The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations
Abstract We study the symbol and the alphabet for two-loop NMHV amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills from the Q ¯ $$ \overline{Q} $$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explai...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-03-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP03(2021)278 |
| _version_ | 1818641530542358528 |
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| author | Song He Zhenjie Li Chi Zhang |
| author_facet | Song He Zhenjie Li Chi Zhang |
| author_sort | Song He |
| collection | DOAJ |
| description | Abstract We study the symbol and the alphabet for two-loop NMHV amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills from the Q ¯ $$ \overline{Q} $$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use Q ¯ $$ \overline{Q} $$ equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities. |
| first_indexed | 2024-12-16T23:28:38Z |
| format | Article |
| id | doaj.art-df99d38741374895b15edc850862e9d9 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-16T23:28:38Z |
| publishDate | 2021-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-df99d38741374895b15edc850862e9d92022-12-21T22:11:57ZengSpringerOpenJournal of High Energy Physics1029-84792021-03-012021314410.1007/JHEP03(2021)278The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equationsSong He0Zhenjie Li1Chi Zhang2School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCASCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of SciencesCAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of SciencesAbstract We study the symbol and the alphabet for two-loop NMHV amplitudes in planar N $$ \mathcal{N} $$ = 4 super-Yang-Mills from the Q ¯ $$ \overline{Q} $$ equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use Q ¯ $$ \overline{Q} $$ equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.https://doi.org/10.1007/JHEP03(2021)278Scattering AmplitudesAnomalies in Field and String TheoriesSupersymmetric Gauge Theory |
| spellingShingle | Song He Zhenjie Li Chi Zhang The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations Journal of High Energy Physics Scattering Amplitudes Anomalies in Field and String Theories Supersymmetric Gauge Theory |
| title | The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations |
| title_full | The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations |
| title_fullStr | The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations |
| title_full_unstemmed | The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations |
| title_short | The symbol and alphabet of two-loop NMHV amplitudes from Q ¯ $$ \overline{Q} $$ equations |
| title_sort | symbol and alphabet of two loop nmhv amplitudes from q ¯ overline q equations |
| topic | Scattering Amplitudes Anomalies in Field and String Theories Supersymmetric Gauge Theory |
| url | https://doi.org/10.1007/JHEP03(2021)278 |
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