The Experts below are selected from a list of 27408 Experts worldwide ranked by ideXlab platform
Emery Sokatchev - One of the best experts on this subject based on the ideXlab platform.
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the super correlator super amplitude duality part ii
Nuclear Physics, 2013Co-Authors: Burkhard Eden, Emery Sokatchev, Paul Heslop, Gregory P. KorchemskyAbstract:Abstract We continue the study of the duality between super-correlators and scattering super-amplitudes in planar N = 4 SYM. We provide a number of further examples supporting the conjectured duality relation between these two seemingly different objects. We consider the five- and six-point one-loop NMHV and the six-point tree-level NNMHV amplitudes, obtaining them from the appropriate correlators of stress-tensor multiplets in N = 4 SYM. In particular, for the rather non-trivial parity-odd sector of the Integrand of the six-point one-loop NMHV amplitude, we find exact agreement between the results obtained from the correlator and from BCFW recurrence relations. Together these results lead to the conjecture that the Integrands of any N k MHV amplitude at any loop order in planar N = 4 SYM can be described by the correlators of stress-tensor multiplets.
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The super-correlator/super-amplitude duality: Part II
Nuclear Physics B, 2013Co-Authors: Burkhard Eden, Paul Heslop, Gregory P. Korchemsky, Emery SokatchevAbstract:We continue the study of the duality between super-correlators and scattering super-amplitudes in planar N=4 SYM. We provide a number of further examples supporting the conjectured duality relation between these two seemingly different objects. We consider the five- and six-point one-loop NMHV and the six-point tree-level NNMHV amplitudes, obtaining them from the appropriate correlators of strength tensor multiplets in N=4 SYM. In particular, we find exact agreement between the rather non-trivial parity-odd sector of the Integrand of the six-point one-loop NMHV amplitude, as obtained from the correlator or from BCFW recursion relations. Together these results lead to the conjecture that the Integrands of any N^kMHV amplitude at any loop order in planar N=4 SYM can be described by the correlators of stress-tensor multiplets.
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constructing the correlation function of four stress tensor multiplets and the four particle amplitude in n 4 sym
Nuclear Physics, 2012Co-Authors: Burkhard Eden, Emery Sokatchev, Paul Heslop, Gregory P. KorchemskyAbstract:We present a construction of the Integrand of the correlation function of four stress-tensor multiplets in N = 4 SYM at weak coupling. It does not rely on Feynman diagrams and makes use of the recently discovered symmetry of the Integrand under permutations of external and integration points. This symmetry holds for any gauge group, so it can be used to predict the Integrand both in the planar and non-planar sectors. We demonstrate the great efficiency of graph-theoretical tools in the systematic study of the possible permutation symmetric Integrands. We formulate a general ansatz for the correlation function as a linear combination of all relevant graph topologies, with arbitrary coefficients. Powerful restrictions on the coefficients come from the analysis of the logarithmic divergences of the correlation function in two singular regimes: Euclidean short-distance and Minkowski light-cone limits. We demonstrate that the planar Integrand is completely fixed by the procedure up to six loops and probably beyond. In the non-planar sector, we show the absence of non-planar corrections at three loops and we reduce the freedom at four loops to just four constants. Finally, the correlation function/amplitude duality allows us to show the complete agreement of our results with the four-particle planar amplitude in N = 4 SYM.
Jaroslav Trnka - One of the best experts on this subject based on the ideXlab platform.
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Building bases of loop Integrands
Journal of High Energy Physics, 2020Co-Authors: Jacob L Bourjaily, Enrico Herrmann, Cameron Langer, Jaroslav TrnkaAbstract:We describe a systematic approach to the construction of loop-Integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of ‘power-counting’ for multi-loop Integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude Integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In particular, we describe complete Integrand bases at two loops sufficient to represent arbitrary-multiplicity amplitudes in four (or fewer) dimensions in any massless quantum field theory with the ultraviolet behavior of the Standard Model or better. We also comment on possible extensions of our framework to arbitrary (including regulated) numbers of dimensions, and to theories with arbitrary mass spectra and charges. At three loops, we describe a basis sufficient to capture all ‘leading-(transcendental-)weight’ contributions of any four-dimensional quantum theory; for maximally supersymmetric Yang-Mills theory, this basis should be sufficient to represent all scattering amplitude Integrands in the theory — for generic helicities and arbitrary multiplicity.
