Konishi form factor at three loops in N=4 supersymmetric Yang-Mills theory

Abstract

We present the first results on the third order corrections to on-shell form factor (FF) of the Konishi operator in N = 4 supersymmetric Yang-Mills theory using Feynman diagrammatic approach in modified dimensional reduction (DR) scheme. We show that it satisfies KG equation in DR scheme while the result obtained in four dimensional helicity (FDH) scheme needs to be suitably modified not only to satisfy the KG equation but also to get the correct ultraviolet (UV) anomalous dimensions. We find that the cusp, soft and collinear anomalous dimensions obtained to third order are same as those of the FF of the half-BPS operator confirming the universality of the infrared (IR) structures of on-shell form factors. In addition, the highest transcendental terms of the FF of Konishi operator are identical to those of half-BPS operator indicating the probable existence of deeper structure of the on-shell FF. We also confirm the UV anomalous dimensions of Konishi operator up to third order providing a consistency check on the both UV and universal IR structures in N = 4. PACS numbers: 12.38BxThe ability to accomplish the challenging job of calculating quantities is of fundamental importance in any potential mathematical theory. In quantum field theory (QFT), this manifests itself in the quest for computing the multi-loop and multi-leg scattering amplitudes under the glorious framework of age-old perturbation theory. The fundamental quantities to be calculated in any gauge theory are the scattering amplitudes or the correlation functions. Recently, there have been surge of interest to study form factors (FFs) as they connect fully on-shell amplitudes and correlation functions. The FFs are a set of quantities which are constructed out of the scattering amplitudes involving on-shell states consisting of elementary fields and an off-shell state described through a composite operator. These are operator matrix elements of the form p σ1 1 , · · · , p σ l l |O|0 where, O represents a gauge invariant composite operator which generates a multiparticle on-shell state |p σ1 1 , · · · , p σ l l upon operating on the vacuum of the theory. p i are the momenta and σ i encapsulate all the other quantum numbers of the particles. More precisely, FFs are the amplitudes of the processes where classical current or field, coupled through gauge invariant composite operator O, produces some quantum state. Studying these quantities not only help to understand the underlying ultraviolet and infrared structures of the theory, but also enable us to calculate the anomalous dimensions of the associated composite operator.The Sudakov FFs (l = 2) in N = 4 maximally supersymmetric Yang-Mills (SYM) theory [1, 2] were initially considered by van Neerven in [3], almost three decades back, where a half-BPS operator belonging to the stressenergy supermultiplet, that contains the conserved currents of N = 4 SYM, was investigated to 2-loop order:Very recently, this was extended to 3-loop in [4]. We will represent scalar and pseudo-scalar fields by φ a m and χ a m ...

Infrared structure of $N$ =4 SYM and Leading trancendentality principle in gauge theory

“…where A i+1 and B i+1 are the cusp [4,5,6,7] and collinear [7] anomalous dimensions respectively. R (i) aa (z) is the regular function as z → 1.…”

“…The three-loop results for the FFs,F I are already known [7], up to the same order the distribution parts of Γ aa (see Eq. (3.1)) can be obtained by using A 3 [6,7] and B 3 [7]. Using f 3 and G 3 (ε) from Eq.…”

“…In above I can be any one of the three operators considered in our current work. Z I (a, ε) is the overall operator renormalization constant, which is unity for I = half-BPS and T operators; however, for I = K , up to three loop, the pertubative coefficients of Z K are available [25,26,27,17,28,7]. F I aa (Q 2 ) is the FF contribution, i.e., the matrix elements of the half-BPS or T or K between the on-shell state aa where a = {λ , g, φ , χ} and vacuum, normalised by the Born contribution, which reads asF…”

“…In the above, we drop all the regular terms resulting from the convolutions and keep only distributions. In [7], the FFs are shown to satisfy the K+G equation [29,30,31,32] and its solution at each order can be expressed in terms of the universal cusp (A I ), soft ( f I ) and collinear anomalous (B I ) dimensions along with some operator dependent contributions [33,24]. ∆ I,SV aa is finite in the limit ε → 0, thus the pole structure of soft distribution function should be similar to that ofF I aa and Γ aa .…”

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.…”