Periodic Reporting for period 3 - KAIROS (Bootstrapping Time: Colliders, Shocks, Strings, and Black Holes)
Reporting period: 2024-01-01 to 2025-06-30
Summary of the context and overall objectives of the project
Quantum field theory (QFT) is a theoretical framework that successfully describes phenomena at small scales as probed by the collider experiments. It describes electromagnetic, weak and strong forces. General relativity, or theory of gravitation, predicts the behavior of planetary systems, galaxies, and the Universe as a whole. Many fundamental questions, both in the realm of QFT and gravity are intractable with the existing computational methods. The most basic one is constructing a unified framework which includes all forces of nature and which allows to make experimental predictions for the current and future experiments. Two basic examples where such methods could be necessary include early stages of the Universe, as well as description of new states of matter made out of strongly interacting particles.
The current project explores and develops new computational tools in the context of both QFT and gravity with a particular emphasis on real-time dynamics and observables which are not accessible using conventional methods. It starts with the known fundamental principles and symmetries and pushes them to their limit, the strategy known as `the bootstrap'. This is in particular applicable to the systems that appear in the high-energy physics due to their higher level of symmetries compared to the systems familiar from the everyday experience. The advantage of such bootstrap methods is that they provide a powerful, model-independent, and mathematically solid exploration framework. The two main arenas considered in the project include bootstrap methods for collider physics, as well as bootstrap methods for gravity and black hole physics.
A culture of curiosity and the quest of understanding nature at its most fundamental level has been enriching our society for centuries. The history of physics has been a journey full of serendipitous discoveries, unexpected technological twists, and profound economic implications. Advancing basic science has proven to be an unrivalled strategy in shaping an open-minded, collaborative, critically thinking society which is ready to adapt and thrive in an ever-changing world. Each solved problem, however theoretical, serves as a backbone for applied sciences and roots breakthroughs in the future.
The current project explores and develops new computational tools in the context of both QFT and gravity with a particular emphasis on real-time dynamics and observables which are not accessible using conventional methods. It starts with the known fundamental principles and symmetries and pushes them to their limit, the strategy known as `the bootstrap'. This is in particular applicable to the systems that appear in the high-energy physics due to their higher level of symmetries compared to the systems familiar from the everyday experience. The advantage of such bootstrap methods is that they provide a powerful, model-independent, and mathematically solid exploration framework. The two main arenas considered in the project include bootstrap methods for collider physics, as well as bootstrap methods for gravity and black hole physics.
A culture of curiosity and the quest of understanding nature at its most fundamental level has been enriching our society for centuries. The history of physics has been a journey full of serendipitous discoveries, unexpected technological twists, and profound economic implications. Advancing basic science has proven to be an unrivalled strategy in shaping an open-minded, collaborative, critically thinking society which is ready to adapt and thrive in an ever-changing world. Each solved problem, however theoretical, serves as a backbone for applied sciences and roots breakthroughs in the future.
Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far
Bootstrap methods for collider physics.
1. Nonperturbative scattering amplitudes. We have initiated a research program and developed tools to explicitly construct relativistic nonperturbative scattering amplitudes that obey all known properties of analyticity, unitarity, and crossing. This result is first of its kind and it opens many directions for future developments. It is based on a series of papers that required using the state-of-the-art analytical and numerical tools in the field of scattering amplitudes. The driving force behind the S-matrix bootstrap is a subtle tension between the fundamental properties of scattering amplitudes that follow from relativistic causality and quantum-mechanical unitarity. In a conventional perturbative approach these properties are implemented using Feynman diagrams. This however results in a divergent perturbative expansion of the scattering amplitude. In contrast we have developed a nonperturbative algorithm which has many of the desired features of the Feynman expansion on one hand, and leads to convergent expansion of the amplitude on the other hand. We have solved this problem for scattering of scalar particles in two, three, and four spacetime dimensions. In the future we would like to apply the same methods to scattering of pions in QCD. Moreover, we plan to develop a publicly available package (possibly using the publicly available machine learning tools) that would allow everyone explore the nonperturbative amplitudes easily.
2. Gravitational and stringy amplitudes. We have established a few basic fundamental facts about gravitational scattering. The first one, concerns the high-energy behavior of gravitational scattering amplitudes. For scattering of massive particles in QFT, it has been known since the 60's that the possible growth of the amplitude with energy is bounded by the fourth power of scattering energy. Using the known properties of gravity at large distances we have established the same bound for scattering in gravity. The significance of this bound comes through the ability to use it to write down dispersion relations which allow one to explore the consequences of consistency of the theory in the UV for the IR observations. Moreover, by studying explicit examples we found new unexpected properties of the known theories (both QFTs coupled to gravity, as well as string theories), as well as existence of deformations of stringy amplitudes that preserve softness at high energy, fixed angle scattering. This work is a part of broader community efforts to improve our understanding of the microscopic origin of gravity.