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Gravity loop Integrands from the ultraviolet
arXiv: High Energy Physics - Theory, 2020Co-Authors: Alex Edison, Enrico Herrmann, Julio Parra-martinez, Jaroslav TrnkaAbstract:We demonstrate that loop Integrands of (super-)gravity scattering amplitudes possess surprising properties in the ultraviolet (UV) region. In particular, we study the scaling of multi-particle unitarity cuts for asymptotically large momenta and expose an improved UV behavior of four-dimensional cuts through seven loops as compared to standard expectations. For N=8 supergravity, we show that the improved large momentum scaling combined with the behavior of the Integrand under BCFW deformations of external kinematics uniquely fixes the loop Integrands in a number of non-trivial cases. In the Integrand construction, all scaling conditions are homogeneous. Therefore, the only required information about the amplitude is its vanishing at particular points in momentum space. This homogeneous construction gives indirect evidence for a new geometric picture for graviton amplitudes similar to the one found for planar N=4 super Yang-Mills theory. We also show how the behavior at infinity is related to the scaling of tree-level amplitudes under certain multi-line chiral shifts which can be used to construct new recursion relations.
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prescriptive unitarity for non planar six particle amplitudes at two loops
Journal of High Energy Physics, 2019Co-Authors: Jacob L Bourjaily, Enrico Herrmann, Cameron Langer, Andrew J Mcleod, Jaroslav TrnkaAbstract:We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, Integrand-level representations of six-particle MHV scattering amplitudes in both maximally supersymmetric Yang-Mills theory and gravity. The Integrand basis we construct is diagonalized on a spanning set of non-vanishing leading singularities that ensures the manifest matching of all soft-collinear singularities in both theories. As a consequence, this Integrand basis naturally splits into infrared-finite and infrared-divergent parts, with hints toward an Integrand-level exponentiation of infrared divergences. Importantly, we use the same basis of Integrands for both theories, so that the presence or absence of residues at infinite loop momentum becomes a feature detectable by inspecting the cuts of the theory. Complete details of our results are provided as sup- plementary material.
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prescriptive unitarity for non planar six particle amplitudes at two loops
arXiv: High Energy Physics - Theory, 2019Co-Authors: Jacob L Bourjaily, Enrico Herrmann, Cameron Langer, Andrew J Mcleod, Jaroslav TrnkaAbstract:We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, Integrand-level representations of six-particle MHV scattering amplitudes in both maximally supersymmetric Yang-Mills theory and gravity. The Integrand basis we construct is diagonalized on a spanning set of non-vanishing leading singularities that ensures the manifest matching of all soft-collinear singularities in both theories. As a consequence, this Integrand basis naturally splits into infrared-finite and infrared-divergent parts, with hints toward an Integrand-level exponentiation of infrared divergences. Importantly, we use the same basis of Integrands for both theories, so that the presence or absence of residues at infinite loop momentum becomes a feature detectable by inspecting the cuts of the theory. Complete details of our results are provided as ancillary files.
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UV cancelations in gravity loop Integrands
Journal of High Energy Physics, 2019Co-Authors: Enrico Herrmann, Jaroslav TrnkaAbstract:In this work we explore the properties of four-dimensional gravity Integrands at large loop momenta. This analysis can not be done directly for the full off-shell Integrand but only becomes well-defined on cuts that allow us to unambiguously specify labels for the loop variables. The ultraviolet region of scattering amplitudes originates from poles at infinity of the loop Integrands and we show that in gravity these integcrands conceal a number of surprising features. In particular, certain poles at infinity are absent which requires a conspiracy between individual Feynman integrals contributing to the amplitude. We suspect that this non-trivial behavior is a consequence of yet-to-be found symmetry or hidden property of gravity amplitudes. We discuss mainly amplitudes in $\mathcal{N}=8$ supergravity but most of the statements are valid for pure gravity as well.
Tiziano Peraro - One of the best experts on this subject based on the ideXlab platform.
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Local Integrands for two-loop all-plus Yang-Mills amplitudes
Journal of High Energy Physics, 2016Co-Authors: Simon Badger, Gustav Mogull, Tiziano PeraroAbstract:We express the planar five- and six-gluon two-loop Yang-Mills amplitudes with all positive helicities in compact analytic form using D-dimensional local Integrands that are free of spurious singularities. The Integrand is fixed from on-shell tree amplitudes in six dimensions using D-dimensional generalised unitarity cuts. The resulting expressions are shown to have manifest infrared behaviour at the Integrand level. We also find simple representations of the rational terms obtained after integration in 4 − 2ϵ dimensions.
