Snowmass TF04 Report Scattering Amplitudes and their Applications. Zvi BernaJaroslav Trnkab

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Snowmass TF04 Report: Scattering Amplitudes and their

Applications.

Zvi Bern,aJaroslav Trnkab

aMani L. Bhaumik Institute for Theoretical Physics,

UCLA Department of Physics and Astronomy, Los Angeles, CA 90095, USA

nCenter for Quantum Mathematics and Physics (QMAP),

Department of Physics, University of California, Davis, CA 95616, USA

Abstract: The ﬁeld of scattering amplitudes plays a central role in elementary-particle physics.

This includes various problems of broader interest for collider physics, gravitational physics, and

fundamental principles underlying quantum ﬁeld theory. We describe various applications and

theoretical advances pointing towards novel descriptions of quantum ﬁeld theories. Comments on

future prospects are included.

arXiv:2210.03146v1 [hep-th] 6 Oct 2022

Contents

1 Introduction 2

1.1 Further Reading 3

2 New Structures from Amplitudes 5

3 Highlights of Amplitudes 6

3.1 Collider Physics 6

3.2 Scattering Amplitudes and Gravitational Waves 7

3.3 Planar N= 4 super-Yang–Mills Amplitudes: from Weak to Strong Coupling 8

3.4 Gravity as a Double Copy of Gauge Theory 9

3.5 String Scattering Amplitudes and World Sheet Models 10

3.6 Web of Theories 10

3.7 Constraints on Eﬀective Field Theories 11

3.8 Positive Geometry and the Amplituhedron 12

3.9 S-Matrix Function Space 12

3.10 Cosmological Bootstrap 13

4 Outlook and Conclusions 13

Executive summary

Virtually everything we have learned about the behavior of elementary particles has been gleaned

from experimental and theoretical studies of scattering processes. The past few decades have taught

us that scattering amplitudes oﬀer remarkable insights into the structure of quantum ﬁeld theories,

as well as eﬃcient routes to precision theoretical results needed to interpret modern experiments.

These insights, including those that follow from novel descriptions of scattering amplitudes suggest

that some of our most cherished notions about quantum theories using the principles of locality

need revision. These novel approaches include those using on-shell approaches, twistor-space and

geometric approaches. It has also become abundantly clear that scattering amplitudes have led to

important progress in issues of interest to the broader community.

Scattering amplitudes have a long history of applications to collider physics, string theory,

supergravity, mathematical physics, and more recently to gravitational-wave physics, summarized

in relevant Snowmass white papers [1–13]. The basic premise of the ﬁeld is a virtuous cycle

between explicitly calculating quantities of experimental or theoretical interest and identifying new

structures that teach us basic facts about quantum ﬁeld theories. These structures in turn lead to

improved methods to carry out out new calculations that then lead to new insights. This positive

feedback loop has continued to infuse the ﬁeld with new ideas and energy informing and guiding

new advances. This has been used to push the state of the art for collider physics, supergravity and

– 1 –

more recently for precision calculations of direct importance to gravitational-wave emission from

binary black holes and neutron stars.

Scattering amplitudes also serve as a wonderful theoretical playground to test new ideas and

connections. This has led to great advances in our understanding of gauge theories, connections to

positive geometry and Amplituhedron, cluster algebras and produced eﬃcient bootstrap methods

for higher-loop amplitudes. We have seen also use of integrability techniques and an intriguing

imprint of AdS/CFT correspondence in the structure of S-matrix at strong coupling. The color-

kinematics duality and related double copy uncovered a deep connection between various quantum

ﬁeld theories, and allows us to construct scattering amplitudes using universal building blocks.

In the coming years we expect that scattering amplitudes will continue to address nontriv-

ial problems in collider physics, gravitation, conformal ﬁeld theories, gravitational-wave physics,

eﬀective ﬁeld theories, and as well as open up new direction such as cosmology and completely un-

expected ones. Tantalizing hints, such as from new geometric approaches to scattering, bootstraps

and unexplained ultraviolet cancellations in extended supergravity theories, suggests that we need

to rethink fundamental principles in quantum ﬁeld theory.

1 Introduction

From Rutherford’s discovery of the atomic nucleus more than a century ago by scattering αparticles

from gold foil, to the much more recent discovery of the Higgs boson at the Large Hadron Collider

(LHC) at CERN, the observation and interpretation of scattering events have been central to our

understanding of elementary-particle interactions. In recent years the ﬁeld of scattering amplitudes

has taken on a renewed vitality, not only because of the continued importance to experimental

and theoretical studies, but also because of the realization that scattering amplitudes oﬀer deep

insight into the very structure of quantum ﬁeld theories. It has had a broad variety of state

of the art applications to collider physics, supergravity, string theory, mathematical physics and

gravitational-wave physics.

