Skip to main content
European Commission logo
English English
CORDIS - EU research results

The unity of scattering amplitudes: gauge theory, gravity, strings and number theory

Periodic Reporting for period 3 - UNISCAMP (The unity of scattering amplitudes: gauge theory, gravity, strings and number theory)

Reporting period: 2022-01-01 to 2023-06-30

The central theme of the project UNISCAMP are scattering amplitudes -- central observables in quantum field theory that provide essential information about the quantum consistency of perturbative gravity. Precise control of the physical and mathematical properties of scattering amplitudes holds the key to long-standing questions on fundamental interactions and the structure of space and time. As a concrete leap in this direction, UNISCAMP addresses predictions in gauge theories, gravity, effective theories and in particular string theory through

- the efficient computation and compact representation of scattering amplitudes and,

- decoding their hidden structures & symmetries and their rich web of connections.

String-theory methods complement conventional approaches to scattering
amplitudes, and UNISCAMP combines the insights from

- the point-particle limit of superstrings & heterotic strings and,

- ambitwistor strings which directly compute field-theory amplitudes.

Both of them naturally incorporate the double-copy relation between gauge-theory & gravity amplitudes and extend the framework to effective field theories including higher-derivative interactions. It is a primary goal of UNISCAMP to pinpoint the unifying principles connecting a wide range of field and string theories.

Moreover, field- and string-theory amplitudes exhibit an intriguing mathematical structure: Their Feynman- and moduli-space integrals yield special functions such as polylogarithms which became a vibrant common theme of high-energy physics, number theory and algebraic geometry. An interdisciplinary goal of UNISCAMP is to

- investigate the low-energy expansion of multiloop string amplitudes and,

- extract an organizing scheme for iterated integrals and modular forms on higher-genus Riemann surfaces.
The workflow and results of UNISCAMP between January 2019 and July 2021 can be roughly divided into (but are not limited to) four areas:

1) Structure of worldsheet integrals in string theories

The first 30 months of the project brought a particularly fruitful flow of ideas and results on the mathematical structure of string interactions, more precisely the moduli-space integrals associated with string worldsheets. For scattering amplitudes of open and closed strings at various loop orders, prototype integrals for punctured disks, spheres, tori, cylinders and higher-genus surfaces have been identified. Based on their differential equations and new methods to perform their low-energy expansions, these prototype integrals offer a streamlined way to extract the mathematical information on a variety of string amplitudes.

At genus zero, the properties of open-string integrals under the so-called motivic Galois coaction were shown to resonate with the diagrammatic coaction of Feynman integrals, establishing a striking bridge between particle physics and string theory. At genus one, worldsheet integrals for open- and closed-string amplitudes were related via formal operations on the ingredients in the respective low-energy expansions which connects ultraviolet-complete gauge and gravity interactions in a novel way. At higher genus, an essential generalization of modular forms to so-called modular graph tensors was identified which for instance underpin relations among simpler, non-tensorial modular forms.

2) Genus-two five-point amplitudes

Based on the RNS- and pure-spinor formulations of the type II superstring, the first explicit result for the complete genus-two five-point amplitude has been presented and thoroughly tested. First, the corresponding two-loop five-point amplitudes in maximally supersymmetric gauge theory and supergravity have been reproduced, along with a string-theory realization of the double-copy structure of perturbative gravity. Second, the low-energy expansion of our result has been lined up with the so-called S-duality of type IIB superstrings. Third, five-point string amplitudes as opposed to four-point ones allow for violations of the R symmetry of supergravity, and we have described the properties of R-symmetry violating processes at genus two.

The genus-two five-point amplitude is a particularly appealing research target because it captures many generic features of multiloop and multileg amplitudes and thus paves the way for higher-point and higher-genus generalizations.

3) Twistors, pure spinors and M theory

Four-dimensional twistor formulations of N=4 super-Yang-Mills scattering amplitudes have given a striking geometric perspective on the interactions of subatomic particles. In a couple of papers we have studied a possible ten-dimensional incarnation of such a framework which revealed strong connections with the pure spinor formalism for superstrings.

In addition, although we understand that a non-perturbative formulation of superstring theory must exist in eleven dimensions, namely M-Theory, our knowledge on its fundamental description is currently very poor. We have studied a possible building block for computing amplitudes of massless M-Theory states in a manifestly super-Poincare covariant manner.

4) Supersymmetric gauge theories

Finally, two important case studies for specific four-dimensional supersymmetric gauge theories were successfully carried out:

- An analysis of the two-loop four-point scattering amplitude in N=2 Super-QCD led to new insights on its infrared properties and mathematical structure. This is the first integrated result at said number of loops and legs for external matter. An infrared subtraction scheme is formulated which cancels explicitly all finite terms up to next-to-maximal transcendentality.

- In planar N=4 super-Yang-Mills theory, techniques from integrability and input from the known symbol-level results at seven points led to a conjecture for the all-order central emission block in this theory. This conjectural result (along with previous work) completely fixes amplitudes in the high-energy limit for all loops and any number of external particles in the kinematic region where all produced particles have negative energy. In turn, these all-order results allow for a full understanding of the function space in this sector including an explicit proof of the maximal-transcendentality principle for such amplitudes.
The above results already greatly advance the state of the art in different areas and set the stage for an even stronger body of results to be obtained from July 2021 until December 2023. The most foreseeable results for the near future include the following:

A) loop-level matrix elements in effective field theory and the gravitational double copy

The studies of worldsheet integrals for superstring loop amplitudes done before July 2021 are tailored to extracting the quantum corrections to the low-energy effective interactions. However, the full loop amplitudes also exhibit the discontinuities of the effective field theory encoded in string amplitudes at lower genus. Systematic methods to determine loop-level matrix elements of effective gauge and gravity interactions are under active investigation, based on synergies between conventional and ambitwistor string theories.

The matrix elements of higher-derivative operators will serve as valuable testing grounds for the multiloop systematics of the gravitational double copy. In particular, the rich dependence on the loop momentum in the integrands of one-loop matrix elements will help to address the challenges from the loop-momentum dependence in gauge and gravity integrands at higher loops. At the same time, the loop-level matrix elements will advance our understanding of the discontinuity structure of full-fledged string amplitudes.

B) elliptic modular graph forms and modular graph tensors

Loop amplitudes of closed superstrings guide the construction of single-valued elliptic polylogarithms in any number of variables by integrating out a subset of the punctures on the worldsheet. At genus one, this led to the framework of elliptic modular graph forms pioneered in a recent publication, and we shall give a characterization in terms of iterated integrals of holomorphic building blocks and their complex conjugates. This not only exposes the intricate network of algebraic and differential relations of elliptic modular graph forms but also paves the way to relate open and closed strings at genus two and beyond.

We have pioneered modular graph tensors as the appropriate framework for the modular forms in higher-genus string amplitudes. It remains to investigate their differential equations and holomorphic building blocks in the near future. The function spaces to be unravelled in this way will be important in other areas of physics including new methods for the evaluation of Feynman integrals, i.e. for precision predictions in particle physics.

C) diagrammatic coaction at genus one

The worldsheet integrals in string tree-level amplitudes were fruitfully connected with the coaction properties of Feynman integrals that evaluate to multiple polylogarithms. This kind of interplay between string theory and particle physics will be generalized to elliptic polylogarithms and beyond. As a crucial first step towards this goal, bases of worldsheet integrals at higher genus will be constructed that close under differentiation in the moduli. This task will be greatly facilitated by the availability of genus-one prototype bases whose generating-series form is a key achievements of the past 30 months.