# Scattering Amplitudes

## Group Members

**Faculty:** Paolo di Vecchia (Nordita), Henrik Johansson, Oliver Schlotterer

**Long-term researcher:** Marco Chiodaroli

**Post.Doc. Fellows:** Francesco Alessio (Nordita), Alexander Edison, Max Guillen, Kays Haddad, Carlo Heissenberg, Martijn Hidding, Sourav Sarkar, Ingrid Vazquez-Holm, Bram Verbeek, Zhewei Yin, Yong Zhang

**PhD Students:** Maor Ben-Shahar, Filippo Balli, Lucile Cangemi, Lucia Garozzo, Carlos Gustavo Rodriguez Fernandez, Paolo Pichini

**Former group members:** Marios Hadjiantonis, Yong Zhang (Postdoc at Perimeter, Waterloo, Canada), Thales Azevedo (Professor at Universidade Federal de Rio de Janeiro, Brazil), Gregor Kälin (Postdoc at DESY, Hamburg, Germany), Gang Chen (postdoctoral research assistant at QMUL, London, UK), Tianheng Wang (Postdoc at Humboldt University, Berlin, Germany), Alessandro Georgoudis (Postdoc at Nordita, Stockholm, Sweden), Oluf Tang Engelund, Gustav Mogull (postdoctoral research assistant at HU Berlin), Fei Teng (postdoctoral research assistant at Pennsylvania State University)

Our research on scattering amplitudes benefits from the close ties between Uppsala University and Nordita through joint appointments, scientific collaborations and common grants. In particular, Paolo Di Vecchia and Henrik Johansson have a joint KAW project “From scattering amplitudes to gravitational waves”. Our group furthermore receives financial support via an ERC starting grant, a VR project grant and a Ragnar Söderberg grant.

## Why scattering amplitudes?

Scattering amplitudes are the central predictions in theories of fundamental interactions. They encode the probabilities for specific scattering processes to happen. In particle physics, scattering amplitudes serve as the bridge between quantum field theory and experiment since the associated probabilities are measured in colliders such as the Large Hadron Collider (LHC) at CERN. In gravitational theories scattering amplitudes provide essential information about their quantum consistency and reveal a variety of surprises about the ultraviolet properties of supersymmetric theories. Moreover, string theory as a candidate theory of quantum gravity was born out of the study of hadronic scattering amplitudes. Finally, the intriguing mathematical structures of scattering amplitudes became a driving force for fruitful interdisciplinary interactions between high-energy physicists and mathematicians.

In the context of both field and string theories, scattering amplitudes turned out to enjoy a hidden simplicity which is inaccessible to the traditional prescriptions to compute them. Hence, by insisting on a manifestation of their striking simplicity, scattering amplitudes can guide us towards radical reformulations of particle physics, gravity, string theory and possibly other approaches to quantum gravity. A ubiquitous example among the hidden structures in string- and field-theory amplitudes is the notion of “double copy”: As will be detailed below, gravity can be viewed as a “square of gauge theories”, and even string amplitudes exhibit the same kind of double-copy structures that have been found in field-theory amplitudes. Our division has played a key role in the discovery of double-copy structures and aims to explore their scope and implications.

## Feynman diagrams versus modern computational methods

The traditional method for computing scattering amplitudes in field theories is based on Feynman rules. They are famous for intuitively assembling all possible histories for a given scattering process such that amplitudes can be obtained from associating mathematical quantities to the diagrams.

Feynman rules have been known for decades and remain the standard textbook method for computing scattering amplitudes. So why are we interested in other methods?

*Figure 1: Samples of Feynman diagrams.*

It turns out that Feynman rules can be rather cumbersome and obscure the simplicity of the final answers. Moreover, the number of Feynman diagrams can often be huge and contain a cornucopia of redundant information that cancels from the short and simple final result. The discrepancy between compact expressions for scattering amplitudes and the spurious complexity in the opening line of the Feynman-diagram prescription suggests that there is an easier way of computing them.