The past few decades of research have revealed remarkable new structures in scattering am-

plitudes that provide striking insights into the structure of modern quantum theories, as well as

eﬃcient routes to carry out theoretical results needed to interpret various experiments. These

insights, including geometric structures in amplitudes, suggest that some of our most cherished

principles about constructing quantum theories using locality need revision. It has also become

abundantly clear in recent years that deep issues in quantum gravity, including its relation to gauge

theories, can be understood through studies of scattering amplitudes.

As illustrated in Fig. 1, a virtuous cycle between new explicit calculations and new identiﬁed

structures that then lead to improved methods is central to progress. Many examples abound,

starting from the n-gluon Parke-Taylor MHV amplitudes [14,15], which was orginally studied in

the context of jet physics at particle colliders. These amplitudes form the basis for many other

advances including the construction of n-point one-loop MHV amplitudes [16], and the Cachazo–

Svrcek–Witten diagrams [17] for obtaining all tree-level helicity amplitudes starting from the MHV

ones, as motivated by twistor-space representations of amplitudes [18,19]. The modern unitary

method [16,20,21] was developed following computations using earlier methods that led to the

simple form of the explictly computed one-loop ﬁve-gluon amplitude of QCD. The Britto–Cachazo–

Feng–Witten (BCFW) on-shell recursion relations [22] were in turn motivated by explicit forms of

– 2 –

new

structures

new

methods

new

calculations

Figure 1. The virtuous cycle between explicit results from calculations, new theoretical structures, and new

methods.

tree amplitudes appearing as coeﬃcients of infrared singularities in one-loop amplitides [23]. The

pace of development of new methods has continued in recent years with examples being new methods

for describing massive states using helicity methods [24], new methods for writing down multi-loop

amplitudes bypassing integration [25] and improved methods to obtain results directly relevant to

precision predictions of gravitational waves from astrophysical sources [26–28]. There are many

other examples of the synergy between explicit results and the development of new methods that

then lead to further new results, with the expectation that the cycle will continue well into the

future.

1.1 Further Reading

Snowmass is a community planning exercise, and the present document aspires to represent the

excitement and interests of the growing community of theorist who work in the area of scattering

amplitudes and topics with direct overlap. We gratefully acknowledge the contributions from the au-

thors of the white papers oﬀering valuable insights and guidance for the future. The following people

have contributed to white papers helpful as input for this summary: Tim Adamo, Nima Arkani-

Hamed, Benjamin Basso, Daniel Baumann, Xavier Bekaert, Nathan Berkovits, Nicolas Boulanger,

Jacob L. Bourjaily, Broedel Broedel, Alessandra Buonanno, Andrea Campoleoni, John Joseph Car-

rasco, Mariana Carrillo-Gonz´alez, Ekta Chaubey, Marco Chiodaroli, Claudia de Rham, Lance J.

Dixon, Claude Duhr, Eric D’Hoker, Henriette Elvang, Fernando Febres Cordero, Dario Francia,

Hjalte Frellesvig, Steven B. Giddings, Walter Goldberger, Daniel Green, Michael B. Green, Maxim

Grigoriev, Martijn Hidding, Henrik Johansson, Austin Joyce, Mohammed Khalil, Martin Kruczen-

ski, Sandipan Kundu, Robin Marzucca, Andrew J. McLeod, Tobias Neumann, Donal O’Connell,

Enrico Pajer, Joao Penedones, Guilherme L. Pimentel, Radu Roiban, Oliver Schlotterer, Ergin Sez-

gin, William Shepherd, Evgeny Skvortsov, Mikhail P. Solon, Marcus Spradlin, Lorenzo Tancredi,

Massimo Taronna,Andrew J. Tolley, Jarosalv Trnka, Matthew Reece, Balt C. van Rees, Charlotte

Sleight, Cristian Vergu, Anastasia Volovich, Matthias Volk, Matt von Hippel, Andreas von Man-

teuﬀel, Stefan Weinzierl, Matthias Wilhelm, Mao Zeng, Chi Zhang, and Shuang-Yong Zhou.

In this summary, due to the large number of papers, we include citations only to a relatively

small number of selected papers and refer readers to the relevant white papers [1–13] for a detailed

list of references. Besides the white papers, readers may also consult various review articles [29–37]

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SnowmassTF04Report:ScatteringAmplitudesandtheirApplications.ZviBern,aJaroslavTrnkabaManiL.BhaumikInstituteforTheoreticalPhysics,UCLADepartmentofPhysicsandAstronomy,LosAngeles,CA90095,USAnCenterforQuantumMathematicsandPhysics(QMAP),DepartmentofPhysics,UniversityofCalifornia,Davis,CA95616,USAAbstract:...

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