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Multi-loop Integrand Reduction via Multivariate Polynomial Division
Proceedings of 11th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology) — PoS(RADCOR 2013), 2014Co-Authors: Hans Van Deurzen, Gionata Luisoni, Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano Peraro, Ulrich SchubertAbstract:We present recent developments on the topic of the Integrand reduction of scattering amplitudes. Integrand-level methods allow to express an amplitude as a linear combination of Master Integrals, by performing operations on the corresponding Integrands. This approach has already been successfully applied and automated at one loop, and recently extended to higher loops. We describe a coherent framework based on simple concepts of algebraic geometry, such as multivariate polynomial division, which can be used in order to obtain the Integrand decomposition of any amplitude at any loop order. In the one-loop case, we discuss an improved reduction algorithm, based on the application of the Laurent series expansion to the Integrands, which has been implemented in the seminumerical library Ninja. At two loops, we present the reduction of ve-point amplitudes inN = 4 SYM, with a unitarity-based construction of the Integrand. We also describe the multi-loop divide-and-conquer approach, which can always be used to nd the Integrand decomposition of any Feynman graph, regardless of the form and the complexity of the Integrand, with purely algebraic operations.
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Multiloop Integrand reduction for dimensionally regulated amplitudes
Physics Letters B, 2013Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano PeraroAbstract:Abstract We present the Integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the Integrand reduction, valid for arbitrary dimensionally regulated loop integrals with any number of loops and external legs, which can be used to obtain the decomposition of any Integrand analytically with a finite number of algebraic operations. The general results are illustrated by applications to two-loop Feynman diagrams in QED and QCD, showing that the proposed reduction algorithm can also be seamlessly applied to Integrands with denominators appearing with arbitrary powers.
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Integrand reduction for two loop scattering amplitudes through multivariate polynomial division
Physical Review D, 2013Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano PeraroAbstract:We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete Integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the Integrand-reduction procedure. By using the division modulo Gr\"obner basis, we can derive a simple Integrand recurrence relation that generates the multiparticle pole decomposition for Integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=8$ supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the Integrand-reduction method to all orders in perturbation theory.
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Integrand reduction of one-loop scattering amplitudes through Laurent series expansion
Journal of High Energy Physics, 2012Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Tiziano PeraroAbstract:We present a semi-analytic method for the Integrand reduction of one-loop amplitudes, based on the systematic application of the Laurent expansions to the Integrand-decomposition. In the asymptotic limit, the coefficients of the master integrals are the solutions of a diagonal system of equations, properly corrected by counterterms whose parametric form is known a priori. The Laurent expansion of the Integrand is implemented through polynomial division. The extension of the Integrand-reduction to the case of numerators with rank larger than the number of propagators is discussed as well.
Pierpaolo Mastrolia - One of the best experts on this subject based on the ideXlab platform.
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Multi-loop Integrand Reduction via Multivariate Polynomial Division
Proceedings of 11th International Symposium on Radiative Corrections (Applications of Quantum Field Theory to Phenomenology) — PoS(RADCOR 2013), 2014Co-Authors: Hans Van Deurzen, Gionata Luisoni, Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano Peraro, Ulrich SchubertAbstract:We present recent developments on the topic of the Integrand reduction of scattering amplitudes. Integrand-level methods allow to express an amplitude as a linear combination of Master Integrals, by performing operations on the corresponding Integrands. This approach has already been successfully applied and automated at one loop, and recently extended to higher loops. We describe a coherent framework based on simple concepts of algebraic geometry, such as multivariate polynomial division, which can be used in order to obtain the Integrand decomposition of any amplitude at any loop order. In the one-loop case, we discuss an improved reduction algorithm, based on the application of the Laurent series expansion to the Integrands, which has been implemented in the seminumerical library Ninja. At two loops, we present the reduction of ve-point amplitudes inN = 4 SYM, with a unitarity-based construction of the Integrand. We also describe the multi-loop divide-and-conquer approach, which can always be used to nd the Integrand decomposition of any Feynman graph, regardless of the form and the complexity of the Integrand, with purely algebraic operations.