3. Event shapes and light-ray operators. Energy correlations between the produced particles are causally measured at the LHC. In the last five years a new connection between these standard collider observables and the theory of light-ray operators in conformal field theories has emerged. It led to the development of the new type of nonperturbative expansion, called the light-ray OPE, that controls the behavior of energy correlations when the angular separation between detectors becomes small. This turns out to be instructive for better understanding of the jet substructure in QCD. Recently we have extended our understanding of the energy correlations in several directions: introducing a nontrivial working-time profile for detectors, as well as understanding general properties of energy correlations in multi-particle states.
Bootstrap methods for holography and black holes.
1. Consistency conditions for gravitational EFTs. It is believed that given a strongly coupled CFT with a large number degrees of freedom ("large gap, large central charge"), its dual description is given by classical gravity. Establishing this conjecture is an open problem. A part of this conjecture that has been proven recently rather rigorously and was explored in this project puts bounds on the Wilson coefficients that enter the four-graviton scattering in terms of the masses of higher spin particles. More precisely, we have analyzed such bounds by considering dispersion relations for graviton scattering in four dimensions. Surprisingly, we observed that in known UV completions of gravity (provided by integrating out particles or strings) certain Wilson coefficients live on small islands which can be located using the `low spin dominance' rule.
2. Black holes, finite temperatures, large charge. In holography black hole physics is mapped to finite temperature physics of the dual theory. Understanding how spacetime emerges in quantum gravity thus requires understanding thermal properties of strongly coupled QFTs. We have initiated a systematic study of the thermal two-point function in holographic theories and derived a few important and novel results in this direction. We have worked out the detailed connection between gravitational orbits around black holes, the so-called double twist operators that appear in the conformal bootstrap, and quantum many-body scars recently discussed in the condensed matter literature. We also used the state-of-the-art results in mathematical physics to find the exact thermal two-point function in terms of the recently discovered Nekrasov functions and to provide the all order solution to the light cone bootstrap. We also have put forward the product formula which expresses the thermal two-point function in terms of the quasi-normal modes and studied its general properties as dictated by the short-distance properties of the correlator. Finally, we have considered a two-point of light-ray operators in heavy, large charge states and proposed a general formula for the energy correlations in such states.
1. Nonperturbative scattering amplitudes. We have initiated a research program and developed tools to explicitly construct relativistic nonperturbative scattering amplitudes that obey all known properties of analyticity, unitarity, and crossing. This result is first of its kind and it opens many directions for future developments. It is based on a series of papers that required using the state-of-the-art analytical and numerical tools in the field of scattering amplitudes. The driving force behind the S-matrix bootstrap is a subtle tension between the fundamental properties of scattering amplitudes that follow from relativistic causality and quantum-mechanical unitarity. In a conventional perturbative approach these properties are implemented using Feynman diagrams. This however results in a divergent perturbative expansion of the scattering amplitude. In contrast we have developed a nonperturbative algorithm which has many of the desired features of the Feynman expansion on one hand, and leads to convergent expansion of the amplitude on the other hand. We have solved this problem for scattering of scalar particles in two, three, and four spacetime dimensions. In the future we would like to apply the same methods to scattering of pions in QCD. Moreover, we plan to develop a publicly available package (possibly using the publicly available machine learning tools) that would allow everyone explore the nonperturbative amplitudes easily.
2. Gravitational and stringy amplitudes. We have established a few basic fundamental facts about gravitational scattering. The first one, concerns the high-energy behavior of gravitational scattering amplitudes. For scattering of massive particles in QFT, it has been known since the 60's that the possible growth of the amplitude with energy is bounded by the fourth power of scattering energy. Using the known properties of gravity at large distances we have established the same bound for scattering in gravity. The significance of this bound comes through the ability to use it to write down dispersion relations which allow one to explore the consequences of consistency of the theory in the UV for the IR observations. Moreover, by studying explicit examples we found new unexpected properties of the known theories (both QFTs coupled to gravity, as well as string theories), as well as existence of deformations of stringy amplitudes that preserve softness at high energy, fixed angle scattering. This work is a part of broader community efforts to improve our understanding of the microscopic origin of gravity.
3. Event shapes and light-ray operators. Energy correlations between the produced particles are causally measured at the LHC. In the last five years a new connection between these standard collider observables and the theory of light-ray operators in conformal field theories has emerged. It led to the development of the new type of nonperturbative expansion, called the light-ray OPE, that controls the behavior of energy correlations when the angular separation between detectors becomes small. This turns out to be instructive for better understanding of the jet substructure in QCD. Recently we have extended our understanding of the energy correlations in several directions: introducing a nontrivial working-time profile for detectors, as well as understanding general properties of energy correlations in multi-particle states.
Bootstrap methods for holography and black holes.
