School on Modern Amplitude Methods for Gauge and Gravity Theories

July 24 – August 4, 2023

São Paulo, Brazil



Gauge theories and gravitational interactions form the basis of our current understanding of the universe. To realize them in a unified framework is a formidable task, yet common properties have been uncovered through a duality between color and kinematics and the subsequent double-copy perspective. More recently, observations of gravitational waves have triggered a surge of research in this field, and modern amplitudes-based techniques have been very useful for the general relativistic two-body problem.

In this advanced school, students will learn modern techniques for computing scattering amplitudes which are not usually seen in standard graduate courses, and will interact with renowned researchers in the field. Selected students will be invited to participate in the “Gravitational Waves meet Amplitudes in the Southern Hemisphere” program which follows the two-week school.

There is no registration fee and limited funds are available for travel and local expenses.


  • Thales Azevedo (IF/UFRJ, Brazil)
  • Nathan Berkovits (ICTP-SAIFR & IFT/UNESP, Brazil)
  • Zvi Bern (UCLA, USA)
  • Gabriel Menezes (DEFIS/UFRRJ, Brazil)

Satisfaction Survey: Here


Week 1:
  • Zvi Bern (UCLA, USA): Generalized unitarity and loops
  • Riccardo Sturani (ICTP-SAIFR/IFT-UNESP, Brazil): Basics of gravitational waves
  • Jaroslav Trnka (UC Davis, USA): Overview of scattering amplitudes
Week 2:
  • Nima Arkani-Hamed (IAS Princeton, USA): Advanced topics in amplitudes
  • Donal  O´Connell (Edinburgh University, Scotland): Double copy approach
  • Radu Roiban (Pennsylvania State University, USA): Gravitational waves from amplitudes

List of Participants: Updated on July 31, 2023


Reading material


Week 1:

Zvi Bern: Here are review articles relevant to my lectures that the students should find useful:
M. L. Mangano and S. J.Parke, “Multiparton amplitudes in gauge theories,” [arXiv:hep-th/0509223 [hep-th]].
L. J. Dixon, “Calculating scattering amplitudes efficiently,” [arXiv:hep-ph/9601359 [hep-ph]].
Z. Bern, L. J. Dixon and D. A. Kosower, “Progress in one loop QCD computations,” [arXiv:hep-ph/9602280 [hep-ph]].
Z. Bern and Y. t. Huang, “Basics of Generalized Unitarity,” [arXiv:1103.1869 [hep-th]].
Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, “The Duality Between Color and Kinematics and its Applications,” [arXiv:1909.01358 [hep-th]].

Riccardo Sturani:
I’ve prepared some lecture notes that should cover a basic introduction to linearized GR, EFT approach to the gravitational two body problem and some phenomenology/data analysis overview.
You can find them at
A somehow detailed list of topics that I intend to cover is (the last topic about NANOgrav is not yet in the notes, writing now about it):
* Astronomical/cosmological scales and GR
* Orbital decay of binary systems
* Linearized GR and counting of degrees of freedom
* Green functions
* EM tensor of GWs
* Multipole expansions at the level of the e.o.m.
* Quadrupole formula and beyond
* Lagrangiand formulation, conservative and dissipative physics
* Near and far zones, gravitational non-linearities
* Interaction with the detector
* Building a GWform, matched filtering
* Elements of cosmology/NANOgrav detection of GWs

Jaroslav Trnka: Here is a (preliminary) description of lectures:
Overview of amplitudes: introduction to on-shell amplitudes, spinor helicity formalism, three-point amplitudes, factorization, Yang-Mills amplitudes, color-ordering, BCFW recursion relations, open challenges, glimpse of the Amplituhedron
Here is a list of some reading but the lectures will be self-contained:
– SAGEX review: and individual overview lectures
– amplitudes book by Elvang and Huang:
– lectures by Lance Dixon:
– lectures by Clifford Cheung:

Week 2:

Radu Roiban: for a “big picture” snapshot as of the middle of last year, I would suggest and
for a slightly more technical one
For explicit use of some of the methods that I will cover, I would suggest looking over and


Recent development revealed a close connection between a certain classical limit of scattering amplitudes in quantum field theory and classical interactions of massive bodies. In these lectures we will review this connection for conservative gravitational dynamics. Topics that will be discussed are:

– perturbative expansions of two-body interactions
– classical interaction potential from scattering amplitudes in nonrelativistic quantum mechanics beyond leading order
– classical regime of scattering amplitudes — tree level
– Hamiltonian and radial action: general picture
– tree-level/leading order/1PM/1PL illustration
– classical regime of scattering amplitudes — loop level: identification of the relevant contributions vs generalized unitarity
– one-loop/next-to-leading order/2PM illustration
– aspects of resummation of the radial action
– comments on beyond 2PM
– other effects — tidal, spin (if time allows)




Online application is now closed


Videos and Files

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School on Modern Amplitude Methods for Gauge and Gravity Theories

Additional Information

School Registration:  ALL participants should register. The registration will be on July 24 (Monday) at the Institute from 08:30 am to 09:30 am.

BOARDING PASS: All participants, whose travel has been provided or will be reimbursed by ICTP-SAIFR, should bring the boarding pass  upon registration. The return boarding pass (PDF, if online check-in, scan or picture, if physical) should be sent to by e-mail.

COVID-19: Brazilians and foreigners no longer have to present proof of vaccination before entering the country.

Visa information: Nationals from several countries in Latin America and Europe are exempt from tourist visa. Nationals from Australia, Canada, Japan and USA are exempt from tourist visa until October 1st, 2023. Please check here which nationals need a tourist visa to enter Brazil.

Accommodation: Participants, whose accommodation will be provided by the institute, will stay at The Universe Flat. Hotel recommendations are available here

How to reach the Institute: The school will be held at ICTP South American Institute, located at IFT-UNESP, which is across the street from a major bus and subway terminal (Terminal Barra Funda). The address which is closer to the entrance of the IFT-UNESP building is R. Jornalista Aloysio Biondi, 120 – Barra Funda, São Paulo. The easiest way to reach us is by subway or bus, please find instructions here.