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Multiloop Integrand reduction for dimensionally regulated amplitudes
Physics Letters B, 2013Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano PeraroAbstract:Abstract We present the Integrand reduction via multivariate polynomial division as a natural technique to encode the unitarity conditions of Feynman amplitudes. We derive a recursive formula for the Integrand reduction, valid for arbitrary dimensionally regulated loop integrals with any number of loops and external legs, which can be used to obtain the decomposition of any Integrand analytically with a finite number of algebraic operations. The general results are illustrated by applications to two-loop Feynman diagrams in QED and QCD, showing that the proposed reduction algorithm can also be seamlessly applied to Integrands with denominators appearing with arbitrary powers.
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Integrand reduction for two loop scattering amplitudes through multivariate polynomial division
Physical Review D, 2013Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano PeraroAbstract:We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete Integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multiparticle cut, whose knowledge is a mandatory prerequisite for applying the Integrand-reduction procedure. By using the division modulo Gr\"obner basis, we can derive a simple Integrand recurrence relation that generates the multiparticle pole decomposition for Integrands of arbitrary multiloop amplitudes. We apply the new reduction algorithm to the two-loop planar and nonplanar diagrams contributing to the five-point scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=8$ supergravity in four dimensions, whose numerator functions contain up to rank-two terms in the integration momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and subtopologies. We obtain the integral basis for the decomposition of each diagram from the polynomial form of the residues. Our approach is well suited for a seminumerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of the Integrand-reduction method to all orders in perturbation theory.
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Integrand reduction of one-loop scattering amplitudes through Laurent series expansion
Journal of High Energy Physics, 2012Co-Authors: Pierpaolo Mastrolia, Edoardo Mirabella, Tiziano PeraroAbstract:We present a semi-analytic method for the Integrand reduction of one-loop amplitudes, based on the systematic application of the Laurent expansions to the Integrand-decomposition. In the asymptotic limit, the coefficients of the master integrals are the solutions of a diagonal system of equations, properly corrected by counterterms whose parametric form is known a priori. The Laurent expansion of the Integrand is implemented through polynomial division. The extension of the Integrand-reduction to the case of numerators with rank larger than the number of propagators is discussed as well.
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scattering amplitudes from unitarity based reduction algorithm at the Integrand level
Journal of High Energy Physics, 2010Co-Authors: Pierpaolo Mastrolia, Giovanni Ossola, T Reiter, Francesco TramontanoAbstract:samurai is a tool for the automated numerical evaluation of one-loop corrections to any scattering amplitudes within the dimensional-regularization scheme. It is based on the decomposition of the Integrand according to the OPP -approach, extended to accommodate an implementation of the generalized d-dimensional unitarity-cuts technique, and uses a polynomial interpolation exploiting the Discrete Fourier Transform. samurai can process Integrands written either as numerator of Feynman diagrams or as product of tree-level amplitudes. We discuss some applications, among which the 6-and 8-photon scattering in QED, and the 6-quark scattering in QCD. samurai has been implemented as a Fortran90 library, publicly available, and it could be a useful module for the systematic evaluation of the virtual corrections oriented towards automating next-to-leading order calculations relevant for the LHC phenomenology.
Jacob L Bourjaily - One of the best experts on this subject based on the ideXlab platform.
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Building bases of loop Integrands
Journal of High Energy Physics, 2020Co-Authors: Jacob L Bourjaily, Enrico Herrmann, Cameron Langer, Jaroslav TrnkaAbstract:We describe a systematic approach to the construction of loop-Integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of ‘power-counting’ for multi-loop Integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude Integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In particular, we describe complete Integrand bases at two loops sufficient to represent arbitrary-multiplicity amplitudes in four (or fewer) dimensions in any massless quantum field theory with the ultraviolet behavior of the Standard Model or better. We also comment on possible extensions of our framework to arbitrary (including regulated) numbers of dimensions, and to theories with arbitrary mass spectra and charges. At three loops, we describe a basis sufficient to capture all ‘leading-(transcendental-)weight’ contributions of any four-dimensional quantum theory; for maximally supersymmetric Yang-Mills theory, this basis should be sufficient to represent all scattering amplitude Integrands in the theory — for generic helicities and arbitrary multiplicity.