1. Consistency conditions for gravitational EFTs. It is believed that given a strongly coupled CFT with a large number degrees of freedom ("large gap, large central charge"), its dual description is given by classical gravity. Establishing this conjecture is an open problem. A part of this conjecture that has been proven recently rather rigorously and was explored in this project puts bounds on the Wilson coefficients that enter the four-graviton scattering in terms of the masses of higher spin particles. More precisely, we have analyzed such bounds by considering dispersion relations for graviton scattering in four dimensions. Surprisingly, we observed that in known UV completions of gravity (provided by integrating out particles or strings) certain Wilson coefficients live on small islands which can be located using the `low spin dominance' rule.
2. Black holes, finite temperatures, large charge. In holography black hole physics is mapped to finite temperature physics of the dual theory. Understanding how spacetime emerges in quantum gravity thus requires understanding thermal properties of strongly coupled QFTs. We have initiated a systematic study of the thermal two-point function in holographic theories and derived a few important and novel results in this direction. We have worked out the detailed connection between gravitational orbits around black holes, the so-called double twist operators that appear in the conformal bootstrap, and quantum many-body scars recently discussed in the condensed matter literature. We also used the state-of-the-art results in mathematical physics to find the exact thermal two-point function in terms of the recently discovered Nekrasov functions and to provide the all order solution to the light cone bootstrap. We also have put forward the product formula which expresses the thermal two-point function in terms of the quasi-normal modes and studied its general properties as dictated by the short-distance properties of the correlator. Finally, we have considered a two-point of light-ray operators in heavy, large charge states and proposed a general formula for the energy correlations in such states.
Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)
Several unexpected beyond the state of the art connections to other fields have been established during the first part of the project:
- Light-ray operators and phenomenology. In the last few years there is a growing body of work using energy correlators to rethink collider observables. This concerns for example the substructure of jets, looking for traces of particles (such as top quarks), as well as probes of the quark-gluon plasma. Remarkably, the developments in the formal conformal field theory (as pursued in this project) have proven to be instructive to phenomenological studies, and vice versa ideas driven by experiments have started to trigger new developments on the formal front. This cross-fertilization will definitely continue in the coming years. An interesting promising avenue in this context is developing theory of light rays in general quantum field theory. We expect this connection to become even more prominent by the end of the project, as well as further developments in our nonperturbative understanding of real-time collider observables and light-ray operators in quantum field theory.
- Nonperturbative S-matrix bootstrap and machine learning. One of the programs pursued in the project is construction of nonperturbative scattering amplitudes. These describe scattering of strongly interacting particles and provide a way to perform computations when the standard methods, such as Feynman diagrams, fail. Very recently we tried to adopt state-of-the-art machine learning techniques to attack this problem and it has proven to be very successful. We expect that by the end of the project these tools will become much more powerful but also standard.
- Thermal correlators, holographic systems, and supersymmetric gauge theories. Thermal correlators are probes of black hole geometry in holographic systems and as such provide an exciting theoretical laboratory. While studying thermal correlators we uncovered an unexpected connection to the recent developments in mathematical physics which allowed us to write down exact expressions for the thermal two-point function in terms of recently discovered special functions originating from supersymmetric localization. We also pushed state of the art analytic computations of thermal correlators in real time. We expect further unexpected insights to emerge from these developments and further progress in computation of thermal correlators using the standard quantum field theory techniques.
- Light-ray operators and phenomenology. In the last few years there is a growing body of work using energy correlators to rethink collider observables. This concerns for example the substructure of jets, looking for traces of particles (such as top quarks), as well as probes of the quark-gluon plasma. Remarkably, the developments in the formal conformal field theory (as pursued in this project) have proven to be instructive to phenomenological studies, and vice versa ideas driven by experiments have started to trigger new developments on the formal front. This cross-fertilization will definitely continue in the coming years. An interesting promising avenue in this context is developing theory of light rays in general quantum field theory. We expect this connection to become even more prominent by the end of the project, as well as further developments in our nonperturbative understanding of real-time collider observables and light-ray operators in quantum field theory.
- Nonperturbative S-matrix bootstrap and machine learning. One of the programs pursued in the project is construction of nonperturbative scattering amplitudes. These describe scattering of strongly interacting particles and provide a way to perform computations when the standard methods, such as Feynman diagrams, fail. Very recently we tried to adopt state-of-the-art machine learning techniques to attack this problem and it has proven to be very successful. We expect that by the end of the project these tools will become much more powerful but also standard.
- Thermal correlators, holographic systems, and supersymmetric gauge theories. Thermal correlators are probes of black hole geometry in holographic systems and as such provide an exciting theoretical laboratory. While studying thermal correlators we uncovered an unexpected connection to the recent developments in mathematical physics which allowed us to write down exact expressions for the thermal two-point function in terms of recently discovered special functions originating from supersymmetric localization. We also pushed state of the art analytic computations of thermal correlators in real time. We expect further unexpected insights to emerge from these developments and further progress in computation of thermal correlators using the standard quantum field theory techniques.