Six-Gluon amplitudes in planar $$ \mathcal{N} $$ = 4 super-Yang-Mills theory at six and seven loops

Abstract

We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.

Origin of the Six-Gluon Amplitude in Planar N=4 Supersymmetric Yang-Mills Theory

“…, based on twoloop results in gauge theory and strong coupling behavior in string theory. The more general behavior (3) for unequal u i was observed through seven loops [13], up to power corrections in the u i . Its structure is reminiscent of Sudakov double-logarithms.…”

“…and two more loops can be found in ref. [13]. Strictly speaking, ln ρ is fixed up to addition of pure even zeta values, which are trivial under the coaction, and in eq.…”

“…Perturbative results in Section 4.2 of ref. [13] yield the numbers in table I for the expansion in g 2 , truncated here to 5 loops due to space limitations, where ζ n = ζ(n) is the Riemann zeta function. Note that Γ oct has an expansion in powers of π 2 only (through 7 loops at least).…”

“…with ρ = ρ(g 2 ) a function of the coupling constant. This function was determined iteratively in [13,33] by demanding that the spaces of functions in which the perturbative amplitudes live obey a coaction principle associated to a cosmic Galois group [48][49][50]. The implementation of this requirement fixes ρ order by order in perturbation theory, ln ρ = 8ζ 2 3 g 6 − 160ζ 3 ζ 5 g 8 + 16(−2ζ 4 ζ 2 3 + 57ζ 2 5 + 105ζ 3 ζ 7 )g 10 + .…”

Conformally-regulated direct integration of the two-loop heptagon remainder

“…Today, this particular quantity-the six-particle remainder function in planar sYM-is known to seven(!) loops; and the seven-particle remainder is known (at least at 'symbol-level') to four loops [38][39][40][41][42][43][44][45][46][47][48][49]. Interestingly, after the two-loop result was found 'the old fashioned way' in [36]-namely, by integrating Feynman integrands-all subsequent results were obtained using methods that made no reference to loop integrands or loop integration whatsoever !…”