2025

Loop-Level Double Copy Relations from Forward Limits

We study double copy relations for loop integrands in gauge theories and gravity based on their constructions from single cuts, which are in turn obtained from forward limits of lower-loop cases. While such a construction from forward limits has been realized for loop integrands in gauge theories, we demonstrate its extension to gravity by reconstructing one-loop gravity integrands from forward limits of trees. Under mild symmetry assumptions on tree-level kinematic numerators (and their forward limits), our method directly leads to double copy relations for one-loop integrands: these include the field-theoretic Kawai-Lewellen-Tye (KLT) relations, whose kernel is the inverse of a matrix with rank $(n{-}1)!$ formed by those in bi-adjoint $\phi^3$ theory, and the Bern-Carrasco-Johansson (BCJ) double copy relations with crossing-symmetric kinematic numerators (we provide local and crossing-symmetric Yang-Mills BCJ numerators for $n=3,4,5$ explicitly). By exploiting the "universal expansion" for one-loop integrands in generic gauge theories, we also obtain an analogous expansion for gravity (including supergravity theories).

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arXiv2025

Superstring amplitudes meet surfaceology

We reformulate tree-level amplitudes in open superstring theory (type-I) in terms of stringy Tr$(\phi^3)$ amplitudes with various kinematical shifts in the "curve-integral" formulation: while the bosonic-string amplitude with $n$ pairs of "scaffolding" scalars comes from a particularly simple shift of the Tr$(\phi^3)$ one (corresponding to $n$ length-$2$ cycles), the analogous superstring amplitude requires "correction" terms given by bosonic-string amplitudes with longer, even-length "cycles", which are also Tr$(\phi^3)$ ones at shifted kinematics dictated by the cycles; in total it is expressed as a sum of $(2n{-}3)!!$ shifted amplitudes originated from the expansion of a reduced Pfaffian. Upon taking $n$ scaffolding residues, this leads to a new formula of the $n$-gluon superstring amplitude, which is manifestly symmetric in $n{-}1$ legs, as a gauge-invariant combination of mixed bosonic string amplitudes with gluons and scalars, which come from length-$2$ cycles and longer ones respectively (the total sum is associated with the expansion a $n\times n$ symmetrical determinant); the corresponding prefactors are nested commutators of $2n$-gon kinematical variables, which nicely become traces of field-strengths for those legs corresponding to scalars in the mixed amplitudes. These interesting linear combinations of bosonic string amplitudes must guarantee the cancellation of tachyon poles and $F^3$ vertices ${\it etc.}$, and they give new relations between the superstring amplitude and its bosonic-string building blocks to all orders in the $\alpha'$ expansion (the first order gives a new formula for gluon amplitudes with a single $F^3$ insertion in terms of Yang-Mills-scalar amplitudes). We provide both the worldsheet and "curve-integral" derivations, and discuss applications to heterotic and type II cases.

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arXiv2025

Form factors from string amplitudes

In this letter, we propose a stringy model for $n$-point tree-level form factor with the off-shell operator in the scalar and gluon theories, from the bosonic string disk amplitude: $n$ open string states and $1$ closed string state scatter on the disk. In the field-theory limit ($\alpha'\to0$), the stringy form factor reduces to the form factor, helps us to investigate the hidden properties of the field-theory form factors, manifest the factorization and soft behaviors, and uncover more non-trivial relations between form factors and scattering amplitudes.

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arXiv2025

All-loop planar integrands in Yang-Mills theory from recursion

In this letter, we generalize the recursion methods based on cut equations arXiv:2412.21027, originally developed for scalar theories, to gluons in pure Yang-Mills theory. In gauge theories, planar loop integrands are subtle to defined and obtained due to the existence of scaleless integrals. A critical challenge arises when constructing higher-loop integrands via recursion methods: integrands with or without all scaleless terms both are incompatible with the reconstruction formalism in arXiv:2412.21027. To address this, we introduce a $\textit{refined}$ integrand by systematically removing specific scaleless contributions, and develop an algorithmic implementation of the recursion to all-loop level. We explicitly demonstrate the framework by three steps and obtain the recursion formula in pure Yang-Mills theory. In the ancillary files, we provide the results up to the two-loop five-point integrand and the simplified result in the large-$D$ limit for the three-loop four-point case.

