Bootstrap Approaches to Scattering Amplitudes

14 Sept 2026, 09:00 → 18 Sept 2026, 18:00 Europe/Berlin

Seminar Room bctp (Bethe Center for Theoretical Physics)

Poster_Bootstrap_klein.pdf

ScheduleBetheForumBootstrap.pdf

Monday 14 September

13:00 Registration

1 An invariant formulation for inflationary observables

Inflationary observables play the double role of setting the
initial condition for the evolution that gives rise to the structure we
observe to day in the sky, and it is a portal to the early universe and
the physics at energy scales 10-11 orders of magniture larger than the
ones at LHC. So, learning how to compute them and to extract physics
out of them is crucial for the understanding of our universe and of the
physics at ultra-high energy. In this talk we report on recent progress
in addressing these issues, concretely we propose a novel geometrical
formulation that is able to capture the full observable - the Bunch-
Davies wavefunction and the in-in correlator - at each given order in
perturbation theory, generalising a previous graph-by-graph
construction named cosmological polytope We provide a gentle
introduction to such a new construction, named cosmotope, and an
example-based illustration of how it allows to understand the analytics
structure of these observables in terms of compatibility conditions
among residues/discontinuities.

Speaker: Paolo Benincasa

15:00 Coffee Break

2 The doubly conformal dispersion relation

Dispersion relations in QFT allow us to reconstruct correlation functions or scattering amplitudes from their discontinuities. This idea, initially exploited in the 1960s S-matrix bootstrap, re-emerges in the modern conformal bootstrap through the conformal dispersion relation (CDR) of Carmi and Caron-Huot, which reconstructs a CFT four-point function as an integral over its double discontinuity weighted by a theory- and dimension-independent kernel. In the flat-space limit of QFT in AdS, the CDR has been argued to reduce to the standard single-variable dispersion relation for scattering amplitudes, providing a genuine bridge between the CFT and S-matrix bootstrap programs.

The CDR has so far lacked a derivation directly from complex analysis, independent of the Lorentzian inversion formula (LIF). In this talk, we bridge this gap. Starting from a single-variable dispersion relation, we uncover a one-dimensional conformal covariance that uniquely fixes the kernel as a 1d CFT conformal block, deriving its closed form purely from complex analysis principles and symmetry properties. With these results in hand, we can take the CDR as the most fundamental relation in the analytic conformal bootstrap. Most of the analytic bootstrap tools, such as the LIF, the Polyakov–Regge blocks and the dispersive sum rules are now derived and well-defined consequences of the CDR, which rely only on simple properties of CFT correlators.

Speaker: Maddalena Ferragatta

17:00 Reception

Tuesday 15 September

3 Bootstrapping Thermal CFTs

We will begin with a review of the thermal bootstrap program both numerical and analytical. And then continue to recent developments including the novel UV/IR relationship between thermal OPE and quasinormal modes.

Speaker: Elli Pomoni

10:30 Coffee Break

4 Multi-trace operators in \mathcal{N}=4 SYM

In this talk, I will discuss recent progress toward understanding the spectrum of multi-trace operators in \mathcal{N}=4 SYM at large 't Hooft coupling and at tree level in the large-N expansion, focusing on triple-trace operators in the R-symmetry singlet sector. While partial information about these operators can be extracted from four-point functions, a complete understanding remains elusive, in part because of their large degeneracy at fixed spin. To extract the anomalous dimensions of these operators, we develop an AdS Hamiltonian approach, i.e., we diagonalize an effective Hamiltonian built from interactions involving the Lagrangian densities L_p, which are superdescendants of the stress-tensor superprimary and its Kaluza–Klein modes. We show that this Hamiltonian reproduces the known spectrum of double-trace operators and makes the hidden 10d symmetry manifest. We then compute its action on triple-trace operators built from three L_p's, determining their anomalous dimensions as a function of spin. If time permits, I will also present preliminary results on higher-trace operators, exploring their eigenvalue statistics and possible signatures of chaotic behavior.

Speaker: Giulia Fardelli

12:00 Lunch Break

5 From nothing to strings: bootstrapping supergravity amplitudes

The nonperturbative S-matrix bootstrap provides a powerful framework for constraining scattering amplitudes using analyticity, unitarity and crossing symmetry, and can lead to precise estimates for bounds on physical observables. In this talk, I will discuss its application to gravitational scattering amplitudes with maximal supersymmetry.
Starting from bounds on the R^4 correction in D=10 2102.02847 and D=11 2212.00151, I will show how to extend the bootstrap to constrain higher order terms in the low energy expansion.
In D=10 the constructed allowed region in the (R^4, D^4 R^4) plane contains the convex hull of the known string theory predictions.
In D=11 the M theory prediction lies close to a kink in the allowed region. The dream is to derive estimates for D^6 R^4 and D^8 R^4, the latter of which has never been computed before.

