Four loop massless propagators: An algebraic evaluation of all master integrals

Abstract

The old "glue-and-cut" symmetry of massless propagators, first established in Ref. [1], leads -after reduction to master integrals is performed -to a host of non-trivial relations between the latter. The relations constrain the master integrals so tightly that they all can be analytically expressed in terms of only few, essentially trivial, watermelon-like integrals. As a consequence we arrive at explicit analytical results for all master integrals appearing in the process of reduction of massless propagators at three and four loops. The transcendental structure of the results suggests a clean explanation of the well-known mystery of the absence of even zetas (ζ 2n ) in the Adler function and other similar functions essentially reducible to massless propagators. Once a reduction of massless propagators at five loops is available, our approach should be also applicable for explicitly performing the corresponding five-loop master integrals.1 The so-called Laporta approach [4-6] seems to be most often utilized but a few other promising methods are being now actively developed [7][8][9][10][11][12].2 At least well-established in practice. See below for an instructive particular example of a class of massless propagators and also [13,14] for an attempt to formalize the concept of the masters integrals and to prove the universality property in general. A related discussion could be found in [15][16][17][18].

Towards QCD running in 5 loops: quark mass anomalous dimension

“…The reduction is based on evaluating sufficiently many terms of the 1/D expansion [31] of the corresponding coefficient functions [32]. The master integrals are known analytically from [33,34].…”

Regularization schemes and higher order corrections

“…All of the four-loop master integrals can be found in Ref. [26]. I find the result of the four-loop calculation to be…”

Evaluating five-loop Konishi in N=4 SYM

“…For the two non-planar integrals, we applied the method of gluing of Refs. [12,15]. I do not believe that it will feasible to evaluate this way master integrals at the next loop level.…”

“…In fact, in the family of the twenty eight master integrals for momentum-space massless propagator integrals (see Ref. [15]), there is at most one master integral in any sector, so that the coordinate-space family of the master integrals 1 is more complicated.…”

“…[15] and took results from that paper in terms of ε expansions up to transcendentality weight seven. For the two non-planar integrals, we applied the method of gluing of Refs.…”