Loop-the-Loop-2: Feynman calculus and its applications to gravity and particle physics

10 Nov 2025, 08:00 → 12 Nov 2025, 19:00 Europe/Rome

“Loop-the-Loop: Feynman Calculus and its Applications to Gravity and Particle Physics” returns for its second edition as a fully online workshop, 10–12 November 2025. The event aims to spark in-depth discussions and strengthen connections across complementary perspectives in the study of scattering amplitudes.

Each day opens with a review lecture by a leading expert and centers on a dedicated theme:

• Applied Mathematics for Feynman Calculus

• Scattering Amplitudes in Gravity

• Scattering Amplitudes in Particle Physics

In addition, the program features two afternoon seminars by leading scientists in gravity and particle physics, and one early career session.

Join Zoom Meeting:
https://unipd.zoom.us/j/89843725109?pwd=Kb81IPJeSKeNSG03bcaeiQFyHrs1gq.1

Meeting ID: 898 4372 5109
Passcode: looploop2

 

Organising Committee

• Giacomo Brunello (SNS Pisa; University of Padova; IPhT-CEA / Université Paris-Saclay; INFN Padova)

• Giulio Crisanti (University of Edinburgh)

• Raj Patil (Max Planck Institute for Gravitational Physics, Potsdam; Humboldt-Universität zu Berlin)

• Sid Smith (University of Padova; University of Edinburgh; INFN Padova)
  
  
  

Characters (C) by Gaia Fontana @qftoons
Poster artwork by Nihar Gupte @mosfet_arts2

 

Past Loop-the-Loop conferences:

• Loop-the-Loop, Youngst@rs MITP, November 12-14 2024 

 

Sponsors

Monday 10 November

Review talk

1 The geometry of Feynman integrals

Precision calculations in quantum field theory rely very often on
perturbation
theory and thus on the computation of Feynman integrals.
Feynman integrals are also fascinating objects from a mathematical point
of view
and show deep connections to algebraic geometry.
In this talk I will review how to extract the geometric information from a
Feynman integral
and how this information can be used to compute more efficiently Feynman
integrals.

Speaker: Stefan Weinzierl

10:20 Break

Applied Mathematics for Feynman Calculus

2 Introduction of novel methods in multi-loop Feynman integral computation

Multi-loop Feynman integral computation has become an important topic in nowadays high energy physics. In this talk, I will introduce recent developments of computational tools including NeatIBP and Kira. I will also introduce their applications.

Speaker: Zihao Wu

3 Uncovering Singularities in Feynman Integrals via Machine Learning

In this talk, I will present a machine-learning framework based on symbolic regression that systematically extracts the complete symbol alphabet of multi-loop Feynman integrals. Rather than relying on singularity analysis, the method directly targets the analytic structure, making it broadly applicable and highly interpretable across different families of integrals. I will begin by outlining the relevant background and then show how the framework successfully reconstructs nontrivial symbol alphabets in practice. Beyond improving individual computations, this approach offers a fresh perspective on the analytic organization of multi-loop amplitudes and points toward new directions for symbolic and data-driven methods in quantum field theory.

Speaker: Yingxuan Xu

11:30 Break

Applied Mathematics for Feynman Calculus

4 Intersection Theory and Relative Cohomology for Feynman Integrals

In this talk, I will briefly review the application of intersection theory to Feynman integrals, covering both the non-relative case and the introduction of relative cohomology. Furthermore, relative cohomology allows the framework of intersection theory to be extended from the traditional Baikov representation to the Lee–Pomeransky representation and possibly beyond. The advantages and limitations of this approach will be discussed, together with illustrative examples.