Recent years have witnessed a variety of modern approaches to simplify the calculation of field-theory amplitudes. For instance, unitarity of the S-matrix and its implications on the analytic structure of scattering amplitudes have considerably extended the computational reach. Furthermore, various flavors of string theories allowed to derive field-theory amplitudes from two-dimensional Riemann surfaces where symmetries and simplifications descend from complex-analysis methods. At the same time, new mathematical techniques for the integration over internal loop momenta in Feynman diagrams have been exploited to attain unprecedented precision in collider predictions or to derive certain amplitudes from a “bootstrap procedure”. In many cases, modern methods exploit previously found results, e.g. by constructing recursion relations, such that amplitudes involving *n+1* particles are written in terms of amplitudes involving *n* or less particles.

## Gauge theories and color-kinematics duality

Gauge theories are essential to our understanding of the fundamental interactions of nature. Quantum Electrodynamics and Quantum Chromodynamics are examples of realistic gauge theories whose predictions have been matched with experiments to outstanding precision. Also, many features of gauge theories are easier to study in presence of supersymmetry, an exchange symmetry between bosons (carriers of fundamental forces) and fermions (matter particles such as quarks and leptons) that has not yet been observed in experiments. In particular, the maximally supersymmetric gauge theory known as N=4 super Yang-Mills became a fruitful laboratory for theoretical studies and a driving force in probing new structures in scattering amplitudes. Many properties of the particularly simple amplitudes in N=4 super Yang-Mills turned out to have an echo in realistic gauge theories and triggered important progress in our understanding of Quantum Chromodynamics.

A particularly broadly applicable hidden symmetry of gauge-theory amplitudes co-discovered by members of our division is known as the *duality between color and kinematics* [1]. In non-abelian gauge theories, external states carry internal or so-called “color” degrees of freedom that enter their scattering amplitudes through the structure constants (or other invariant tensors) of the gauge group. The term “kinematics” refers to any other quantity entering a non-abelian-gauge-theory amplitude such as polarizations and momenta. According to the color-kinematics duality, gauge-theory amplitudes admit representations, where color and kinematic degrees of freedom enter on completely symmetric footing and can be freely interchanged. In particular, the group-theoretic Jacobi identities of color factors are conjectured to apply to kinematic factors in a color-kinematics dual amplitude representation.

*Figure 2: Color-kinematics duality in gauge-theory amplitudes.*

A wide range of evidence has accumulated for the conjectural color-kinematics duality – it has been manifested in quantum corrections to both supersymmetric and non-supersymmetric gauge-theory amplitudes. Moreover, the scope of the color-kinematics duality is not limited to gauge theories – it has been found for effective theories of scalars (such as pion interactions) and in string amplitudes. Like this, the color-kinematics duality points to universal structures connecting different areas of high-energy physics.

## Gravity versus double copies of gauge theories

The color-kinematics duality becomes particularly powerful in connection with the notion of “double copy”. This has been firstly exemplified in the study of perturbative (super-)gravity where Feynman rules for scattering of gravitons are derived from the Einstein–Hilbert Lagrangian (possibly along with supersymmetric extensions). The resulting gravitational amplitudes have been infamous for the hopeless complexity of their Feynman diagrams and for the naive power-counting expectation that their quantum corrections should be plagued by an unacceptable pattern of ultraviolet divergences.

*Figure 3: The double-copy structure of gravity amplitudes.*

However, both of these pessimistic expectations have been elegantly countered by the color-kinematics duality and the closely related double copy: A variety of gravitational amplitudes can be assembled from squares of gauge-theory kinematic factors provided that the latter manifest their duality to color [2]. This kind of double-copy construction applies to high orders in the quantum corrections of both supergravity and to non-supersymmetric Einstein gravity. Thanks to the gravitational double copy, numerous amplitudes became accessible to pen-and-paper calculations which for instance sidestep the more than 10^{18} terms in a Feynman-diagram approach to the following graphs.

*Figure 4: The double-copy representation of the 3-loop 4-point amplitude in so-called N=8 supergravity can be encoded in 12 graphs each of which represents the square of simple gauge-theory building blocks.*

Furthermore, the compact double-copy representations of gravitational amplitudes revealed stunning surprises about their ultraviolet behaviour and clarified a multitude of cancellations among the traditionally expected divergences. On the longer run, the double copy is a promising key tool to assess the viability of pointlike theories of quantum gravity.

