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Andrzej Pokraka

Publications and source records attributed to Andrzej Pokraka.

At least 19 recordsLinked to original sources
arXiv2026

A Graphical Coaction for FRW Wavefunction Coefficients

We show that the wavefunction of the universe in theories of conformally coupled scalars in power-law Friedmann-Robertson-Walker (FRW) cosmologies satisfies a graphical coaction, by means of which we can understand its complete analytic structure in terms of the acyclic minors of Feynman graphs. Our construction extends to all particle multiplicities and any loop order, and if we isolate certain weight-one contributions, it reproduces the ``kinematic flow'' that encodes the differential equation of the wavefunction coefficients. Similarly, any discontinuity of the wavefunction coefficient is easily extracted from the coaction.

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arXiv2026

Magic Relations and Critical Varieties of Feynman Integrals

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

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arXiv2026

Direct-to-Symbol Integration from Landau Analysis

We propose a Landau-analysis-inspired strategy for directly computing the symbol of integrals that evaluate to multiple polylogarithms in twisted cohomology. The central idea is to identify all ways in which singular hypersurfaces, twisted hypersurfaces, and integration boundaries can interact when external parameters are varied, giving rise to logarithmic or algebraic branch points. By tracking how an integral is modified when analytically continued around each of its branch points, we can recursively construct its symbol. We illustrate this approach by outlining an efficient algorithm for computing the symbol of finite integrals over twisted hyperplane arrangements, when they are expanded around special values of the twist parameter. This algorithm can be used to compute the (transcendental) integrands of cosmological correlators in conformally coupled theories to any order in the twist expansion.

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arXiv2026

A Graphical Coaction for FRW Integrals from Partial/Relative Twisted (Co)homology

We construct a graphical coaction for Friedmann-Robertson-Walker (FRW) integrals at all loop orders in conformally-coupled scalar theories with non-conformal polynomial interactions. Our construction makes use of intersection theory in the context of (partial/relative) twisted (co)homology, which we use to decompose FRW integrals (and their discontinuities and derivatives) into building blocks that can be represented as decorations of the original Feynman diagram. This facilitates a purely graphical description of the coaction, up to rational prefactors that can be read off from the graph. Our construction provides a comprehensive combinatorial framework for dissecting the analytic properties of cosmological observables; in particular, we demonstrate that the combinatorics of the differential equations that govern FRW integrals -- their so-called kinematic flow -- is a natural consequence of our coaction. We have also developed a user-friendly web application that computes the graphical coaction of any graph: https://frwcoaction.ca. Whenever possible, the web application also computes the differentials and discontinuities. A Mathematica notebook with the same functionality is also hosted at on a public GitHub repository.

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arXiv2026

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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arXiv2025

Toward the Analytic Bootstrap of Energy Correlators

In this paper, we present a framework for the analytic bootstrap of three-point energy correlators, a crucial observable in $\mathcal{N}=4$ super Yang-Mills theory and quantum chromodynamics (QCD). Our approach combines spherical contour techniques, general physical constraints such as pole cancellations, and power correction data in the singular limits to determine its analytic expression. In contrast to previous bootstrap studies restricted to scattering amplitudes for supersymmetric theories, our framework makes use of the properties of Feynman integrals, marking a significant step toward bootstrapping realistic QCD observables. Using this method, we derive analytic expressions for leading-order three-point energy correlators with equal and unequal energy weights, where the latter are crucial ingredients for projected $N$-point energy correlators. We also apply the recently developed technique of analytic regression with lattice reduction as a way to bypass needing explicit expressions for the singular limits. Bridging theoretical advances in scattering amplitudes with the renewed interest in weighted cross-sections, our work opens the door to precision tests of QCD dynamics through analytic event-shape predictions.

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arXiv2025

A double copy from twisted (co)homology at genus g

We study a family of generalized hypergeometric integrals defined on punctured Riemann surfaces of genus g. These integrals are closely related to g-loop string amplitudes in chiral splitting, where one leaves the loop-momenta, moduli and all but one puncture un-integrated. We study the twisted homology groups associated to these integrals, and determine their intersection numbers. We make use of these homology intersection numbers to write a double-copy formula for the "complex" version of these integrals -- their closed-string analogues. To verify our findings, we develop numerical tools for the evaluation of the integrals in this work. This includes the recently introduced Enriquez kernels -- integration kernels for higher-genus polylogarithms.

