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Piotr Tourkine - One of the best experts on this subject based on the ideXlab platform.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:Financial support from EPSRC Grant No. EP/K032208/1 during the program GTA 2016. Y. G. is supported by the EPSRC Doctoral Prize Scheme EP/M508111/1, LJM by the EPSRC Grant No. EP/M018911/1, and the work of P. T. is supported by STFC Grant No. ST/L000385/1.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations give striking formulas for massless scattering amplitudes at tree level and, as shown recently, at one loop. The progress at loop level was based on ambitwistor-string theory, which naturally yields the scattering equations. We proposed that, for ambitwistor strings, the standard loop expansion in terms of the genus of the world sheet is equivalent to an expansion in terms of nodes of a Riemann Sphere, with the nodes carrying the loop momenta. In this paper, we show how to obtain two-loop scattering equations with the correct factorization properties. We adapt genus-two integrands from the ambitwistor string to the nodal Riemann Sphere and show that these yield correct answers, by matching standard results for the four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills theory. In the Yang-Mills case, this requires the loop analogue of the Parke-Taylor factor carrying the color dependence, which includes nonplanar contributions.
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one loop amplitudes on the Riemann Sphere
Journal of High Energy Physics, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations provide a powerful framework for the study of scattering amplitudes in a variety of theories. Their derivation from ambitwistor string theory led to proposals for formulae at one loop on a torus for 10 dimensional supergravity, and we recently showed how these can be reduced to the Riemann Sphere and checked in simple cases. We also proposed analogous formulae for other theories including maximal super-Yang-Mills theory and supergravity in other dimensions at one loop. We give further details of these results and extend them in two directions. Firstly, we propose new formulae for the one-loop integrands of Yang-Mills theory and gravity in the absence of supersymmetry. These follow from the identification of the states running in the loop as expressed in the ambitwistor-string correlator. Secondly, we give a systematic proof of the non-supersymmetric formulae using the worldsheet factorisation properties of the nodal Riemann Sphere underlying the scattering equations at one loop. Our formulae have the same decomposition under the recently introduced Q-cuts as one-loop integrands and hence give the correct amplitudes.
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Loop Integrands from the Riemann Sphere
2015Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations on the Riemann Sphere give rise to remarkable formulae for tree-level gauge theory and gravity amplitudes. Adamo, Casali and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann Sphere. What emerges is a framework for loop integrands on the Riemann Sphere that promises to have wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulae, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.
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loop integrands for scattering amplitudes from the Riemann Sphere
Physical Review Letters, 2015Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations on the Riemann Sphere give rise to remarkable formulas for tree-level gauge theory and gravity amplitudes. Adamo, Casali, and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann Sphere. What emerges is a framework for loop integrands on the Riemann Sphere that promises to have a wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulas, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.
Yvonne Geyer - One of the best experts on this subject based on the ideXlab platform.
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Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere
Journal of High Energy Physics, 2018Co-Authors: Yvonne Geyer, Ricardo MonteiroAbstract:We derive from ambitwistor strings new formulae for two-loop scattering amplitudes in supergravity and super-Yang-Mills theory, with any number of particles. We start by constructing a formula for the type II ambitwistor string amplitudes on a genus-two Riemann surface, and then study the localisation of the moduli space integration on a degenerate limit, where the genus-two surface turns into a Riemann Sphere with two nodes. This leads to scattering amplitudes in supergravity, expressed in the formalism of the two-loop scattering equations. For super-Yang-Mills theory, we import `half' of the supergravity result, and determine the colour dependence by considering a current algebra on the nodal Riemann Sphere, thereby completely specifying the two-loop analogue of the Parke-Taylor factor, including non-planar contributions. We also present in appendices explicit expressions for the Szego kernels and the partition functions for even spin structures, up to the relevant orders in the degeneration parameters, which may be useful for related investigations in conventional superstring theory.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:Financial support from EPSRC Grant No. EP/K032208/1 during the program GTA 2016. Y. G. is supported by the EPSRC Doctoral Prize Scheme EP/M508111/1, LJM by the EPSRC Grant No. EP/M018911/1, and the work of P. T. is supported by STFC Grant No. ST/L000385/1.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations give striking formulas for massless scattering amplitudes at tree level and, as shown recently, at one loop. The progress at loop level was based on ambitwistor-string theory, which naturally yields the scattering equations. We proposed that, for ambitwistor strings, the standard loop expansion in terms of the genus of the world sheet is equivalent to an expansion in terms of nodes of a Riemann Sphere, with the nodes carrying the loop momenta. In this paper, we show how to obtain two-loop scattering equations with the correct factorization properties. We adapt genus-two integrands from the ambitwistor string to the nodal Riemann Sphere and show that these yield correct answers, by matching standard results for the four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills theory. In the Yang-Mills case, this requires the loop analogue of the Parke-Taylor factor carrying the color dependence, which includes nonplanar contributions.