In fact, as visualized in the web of theories below, the scope of the double copy goes well beyond (super-)gravity: Amplitude representations in double-copy form have been found for Born–Infeld electrodynamics and for Galileons [3] – certain low-energy modifications of gravity – but also for open and closed strings [4].

*Figure 5: The web of theories connected by double copy. Taken from [C].*

Finally, it is natural to look for applications of double-copy methods to perturbative gravity calculations in different contexts. A promising arena is provided by the computations of gravitational-wave templates that are employed to match data from LIGO and Virgo with theoretical predictions, particularly in the so-called inspiral phase of the merger. Since these calculations presents close analogies with amplitude computations, one of the goals of the group is to generalize the double copy to streamline calculations relevant to gravitational wave physics.

## String-theory amplitudes

The first relation between gravitational amplitudes and squares of gauge-theory building blocks has been derived from the point-particle limit of string theories by Kawai, Lewellen and Tye in 1986 [5]. While non-abelian gauge theories arise from the ground-state vibrations of open strings, gravity and its supersymmetric extensions descend from massless closed-string excitations. In the low-energy limit, the double-copy structure of gravity amplitudes then lines up with the geometric intuition that closed strings can be formed by joining the endpoints of two open strings. At leading orders in the quantum corrections, the color-kinematics duality of gauge theory and double-copy representations of gravity amplitudes have been derived from string theories [6]. Moreover, the systematic extension to higher orders belongs to the central research goals of our group.

*Figure 6: Closed-string excitations including gravitons are double copies of open-string excitations including gauge bosons.*

Recent work involving members of our division revealed that double-copy structures even extend to *open-*string amplitudes [4]. Double-copy representations of this type efficiently package all the polarization dependence into field-theory building blocks and arrange their string corrections to solely depend on the momenta. These simplifications rely on reorganizations of the underlying conformal-field-theory techniques and are believed to extend to higher orders in string perturbation theory.

*Figure 7: The double-copy structure of string tree-level amplitudes.*

Finally, string amplitudes exhibit an intriguing mathematical structure: They involve moduli-space integrals over punctured Riemann surfaces which evaluate to special functions such as polylogarithms, multiple zeta values and their elliptic generalizations. Modern number-theoretic concepts including motives, associators and single-valued maps played an essential role in unravelling the hidden simplicity of string amplitudes [7]. Similar mathematical structures appear in the Feynman integrals of field-theory amplitudes and became a vibrant common theme of high-energy physics and number theory. As a particular quality of string amplitudes, they tend to introduce various flavours of polylogarithms in particularly pure contexts and therefore offer a rewarding laboratory to explore the properties of new classes of functions. For instance, one-loop open-string amplitudes gave rise to the first appearance of elliptic multiple zeta values in physics [8].

*Figure 8: Worldsheet of interacting closed strings can be mapped to punctured Riemann surfaces of various genera.*

## Conclusion

In summary, a wealth of intriguing structures has been recently discovered in scattering amplitudes of various field and string theories. Many of these hidden structures play a key role in unifying seemingly unrelated areas of physics as visualized in the above web of double-copy relations [Figure 5]. Also, the crosstalk between string theorists, particle phenomenologists and mathematicians became increasingly fertile for all sides. All of this is a major motivation for our group to stay focused on amplitudes and to work on ambitious research objectives such as

- Understanding the origin of the color-kinematics duality through a kinematic algebra, and systematic methods to manifest the duality in a multiloop context.
- Unravelling the ultraviolet properties of supergravity theories with different degrees of supersymmetry.
- Identifying loop-level double-copy structure in string amplitudes.
- Applying the color-kinematics duality and double copy to high-precision calculations in QCD.
- Exploring the mathematical properties – in particular the transcendentality structure – in multiloop amplitudes of string- and field theories.
- Computing gravitational-wave templates via double-copy techniques.