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arXiv2025

Kinematic flow from the flow of cuts

The wavefunction coefficients of conformally coupled scalars in power-law FRW cosmologies satisfy differential equations governed by a set of simple combinatorial rules known as the kinematic flow. In this paper we derive the kinematic flow, expressed using a set of differential forms referred to as the cut basis, from a geometric perspective, relying solely on the cosmological hyperplane arrangement and without invoking bulk physics. Each element of the cut basis corresponds to the positive geometry associated to an independent cut of the physical FRW-form and can be labeled by decorating (minors of) the truncated Feynman graph with an acyclic orientation. We provide a straightforward prescription to associate a logarithmic differential form to each element of the cut basis by considering its corresponding decorated graph. Moreover, we show that the residues of the physical FRW-form are canonical forms of certain graphical zonotopes labeled by the same set of decorated graphs. These zonotopes control the cut combinatorics -- flow of cuts -- of the physical FRW-form and the cut basis (by construction). Using the theory of relative twisted cohomology and intersection theory, we derive a closed form formula for the differential equations of the cut basis. We also introduce combinatorial rules that compute the kinematic differential of any basis element without explicit calculation. The combinatorics of our differential equations is a natural consequence of the flow of cuts and is equivalent (up to rescaling) to the kinematic flow for the recently studied time integral basis. In particular, our differential equations decouple into exponentially many sectors, one for each way of cutting a subset of edges of the graph.

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arXiv2025

Hidden Zeros of the Cosmological Wavefunction

Motivated by the recent discovery of hidden zeros in particle and string amplitudes, we characterize zeros of individual graph contributions to the cosmological wavefunction of a scalar field theory. We demonstrate that these contributions factorize near these zeros for all tree graphs and provide evidence that this extends to loop graphs as well. We explicitly construct polytopal realizations of the relevant graph associahedra and show that the cosmological zeros have natural geometric and physical interpretations. As a byproduct, we establish an equivalence between the wavefunction coefficients of chain graphs and flat-space Tr$(ϕ^3)$ amplitudes, enabling us to leverage the cosmological zeros to uncover the recently discovered hidden zeros of colored amplitudes.

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arXiv2025

A physical basis for cosmological correlators from cuts

Significant progress has been made in our understanding of the analytic structure of FRW wavefunction coefficients, facilitated by the development of efficient algorithms to derive the differential equations they satisfy. Moreover, recent findings indicate that the twisted cohomology of the associated hyperplane arrangement defining FRW integrals overestimates the number of integrals required to define differential equations for the wavefunction coefficient. We demonstrate that the associated dual cohomology is automatically organized in a way that is ideal for understanding and exploiting the cut/residue structure of FRW integrals. Utilizing this understanding, we develop a systematic approach to organize compatible sequential residues, which dictates the physical subspace of FRW integrals for any $n$-site, $\ell$-loop graph. In particular, the physical subspace of tree-level FRW wavefunction coefficients is populated by differential forms associated to cuts/residues that factorize the integrand of the wavefunction coefficient into only flat space amplitudes. After demonstrating the validity of our construction using intersection theory, we develop simple graphical rules for cut tubings that enumerate the space of physical cuts and, consequently, differential forms without any calculation.

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arXiv2024

Cosmology meets cohomology

The cosmological polytope and bootstrap programs have revealed interesting connections between positive geometries, modern on-shell methods and bootstrap principles studied in the amplitudes community with the wavefunction of the Universe in toy models of FRW cosmologies. To compute these FRW correlators, one often faces integrals that are too difficult to evaluate by direct integration. Borrowing from the Feynman integral community, the method of (canonical) differential equations provides an efficient alternative for evaluating these integrals. Moreover, we further develop our geometric understanding of these integrals by describing the associated \emph{relative} twisted cohomology. Leveraging recent progress in our understanding of relative twisted cohomology in the Feynman integral community, we give an algorithm to predict the basis size and simplify the computation of the differential equations satisfied by FRW correlators.

