Registration Room: Auditorium (A.1.01/03 Alps)

Welcome Room: Auditorium (A.1.01/03 Alps)

Quantum polylogarithms

• Alexander Goncharov ((Yale University))

Room: Auditorium (A.1.01/03 Alps) Scattering amplitudes can be (often) calculated via certain special functions called multiple polylogarithms / their generalizations. I will introduce h-deformation of these functions. The simplest of them is the quantum dilogarithm function, which comes with the quantum deformation of the Abel's five term relation, and has many applications in Math and Physics. These h-deformations suggest that the scattering amplitudes in general should also have natural h-deformations.

Two-Loop Amplitudes for Top–Quark Pair Production with a Jet

• Simone Zoia

Room: Auditorium (A.1.01/03 Alps) The production of a top-quark pair with a jet is a key process in the physics programme of the Large Hadron Collider (LHC). Its sensitivity to fundamental Standard Model parameters and the rising precision of LHC data demand that this process is computed to at least next-to-next-to-leading order in QCD. The main obstacle are the relevant two-loop five-particle amplitudes, due to the appearance of elliptic functions coupled with severe algebraic complexity. In this talk, I will present an analytic computation of these amplitudes in the leading colour approximation. Building on the symbol formalism for transcendental functions and on finite-field arithmetic to handle the algebraic complexity, our new techniques allow us to obtain results that can be directly deployed in phenomenology.

Cluster Charts for Massive Kinematic Spaces

• Dani Kaufman

Room: Auditorium (A.1.01/03 Alps)

QCD evolution - insights from conformal symmetry

• Sven-Olaf Moch

We have computed the even-N moments of splitting functions at four-loop order in perturbative QCD, using off-shell operator renormalization for a general gauge group. These results agree with previous data and, combined with endpoint constraints, allowed us to develop precise approximations for the four-loop splitting functions, reducing uncertainties to about 1\% for relevant collider energies. Additionally, we addressed the mixing of gauge-invariant with gauge-variant operators using generalized BRST symmetry, an essential step for four-loop operator renormalization. A key innovation is our method for reconstructing all-N analytic expressions by solving systems of Diophantine equations, guided by conformal symmetry and reciprocity properties. This approach enables the derivation of high-order splitting functions from finite moments efficiently and systematically, advancing the theoretical precision of QCD evolution predictions for collider physics.

Constraining the symbol from singularities

• Cristian Vergu

I will review how to constrain the symbol and the beyond the symbol terms for various integrals by using the notion of Pham-Steinmann compatibility of Landau singularities.

Cluster Symbols and Recurrent Patterns for Amplitudes

• Song He

Gong Show

Panel discussion: Lance Dixon, Thomas Gehrmann, Gregory Korchemsky "15 years of Symbology and beyond"

From symbols to cross sections: precision theory predictions for high energy collider physics

• Thomas Gehrmann

Room: Auditorium (A.1.01/03 Alps) The advances in the analytical computation of loop (and phase space) integrals enable highly precise predictions for collider processes. Combined with accurate measurements from the LHC and earlier particle colliders, these predictions are used in precision determinations of Standard Model parameters and in indirect searches for new physics effects. We describe techniques and recent results for precision calculations and discuss their impact for particle phenomenology on selected examples.

The Magnus expansion in quantum field theory

• Andreas Brandhuber

Room: Auditorium (A.1.01/03 Alps) The Magnus expansion provides an interesting alternative to the celebrated Dyson expansion in quantum field theory, and found early applications in quantum mechanics, atomic and molecular physics, and numerical solutions of matrix linear differential equations. Applications to field theory are more recent and are linked to the rewriting of the S-matrix as the exponential of the N-operator, namely S=exp(i N), which is particularly relevant in the computation of classical observables in general relativity. I will describe direct methods to compute matrix elements of the N-operator, also called Magnus amplitudes, bypassing scattering amplitudes. At tree level, Magnus amplitudes are expressed in terms of retarded and advanced propagators, with each diagram weighted by factors that we identify as Murua coefficients. At loop level this structure is augmented by the Hadamard cut function, and we establish remarkable relations between loop and tree Magnus amplitudes via forward limits. I describe this in the context of a scalar field theory, but our methods are applicable to general theories as well as to integral functions appearing in eg. gravitational-wave computations. Magnus amplitudes are manifestly free of hyper-classical terms, and combined with known relations between Magnus amplitudes and the radial action, our results lay the groundwork for systematic and efficient calculations of classical observables from quantum field theory.