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one loop amplitudes on the Riemann Sphere
Journal of High Energy Physics, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations provide a powerful framework for the study of scattering amplitudes in a variety of theories. Their derivation from ambitwistor string theory led to proposals for formulae at one loop on a torus for 10 dimensional supergravity, and we recently showed how these can be reduced to the Riemann Sphere and checked in simple cases. We also proposed analogous formulae for other theories including maximal super-Yang-Mills theory and supergravity in other dimensions at one loop. We give further details of these results and extend them in two directions. Firstly, we propose new formulae for the one-loop integrands of Yang-Mills theory and gravity in the absence of supersymmetry. These follow from the identification of the states running in the loop as expressed in the ambitwistor-string correlator. Secondly, we give a systematic proof of the non-supersymmetric formulae using the worldsheet factorisation properties of the nodal Riemann Sphere underlying the scattering equations at one loop. Our formulae have the same decomposition under the recently introduced Q-cuts as one-loop integrands and hence give the correct amplitudes.
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Loop Integrands from the Riemann Sphere
2015Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations on the Riemann Sphere give rise to remarkable formulae for tree-level gauge theory and gravity amplitudes. Adamo, Casali and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann Sphere. What emerges is a framework for loop integrands on the Riemann Sphere that promises to have wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulae, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.
Ricardo Monteiro - One of the best experts on this subject based on the ideXlab platform.
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Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere
Journal of High Energy Physics, 2018Co-Authors: Yvonne Geyer, Ricardo MonteiroAbstract:We derive from ambitwistor strings new formulae for two-loop scattering amplitudes in supergravity and super-Yang-Mills theory, with any number of particles. We start by constructing a formula for the type II ambitwistor string amplitudes on a genus-two Riemann surface, and then study the localisation of the moduli space integration on a degenerate limit, where the genus-two surface turns into a Riemann Sphere with two nodes. This leads to scattering amplitudes in supergravity, expressed in the formalism of the two-loop scattering equations. For super-Yang-Mills theory, we import `half' of the supergravity result, and determine the colour dependence by considering a current algebra on the nodal Riemann Sphere, thereby completely specifying the two-loop analogue of the Parke-Taylor factor, including non-planar contributions. We also present in appendices explicit expressions for the Szego kernels and the partition functions for even spin structures, up to the relevant orders in the degeneration parameters, which may be useful for related investigations in conventional superstring theory.
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Space-time CFTs from the Riemann Sphere
Journal of High Energy Physics, 2017Co-Authors: Tim Adamo, Ricardo Monteiro, Miguel F. PaulosAbstract:We consider two-dimensional chiral, first-order conformal field theories governing maps from the Riemann Sphere to the projective light cone inside Minkowski space -- the natural setting for describing conformal field theories in two fewer dimensions. These theories have a SL(2) algebra of local bosonic constraints which can be supplemented by additional fermionic constraints depending on the matter content of the theory. By computing the BRST charge associated with gauge fixing these constraints, we find anomalies which vanish for specific target space dimensions. These critical dimensions coincide precisely with those for which (biadjoint) cubic scalar theory, gauge theory and gravity are classically conformally invariant. Furthermore, the BRST cohomology of each theory contains vertex operators for the full conformal multiplets of single field insertions in each of these space-time CFTs. We give a prescription for the computation of three-point functions, and compare our formalism with the scattering equations approach to on-shell amplitudes.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:Financial support from EPSRC Grant No. EP/K032208/1 during the program GTA 2016. Y. G. is supported by the EPSRC Doctoral Prize Scheme EP/M508111/1, LJM by the EPSRC Grant No. EP/M018911/1, and the work of P. T. is supported by STFC Grant No. ST/L000385/1.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations give striking formulas for massless scattering amplitudes at tree level and, as shown recently, at one loop. The progress at loop level was based on ambitwistor-string theory, which naturally yields the scattering equations. We proposed that, for ambitwistor strings, the standard loop expansion in terms of the genus of the world sheet is equivalent to an expansion in terms of nodes of a Riemann Sphere, with the nodes carrying the loop momenta. In this paper, we show how to obtain two-loop scattering equations with the correct factorization properties. We adapt genus-two integrands from the ambitwistor string to the nodal Riemann Sphere and show that these yield correct answers, by matching standard results for the four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills theory. In the Yang-Mills case, this requires the loop analogue of the Parke-Taylor factor carrying the color dependence, which includes nonplanar contributions.
