arXiv subjects

Xuhang Jiang

Publications and source records attributed to Xuhang Jiang.

16 recordsLinked to original sources

arXiv2026

Notes on the bootstrap of four-point conformal integrals

We set up a bootstrap workflow to study four-point conformal integrals in position space, using leading singularities, single-valued multiple polylogarithmic ansätze and boundary data from expansion by regions. These four-point conformal integrals are general in the sense that they are generated by the four-point projections of all possible $f$-graphs, including all non-planar $f$-graph sectors. For three-loop cases, fourteen of the fifteen inequivalent integrand basis can be directly calculated by \texttt{HyperlogProcedures} and the last one is fixed by Gram identity. Then we concentrate on how far the bootstrap workflow can go for four-loop cases, though it works for three-loop cases as well. We show that integrals with several leading singularities can be made tractable by decomposing them into pieces with simpler cut structure. Some four-loop integrals which can not be calculated or very hard to be calculated by other methods for now are obtained in this way. We also provide a package with skill files which is suitable to be read and used by current AI models.

hep-th↗

arXiv2026

On Solving Dual Conformal Integrals in Coulomb-branch Amplitudes and Their Periods

We define and study infinite families of all-loop planar, dual conformal invariant (DCI) integrals, which contribute to four-point Coulomb-branch amplitudes and correlators in ${\cal N}=4$ supersymmetric Yang-Mills theory, by solving ``boxing'' differential equations via \texttt{HyperlogProcedures}~\cite{hyperlogprocedures}; The resulting single-valued harmonic polylogarithmic functions (SVHPL) are nicely labeled by ``binary'' strings of $0$ and $1$ without consecutive $1$'s. These functions are special cases of the so-called generalized ladders studied in~\cite{Drummond:2012bg}, where extended Steinmann relations (no consecutive $1$'s) are imposed due to planarity. Our results can be viewed as ``two-dimensional'' extensions of the well-known ladder integrals to many more infinite families of DCI integrals: the ladders have strings with a single $1$ followed by all $0$'s, and the other extreme, which nicely evaluate to the ``zigzag'' SVHPL functions with alternating $1$'s and $0$'s, are nothing but the four-point DCI integrals from the very special family of anti-prism $f$-graphs (while all other binary DCI integrals lie in between these two extreme cases). We also study periods of these integrals: while their periods are in general complicated single-valued multiple zeta values (SVMZV), the ``zigzag'' DCI integrals from anti-prism gives exactly the famous ``zigzag'' periods proportional to $ζ_{2L{+}1}$, and empirically it provides a numerical lower-bound for $L$-loop periods of any binary string, with the upper-bound given by that of the ladder (also proportional to $ζ_{2L{+}1}$). Based on $f$-graphs as a tool for studying these periods, we discuss several interesting facts and observations about these (motivic) SVMZV and relations among them to all loops, and enumerate a basis for them up to $L=10$.

hep-th↗

arXiv2025

Notes on Fourier transform and its application to three-point momentum-space integrals

The Fourier transform of two-point momentum-space Feynman integrals with massless propagators and two off-shell legs can be used to prove identities between their periods, exemplified by the glue-and-cut identity. We generalize this framework to massless momentum-space Feynman integrals with three off-shell legs and obtain a similar family of identities that can be used to calculate these integrals, especially for a non-planar subset of them, which naturally arise in the off-shell Sudakov form factors.

hep-th↗

arXiv2025

Heptagon Symbols at Five Loops and All-Loop Sequences

We revisit the symbol bootstrap program for the seven-particle MHV and NMHV amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills (SYM) based on the alphabet associated with the $E_6$ cluster algebra. After imposing integrability, cluster adjacency (or extended Steinmann), first- and last-entry conditions, the solution space is already highly restrictive: e.g. for MHV case there are exactly $1,1,2,3,4$ parity-invariant solutions for $L=1,2,\cdots, 5$, which automatically satisfy dihedral symmetry. Remarkably, after further requiring a well-defined collinear limit, we find a unique solution for both MHV and NMHV sectors where all coefficients (e.g. more than $3.1\times 10^{10}$ for MHV at $L=5$) turn out to be integers. Furthermore, we observe recurrent patterns for coefficients of special words in $E_6$ symbols mirroring those found for the $C_2$ symbol of three-point form factors, which lead to numerous predictions in the form of all-loop sequences. As an initial application, we show that these sequences uniquely fix the MHV symbol through five loops without input from collinear limits. Given the simplicity and surprisingly strong constraining power of both physical constraints and the newly observed sequences, we conjecture that there is a unique symbol satisfying these constraints at any loop order.

hep-th↗

arXiv2025

Notes on conformal integrals: Coulomb branch amplitudes, magic identities and bootstrap

