Quantum Fields and Strings Group

Research

Our gourp’s research is broadly focused on High-Energy Physics Theory, especially on the structures of Scattering Amplitudes in the QFT and String Theory.

A brief historical review of scattering amplitude research and its major developments:

  1. The spinor-helicity formalism, introduced creatively in the 1980s by three Chinese physicists（Zhan Xu, Dahua Zhang, and Lee Chang）, greatly simplified the structure of scattering amplitudes. This method became known as the “Chinese Magic.” A prominent example is the MHV tree-level amplitude in pure Yang–Mills theory obtained by Parke and Taylor:

    \[A_{n}\left[1^{+} \ldots i^{-} \ldots j^{-} \ldots n^{+}\right]=\frac{\langle i j\rangle^{4}}{\langle 12\rangle\langle 23\rangle \cdots\langle n 1\rangle}\]

    Around the same time, Witten recognized the underlying mathematical structure and introduced twistor methods into scattering amplitudes, formulating the twistor string model and significantly advancing our understanding of the physics behind amplitudes.

  2. In the 1990s, the unitarity-cut method, pioneered by Bern, Dixon, and Kosower, emerged and successfully resolved many one-loop amplitude computations. This pushed the precision of theoretical predictions for observable quantities to the next-to-leading order.

  3. Around 2005, on-shell methods experienced major breakthroughs. The Britto–Cachazo–Feng–Witten (BCFW) recursion extended external momenta into the complex plane and used Cauchy’s residue theorem to fully reveal the analytic structure of tree amplitudes, expressing them via recursion relations. This greatly simplified amplitude computations. Combined with the forward limit, BCFW can also be applied to loop integrands. BCFW additionally leads to the Cachazo–Svrcek–Witten (CSW) off-shell formalism, which decomposes amplitudes into recursive structures built from MHV amplitudes.

  4. Around 2008, Bern, Carrasco, and Johansson (BCJ) proposed the BCJ relations—also known as color-kinematics duality—which further constrained the freedom of tree-level amplitudes and revealed deep connections between gauge theory and gravity amplitudes. As a mathematician, Hodges made significant contributions during this period: he wrote the MHV gravity tree amplitude in determinant form, introduced momentum twistors, and applied polytopes to describe amplitudes, resolving spurious poles in BCFW recursion. This inspired later work by Nima et al., including the use of Grassmannians to describe tree amplitudes in \mathcal{N}=4 SYM.

  5. The year 2013 marked an important milestone: Cachazo, He, and Yuan (CHY) introduced the CHY formalism, expressing arbitrary-dimensional tree amplitudes in a remarkably elegant form based on integrals over the Riemann sphere. Meanwhile, Nima Arkani-Hamed and collaborators sought the mathematics behind scattering amplitudes and introduced the Amplituhedron—a geometric object whose volume encodes tree amplitudes and one-loop integrands in \mathcal{N}=4 SYM.

  6. In 2017, Sebastian Mizera introduced algebraic topology into scattering amplitudes, interpreting certain amplitudes as intersection numbers in topology and computing the number of master integrals in loop computations. Nima and collaborators further elevated the amplituhedron to a broader mathematical framework: positive geometry, extending amplitude ideas to many other contexts (cosmological wavefunctions, scalar field theories, etc.).

  7. ……

In summary, current scattering amplitude research branches into several major directions. Some researchers focus on high-loop computations, which are extremely complex and often rely on bootstrap techniques and symmetry principles (color-kinematics duality, dual conformal symmetry, etc.) to compute integrands, integrals, or their symbols. Others focus on the mathematical structures behind amplitudes, such as Hopf algebras and coproducts. Still others pursue geometric interpretations of amplitudes, including positive geometry and related frameworks.

Below is a brief overview of our gourp’s research interests and some suggested introductory references for students and collaborators.

0. Scattering Amplitudes in Gauge Theory and Gravity

We are interested in the modern formulation of scattering amplitudes in gauge theory and gravity, including on-shell methods, recursion relations, and hidden structures such as dual conformal symmetry and color–kinematics duality.

Typical questions include:

Recommended reading (introductory level)

1. Hidden Properties and Relations of Scattering Amplitudes

Even though the study of scattering amplitudes is already well developed, many hidden properties and relations (such as the zeros, splitting, and universial expansion relations) have only been discovered recently. There remain mysteries yet to be uncovered, and we continue to seek a deeper understanding of scattering amplitudes.

Recommended reading

2. Color-Kinematics Duality and Double Copy

One of the most remarkable discoveries in the study of scattering amplitudes is the color–kinematics (CK) duality and its associated double copy. These structures appear not only in field theory but also in string theory. The CK duality and the double copy reveal hidden relations between gauge theories and gravity, providing powerful insights into the fundamental structure of gravity.

Recommended reading (introductory level)

Further reading

3. String Amplitudes and Worldsheet Formula

Field-theory amplitudes can be obtained from string amplitudes in the $\alpha’\to0$ limit, so studying string amplitudes allows us to uncover information that goes beyond field theory . Moreover, many string-inspired methods have achieved remarkable success, such as worldsheet formulations and the Cachazo–He–Yuan (CHY) formula, which elegantly capture the underlying structures of tree-level field-theory amplitudes.

Recommended reading (introductory level)

Further reading

4. Geometry Behind Scattering Amplitudes

A particularly remarkable development in the study of scattering amplitudes is the emergence of underlying geometric structures. For example, amplitudes in $\mathcal{N}=4$ Super–Yang–Mills theory are elegantly encoded in the positive Grassmannian. In the simplest scalar theory, $\mathrm{Tr}(\phi^3)$, tree-level amplitudes are associated with the associahedron, while higher-loop amplitudes correspond to the surfacehedron. These insights have inspired the recent “surfaceology” program, which aims to uncover all-loop geometric patterns in more general theories, such as Yang–Mills and the Nonlinear Sigma Model.

Recommended reading (introductory level)

Further reading

5. Observables in Curved Spacetime

Our group is also interested in the study of observables in curved spacetime. There are two main directions: 1. Anti–de Sitter (AdS) spacetime: Owing to the holographic principle, AdS is dual to a conformal field theory (CFT) living on its boundary. We investigate observables in the AdS bulk, particularly scattering amplitudes, and frequently make use of the duality to study CFT correlation functions through their Mellin-space representations. 2. de Sitter (dS) spacetime: In this setting, our focus shifts to the wavefunction coefficients of the universe in dS spacetime, which play an analogous role to scattering amplitudes and encode rich cosmological information.

Recommended reading (introductory level)

Further reading

If you are a student or potential collaborator interested in these topics and would like to discuss projects or reading directions, feel free to contact us by email.