Speaker: Mingming Lu

5 Feynman Integral Reductions Via Priority Function

Feynman integrals are central to scattering amplitudes and precision collider physics, yet their evaluation is often bottlenecked by the combinatorial complexity of integration-by-parts (IBP) reductions. There has been quite rapid progress in recent years, with developments such as finite field techniques, symbolic reduction rules, syzygies, intersection theory, improved seeding, and various combinations of them.
In this talk, we present a new reduction method based on a priority function. The goal of this method is to reduce the number of required seeding integrals and thus accelerate the computations. An effective priority function can be determined for specific integral topologies using AI techniques—specifically, we used Genetic Algorithms and Large Language Models—or by human heuristics, or a combination of AI and human input. We found that some priority functions work well for specific topologies but not for others, while some work relatively well for many topologies, including those with a large number of loops. In some examples, the number of seeding integrals is reduced by a factor of 3000, showing the potential of the priority function method.

Speaker: Tong-Zhi Yang

12:40 Lunch

Applied Mathematics for Feynman Calculus

6 Relative twisted cohomology in QFT and cosmology

Speaker: Andrzej Pokraka

7 Two-loop integrals for top-pair production plus a W boson

The associated production of a top-antitop quark pair with a W boson is one of the heaviest signatures probed at the LHC. The corresponding rates have been found to be consistently higher than the Standard Model predictions, calling for improved theoretical predictions.
In this talk I will discuss one of the main bottlenecks for the exact computation of the two-loop QCD amplitude, namely the Feynman integrals. I will discuss a method for evaluating the integrals by computing and solving the systems of differential equations they satisfy. I will present strategies to address the complexity of the computation, which involves complicated analytic structures, such as nested square roots and elliptic functions, and expressions with an high degree of algebraic complexity.

Speaker: Mattia Pozzoli

15:30 Break

Applied Mathematics for Feynman Calculus

8 Differential Space of Feynman Integrals

We present a novel algorithm for constructing differential operators with respect to external variables that annihilate Feynman-like integrals and give rise to the associated D-modules, based on Griffiths–Dwork reduction. By leveraging the Macaulay matrix method, we derive corresponding relations among partial differential operators, including systems of Pfaffian equations and Picard-Fuchs operators. For the studied examples, we observe that the holonomic rank of the D-modules coincides with the dimension of the corresponding de Rham co-homology groups.

Speaker: Wojciech Flieger

16:40 Break

Special seminar

9 From Quantum Scattering to Gravitational Waves

I will introduce several key concepts from the
modern theory of quantum scattering amplitudes. These ideas have found
wide-ranging applications, including in collider physics,
supersymmetric gauge theory, supergravity, and gravitational-wave physics. I will show that, within
perturbation theory, gauge theories of the type used in collider
physics are directly connected to theories of gravity, including
Einstein’s general relativity, via a construction known as the double copy
As a particularly recent development, I
will explain how this understanding of gravity provides a
useful starting point for obtaining state-of-the-art precision
predictions for binary black holes and other compact astrophysical
systems in the context of gravitational-wave physics.

Speaker: Zvi Bern

Tuesday 11 November

Review talk

10 Julio Para-martinez

10:20 Break

Scattering Amplitudes in Gravity

11 Post-Minkowskian Integrals at 4 Loops: 0SF, 1SF, and 2SF

The application of quantum field theory methods to the relativistic two-body problem in gravity has led to extraordinary advances in recent years, with applications to real gravitational wave physics. I will demonstrate how the WQFT formalism leads to particularly interesting Feynman integrals that at 4 loops fit neatly into a hierarchy of increasing difficulty: 0SF, 1SF, and 2SF. I will review results on the first two orders and discuss additional challenges at 2SF.

Speaker: Mathias Driesse

12 Binary Kerr black-hole scattering at 2PM from quantum higher-spin Compton

In this talk, I’ll discuss how the dynamics of spinning sources are incorporated when computing post-Minkowskian (PM) observables within the quantum field theory framework. Our goal is to capture the dynamics of a Kerr black hole rather than a generic compact object. I will present our candidate tree-level amplitude for gravitational Compton scattering, fixed using tools from massive higher-spin quantum field theory. Using this as a building block, we compute the classical 2PM impulse and eikonal phase, obtaining new all-orders-in-spin results for certain contributions, and the remaining ones up to O(S^11). Since Kerr 2PM dynamics beyond O(S^5) remain unsettled, our results provide a useful reference point for future work.