## Suggested Reading

A first overview of the recent developments in scattering amplitudes can be obtained from the textbook [A] and lecture notes such as [B]. In particular, a comprehensive review on the color-kinematics duality coauthored by Marco Chiodaroli and Henrik Johansson appeared in September 2019 [C]. Moreover, popular scientific articles on scattering amplitudes include [D] on ultraviolet properties of supergravity, [E] on the modern amplitudes program, [F] on the future of quantum gravity, [G] on the number-theoretic properties of amplitudes. A selection of video lectures on scattering amplitudes is given below.

[A] H. Elvang and Y. t. Huang, “Scattering Amplitudes in Gauge Theory and Gravity,” Cambridge University Press (2015), ISBN: 9781107706620

[B] C. Cheung, “TASI Lectures on Scattering Amplitudes,” [arXiv:1708.03872 [hep-ph]]; S. Weinzierl, “Tales of 1001 Gluons,” Phys. Rept. **676** (2017) 1 [arXiv:1610.05318 [hep-th]]; J. J. M. Carrasco, “Gauge and Gravity Amplitude Relations,” [arXiv:1506.00974 [hep-th]].

[C] 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]].

[D] H. Nicolai, “Vanquishing infinity,” Physics 2 (2009), [https://physics.aps.org/articles/v2/70].

[E] Z. Bern, L. J. Dixon and D. A. Kosower, Scientific American (2013), “Loops, Trees and the Search for New Physics,” [https://www.scientificamerican.com/article/loops-trees-and-the-search-for-new-physics-extreme-physics-special/].

[F] N. Wolchover, “Betting on the Future of Quantum Gravity,” Quanta Magazine (2014), [https://www.quantamagazine.org/betting-on-the-future-of-quantum-gravity-20140314/].

[G] K. Hartnett, “Strange Numbers Found in Particle Collisions,” Quanta Magazine (2016), [https://www.quantamagazine.org/strange-numbers-found-in-particle-collisions-20161115/].

## Video lectures

- John-Joseph Carrasco, Johannes Henn, Marcus Spradlin, Jaroslav Trnka and Stefan Weinzierl, Amplitudes Summer School 2017, Higgs Centre for Theoretical Physics, Edinburgh, UK
- Lance Dixon, Amplitudeology, Lectures at the TASI Summer School 2017, Boulder, Colorado, US
- Marco Chiodaroli, Yang–Mills–Einstein amplitudes from the double copy, Talk at the program “Scattering Amplitudes and Beyond” at KITP, Santa Barbara, US
- Freddy Cachazo, Oliver Schlotterer et al., Scattering amplitudes in QFT and string theory, 15-hour course of the PSI master program (2017/18) at the Perimeter Institute, Waterloo, Canada
- Nima Arkani-Hamed, Zvi Bern, Jacob Bourjaily, Claude Duhr, Eric D’Hoker, Song He, Yu-tin Huang and Alexander Postnikov, Amplitudes Summer School 2018, Center for Quantum Mathematics and Physics (QMAP), Davis, US
- Henrik Johansson, Color-kinematics duality and double copy, lecture at the “MITP Summer School 2018: Towards the Next Quantum Field Theory of Nature”, Mainz, Germany
- Oliver Schlotterer, Superstring Amplitudes in RNS and Pure Spinor Formalism, Training week of the program “String Theory from a Worldsheet Perspective” at the GGI Florence, Italy
- Gleb Arutyunov, Claude Duhr, Johannes Henn, David Kosower and Oliver Schlotterer, 1st SAGEX scientific school at DESY, Hamburg, Germany
- Jacob Bourjaily, Johannes Broedel, Lance Dixon, Alexander Ochirov and Erik Panzer, 2nd SAGEX scientific school at Humboldt University, Berlin, Germany
- Claude Duhr, Ben Page, Duncan Pettengill and Yang Zhang, online SAGEX Mathematica and Maple school hosted by CEA Saclay, Paris, France
- Jacob Bourjaily, Ruth Britto, Lance Dixon, Claude Duhr, Henriette Elvang, Yvonne Geyer, Alexander Huss, Donal O’Connell, Erik Panzer, Ira Rothstein and Oliver Schlotterer, online Summer School Amplitudes Games hosted by MITP, Mainz, Germany
- Tomasz Lukowski, Sabrina Pasterski, Chia-Hsien Shen and Kai Yan, online SAGEX PhD school in Amplitudes hosted by NBI, Copenhagen, Denmark

