Cutting through form factors and cross sections of non-protected operators in N = 4 $$ \mathcal{N}=4 $$ SYM

Abstract

Abstract:We study the form factors of the Konishi operator, the prime example of nonprotected operators in N = 4 SYM theory, via the on-shell unitarity method. Since the Konishi operator is not protected by supersymmetry, its form factors share many features with amplitudes in QCD, such as the occurrence of rational terms and of UV divergences that require renormalization. A subtle point is that this operator depends on the spacetime dimension. This requires a modification when calculating its form factors via the on-shell unitarity method. We derive a rigorous prescription that implements this modification to all loop orders and obtain the two-point form factor up to two-loop order and the three-point form factor to one-loop order. From these form factors, we construct an IR-finite crosssection-type quantity, namely the inclusive decay rate of the (off-shell) Konishi operator to any final (on-shell) state. Via the optical theorem, it is connected to the imaginary part of the two-point correlation function. We extract the Konishi anomalous dimension up to two-loop order from it.

On-shell methods for form factors in $$\mathcal{N}=4$$ SYM and their applications

“…The main point here is that one can use modern on-shell methods to compute form factors, and then also compute the UV information. Such a strategy has been used in [68,69,70,71,72,73]. We will review this strategy with explicit examples below.…”

“…The unitarity computation of form factors of BPS operators was pursued in [66,67]. Form factors of non-protected operators and their application to the anomalous dimension problem in N = 4 SYM theory have been studied in [68,69,70,71,72,73]. Form factors with multi-operator insertions have been studied in [74,75,76] (see also a try at strong coupling in [49]).…”

Two-loop doubly massive four-point amplitude involving a half-BPS and Konishi operator

“…The results of the anomalous dimensions up to five loops are present in the literature [15][16][17][18][19][20][21][22][23][24][25]. The two-point FF to two-loop and three-point to one-loop were computed in [26] where the first one was later extended by us in [11] to three-loop and the latter one to two-loop by one of us in [27]. In this article, for the first time, we focus on a four-point amplitude of two different composite operators: half-BPS and Konishi.…”

“…In addition, though the N = 4 SYM theory is ultraviolet (UV) finite in 4-dimensions, there can be UV m = 0 m = m 1 m = m 2 m = 0 divergences in the FF beyond leading order because of the composite operators. In [26], it was shown that the FF of unprotected operators like Konishi calculated in four dimensional helicity (FDH) scheme [33,34] fail to produce the correct anomalous dimensions, instead in modified dimensional reduction (DR) [35,36] it indeed gives the correct results. Due to the similarity between the latter scheme with dimensional regularisation [37] which is mostly used for the radiative corrections following Feynman diagrammatic approach, it is much more convenient to employ the DR scheme.…”

“…Inspired by these observations, a long-standing question is floating around: is the UT property a generic feature of N = 4 SYM amplitudes? In [11,26], it has been shown that UT property breaks down for the Sudakov FF of non-protected operators like Konishi. In this article, we address this question in the context of four-point amplitude involving two different color singlet states described by a half-BPS and Konishi.…”