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one loop amplitudes on the Riemann Sphere
Journal of High Energy Physics, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations provide a powerful framework for the study of scattering amplitudes in a variety of theories. Their derivation from ambitwistor string theory led to proposals for formulae at one loop on a torus for 10 dimensional supergravity, and we recently showed how these can be reduced to the Riemann Sphere and checked in simple cases. We also proposed analogous formulae for other theories including maximal super-Yang-Mills theory and supergravity in other dimensions at one loop. We give further details of these results and extend them in two directions. Firstly, we propose new formulae for the one-loop integrands of Yang-Mills theory and gravity in the absence of supersymmetry. These follow from the identification of the states running in the loop as expressed in the ambitwistor-string correlator. Secondly, we give a systematic proof of the non-supersymmetric formulae using the worldsheet factorisation properties of the nodal Riemann Sphere underlying the scattering equations at one loop. Our formulae have the same decomposition under the recently introduced Q-cuts as one-loop integrands and hence give the correct amplitudes.
Lionel Mason - One of the best experts on this subject based on the ideXlab platform.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:Financial support from EPSRC Grant No. EP/K032208/1 during the program GTA 2016. Y. G. is supported by the EPSRC Doctoral Prize Scheme EP/M508111/1, LJM by the EPSRC Grant No. EP/M018911/1, and the work of P. T. is supported by STFC Grant No. ST/L000385/1.
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two loop scattering amplitudes from the Riemann Sphere
Physical Review D, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations give striking formulas for massless scattering amplitudes at tree level and, as shown recently, at one loop. The progress at loop level was based on ambitwistor-string theory, which naturally yields the scattering equations. We proposed that, for ambitwistor strings, the standard loop expansion in terms of the genus of the world sheet is equivalent to an expansion in terms of nodes of a Riemann Sphere, with the nodes carrying the loop momenta. In this paper, we show how to obtain two-loop scattering equations with the correct factorization properties. We adapt genus-two integrands from the ambitwistor string to the nodal Riemann Sphere and show that these yield correct answers, by matching standard results for the four-point two-loop amplitudes of maximal supergravity and super-Yang-Mills theory. In the Yang-Mills case, this requires the loop analogue of the Parke-Taylor factor carrying the color dependence, which includes nonplanar contributions.
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one loop amplitudes on the Riemann Sphere
Journal of High Energy Physics, 2016Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations provide a powerful framework for the study of scattering amplitudes in a variety of theories. Their derivation from ambitwistor string theory led to proposals for formulae at one loop on a torus for 10 dimensional supergravity, and we recently showed how these can be reduced to the Riemann Sphere and checked in simple cases. We also proposed analogous formulae for other theories including maximal super-Yang-Mills theory and supergravity in other dimensions at one loop. We give further details of these results and extend them in two directions. Firstly, we propose new formulae for the one-loop integrands of Yang-Mills theory and gravity in the absence of supersymmetry. These follow from the identification of the states running in the loop as expressed in the ambitwistor-string correlator. Secondly, we give a systematic proof of the non-supersymmetric formulae using the worldsheet factorisation properties of the nodal Riemann Sphere underlying the scattering equations at one loop. Our formulae have the same decomposition under the recently introduced Q-cuts as one-loop integrands and hence give the correct amplitudes.
