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

Abstract

We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar $$ \mathcal{N} $$ N = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for n = 6, 7 to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for MHV amplitudes up to 9 points derived from $$ \overline{Q} $$ Q ¯ equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.

A nice two-loop next-to-next-to-MHV amplitude in $$ \mathcal{N} $$ = 4 super-Yang-Mills

“…, 8, treated as 2 length-4 orbits above), and 3 length-4 orbits generated by seeds 1245 , 1256 and 1267 . The first two entries respect Steinmann relations, and we find that all first two entries are consistent with the general prediction of [70]: they are either log log terms respecting Steinmann relations, or the symbol of one-loop box functions i.e. I (1) (x 1 , x 3 , x 5 , x 7 ) and finite part of its lower-mass degenerations In particular, those terms with 3 odd letters are very simple: the first-two entries can only be the symbol of the four-mass box, eq.…”

The Three-loop MHV Octagon from $\bar{Q}$ equations

“…It is highly desirable to understand how the alphabets for MHV octagons differ at different loop orders. Furthermore, as reported in [57], instead of the usual BDS-subtracted amplitudes, the three-loop MHV octagon with minimal subtraction [58] fulfills the (extended) Steinmann relation. There are other arguments [35,59] that suggest larger alphabets for octagons, however, it is worthwhile to survey the bootstrap program for octagons already.…”

“…Besides the application in the computations of MHV/NMHV amplitudes, another major application of the Q equations is to constrain the last entries and the last two entries [57] for amplitudes. On the one hand, it is worth exploring the relation of such constraints with extended Steinmann/cluster adjacency, on the other hand, similar constraints also appear in the bootstrap of certain form factors [60,61], it would be interesting to generalize the Q equations to these cases and investigate other non-perturbative applications of the Q equations.…”

A study of Feynman integrals with uniform transcendental weights and their symbology

“…Therefore, Steinmann relations for X also guarantee the extended Steinmann relations of I (w) a . In [39], it's conjectured from first entry condition and Steinmann relations that the first-two entries of DCI integrals can only be linear combination of one-loop box functions and some trivial log log functions. For any order of of I (w) a , we also find that the first-two entries of I i can only be linear combinations of…”

“…The symbology of DCI integrals, strictly in D = 4, is much better understood by explicit computations [30][31][32][33][34][35] as well as by Landau analysis [12,13,36,37], considerations based on cluster algebras [18][19][20]38] etc. Moreover, properties of the symbols such as conditions on first two entries and (extended) Steinmann relations have also been studied more extensively for DCI integrals (see [39] and references therein). By sending a generic dual point, which are not null separated from adjacent points, to infinity,…”