Truncated cluster algebras and Feynman integrals with algebraic letters

Abstract

We propose that the symbol alphabet for classes of planar, dual-conformal-invariant 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 D4, D5 and D6 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 D4 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.

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

“…Note that a priori these cluster adjacency conditions based on (conjectural) Sklyanin bracket are different from the notion of adjacency for any pair of F polynomials in a clusteralgebra alphabet as studied in [59]. However, we find that for the D 4 case, the latter is a consequence of the former!…”

“…Another mysterious property of some Feynman integrals and scattering amplitudes (in and beyond planar N = 4 SYM) is that their symbol alphabet seems to fit into a cluster algebra (or truncated ones [59] with algebraic letters). In the presence of such cluster algebras, ES relations become closely related to cluster adjacency conditions, and we have established cluster adjacency conditions using A-coordinates in all these integrals and amplitudes, modulo subtlety with algebraic letters.…”

“…Next we move to the class of integrals with more non-trivial algebraic letters, namely double-pentagon ladder I (L) 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.…”

“…What we have found is that the only minor violation in the "rational" part involves rational letters 1256 and 3478 in consecutive entries; in fact, they have only appeared next to each other at second and third entries. This is good news since the "irrational" part only contains algebraic letters at these two entries [59], which also contribute to double discontinuities involving this pair. Recall that the 5 algebraic letters are of the form (z − a)/(z − a) for…”

“…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 , .…”