Speaker: Lucile Cangemi

11:30 Break

Scattering Amplitudes in Gravity

13 Non linear memory from reverse unitarity

I will present a recent computation of the nonlinear gravitational-wave memory in two-body scattering, using scattering amplitudes and soft theorems in the weak-field post-Minkowskian regime. The effect first appears at NNLO PM, and I will show how its evaluation reduces to simpler, lower-point cut two-loop integrals.

Speaker: Alessandro Georgoudis

14 High Precision Kerr Black Hole Scattering

Scattering amplitudes have emerged as a powerful tool for predicting physical observables in two-body gravitational interactions. In this talk, I will present state-of-the-art results for the scattering of a spinning and non-spinning black hole, modelled as point particles in fixed-spin representations. A key complication in this approach is the mixing of classical and quantum contributions, introduced by the finite-spin description. We resolve this ambiguity using the recently developed spin interpolation method, which cleanly isolates the classical information. With this advance, we obtain the first computation of the classical two-loop scattering amplitude including terms up to quartic order in spin. From this amplitude we derive the radial action, enabling the extraction of observables such as the linear and angular impulses. Remarkably, the resulting amplitude reveals a novel spin-shift symmetry in the probe limit: it remains invariant under a shift of the black hole spin by the momentum transfer in the scattering process.

Speaker: Dogan Akpinar

12:40 Lunch

Scattering Amplitudes in Gravity

15 The multipolar post-Minkowskian iteration and tails of memory

In traditional post-Newtonian methods, the spacetime is split into a near zone and an exterior vacuum zone. In the near zone, the metric is solved by iterating the Einstein equations in a small post-Newtonian (PN) parameter, either the relative velocity of the binary or the relative inverse separation; the power counting is done in powers of c. In the exterior vacuum zone, no assumptions are made about the binary, and one performs the weak-field post-Minkowskian (PM) expansion, counted in powers of G. Since the metric is decomposed into multipolar moments (to be matched to the near zone), this iteration is called "multi-polar post-Minkowskian" (MPM). At quadratic order in the iteration, one encounters integrals involving a static mass and a quadrupole (the tails, which enter at 1.5PN in the waveform and energy flux), as well as two quadrupoles (which give rise to the memory, which enters at 2.5PN). At cubic order: two static masses and a quadrupole give rise to tails-of-tails at 3PN; one static mass, one static angular momentum and one quadrupole moment give rise to the "spin-quadrupole tails" at 4PN; and finally, one static mass and two dynamical quadrupoles give rise at 4PN to the "tails of memory". The latter interaction is the hardest to compute due to two interacting quadrupoles: one has to deal with nested integrals and complicated kernels involving polylogarithms, which eventually simplify. In multiloop language, they would correspond to 3-loop two-point massive/massless integrals. In this talk, I will discuss the integration techniques developed to obtain these terms.

Speaker: David Trestini

16 Post-Newtonian tidal effects in the gravitational waveform from binary inspirals in scalar-tensor theories

With the advent of third-generation gravitational-wave detectors, accurately modelling signals from binary neutron stars is essential to disentangle matter effects from possible deviations from general relativity. In this talk, I will present recent results on tidal effects in scalar–tensor theories, which are expected to be more prominent than in general relativity due to the presence of a time-varying scalar dipole moment inducing scalar-induced tidal deformations of neutron stars.

Using the post-Newtonian multipolar post-Minkowskian (PN–MPM) formalism, we compute tidal corrections in the radiative sector up to next-to-next-to-leading (relative 2PN) order, to reach a level where gravitationally induced tidal deformations start contributing. This includes the derivation of the emitted energy flux and to the waveform phasing. We further obtain the tidal contributions to the waveform amplitude modes up to relative 1.5PN order. Finally, the PN–MPM framework naturally incorporates hereditary (nonlocal-in-time) effects, such as tails and memories, which we show also carry tidal corrections relevant for high-precision waveform modelling in scalar–tensor gravity.

