The S-matrix bootstrap II: two dimensional amplitudes

104

167

We study analytically the constraints of the conformal bootstrap on the lowlying spectrum of operators in field theories with global conformal symmetry in one and two spacetime dimensions. We introduce a new class of linear functionals acting on the conformal bootstrap equation. In 1D, we use the new basis to construct extremal functionals leading to the optimal upper bound on the gap above identity in the OPE of two identical primary operators of integer or half-integer scaling dimension. We also prove an upper bound on the twist gap in 2D theories with global conformal symmetry. When the external scaling dimensions are large, our functionals provide a direct point of contact between crossing in a 1D CFT and scattering of massive particles in large AdS 2 . In particular, CFT crossing can be shown to imply that appropriate OPE coefficients exhibit an exponential suppression characteristic of massive bound states, and that the 2D flat-space S-matrix should be analytic away from the real axis.

Section: Future Directionssupporting

confidence: 74%

“…Coordinates x and θ are related through 23) and the Fourier coefficients can be obtained from the integral kernel h(x)(1 − x) −2∆ ψ via…”

Section: Fixing the Remaining Redundancymentioning

confidence: 99%

Analytic bounds and emergence of AdS2 physics from the conformal bootstrap

Mazáč

2017

104

167

“…This makes it natural to impose the vanishing residue conditions independently for each partial wave . 6 This feature of the Mellin space approach to the conformal bootstrap makes it very close in spirit to the flat space S-matrix bootstrap (see [74,75] for one recently proposed way of connecting the two).…”

Section: Jhep05(2017)027mentioning

confidence: 81%

A Mellin space approach to the conformal bootstrap

Gopakumar

Kaviraj

Sen

et al. 2017

175

285

We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CF T d four point functions and expands them in terms of crossing symmetric combinations of AdS d+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O( 3 ) rather straightforwardly. This approach also yields new analytic predictions for OPE coefficients to the same order which fit nicely with recent numerical estimates for the Ising model (at = 1). We will also mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.

“…This clarifies a long standing puzzle. It was thus far stated as a mystery why was unitarity saturated at the boundary of the physical S-matrix space in many different contexts [4,5,8,9,[11][12][13]. This dual problem, with its associated zero duality gap theorems, provides a clean explanation in the two dimensional examples.…”

Section: Introductionmentioning

confidence: 99%

The O(N) S-matrix monolith

Córdova

Kruczenski

et al. 2020

Self Cite

We consider the scattering matrices of massive quantum field theories with no bound states and a global O(N ) symmetry in two spacetime dimensions. In particular we explore the space of two-to-two S-matrices of particles of mass m transforming in the vector representation as restricted by the general conditions of unitarity, crossing, analyticity and O(N ) symmetry. We found a rich structure in that space by using convex maximization and in particular its convex dual minimization problem. At the boundary of the allowed space special geometric points such as vertices were found to correspond to integrable models. The dual convex minimization problem provides a novel and useful approach to the problem allowing, for example, to prove that generically the S-matrices so obtained saturate unitarity and, in some cases, that they are at vertices of the allowed space. arXiv:1909.06495v2 [hep-th]