Truncated cluster algebras and Feynman integrals with algebraic letters

Abstract

We propose that the symbol alphabet for classes of planar, dual-conformalinvariant Feynman integrals can be obtained as truncated cluster algebras purely from their kinematics, which correspond to boundaries of (compactifications of) G + (4, n)/T for the n-particle massless kinematics. For one-, two-, three-mass-easy hexagon kinematics with n = 7, 8, 9, we find finite cluster algebras D 4 , D 5 and D 6 respectively, in accordance with previous result on alphabets of these integrals. As the main example, we consider hexagon kinematics with two massive corners on opposite sides and find a truncated affine D 4 cluster algebra whose polytopal realization is a co-dimension 4 boundary of that of G + (4, 8)/T with 39 facets; the normal vectors for 38 of them correspond to g-vectors and the remaining one gives a limit ray, which yields an alphabet of 38 rational letters and 5 algebraic ones with the unique four-mass-box square root. We construct the space of integrable symbols with this alphabet and physical first-entry conditions, whose dimension can be reduced using conditions from a truncated version of cluster adjacency. Already at weight 4, by imposing last-entry conditions inspired by the n = 8 double-pentagon integral, we are able to uniquely determine an integrable symbol that gives the algebraic part of the most generic double-pentagon integral. Finally, we locate in the space the n = 8 double-pentagon ladder integrals up to four loops using differential equations derived from Wilson-loop d log forms, and we find a remarkable pattern about the appearance of algebraic letters.

Notes on worldsheet-like variables for cluster configuration spaces

“…For cluster configuration spaces, this may be related to the possible meaning of cluster string integrals: for D n case, the stringy integral has not only correct α → 0 limit as one-loop planar φ 3 integrand, but also correct factorization at massless poles for finite α ; it would be highly desirable if we can understand better such stringy integrals using these z-variables. As we have seen in [9,27,28], the letters of D n using z-variables (for n ≤ 6) have nicely appeared in the symbol of "ladder-type" Feynman integrals to all loops, where the z's are nothing but all the last entries. This has opened up another direction for studying possible physical meanings of such worldsheet-like variables.…”

“…It would be interesting to compare the topological properties of their configuration spaces including the Euler characteristic [14,15,[44][45][46]), their ABHY realizations and even applications to the symbol alphabets that appear in higher-loop integrals of SYM etc. [13,28,47,48].…”

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

“…dp (1,4,5,8), which have been computed up to L = 4 with 25 rational letters and 5 independent algebraic letters [59]. First of all, since the algebraic letters only contain the square root as that of four-mass box I 2468 , which only involve one pair of overlapping channels x 2 26 and x 2 48 as above.…”

“…The knowledge of alphabet and more refined information can be used for bootstrapping Feynman integrals [50,57] (see also [27]). In [58,59], we have identified (truncated) cluster algebras for the alphabets of a class of finite, dual conformal invariant (DCI) [60,61] Feynman integrals to high loops, based on recently-proposed Wilson-loop d log representation [62,63]. For ladder integrals with possible "chiral pentagons" on one or both ends (without any square roots), we find a sequence of cluster algebras D 2 , D 3 , • • • , D 6 for their alphabets, depending on n and the kinematic configurations; for cases with square root, such as the n = 8 double-penta ladder integrals, we find a truncated affine D 4 cluster algebra which (minimally) contain 25 rational letters and 5 algebraic ones.…”

Kinematics, cluster algebras and Feynman integrals

“…There has been significant progress in computing and studying finite, conformal Feynman integrals contributing to amplitudes [20][21][22][23][24][25][26][27][28][29]. Remarkably, cluster algebras and adjacency properties seem to apply to individual Feynman integrals in N = 4 SYM (for n = 6, 7 [14,30], as well as n ≥ 8 [31][32][33]). Remarkably, these structures have also been found for more general Feynman integrals [34] and for other quantities such as form factors [35,36].…”