Periodic Reporting for period 4 - AMPLITUDES (Manifesting the Simplicity of Scattering Amplitudes)
Berichtszeitraum: 2022-08-01 bis 2023-01-31
The AMPLITUDES project's goal was to advance our understanding of the most fundamental frameworks of modern science—quantum field theory. Quantum field theory is a modern-day version of "force equals mass times acceleration"—it is not any particular theory, but a general mathematical framework which allows physicists to make predictions for experiment taking into account both (special) relativity and quantum-mechanical uncertainty. It is a set of standard tools—not unlike the Calculus of classical physics—built to translate any particular physical "law" (real or imagined, fundamental or approximate) into predictions. These predictions often take the form of "scattering amplitudes" which encode the predicted relative probabilities of the outcomes of any experiment as a function of all the observable information about the initial and final states involved. Scattering amplitudes are often computed perturbatively—with increasing orders of precision/approximation requiring increasingly difficult computations by the physicist.
Although quantum field theory is extremely well established and is arguably the most precisely-tested idea in all of science, it remains extremely difficult to use. In particular, the standard methods (found in textbooks, say) render all but the simplest—those involving the fewest numbers of particles or the lowest orders of approximation—either computationally intractable or theoretically inscrutable.
The AMPLITUDES project has succeeded in this program of research by pushing the limits of what is known about scattering amplitudes—by making predictions (often in toy-model theories) that were considered inconceivably difficult only years or months before; and by pushing these limits, generating new landscapes of "theoretical data" about the (mathematical) nature of scattering amplitudes that will help us uncover new simplicities and mathematical form, feeding the tools of the future.
The project has achieved important inroads into many areas of our understanding of quantum field theory. It has resulted in 78 scientific publications (including preprints) to date, and has generated enormous interest in a wide array of communities interested in both the mathematical structure being uncovered and the applications of the new techniques being developed.
Although quantum field theory is extremely well established and is arguably the most precisely-tested idea in all of science, it remains extremely difficult to use. In particular, the standard methods (found in textbooks, say) render all but the simplest—those involving the fewest numbers of particles or the lowest orders of approximation—either computationally intractable or theoretically inscrutable.
The AMPLITUDES project has succeeded in this program of research by pushing the limits of what is known about scattering amplitudes—by making predictions (often in toy-model theories) that were considered inconceivably difficult only years or months before; and by pushing these limits, generating new landscapes of "theoretical data" about the (mathematical) nature of scattering amplitudes that will help us uncover new simplicities and mathematical form, feeding the tools of the future.
The project has achieved important inroads into many areas of our understanding of quantum field theory. It has resulted in 78 scientific publications (including preprints) to date, and has generated enormous interest in a wide array of communities interested in both the mathematical structure being uncovered and the applications of the new techniques being developed.
Among the most significant results obtained through AMPLITUDES are:
1. An all-loop, non-planar, non-supersymmetric generalization of the prescriptive unitarity program. This culminated in the recent work “Building Bases of Loop Integrands”, co-authored with Enrico Herrmann, Cameron Langer, and Jaroslav Trnka, which included an all-orders proposal for a power-counting stratification of loop integrands (suitable for unitarity-based reconstruction of amplitudes) in any renormalizable quantum field theory in any number of dimensions, and an explicit enumeration of a complete basis suitable for the leading “transcendental-weight” part of any—involving arbitrary numbers and species of external states—three-loop scattering amplitude in any four-dimensional quantum field theory.
This work built upon a series of earlier results which were also completed as part of this project. In particular, this included the first closed formula for all-multiplicity amplitudes in any four-dimensional quantum field theory at two loops, which was published in the Physical Review Letters (PRL) in the article “All-Multiplicity Nonplanar Amplitude Integrands in sYM at Two Loops”, which followed very shortly after the result for six particles, published in JHEP, “Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops”, and the previous work on non-planar integrands “Amplitudes at Infinity”, published in PRL.
2. Development, refinement, and exploitation of new technology for loop integration. This included publications “Rationalizing Loop Integration”, “Manifestly Dual-Conformal Loop Integration”, “Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory”, “Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder”, and “All-Mass n-gon Integrals in n Dimensions”, all of which were published in JHEP.
3. The discovery and exploration of novel structures in the analytic form of non-polylogarithmic Feynman integrals. Specifically, my collaborators and I uncovered several infinite classes of finite Feynman integrals (defined in massless phi^4 theory in 4 dimensions) that involve period integrals Calabi-Yau manifolds of dimension increasing with loop order. These results were published in “Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms” and “A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries”, both published in PRL, and “Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space”, “Positive Geometries and Differential Forms with Non-Logarithmic Singularities, Part 1”, and “Traintrack Calabi-Yaus from Twistor Geometry”, published in JHEP.