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arXiv2024

On one-loop amplitudes in gauge theories

We propose a new ``universal expansion" for one-loop amplitudes with arbitrary number of gluons in $D$ dimensions, which holds for general gauge theories with gluons/fermions/scalars in the loop, including pure and supersymmetric Yang-Mills theories. It expresses the $n$-gluon amplitudes as a linear combination of universal scalar-loop amplitudes with $n{-}m$ gluons and $m$ scalars, multiplied by gauge-invariant building blocks (defined for general gauge theories); the integrands of these scalar-loop amplitudes are given in terms of tree-level objects attached to the scalar loop, or by differential operators acting on the most important part which is proportional to $D$ (with $m=0$). We present closed-formula for these one-loop integrands and prove them by showing that the single cuts are correctly reproduced by the gluing of an additional pair of gluons (fermions/scalars) in the forward limit, plus $n$ gluons in a tree amplitude.

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arXiv2024

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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arXiv2024

Supergluon scattering in AdS: constructibility, spinning amplitudes, and new structures

We elaborate on a new recursive method proposed in arXiv:2312.15484 for computing tree-level $n$-point supergluon amplitudes as well as those with one gluon, i.e., spinning amplitudes, in ${\rm AdS}_5 \times S^3$. We present an improved proof for the so-called "constructibility" of supergluon and spinning amplitudes based on their factorizations and flat-space limit, which allows us to determine these amplitudes in Mellin space to all $n$. We present explicit and remarkably simple expressions for up to $n=7$ supergluon amplitudes and $n=6$ spinning amplitudes, which can be viewed as AdS generalizations of the scalar-scaffolded gluon amplitudes proposed recently. We then reveal a series of hidden structures of these AdS amplitudes including (1) an understanding of general pole structures especially the precise truncation on descendent poles (2) a derivation of simple "Feynman rules" for the all-$n$ amplitudes with the simplest R-symmetry structures, and (3) certain universal behavior analogous to the soft/collinear limit of flat-space amplitudes.

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arXiv2024

On universal splittings of tree-level particle and string scattering amplitudes

In this paper, we study the newly discovered universal splitting behavior for tree-level scattering amplitudes of particles and strings~\cite{Cao:2024gln}: when a set of Mandelstam variables (and Lorentz products involving polarizations for gluons/gravitons) vanish, the $n$-point amplitude factorizes as the product of two lower-point currents with $n{+}3$ external legs in total. We refer to any such subspace of the kinematic space of $n$ massless momenta as "2-split kinematics", where the scattering potential for string amplitudes and the corresponding scattering equations for particle amplitudes nicely split into two parts. Based on these, we provide a systematic and detailed study of the splitting behavior for essentially all ingredients which appear as integrands for open- and closed-string amplitudes as well as Cachazo-He-Yuan (CHY) formulas, including Parke-Taylor factors, correlators in superstring and bosonic string theories, and CHY integrands for a variety of amplitudes of scalars, gluons and gravitons. These results then immediately lead to the splitting behavior of string and particle amplitudes in a wide range of theories, including bi-adjoint $\phi^3$ (with string extension known as $Z$ and $J$ integrals), non-linear sigma model, Dirac-Born-Infeld, the special Galileon, etc., as well as Yang-Mills and Einstein gravity (with bosonic and superstring extensions). Our results imply and extend some other factorization behavior of tree amplitudes considered recently, including smooth splittings~\cite{Cachazo:2021wsz} and factorizations near zeros~\cite{Arkani-Hamed:2023swr}, to all these theories. A special case of splitting also yields soft theorems for gluons/gravitons as well as analogous soft behavior for Goldstone particles near their Adler zeros.

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arXiv2024

A universal splitting of string and particle scattering amplitudes

We propose a new splitting behavior of tree-level string/particle amplitudes for scalars, gluons and gravitons. We identify certain subspaces in the space of Mandelstam variables, where the universal Koba-Nielsen factor splits into two parts (each with an off-shell leg). Both open- and closed-string amplitudes with Parke-Taylor factors naturally factorize into two stringy ${\it currents}$, which implies the splitting of bi-adjoint $\phi^3$ amplitudes and via a simple deformation to unified stringy amplitudes, the splitting of amplitudes in the non-linear sigma model and Yang-Mills-scalar theory; the same splitting holds for scalar amplitudes without color such as the special Galileon. Remarkably, if we impose similar constraints on Lorentz products involving polarizations, gluon and graviton amplitudes in bosonic string and superstring theories also split into two (stringy) currents. A special case of the splitting implies soft theorems, and more generally it extends recently proposed smooth splittings and new factorizations near zeros to all these theories.