Reciprocal A-Determinants and Banana Singularities

• Simon Telen

Room: Auditorium (A.1.01/03 Alps) We introduce the reciprocal A-determinant of a reciprocal linear space as an analog of the principal A-determinant of a toric variety. For an appropriate choice of the matrix A, the zero locus of the reciprocal A-determinant consists of the Landau singularities of a banana integral. I will explain the definition and illustrate it with examples. I will also introduce an analog of the GKZ A-hypergeometric system which annihilates the integral, and whose singular locus is our determinant. This is joint work with Saiei-Jaeyeong Matsubara-Heo.

Symbology for classical and multiple polylogarithms--a retrospective

• Herbert Gangl

Room: Auditorium (A.1.01/03 Alps) We will try to outline the journey of the first polylogarithmic "symbol'' as we perceived it personally, from its inception in algebraic K-theory to its heyday when Goncharov single-handedly developed the multiple polylog version, and highlight some of its striking successes.

Polylogs and more in classical field theory

• Julio Parra-Martínez

Room: Auditorium (A.1.01/03 Alps) It was recently understood that our favorite classes of special functions are not exclusive to quantum field theory, but also feature in perturbative computations in classical physics. I will explain how this comes about and provide a brief survey of the appearance of various classes of special functions in computations in perturbative classical General Relativity.

MPP Colloquium: The DNA of Particle Scattering

• Lance Dixon (SLAC/Stanford University)

Room: Auditorium (A.1.01/03 Alps) At the Large Hadron Collider, the copious scattering of quarks and gluons in quantum chromodynamics (QCD) produces Higgs bosons as well as many backgrounds to searches for new physics. Better theoretical precision for Standard Model cross sections is needed to match experimental improvements with the high-luminosity LHC upgrade. Quark and gluon scattering in QCD can be evaluated in perturbation theory and leads to highly intricate, multivariate mathematical functions. To gain further insight into these functions, one can study a simpler cousin of QCD called planar N=4 SYM. The structural features of these intricate results can be decoded -- using "symbology" -- in a way analogous to sequencing DNA. Each derivative reads off a letter, like the A,T,G,C letters of the DNA code. Understanding the alphabet, and then reading the code, exposes the physics and mathematics of quantum scattering. Bizarre new symmetries have been unveiled by humans staring at this theoretical data. For example, two different scattering amplitudes are secretly related to each other by reading the code backwards. The next hidden symmetries may be revealed by machine learning models "staring at" the data.

Analytical Regression and the Semi-Numerical Landau Bootstrap

• Matthew Schwartz

Room: Auditorium (A.1.01/03 Alps) The Landau bootstrap program attempts to completely determine a loop integral from its singular behavior. A finite number of possible singularities determine a finite alphabet and a finite set of symbol terms by which the integral can be expressed. Then increasingly subtle information about location and relations among singularities, such as their potential adjacency in the symbol, can be used to systematically whittle down the number of terms. Complementary to this top down approach is a bottom up semi-numerical approach where the integral and the symbol terms are evaluated numerically and then fit by analytic regression. This talk will review these two approaches and how they can be combined to determine some Feynman integrals in end-to-end automatable fashion.

Progress on IR singularities 
of multileg QCD amplitudes

• Einan Gardi

Room: Auditorium (A.1.01/03 Alps) I review the state-of-the-art knowledge of IR singularities in multileg QCD amplitudes, identifying the key reasons for the remarkable simplicity of the soft anomalous dimension. I then present a novel strategy to compute this quantity using a lightcone expansion of correlators of semi-infinite Wilson lines by the method of regions. Recently, this strategy allowed us to determine the three-loop soft anomalous dimension, accounting for interactions between a single massive coloured particle and any number of massless ones. Finally, I discuss some of the key properties of the soft anomalous dimension, associated with its analytic structure and with special kinematic limits. These provide not only powerful checks of the computation, but also an alternative avenue to determine similar quantities by bootstrap.

Mellin Amplitudes for Energy Correlators

• Kai Yan

Room: Auditorium (A.1.01/03 Alps) The N-point Energy Correlators (ENC) are locally finite observables in N=4 super Yang Mills theory. In the multi-collinear limit, these are Chen-iterated integrals over state-summed tree-level squared amplitudes. In this talk we establish mapping relations between (N+1)-point dual conformal feynman loop integrals and ENC by introducing Mellin amplitudes for the latter, exploiting Symanzik star formula. Given the Mellin amplitudes, we build differential equations relating the ENC to even-point one-loop feynman integrals in even dimension, unraveling its analytic structures bypassing integration by parts. Our study aims to predict the space of integrated functions from singularities present in the squared amplitudes, offering insights for a direct bootstrap program for event-shape observables in N=4 sYM and QCD.