Speaker: Eve Dones

15:30 Break

Scattering Amplitudes in Gravity

17 High precision scattering waveforms from the Self Force approach

In this talk I will review how to get analytical information on scattering waveforms by employing the Self Force approach, which contains resumed contributions from both the weak field approximation, or post-Minkowskian (PM), and the low velocity limit, i.e. post-Newtonian (PN) approximation. By leveraging this feature it would be possible to get analytical information at very high order in both PM and PN, which will allow us to discuss systematically the functional forms of the time domain waveform. Some final comments on the frequency domain waveform will also be provided, together with some prospects for the future.

Speaker: Davide Usseglio

18 Scalar Black Hole Love numbers to O(G^7)

At wavelengths large compared to the source size, any compact object admits a point-particle EFT whose finite-size effects are encoded by Love numbers. In this talk I will discuss a novel method to compute gravitational wave amplitudes by reinterpreting the Feynman diagram expansion as a Born series, solution of an effective wave equation. This method enables efficient, systematic calculations of scattering amplitudes off any compact object, yielding new predictions for scalar black-hole Love numbers and their renormalization-group equations up to O(G^7).

Speaker: Giulia Isabella

16:40 Break

Special seminar

19 Alessandra Buonanno

Wednesday 12 November

Review talk

20 Claude Duhr

10:20 Break

Scattering Amplitudes in Particle Physics

21 One-loop Amplitudes for t tbar jet and t tbar photon Productions at the LHC

We present analytic expressions for the one-loop QCD helicity amplitudes contributing to top-quark
pair production in association with a photon or a jet at the Large Hadron Collider (LHC), evaluated
through O(ϵ^2) in the dimensional regularisation parameter, ϵ.
These amplitudes are required to construct the two-loop hard functions that enter the NNLO QCD computation.
The helicity amplitudes are expressed as linear combinations of algebraically independent components of the ϵ-expanded
master integrals, with the corresponding rational coefficients written in terms of momentum-twistor variables.
We derive differential equations for the pentagon functions, which enable efficient numerical evaluation via generalised power series expansion method.

Speaker: Souvik Bera

22 On Loop Integral Numerical Evaluation with LINE

The future of precision phenomenology depends on accurate and efficient evaluations of Feynman integrals, essential for higher-order radiative corrections. As experimental precision advances, the need for high-performance computational tools capable of handling complex multi-loop calculations becomes increasingly urgent. Efficient and scalable codes are key to making state-of-the-art theoretical predictions feasible for phenomenological studies. The differential equation method for master integral evaluation has recently emerged as a powerful and versatile approach, which can be adapted for either numerical or analytical approaches. In this talk, I will present LINE (Loop Integral Numerical Evaluation), an open-source C code devoted to the numerical evaluations of Feynman integrals. LINE unifies several functionalities into a single, coherent framework, enabling both direct numerical evaluations and propagation through differential equations with specified boundary conditions. LINE is written in C, with the goal of making it efficient and suited parallelized runs.

Speaker: Jonathan Ronca

11:30 Break

Scattering Amplitudes in Particle Physics

23 The geometric bookkeeping guide for 𝜀-factorised differential equations

Precision predictions for high-energy experiments rely on accurately evaluating multi‑loop, multi‑scale Feynman integrals in dimensional regularisation. The method of differential equations is now the standard tool for this task, but its full power is only realised when the system can be brought into an 𝜀-factorised form. In this talk, we present an algorithmic framework that constructs such 𝜀-factorised differential equations for arbitrary integral families without depending on the underlying geometry.

Speaker: Antonela Matijašić

24 Electroweak double-box integrals for Møller scattering with three Z bosons

In this work, we have computed the planar and non-planar contributions to the Møller scattering with three massive Z bosons exchanged between the fermion lines. These cases involve complicated geometries like curves of genus 2 and K3 surfaces, which until recently remained inaccessible. We demonstrate how these can be tamed with a newly developed technique for epsilon-factorization of Feynman Integrals.