## References

[1] Z. Bern, J. J. M. Carrasco and H. Johansson, “New Relations for Gauge-Theory Amplitudes,” Phys. Rev. D **78** (2008) 085011 [arXiv:0805.3993 [hep-ph]]

[2] Z. Bern, J. J. M. Carrasco and H. Johansson, “Perturbative Quantum Gravity as a Double Copy of Gauge Theory,” Phys. Rev. Lett. **105** (2010) 061602 [arXiv:1004.0476 [hep-th]; Z. Bern, J. J. Carrasco, W. M. Chen, H. Johansson and R. Roiban, “Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes,” Phys. Rev. Lett. **118** (2017) no.18, 181602 [arXiv:1701.02519 [hep-th]]

[3] F. Cachazo, S. He and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,” JHEP **1507** (2015) 149 [arXiv:1412.3479 [hep-th]]

[4] J. Broedel, O. Schlotterer and S. Stieberger, “Polylogarithms, Multiple Zeta Values and Superstring Amplitudes,” Fortsch. Phys. **61** (2013) 812 [arXiv:1304.7267 [hep-th]]; J. J. M. Carrasco, C. R. Mafra and O. Schlotterer, “Abelian Z-theory: NLSM amplitudes and alpha'-corrections from the open string,” JHEP **1706** (2017) 093 [arXiv:1608.02569 [hep-th]]; C. R. Mafra and O. Schlotterer, “Double-Copy Structure of One-Loop Open-String Amplitudes,” Phys. Rev. Lett. **121** (2018) no.1, 011601 [arXiv:1711.09104 [hep-th]]; T. Azevedo, M. Chiodaroli, H. Johansson and O. Schlotterer, “Heterotic and bosonic string amplitudes via field theory,” JHEP **1810** (2018) 012 [arXiv:1803.05452[hep-th]]

[5] H. Kawai, D. C. Lewellen and S. H. H. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl. Phys. B **269** (1986) 1

[6] C. R. Mafra, O. Schlotterer and S. Stieberger, “Explicit BCJ Numerators from Pure Spinors,” JHEP **1107** (2011) 092 [arXiv:1104.5224 [hep-th]]; C. R. Mafra and O. Schlotterer, “Towards one-loop SYM amplitudes from the pure spinor BRST cohomology,” Fortsch. Phys. **63** (2015) no.2, 105 [arXiv:1410.0668 [hep-th]]; C. R. Mafra and O. Schlotterer, “Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace,” JHEP **1510** (2015) 124 [arXiv:1505.02746[hep-th]]; S. He, O. Schlotterer and Y. Zhang, “New BCJ representations for one-loop amplitudes in gauge theories and gravity,” Nucl. Phys. B **930** (2018) 328 [arXiv:1706.00640 [hep-th]]

[7] O. Schlotterer and S. Stieberger, “Motivic Multiple Zeta Values and Superstring Amplitudes,” J. Phys. A **46** (2013) 475401 [arXiv:1205.1516 [hep-th]]. J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, “All order alpha'-expansion of superstring trees from the Drinfeld associator,” Phys. Rev. D **89** (2014) no.6, 066014 [arXiv:1304.7304 [hep-th]]; O. Schlotterer and O. Schnetz, “Closed strings as single-valued open strings: A genus-zero derivation,” J. Phys. A **52** (2019) no.4, 045401 [arXiv:1808.00713 [hep-th]]; F. Brown and C. Dupont, “Single-valued integration and superstring amplitudes in genus zero,” [arXiv:1910.01107 [math.NT]]

[8] J. Broedel, C. R. Mafra, N. Matthes and O. Schlotterer, “Elliptic multiple zeta values and one-loop superstring amplitudes,” JHEP **1507** (2015) 112 [arXiv:1412.5535 [hep-th]]