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arXiv2024

Surfaceology for Colored Yukawa Theory

Arkani-Hamed and collaborators have recently shown that scattering amplitudes for colored theories can be expressed as integrals over combinatorial objects simply constructed from surfaces decorated by kinematic data. In this paper we extend the curve integral formalism to theories with colored fermionic matter and present a compact formula for the all-loop, all-genus, all-multiplicity amplitude integrand of a colored Yukawa theory. The curve integral formalism makes certain properties of the amplitudes manifest and repackages non-trivial numerators into a single combinatorial object. We also present an efficient formula for $L$-loop integrated amplitudes in terms of a sum over $2^L$ combinatorial determinants.

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arXiv2024

A double copy from twisted (co)homology at genus one

We study the twisted (co)homology of a family of genus-one integrals -- the so called Riemann-Wirtinger integrals. These integrals are closely related to one-loop string amplitudes in chiral splitting where one leaves the loop-momentum, modulus and all but one puncture un-integrated. While not actual one-loop string integrals, they share many properties and are simple enough that the associated twisted (co)homologies have been completely characterized [Goto2022,arXiv:2206.03177]. Using intersection numbers -- an inner product on the vector space of allowed differential forms -- we derive the Gauss-Manin connection for two bases of the twisted cohomology providing an independent check of [Mano&Watanabe2012]. We also use the intersection index -- an inner product on the vector space of allowed contours -- to derive a double-copy formula for the closed-string analogues of Riemann-Wirtinger integrals (one-dimensional integrals over the torus). Similar to the celebrated KLT formula between open- and closed-string tree-level amplitudes, these intersection indices form a genus-one KLT-like kernel defining bilinears in meromorphic Riemann-Wirtinger integrals that are equal to their complex counterparts.

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arXiv2024

Five-dimensional spinor helicity for all masses and spins

We develop a spinor helicity formalism for five-dimensional scattering amplitudes of any mass and spin configuration. While five-dimensional spinor helicity variables have been previously studied in the context of N=2,4 supersymmetric Yang-Mills scattering amplitudes with spin less than two arXiv:2202.08257, we propose an alternative viewpoint that stems from d-dimensional spinor helicity variables avoiding the use of the exceptional low-dimensional isomorphism $SO(4,1) \cong USp(2,2)$ and the decomposition of a massive momentum into the sum of two massless momenta. By enumerating all possible independent little group tensors, we systematically build the full space of five-dimensional three-point tree-level scattering amplitudes for any configuration of spins and masses. Furthermore, we provide a prescription for computing the high energy limit of scattering amplitudes written in our spinor helicity variables. We also expect that our formalism will be applicable to effective field theories with higher spin, in particular, the scattering of highly spinning black holes in five dimensions.

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arXiv2024

The soaring kite: a tale of two punctured tori

We consider the 5-mass kite family of self-energy Feynman integrals and present a systematic approach for constructing an epsilon-form basis, along with its differential equation pulled back onto the moduli space of two tori. Each torus is associated with one of the two distinct elliptic curves this family depends on. We demonstrate how the locations of relevant punctures, which are required to parametrize the full image of the kinematic space onto this moduli space, can be extracted from integrals over maximal cuts. A boundary value is provided such that the differential equation is systematically solved in terms of iterated integrals over g-kernels and modular forms. Then, the numerical evaluation of the master integrals is discussed, and important challenges in that regard are emphasized. In an appendix, we introduce new relations between g-kernels.

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arXiv2023

Two-point sum-rules in three-dimensional Yang-Mills theory

We compute the stress-tensor two-point function in three-dimensional Yang-Mills theory to three-loops in perturbation theory. Using its calculable shape at high momenta, we test the notion that its Borel transform is saturated at low energies by the lowest glueball state(s). This assumption provides relatively stable estimates for the mass of the lightest glueball that we compare with lattice simulations. We also provide estimates for the coupling of the lightest glueball to the stress tensor. Along the way, we comment on the extent that such estimates are non-rigorous. Lastly, we discuss the possibility of applying the sum-rule analysis to two-point functions of higher-spin operators and obtain a crude approximation for the glueball couplings to these operators.

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arXiv2023

Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an epsilon-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of $q$-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known epsilon-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a "good" loop-by-loop basis at three-loop.

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