We study multi-loop conformal integrals for four-point correlators of planar ${\cal N}=4$ super-Yang-Mills theory, and in particular those contributing to Coulomb branch amplitudes in the ten-dimensional lightlike limit, where linear combinations of such integrals are determined by the large R-charge octagons exactly known from integrability. Exploiting known results for integrands, we review those combinations of dual conformal invariant (DCI) integrals that must evaluate to determinants of ladders, generalizing the simplest cases of Basso-Dixon fishnet integrals; in this way, we summarize all-loop predictions for the integrands (which are extracted from $f$-graphs) contributing to components of Coulomb branch amplitudes, such as next-to-fishnet integrals. Moreover, this exercise produces new ``magic identities", {\it i.e.} certain combinations of DCI integrals equal zero, and we enumerate and simplify such identities up to six loops explicitly. On the other hand, most of these individual integrals have not been computed beyond three loops, and as a first step we consider a bootstrap program for DCI integrals based on their leading singularities and the space of pure functions. We bootstrap the $3$ non-trivial DCI integrals for four-loop Coulomb branch amplitudes (providing an independent verification of the four-loop magic identity), which all take remarkably simple form as weight-$8$ single-valued harmonic polylogarithms. We also compute all leading singularities and a large portion of the pure functions for the $34$ DCI integrals contributing to five-loop amplitudes, where not only some integrals evaluate to functions beyond harmonic polylogarithms but they also contain lower-weight pieces individually.

hep-th↗

arXiv2025

Symbol letters of Feynman integrals from Gram determinants

Symbol letters are crucial for analytically calculating Feynman integrals in terms of iterated integrals. We present a novel method to construct the symbol letters for a given integral family without prior knowledge of the canonical differential equations. We provide a program package implementing our algorithm, and demonstrate its effectiveness in solving non-trivial problems with multiple loops and legs. Using our method, we successfully bootstrap the canonical differential equations for a two-loop five-point family with two external masses and for a three-loop four-point family with two external masses, which were previously unknown in the literature. We anticipate that our method can be applied to a wide range of cutting-edge calculations in the future.

hep-ph↗

arXiv2024

Landau-based Schubert analysis

We revisit the conjectural method called Schubert analysis for generating the alphabet of symbol letters for Feynman integrals, which was based on geometries of intersecting lines associated with corresponding cut diagrams. We explain the effectiveness of this somewhat mysterious method by relating such geometries to the corresponding Landau singularities, which also amounts to ``uplifting" Landau singularities of a Feynman integral to its symbol letters. We illustrate this {\it Landau-based Schubert analysis} using various multi-loop Feynman integrals in four dimensions and present an automated {\ttfamily Mathematica} notebook for it. We then apply the method to a simplified problem of studying alphabets of physical quantities such as scattering amplitudes and form factors in planar ${\cal N}=4$ super-Yang-Mills. By focusing on a small set of Landau diagrams (as opposed to all relevant Feynman integrals), we show how this method nicely produces the two-loop alphabet of $n$-point MHV amplitudes and that of the $n=4$ MHV form factors. A byproduct of our analysis is an explicit representation of any symbol alphabet obtained this way as the union of various type-$A$ cluster algebras.

hep-th↗

arXiv2024

From squared amplitudes to energy correlators

The leading order $N$-point energy correlators of maximally supersymmetric Yang-Mills theory in the limit where the $N$ detectors are collinear can be expressed as an integral of the $1\to N$ splitting function, which is given by the $(N{+}3)$-point squared super-amplitudes at tree level. This provides yet another example that the integrand of certain physical observable -- $N$-point energy correlator -- is computed by the canonical form of a positive geometry -- the (tree-level) "squared amplituhedron". By extracting such squared amplitudes from the $f$-graph construction, we compute the integrand of energy correlators up to $N=11$ and reveal new structures to all $N$; we also show important properties of the integrand such as soft and multi-collinear limits. Finally, we take a first look at integrations by studying possible residues of the integrand: our analysis shows that while this gives prefactors in front of multiple polylogarithm functions of $N=3,4$, the first unknown case of $N=5$ already involves elliptic polylogarithmic functions with many distinct elliptic curves, and more complicated curves and higher-dimensional varieties appear for $N>5$.

