On form factors in $ \mathcal{N} = 4 $ SYM

Abstract

In this paper we study the form factors for the half-BPS operators O (n) I and the N = 4 stress tensor supermultiplet current T AB up to the second order of perturbation theory and for the Konishi operator K at first order of perturbation theory in the N = 4 SYM theory at weak coupling. For all the objects we observe the exponentiation of the IR divergences with two anomalous dimensions: the cusp anomalous dimension and the collinear anomalous dimension. For the IR finite parts we obtain a similar situation as for the gluon scattering amplitudes, namely, apart from the case of T AB and K the finite part has some remainder function which we calculate up to the second order. It involves the generalized Goncharov polylogarithms of several variables. All the answers are expressed in terms of the integrals related to the dual conformal invariant ones which might be a signal of integrable structure standing behind the form factors.Keywords: N = 4 Super Yang-Mills Theory, form factors, N = 1 superspace.

Form factor and boundary contribution of amplitude

“…, n) under such shifting. Following [56], we find that at the large N limit, the leading interaction part V is given by 19) where T αβ is defined through Φ α = T αβ Φ † β , and α 1 = α n+1 , α 2 = α n+2 , indicating that the shifted fields Φ n+1 , Φ n+2 are the two fields of Tr(Φ α 1 Φ α 2 ) in (2.14) with specific field type. In general, the OPE of shifted fields has the form [56] …”

“…Intensive discussion on the recursion relation of form factor is provided later in [18]. A generalization to the form factor of full stress tensor multiplet is discussed in [17] and [19], where in the former one, supersymmetric version of BCFW recursion relation is pointed out to be applicable to super form factor. Shortly after, the color-kinematic duality is implemented in the context of form factor [20], both at tree and loop-level, to generate the integrand of form factor.…”

“…The three-point two-loop form factor of half-BPS operator is achieved in [29], and the general n-point form factor as well as the remainder functions in [30]. The scalar operator with arbitrary number of scalars is discussed in [19,30,31]. Beyond the half-BPS operators, form factors of non-protected operators, such as dilatation operator [32], Konishi operator [33], operators in the SU(2) sectors [34], are also under investigation.…”

Grassmannians and form factors with q 2 = 0 in N $$ \mathcal{N} $$ =4 SYM theory

“…The unique investigation of form factors of non-gauge invariant operators build from single field (off-shell currents) was made in [31]. After nearly a decade the investigation of 1/2-BPS form factors was again initiated in [32,33]. Later the form factors of operators from 1/2-BPS and Konishi operator supermultiplets were intensively investigated both at weak [34][35][36][37] and strong couplings [38,39].…”

CHAPLIN—Complex Harmonic Polylogarithms in Fortran

“…In this paper we concentrate exclusively on harmonic polylogarithms (HPL's) up to weight four, which since their introduction have found many applications in computations up to two-loop order in the perturbative expansion, e.g., [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. In order to confront the theoretical next-to-next-to-leading order (NNLO) predictions to experiment, it is mandatory to be able to evaluate HPL's numerically in a fast and accurate way.…”