Generalized unitarity and the worldsheet S matrix in AdS_n x S^n x M^(10-2n)

TL;DR

This work develops a two-dimensional generalized unitarity framework to compute the logarithmic parts of the worldsheet S matrix for integrable strings on AdS backgrounds up to two loops. By building loop corrections from tree-level amplitudes and carefully handling regularization, the authors show that only logarithmic momentum-dependent terms appear at these orders and that double logarithms cancel, consistent with exponentiation of the one-loop dressing phase across multiple backgrounds. The diagonal, symmetry-determined rational terms are fixed, while off-diagonal rational pieces follow from symmetry, enabling a complete construction of the logarithmic S-matrix parts and providing strong evidence for perturbative integrability. The results illuminate the dressing-phase structure and regularization issues, setting the stage for higher-loop analyses and broader applications in gauge/string dualities.

Abstract

The integrability-based solution of string theories related to AdS(n)/CFT(n-1) dualities relies on the worldsheet S matrix. Using generalized unitarity we construct the terms with logarithmic dependence on external momenta at one- and two-loop order in the worldsheet S matrix for strings in a general integrable worldsheet theory. We also discuss aspects of calculations at higher orders. The S-matrix elements are expressed as sums of integrals with coefficients given in terms of tree-level worldsheet four-point scattering amplitudes. One-loop rational functions, not determined by two-dimensional unitarity cuts, are fixed by symmetry considerations. They play an important role in the determination of the two-loop logarithmic contributions. We illustrate the general analysis by computing the logarithmic terms in the one- and two-loop four-particle S-matrix elements in the massive worldsheet sectors of string theory in AdS_5 x S^5, AdS_4 x CP^3, AdS_3 x S^3 x S^3 x S^1 and AdS_3 x S^3 x T^4. We explore the structure of the S matrices and provide explicit evidence for the absence of higher-order logarithms and for the exponentiation of the one-loop dressing phase.

Figures (6)

• Figure 1: Integrals with fields that are truncated away at the classical level. The external momentum configuration guarantees that in (a) and (b) there are $\varphi$ states crossing the cut. The one- and two-loop integrals are constants, independent of external momenta. The three-loop integral depends on both external momenta and therefore need not be a rational function.
• Figure 2: The integrals appearing in the one-loop four-point amplitudes. Tensor integrals can be reduced to them as well as to tadpole integrals, which are momentum-independent.
• Figure 3: Two-particle cuts of the one-loop four-point amplitudes
• Figure 4: The integrals appearing in the two-loop four-point amplitudes. Each cut in fig. \ref{['2loop_i2p_cut']} determines the coefficient of one of these integrals. There exist, of course, other two-loop four-point integrals; the structure of the Lagrangian suggests that integrals with vertices with an odd number of edges cannot appear while the integral with a six-point vertex is momentum-independent and thus it can contribute only to terms with rational momentum dependence.
• Figure 5: Iterated two-particle cuts of two-loop four-point amplitudes. They are all maximal cuts (in two dimensions). It is not possible to relax the cut condition on any propagator either because the corresponding tree-level amplitude does not exist or because the resulting higher-point tree amplitude has an on-shell propagator as a consequence of integrability and S-matrix factorization. As discussed in sec. \ref{['worldsheetPT']} all cuts of a four-point two-loop amplitude which is a product of tree amplitudes is equivalent to a sum of the cuts shown here.