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prescriptive unitarity for non planar six particle amplitudes at two loops
Journal of High Energy Physics, 2019Co-Authors: Jacob L Bourjaily, Enrico Herrmann, Cameron Langer, Andrew J Mcleod, Jaroslav TrnkaAbstract:We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, Integrand-level representations of six-particle MHV scattering amplitudes in both maximally supersymmetric Yang-Mills theory and gravity. The Integrand basis we construct is diagonalized on a spanning set of non-vanishing leading singularities that ensures the manifest matching of all soft-collinear singularities in both theories. As a consequence, this Integrand basis naturally splits into infrared-finite and infrared-divergent parts, with hints toward an Integrand-level exponentiation of infrared divergences. Importantly, we use the same basis of Integrands for both theories, so that the presence or absence of residues at infinite loop momentum becomes a feature detectable by inspecting the cuts of the theory. Complete details of our results are provided as sup- plementary material.
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prescriptive unitarity for non planar six particle amplitudes at two loops
arXiv: High Energy Physics - Theory, 2019Co-Authors: Jacob L Bourjaily, Enrico Herrmann, Cameron Langer, Andrew J Mcleod, Jaroslav TrnkaAbstract:We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, Integrand-level representations of six-particle MHV scattering amplitudes in both maximally supersymmetric Yang-Mills theory and gravity. The Integrand basis we construct is diagonalized on a spanning set of non-vanishing leading singularities that ensures the manifest matching of all soft-collinear singularities in both theories. As a consequence, this Integrand basis naturally splits into infrared-finite and infrared-divergent parts, with hints toward an Integrand-level exponentiation of infrared divergences. Importantly, we use the same basis of Integrands for both theories, so that the presence or absence of residues at infinite loop momentum becomes a feature detectable by inspecting the cuts of the theory. Complete details of our results are provided as ancillary files.
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scattering amplitudes and the positive grassmannian
arXiv: High Energy Physics - Theory, 2012Co-Authors: Nima Arkanihamed, Jacob L Bourjaily, Freddy Cachazo, Jaroslav Trnka, Alexander Postnikov, A B GoncharovAbstract:We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop Integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion of a simplex in projective space. All physically important operations involving on-shell diagrams map to canonical operations on permutations; in particular, BCFW deformations correspond to adjacent transpositions. Each cell of the positive Grassmannian is naturally endowed with positive coordinates and an invariant measure which determines the on-shell function associated with the diagram. This understanding allows us to classify and compute all on-shell diagrams, and give a geometric understanding for all the non-trivial relations among them. Yangian invariance of scattering amplitudes is transparently represented by diffeomorphisms of G(k,n) which preserve the positive structure. Scattering amplitudes in (1+1)-dimensional integrable systems and the ABJM theory in (2+1) dimensions can both be understood as special cases of these ideas. On-shell diagrams in theories with less (or no) supersymmetry are associated with exactly the same structures in the Grassmannian, but with a measure deformed by a factor encoding ultraviolet singularities. The Grassmannian representation of on-shell processes also gives a new understanding of the all-loop Integrand for scattering amplitudes, presenting all Integrands in a novel dLog form which directly reflects the underlying positive structure.
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local integrals for planar scattering amplitudes
Journal of High Energy Physics, 2012Co-Authors: Nima Arkanihamed, Jacob L Bourjaily, Freddy Cachazo, Jaroslav TrnkaAbstract:Recently, an explicit, recursive formula for the all-loop Integrand of planar scattering amplitudes in $ \mathcal{N} = {4} $ SYM has been described, generalizing the BCFW formula for tree amplitudes, and making manifest the Yangian symmetry of the theory. This has made it possible to easily study the structure of multi-loop amplitudes in the theory. In this paper we describe a remarkable fact revealed by these investigations: the Integrand can be expressed in an amazingly simple and manifestly local form when represented in momentum-twistor space using a set of chiral integrals with unit leading singularities. As examples, we present very-concise expressions for all 2- and 3-loop MHV Integrands, as well as all 2-loop NMHV Integrands. We also describe a natural set of manifestly IR-finite integrals that can be used to express IR-safe objects such as the ratio function. Along the way we give a pedagogical introduction to the foundations of the subject. The new local forms of the Integrand are closely connected to leading singularities — matching only a small subset of all leading singularities remarkably suffices to determine the full Integrand. These results strongly suggest the existence of a theory for the Integrand directly yielding these local expressions, allowing for a more direct understanding of the emergence of local spacetime physics.
