Symbol alphabets from tensor diagrams

Abstract

We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar $$ \mathcal{N} $$ N = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope $$ {\mathcal{C}}^{\dagger}\left(4,9\right) $$ C † 4 9 encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.

Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

“…Letters of the type i(jk)∩(lmn)(opq)∩ (rst) are not at all contained in the alphabet of the two-loop NMHV nine-particle amplitude. Additional evidence in support of our proposal for the rational part of the alphabet is that it agrees with the one obtained by other means in [74], appearing simultaneously with this article.…”

Comments on all-loop constraints for scattering amplitudes and Feynman integrals

“…There are 70 allowed unfrozen letters before 1389 and 71 before 1489 . As discussed in [41,42] using the tropical Grassmannian, naively there can be thousands of rational letters before each last entry. Our computation, however, drastically reduces this number, showing that the possible pairs are in fact very limited.…”

“…As already seen for one-loop N 2 MHV, amplitudes with n ≥ 8 generally involve letters that cannot be expressed as rational functions of Plücker coordinates of the kinematics G(4, n)/T ; more non-trivial algebraic letters appear in computations based on Q equations [31] for two-loop NMHV amplitudes for n = 8 and n = 9 [32,33], requiring extension of Grassmannian cluster algebras to include algebraic letters. Solutions to both problems have been proposed using tropical positive Grassmannian [34] and related tools for n = 8 [35][36][37][38][39][40] and n = 9 [41,42], as well as using Yangian invariants or the associated collections of plabic graphs [43][44][45][46].…”

Symbol alphabets from plabic graphs

“…We are grateful to L. Ren for collaboration on the related work [15] and to N. Arkani-Hamed for numerous encouraging conversations. This work was supported in part by the US Department of Energy under contract DE-SC0010010 Task A and by Simons Investigator Award #376208 (AV).…”

“…The fact that cluster algebras [1][2][3][4] govern the symbol alphabets [5] of multiloop nparticle amplitudes in planar maximally supersymmetric Yang-Mills (SYM) theory is by now well-established for n = 6, 7 [6] (see [7] for a review of recent progress on the computation of these amplitudes via bootstrap). Starting at n = 8 qualitatively new features arise, which have been studied via several different approaches (see for example [8][9][10][11][12][13][14][15][16]).…”