hep-th↗

arXiv2024

Differential equations and recursive solutions for cosmological amplitudes

Recently considerable efforts have been devoted to computing cosmological correlators and the corresponding wavefunction coefficients, as well as understanding their analytical structures. In this note, we revisit the computation of these ``cosmological amplitudes" associated with any tree or loop graph for conformal scalars with time-dependent interactions in the power-law FRW universe, directly in terms of iterated time integrals. We start by decomposing any such cosmological amplitude (for loop graph, the ``integrand" prior to loop integrations) as a linear combination of {\it basic time integrals}, one for each {\it directed graph}. We derive remarkably simple first-order differential equations involving such time integrals with edges ``contracted" one at a time, which can be solved recursively and the solution takes the form of Euler-Mellin integrals/generalized hypergeometric functions. By combining such equations, we then derive a complete system of differential equations for all time integrals needed for a given graph. Our method works for any graph: for a tree graph with $n$ nodes, this system can be transformed into the {\it canonical differential equations} of size $4^{n{-}1}$ quivalent to the graphic rules derived recently%so-called ``kinematic flow", and we also derive the system of differential equations for loop integrands {\it e.g.} of all-loop two-site graphs and one-loop $n$-gon graphs. Finally, we show how the differential equations truncate for the de Sitter (dS) case (in a way similar to differential equations for Feynman integrals truncate for integer dimensions), which immediately yields the complete symbol for the dS amplitude with interesting structures {\it e.g.} for $n$-site chains and $n$-gon cases.

hep-th↗

arXiv2024

The recursive structure of Baikov representations II: the top-down reduction with intersection theory

Following our previous study of the recursive structure of Baikov representations, we discuss its application in the integration-by-parts reduction of Feynman integrals. We combine the top-down reduction approach with the recursive structure, which can greatly simplify the calculation for each sector in many cases. We introduce a new concept called the top-sector ISP reduction, which generalizes the maximal-cut reduction by retaining the sub-sector information. After subtracting the top-sector components, we provide a general method to transform the remaining integrand explicitly to sub-sectors, such that the reduction procedure can be carried out recursively. In this work, we use the intersection theory to demonstrate our method, although it can be applied to any implementation of the integration-by-parts reduction.

hep-ph↗

arXiv2023

The recursive structure of Baikov representations I: generics and application to symbology

In this paper, we explore the recursive structure of Baikov representations for Feynman integrals. We demonstrate that the various Baikov representations for all sectors of an integral family can be organized in a tree-like structure. Using this structure, we show that the symbol letters of one-loop Feynman integrals can be written in terms of minors of a matrix associated with the top sector. Non-trivial relations among these symbol letters can then be easily discovered using results from linear algebra.

hep-ph↗

arXiv2023

On symbology and differential equations of Feynman integrals from Schubert analysis

We take the first step in generalizing the so-called "Schubert analysis", originally proposed in twistor space for four-dimensional kinematics, to the study of symbol letters and more detailed information on canonical differential equations for Feynman integral families in general dimensions with general masses. The basic idea is to work in embedding space and compute possible cross-ratios built from (Lorentz products of) maximal cut solutions for all integrals in the family. We demonstrate the power of the method using the most general one-loop integrals, as well as various two-loop planar integral families (such as sunrise, double-triangle and double-box) in general dimensions. Not only can we obtain all symbol letters as cross-ratios from maximal-cut solutions, but we also reproduce entries in the canonical differential equations satisfied by a basis of dlog integrals.

hep-th↗

arXiv2023

$\varepsilon$-forms for non-planar triangles with elliptic curves at two loops

In this talk, we discuss how to generalize ideas developed for Banana integrals to two two-loop non-planar triangle Feynman integrals involving elliptic curves, which have non-trivial sub-sectors and whose Picard-Fuchs operators share less symmetry than Banana integrals, to obtain the canonical differential equations and to solve them with suitable boundary conditions.

hep-th↗

arXiv2023

$\varepsilon$-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves

In this paper, we investigate two-loop non-planar triangle Feynman integrals involving elliptic curves. In contrast to the Sunrise and Banana integral families, the triangle families involve non-trivial sub-sectors. We show that the methodology developed in the context of Banana integrals can also be extended to these cases and obtain $\varepsilon$-factorized differential equations for all sectors. The letters are combinations of modular forms on the corresponding elliptic curves and algebraic functions arising from the sub-sectors. With uniform transcendental boundary conditions, we express our results in terms of iterated integrals order-by-order in the dimensional regulator, which can be evaluated efficiently. Our method can be straightforwardly generalized to other elliptic integral families and have important applications to precision physics at current and future high-energy colliders.

hep-th↗

arXiv2022

Baikov representations, intersection theory, and canonical Feynman integrals

The method of canonical differential equations is an important tool in the calculation of Feynman integrals in quantum field theories. It has been realized that the canonical bases are closely related to $d$-dimensional $d\log$-form integrands. In this work, we introduce the concept of generalized loop-by-loop Baikov representation, and clarify its relation and difference with Feynman integrals using the language of intersection theory. We then utilize the generalized Baikov representation to construct $d$-dimensional $d\log$-form integrands, and discuss how to convert them to Feynman integrals. We describe the technical details of our method, in particular how to deal with the difficulties encountered in the construction procedure. Our method provides a constructive approach to the problem of finding canonical bases of Feynman integrals, and we demonstrate its applicability to complicated scattering amplitudes involving multiple physical scales.

hep-th↗

arXiv2021

Constructing Canonical Feynman Integrals with Intersection Theory

Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct $d\log$-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales.

hep-th↗