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arXiv2024

NLSM $\subset$ Tr$(\phi^3)$

Scattering amplitudes for the simplest theory of colored scalar particles - the Tr($\Phi^3$) theory - have recently been the subject of active investigations. In this letter we describe an unanticipated wider implication of this work: the Tr($\Phi^3$) theory secretly contains Non-linear Sigma Model (NLSM) amplitudes to all loop orders. The NLSM amplitudes are obtained from Tr$(\Phi^3)$ 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($\Phi^3$) scalars, most of which are not interpretable from the Lagrangian description.

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arXiv2023

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 $\phi^3$ 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 $\phi^3$ 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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arXiv2023

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$(\phi^3)$ 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$(\phi^3)$ 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$(\phi^3)$ 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$(\phi^3)$ 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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arXiv2023

Constructibility of AdS supergluon amplitudes

We prove that all tree-level $n$-point supergluon (scalar) amplitudes in AdS$_5$ can be recursively constructed, using factorization and flat-space limit. Our method is greatly facilitated by a natural R-symmetry basis for planar color-ordered amplitudes, which reduces the latter to "partial amplitudes" with simpler pole structures and factorization properties. Given the $n$-point scalar amplitude, we first extract spinning amplitudes with $n{-}2$ scalars and one gluon by imposing "gauge invariance", and then use a special "no-gluon kinematics" to determine the $(n{+}1)$-point scalar amplitude completely (which in turn contains the $n$-point single-gluon amplitude). Explicit results of up to 8-point scalar amplitudes and up to 6-point single-gluon amplitudes are included as Supplemental Material.

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arXiv2023

Cutting the traintracks: Cauchy, Schubert and Calabi-Yau

In this note we revisit the maximal-codimension residues, or leading singularities, of four-dimensional $L$-loop traintrack integrals with massive legs, both in Feynman parameter space and in momentum (twistor) space. We identify a class of "half traintracks" as the most general degenerations of traintracks with conventional (0-form) leading singularities, although the integrals themselves still have rigidity $\lfloor\frac{L-1}2\rfloor$ due to lower-loop "full traintrack'' subtopologies. As a warm-up exercise, we derive closed-form expressions for their leading singularities both via (Cauchy's) residues in Feynman parameters, and more geometrically using the so-called Schubert problems in momentum twistor space. For $L$-loop full traintracks, we compute their leading singularities as integrals of $(L{-}1)$-forms, which proves that the rigidity is $L{-}1$ as expected; the form is given by an inverse square root of an irreducible polynomial quartic with respect to each variable, which characterizes an $(L{-}1)$-dim Calabi-Yau manifold (elliptic curve, K3 surface, etc.) for any $L$. We also briefly comment on the implications for the "symbology" of these traintrack integrals.

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arXiv2022

Covariant color-kinematics duality, Hopf algebras and permutohedra

Based on the covariant color-kinematics duality, we investigate combinatorial and algebraic structures underlying their Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes in Yang-Mills-scalar (YMS) theory. The closed-formulae for BCJ numerators of YMS amplitudes and the pure-Yang-Mills (YM) ones exhibit nice quasi-shuffle Hopf algebra structures, and interestingly they can be viewed as summing over boundaries of all dimensions of a combinatorial permutohedron. In particular, the numerator with two scalars and $n{-}2$ gluons contains Fubini number ( $F_{n{-}2}$ ) of terms in one-to-one correspondence with boundaries of a $(n{-}3)$-dimensional permutohedron, and each of them has its own spurious-pole structures and a gauge-invariant numerator (both depending on reference momenta). From such Hopf algebra or permutohedron structure, we derive new recursion relations for the numerators and intriguing "factorization" on each spurious pole/facet of the permutohedron. Similar results hold for general YMS numerators and the pure-YM ones. Finally, with a special choice of reference momenta, our results imply BCJ numerators in a heavy-mass effective field theory with two massive particles and $n{-}2$ gluons/gravitons: we observe highly nontrivial cancellations in the heavy-mass limit, leading to new formulae for the effective numerators which resemble those obtained in recent works.

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