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Loop Integrands from the Riemann Sphere
2015Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations on the Riemann Sphere give rise to remarkable formulae for tree-level gauge theory and gravity amplitudes. Adamo, Casali and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann Sphere. What emerges is a framework for loop integrands on the Riemann Sphere that promises to have wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulae, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.
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loop integrands for scattering amplitudes from the Riemann Sphere
Physical Review Letters, 2015Co-Authors: Yvonne Geyer, Lionel Mason, Ricardo Monteiro, Piotr TourkineAbstract:The scattering equations on the Riemann Sphere give rise to remarkable formulas for tree-level gauge theory and gravity amplitudes. Adamo, Casali, and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann Sphere. What emerges is a framework for loop integrands on the Riemann Sphere that promises to have a wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulas, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.
Julio C Maganacaceres - One of the best experts on this subject based on the ideXlab platform.
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classification of rational 1 forms on the Riemann Sphere up to text psl 2 mathbb c
Boletin De La Sociedad Matematica Mexicana, 2019Co-Authors: Julio C MaganacaceresAbstract:We study the family \(\varOmega ^1(-1^{s})\) of rational 1-forms on the Riemann Sphere, having exactly \(-s \le -2\) simple poles. Three equivalent \((2s-1)\)-dimensional complex atlases on \(\varOmega ^1(-1^{s})\), using coefficients, zeros–poles and residues–poles of the 1-forms, are recognized. A rational 1-form is called isochronous when all their residues are purely imaginary. We prove that the subfamily \(\mathcal {RI}\varOmega ^1(-1^{s})\) of isochronous 1-forms is a \((3s-1)\)-dimensional real analytic submanifold in the complex manifold \(\varOmega ^1(-1^{s})\). The complex Lie group \(\text {PSL}(2,\mathbb {C})\) acts holomorphically on \(\varOmega ^1(-1^{s})\). For \(s \ge 3\), the \(\text {PSL}(2,\mathbb {C})\)-action is proper on \(\varOmega ^1(-1^{s})\) and \(\mathcal {RI}\varOmega ^1(-1^{s})\). Therefore, the quotients \(\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})\) and \(\mathcal {RI}\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})\) admit a stratification by orbit types. Realizations for the quotients \(\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})\) and \(\mathcal {RI}\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})\) are given, using an explicit set of \(\text {PSL}(2,\mathbb {C})\)-invariant functions.
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classification of rational 1 forms on the Riemann Sphere up to text psl 2 mathbb c psl 2 c
Boletin De La Sociedad Matematica Mexicana, 2019Co-Authors: Julio C MaganacaceresAbstract:We study the family $$\varOmega ^1(-1^{s})$$ of rational 1-forms on the Riemann Sphere, having exactly $$-s \le -2$$ simple poles. Three equivalent $$(2s-1)$$ -dimensional complex atlases on $$\varOmega ^1(-1^{s})$$ , using coefficients, zeros–poles and residues–poles of the 1-forms, are recognized. A rational 1-form is called isochronous when all their residues are purely imaginary. We prove that the subfamily $$\mathcal {RI}\varOmega ^1(-1^{s})$$ of isochronous 1-forms is a $$(3s-1)$$ -dimensional real analytic submanifold in the complex manifold $$\varOmega ^1(-1^{s})$$ . The complex Lie group $$\text {PSL}(2,\mathbb {C})$$ acts holomorphically on $$\varOmega ^1(-1^{s})$$ . For $$s \ge 3$$ , the $$\text {PSL}(2,\mathbb {C})$$ -action is proper on $$\varOmega ^1(-1^{s})$$ and $$\mathcal {RI}\varOmega ^1(-1^{s})$$ . Therefore, the quotients $$\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})$$ and $$\mathcal {RI}\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})$$ admit a stratification by orbit types. Realizations for the quotients $$\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})$$ and $$\mathcal {RI}\varOmega ^1(-1^{s})/\text {PSL}(2,\mathbb {C})$$ are given, using an explicit set of $$\text {PSL}(2,\mathbb {C})$$ -invariant functions.
