Rooting out letters: octagonal symbol alphabets and algebraic number theory
Abstract It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Re- cent advances in loop integration te...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2020-02-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP02(2020)025 |
| _version_ | 1818920356189044736 |
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| author | Jacob L. Bourjaily Andrew J. McLeod Cristian Vergu Matthias Volk Matt von Hippel Matthias Wilhelm |
| author_facet | Jacob L. Bourjaily Andrew J. McLeod Cristian Vergu Matthias Volk Matt von Hippel Matthias Wilhelm |
| author_sort | Jacob L. Bourjaily |
| collection | DOAJ |
| description | Abstract It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Re- cent advances in loop integration technology have made this an ‘experimentally testable’ hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is — surprisingly — rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters. |
| first_indexed | 2024-12-20T01:20:27Z |
| format | Article |
| id | doaj.art-075fa295b27f41d7973875b790f8f285 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-20T01:20:27Z |
| publishDate | 2020-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-075fa295b27f41d7973875b790f8f2852022-12-21T19:58:27ZengSpringerOpenJournal of High Energy Physics1029-84792020-02-012020212310.1007/JHEP02(2020)025Rooting out letters: octagonal symbol alphabets and algebraic number theoryJacob L. Bourjaily0Andrew J. McLeod1Cristian Vergu2Matthias Volk3Matt von Hippel4Matthias Wilhelm5Niels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of CopenhagenNiels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of CopenhagenNiels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of CopenhagenNiels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of CopenhagenNiels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of CopenhagenNiels Bohr International Academy and Discovery Center, Niels Bohr Institute, University of CopenhagenAbstract It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Re- cent advances in loop integration technology have made this an ‘experimentally testable’ hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is — surprisingly — rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.https://doi.org/10.1007/JHEP02(2020)025Scattering AmplitudesSupersymmetric Gauge Theory |
| spellingShingle | Jacob L. Bourjaily Andrew J. McLeod Cristian Vergu Matthias Volk Matt von Hippel Matthias Wilhelm Rooting out letters: octagonal symbol alphabets and algebraic number theory Journal of High Energy Physics Scattering Amplitudes Supersymmetric Gauge Theory |
| title | Rooting out letters: octagonal symbol alphabets and algebraic number theory |
| title_full | Rooting out letters: octagonal symbol alphabets and algebraic number theory |
| title_fullStr | Rooting out letters: octagonal symbol alphabets and algebraic number theory |
| title_full_unstemmed | Rooting out letters: octagonal symbol alphabets and algebraic number theory |
| title_short | Rooting out letters: octagonal symbol alphabets and algebraic number theory |
| title_sort | rooting out letters octagonal symbol alphabets and algebraic number theory |
| topic | Scattering Amplitudes Supersymmetric Gauge Theory |
| url | https://doi.org/10.1007/JHEP02(2020)025 |
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