4. The discovery of non-algebraic Yangian-invariants associated with on-shell functions integrated over elliptic (or higher-dimensional) contour integrals, described in "An Elliptic Yangian-Invariant `Leading Singularity' in PRL; the connection between this generalization of leading singularities and coefficients of loop integrands appearing in generalized unitarity was made clear in the subsequent work "Prescriptive Unitarity with Elliptic Leading Singularities".
5. The exploration of how integrand bases can be organized (`stratified') according to particular physical structure, including rigidity (or non-polylogarithmicity), IR/UV finiteness, etc, with applications to non-supersymmetric theories at one and two loops. In particular, in the work "The Stratification of Rigidity" published in JHEP, we showed that all leading-weight contributions to planar scattering amplitudes in four dimensions integrand bases could be represented in terms of integrals that were individually pure and polylogarithmic or pure and elliptic-polylogarithmic, with each of the later depending on a single elliptic curve geometry.
1. An all-loop, non-planar, non-supersymmetric generalization of the prescriptive unitarity program. This culminated in the recent work “Building Bases of Loop Integrands”, co-authored with Enrico Herrmann, Cameron Langer, and Jaroslav Trnka, which included an all-orders proposal for a power-counting stratification of loop integrands (suitable for unitarity-based reconstruction of amplitudes) in any renormalizable quantum field theory in any number of dimensions, and an explicit enumeration of a complete basis suitable for the leading “transcendental-weight” part of any—involving arbitrary numbers and species of external states—three-loop scattering amplitude in any four-dimensional quantum field theory.
This work built upon a series of earlier results which were also completed as part of this project. In particular, this included the first closed formula for all-multiplicity amplitudes in any four-dimensional quantum field theory at two loops, which was published in the Physical Review Letters (PRL) in the article “All-Multiplicity Nonplanar Amplitude Integrands in sYM at Two Loops”, which followed very shortly after the result for six particles, published in JHEP, “Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops”, and the previous work on non-planar integrands “Amplitudes at Infinity”, published in PRL.
2. Development, refinement, and exploitation of new technology for loop integration. This included publications “Rationalizing Loop Integration”, “Manifestly Dual-Conformal Loop Integration”, “Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory”, “Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder”, and “All-Mass n-gon Integrals in n Dimensions”, all of which were published in JHEP.
3. The discovery and exploration of novel structures in the analytic form of non-polylogarithmic Feynman integrals. Specifically, my collaborators and I uncovered several infinite classes of finite Feynman integrals (defined in massless phi^4 theory in 4 dimensions) that involve period integrals Calabi-Yau manifolds of dimension increasing with loop order. These results were published in “Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms” and “A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries”, both published in PRL, and “Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space”, “Positive Geometries and Differential Forms with Non-Logarithmic Singularities, Part 1”, and “Traintrack Calabi-Yaus from Twistor Geometry”, published in JHEP.
4. The discovery of non-algebraic Yangian-invariants associated with on-shell functions integrated over elliptic (or higher-dimensional) contour integrals, described in "An Elliptic Yangian-Invariant `Leading Singularity' in PRL; the connection between this generalization of leading singularities and coefficients of loop integrands appearing in generalized unitarity was made clear in the subsequent work "Prescriptive Unitarity with Elliptic Leading Singularities".
5. The exploration of how integrand bases can be organized (`stratified') according to particular physical structure, including rigidity (or non-polylogarithmicity), IR/UV finiteness, etc, with applications to non-supersymmetric theories at one and two loops. In particular, in the work "The Stratification of Rigidity" published in JHEP, we showed that all leading-weight contributions to planar scattering amplitudes in four dimensions integrand bases could be represented in terms of integrals that were individually pure and polylogarithmic or pure and elliptic-polylogarithmic, with each of the later depending on a single elliptic curve geometry.
Most of the discoveries and much of the progress made during the course of the AMPLITUDES project pushed the state of the art in our ability to compute, represent, and understand scattering amplitudes in an increasing variety of quantum field theories---including those relevant to real-world applications. A number of concrete examples illustrating novel phenomena (such as the maximally rigid series of Feynman integrals in scalar field theory) helped to sharpen our understanding of the class of special functions that arise in perturbative quantum field theory, and orient the field toward greater understanding. And the structural ideas about the organization of scattering amplitude integrands in terms of master integral bases according to UV/IR finiteness, rigidity, and prescriptivity helped lead the way toward a more systematic understanding of perturbation theory in general theories---and lead directly to the applications including the first all-multiplicity result in a non-planar theory at two loop order.