Speaker: Dmytro Melnichenko

25 Bootstrapping Gravity with Crossing Symmetric Dispersion Relations

I will discuss how to derive bounds on Wilson coefficients in gravitational effective field theories using fully crossing symmetric dispersion relations. These sum rules naturally isolate finite subsets of low-energy couplings without relying on the forward limit or specific high-energy completions. I will show how we validate our method by matching bounds computed previously for scalar scattering with gravity as well as for supergraviton scattering. For graviton scattering we use crossing symmetric functions that combine various helicity combinations for the maximal-helicity violating amplitude. We also derive new bounds on the coupling of gravitons to a massive spin-4 state at tree level. These results demonstrate the power of crossing symmetric sum rules as a tool in the S-matrix bootstrap.

Speaker: Celina Pasiecznik

13:10 Lunch

Scattering Amplitudes in Particle Physics

26 Bootstrapping the QCD Soft Anomalous Dimension

The soft anomalous dimension encodes infrared divergences that arise from soft virtual particles. Crucially, this object – a matrix in colour space – is universal; it need only be calculated once then can be used for all QCD amplitudes. While the soft anomalous dimension has been directly calculated up to three loops in the massless case, the three loop massive and four loop massless remain out of reach. In this talk, I will discuss the symbol bootstrap approach to calculating the soft anomalous dimension and ways in which the ansatz can be constrained.

Speaker: Laura Walsh

27 Positivity properties of five-point two-loop Wilson loops with Lagrangian insertion

I will present the geometric integrand expansion, developed by Arkani-Hamed, Henn, and Trnka, for the pentagonal Wilson loop with a Lagrangian insertion in maximally supersymmetric Yang–Mills theory. The discussion will focus on integrated results corresponding to an all-loop class of ladder-type geometries, with particular emphasis on the two-loop observable examined through this geometric framework. We analytically evaluate the new two-loop integrals arising from the negative-geometry contribution using the canonical differential equations method. The analytic results reveal numerical evidence that each component exhibits uniform sign behaviour within the Amplituhedron region. Finally, I will report recent progress in bootstrapping ladder-type geometries via geometric Landau analysis, which identifies the physically relevant singularities of the integrals. This method successfully determines the symbol of the six-point two-loop and five-point three-loop ladder negative geometries, with the latter introducing novel pentagon alphabet letters also appearing in planar three-loop Feynman integrals.

Speaker: Shun-Qing Zhang

15:30 Break

Scattering Amplitudes in Particle Physics

28 Jet production at lepton colliders: an amplitude perspective

Modern amplitude technology is a crucial ingredient for the physics programme of future particle colliders. In this talk, I will focus on a classic test of QCD, jet production at an electron-positron collider. I will report on the calculation of a relevant three-loop matrix element, and on the development of an N3LO infrared subtraction scheme for final-state radiation.

Speaker: Petr Jakubčík

29 Parametric annihilators for integral reduction & differential equations

We elaborate on the method of parametric annihilators for deriving relations among integrals. Annihilators are differential operators that annihilate multi-valued integration kernels appearing in suitable integral representations of special functions and Feynman integrals. We describe a method for computing parametric annihilators based on efficient linear solvers and show how to use them to derive relations between a wide class of special functions. These include hypergeometric functions, Feynman integrals relevant to high-energy physics and duals of Feynman integrals. We finally present the public Mathematica package CALICO for computing parametric annihilators and its usage in several examples.

Speaker: Gaia Fontana

16:40 Break

Special seminar

30 Funding Opportunities for Early Career Scientists

The aim of this session is to explore career opportunities for early-career scientists, with a focus on postdoctoral positions and fellowship applications. I will present available fellowships across different countries and share tips on applying for the European Marie Curie Fellowship. The session will be interactive, and early-career scientists are encouraged to ask questions based on their experiences related to job applications.

Speaker: Elisa Maggio