Topological and Algebraic Structures behind Feynman Integrals: Intersection Theory meets D-Modules

• Pierpaolo Mastrolia

Room: Auditorium (A.1.01/03 Alps)

Symbolic AI models as a tool for discovery - Examples from physics and mathematics

• François Charton

Room: Auditorium (A.1.01/03 Alps) Training AI models on instances of hard problems can provide insights on the underlying phenomenon. I will present two examples from theoretical physics and pure mathematics: learning the bootstrapped amplitude symbols of the 3-gluon form factor, in planar N=4 supersymmetric Yang Mills, and learning interesting properties of the Collatz sequence.

Symbology meets enumerative combinatorics

• Gregory Korchemsky

Room: Auditorium (A.1.01/03 Alps)

Symbols for the Large Hadron Collider

• Andreas von Manteuffel

Room: Auditorium (A.1.01/03 Alps) Advances in the understanding of Feynman integrals and their associated special functions have enabled more precise predictions for high-energy experiments. In this talk, I review calculations of phenomenologically relevant scattering amplitudes that have greatly benefited from the symbol calculus for multiple polylogarithms. In particular, I discuss cases involving algebraic alphabets.

Conference dinner for speakers and chairs By invitation only.

Cluster adjacency and scattering amplitudes

• James Drummond

Room: Auditorium (A.1.01/03 Alps) I will provide a review of cluster adjacency properties of scattering amplitudes in planar N=4 super Yang-Mills theory. I will then discuss the extent to which similar properties appear in more general massless scattering amplitudes.

Yangian symmetry and the symbol

• Anne Spiering

Room: Auditorium (A.1.01/03 Alps) Certain classes of Feynman integrals show integrable structures through Yangian differential equations they satisfy. I will present results of ongoing work on the implications at the level of the symbol, focusing on one-loop n-gon integrals in n dimensions.

Scattering Amplitudes beyond Polylogarithms

• Lorenzo Tancredi

Room: Auditorium (A.1.01/03 Alps) Our understanding of scattering amplitudes in Gauge theories is tightly intertwined with our knowledge of polylogarithms. In this context, the symbol map has not only allowed us to make sense of their analytic properties but also to devise new calculational techniques. Building on our understanding of polylogarithms, this talk will illustrate how the emergence of new geometries in scattering amplitudes necessitates the generalisation of many concepts familiar in the polylog world. I will discuss which concepts can be easily generalised and which ones require care, providing important building blocks for our understanding of scattering amplitudes defined on elliptic curves, Calabi-Yau geometries, and higher-genus hypersurfaces.

TBA

• Nima Arkani-Hamed

Room: Auditorium (A.1.01/03 Alps)

Single-valued elliptic polylogarithms from zeta generators

• Oliver Schlotterer

Room: Auditorium (A.1.01/03 Alps) Multiple polylogarithms at genus zero are by themselves multi-valued functions but can be completed to single-valued polylogarithms by adding suitable combinations of their complex conjugates and multiple zeta values. As a generalization to genus one, this talk presents an explicit construction of single-valued elliptic polylogarithms depending on one point on the torus where the monodromies from the homology cycles and the singular point at the origin cancel. The construction is carried out at the level of Lie-algebra valued generating series where the combinations of elliptic polylogarithms, their complex conjugates and (elliptic) multiple zeta values are controlled by certain operations on the generators. The series in single-valued polylogarithms at genus zero and genus one exhibit striking parallels since the appearance of multiple zeta values is controlled by the action of so-called zeta generators on the Lie-algebra setup of both cases.

Origin limits of large-charge correlators in planar N=4 SYM

• Benjamin Basso

Room: Auditorium (A.1.01/03 Alps) I will discuss the behavior of large-charge correlation functions in planar N=4 SYM in the so-called Origin limits, where many cross ratios simultaneously approach zero - corresponding to corners of the positive kinematic region. Focusing on two-dimensional kinematics, I’ll explain how these limits can be explored using tools from cluster algebras and the hexagon formalism. I will also present a conjecture for the all-order behavior of correlation functions in this regime, based on a tilted version of the cusp anomalous dimension.

Marcus Spradlin "Symbology@15: The Final Entry" Room: Auditorium (A.1.01/03 Alps)