Speaker: Andrea Guerrieri

15:00 Coffee Break

6 From Partons to Strings: Scattering on the Coulomb branch of N = 4 SYM

We study scattering on the Coulomb branch of planar N=4 SYM.
At weak 't Hooft coupling scattering amplitudes can be computed using Feynman diagrams and the relevant degrees of freedom are quarks and gluons. At strong 't Hooft coupling the dynamics is captured by string scattering in flat space. Dual conformal symmetry makes it possible to constrain the unitarity properties and the spectrum of the amplitude: bound states are encoded in the cusp anomalous dimension of a Maldacena-Wilson line with operators inserted at the cusp, which is fixed by integrability. In this talk, we show how to combine dual conformal invariance, integrability, worldsheet bootstrap and the S-matrix bootstrap to explore scattering at finite 't Hooft coupling. [Based on 2510.19909 + work in progress]

Speaker: Elisabetta Armanini

Wednesday 16 September

7 One-loop amplitudes in AdS_3 x S^3 and the (un)reasonable effectiveness of n=21

The one-loop amplitude for four tensor multiplets in six-dimensional (2,0) supergravity coupled to n tensor multiplets exhibits a remarkable ultraviolet cancellation when the number of tensor multiplets is tuned to n=21. This value is precisely realised by the low-energy limit of type IIB string theory compactified on K3. In this talk, I will discuss how the same distinguished value appears when the theory is considered on an AdS_3 x S^3 background, where it admits a well-known CFT holographic dual. In particular, I will show how n=21 naturally emerges from bootstrapping the four-tensor correlator at one loop using crossing symmetry and consistency of the OPE. Finally, I will demonstrate the consistency of the AdS correlator with the corresponding flat-space amplitude via the flat-space limit. The talk is based on recent work in collaboration with Francesco Aprile and Hynek Paul.

Speaker: Michele Santagata

10:30 Coffee Break

8 Positivity at finite coupling

I will review how causality and unitarity impose positivity bounds on effective field theories via dispersion relations on the 2-2 scattering amplitude, and I will show what happens in contexts like QED or GR, where IR effects affect the amplitude picture.

Speaker: Francesco Riva

12:00 Lunch Break

14:00 Free Afternoon

18:30 Dinner Restaurant Bönnsch

Thursday 17 September

9 The Landau/Regge correspondence

Regge theory was originally developed to capture the high-energy behavior of scattering amplitudes. In this talk, I will discuss how techniques from Regge theory can be used to predict the Landau singularities of scattering amplitudes. We verify these predictions by numerically solving the Landau equations. This talk is based on ongoing work with Miguel Correia and Michele Ligorio.

Speaker: Sasha Zhiboedov

10:30 Coffee Break

10 Solving strong QCD dynamics with Gauge Theory Bootstrap

The Gauge Theory Bootstrap was proposed as a way to solve the strongly coupled dynamics of asymptotically free gauge theories like QCD. The method considers a general set of scattering amplitudes, form factors and spectral densities and matches them with weakly coupled pions at low energy and perturbative QCD calculations at high energy. With the basic inputs N_f, N_c, alpha_s, m_q and m_pi, f_pi, generic constraints of analyticity, crossing and unitarity combined with dynamical UV information restricts the strongly coupled physics at intermediate energies and generate resonances such as sigma, rho, f_2. We use this framework to explore gauge theories with varying quark mass m_q and number of colors N_c.

Speaker: Yifei He

12:00 Lunch Break

11 The S-matrix Bootstrap for a Custodially-Symmetric Higgs

The S-matrix bootstrap uses analyticity, unitarity, and crossing symmetry to place nonperturbative bounds on 4-point scattering amplitudes. In my talk, I describe the application of the bootstrap to the scattering of O(4) symmetric scalars, an approximation to the Higgs boson in the custodial limit, explain what we are able to learn about amplitudes in the allowed region, and discuss how current experimental constraints on Higgs self-interactions can manifest in the bootstrap space.

Speaker: Justin Berman

15:30 Coffee Break

12 Bethe Colloquium: BMS particles

The concept of a particle is one of the basic building blocks of physics. In flat spacetime, particles are classified according to the symmetries of special relativity – the Poincaré group – and are characterized by familiar quantities such as mass and spin. Gravity, however, changes this picture in a subtle way. Even far away from all matter, spacetime possesses an infinite-dimensional symmetry known as the Bondi-Metzner-Sachs (BMS) group. In this talk, I will introduce the notion of a “BMS particle”: a quantum state classified by the BMS group rather than only by the Poincaré group. I will explore how this broader notion offers a new perspective on the infrared physics of gauge theories and gravity, and sheds new light on the holographic description of spacetime.

Speaker: Laura Donnay

Friday 18 September

13 Bootstrapping Gauge Theory Observables with Lattice Path Combinatorics

A broad class of observables in four-dimensional superconformal gauge theories admits a universal representation as Fredholm determinants of semi-infinite Bessel matrices. Their weak coupling expansion is naturally organized in terms of Chen iterated integrals of uniform transcendental weight, accompanied by universal positive-integer coefficients. Remarkably, these integers have an enumerative interpretation: they count weighted lattice paths confined to a nontrivial domain, generalizing classical Dyck paths. This correspondence reveals an unexpected connection between integrability, scattering amplitudes and enumerative combinatorics, and suggests a new approach to bootstrapping finite-coupling observables in gauge theories.

Speaker: Grisha Korchemsky

10:30 Coffee Break

14 Symbol Bootstrap: From N=4 sYM to QCD

In this talk, we report two advances that extend the symbol bootstrap programme from planar N=4 super Yang–Mills to QCD. In the first part, we bootstrap the four-point NMHV stress-tensor form factor in N=4 sYM, where symmetry conditions together with multi-collinear consistency conditions uniquely fix the symbol through three loops. We find that the known 88 letter alphabet for MHV form factors captures the alphabet for the NMHV sector as well. This provides the first multi loop non MHV form factor data and confirms the alphabet’s universality beyond the MHV sector. In the second part, we perform the first bootstrap computation of two loop six gluon MHV amplitudes in massless planar QCD at the symbol level, by concentrating on its maximal transcendentality part. We compute the prefactors using on shell diagrams, and establish the connection between the transcendental functions and prescriptive unitarity integrals, both imported from N=4 sYM. Finally, we discuss a generalized version of the maximal transcendentality principle, which provides a systematic bridge for amplitudes between supersymmetric theories and practical QCD.

Speaker: Qinglin Yang

12:00 The End