The S-matrix bootstrap IV: multiple amplitudes

Abstract

We explore the space of consistent three-particle couplings in Z 2 -symmetric twodimensional QFTs using two first-principles approaches. Our first approach relies solely on unitarity, analyticity and crossing symmetry of the two-to-two scattering amplitudes and extends the techniques of [2] to a multi-amplitude setup. Our second approach is based on placing QFTs in AdS to get upper bounds on couplings with the numerical conformal bootstrap, and is a multi-correlator version of [1]. The space of allowed couplings that we carve out is rich in features, some of which we can link to amplitudes in integrable theories with a Z 2 symmetry, e.g., the three-state Potts and tricritical Ising field theories. Along a specific line our maximal coupling agrees with that of a new exact S-matrix that corresponds to an elliptic deformation of the supersymmetric Sine-Gordon model which preserves unitarity and solves the Yang-Baxter equation.

S-matrix bootstrap: Supersymmetry, Z2 , and Z4 symmetry

“…While if the theory has no bound states we can measure the two-to-two S-matrix elements at some off shell points, thus defining effective off shell four point couplings. By picking appropriate linear functionals and S-matrix Anstaze, we thus explore the possible S-matrix space sections compatible with crossing and unitarity following [1,2]. In this paper we will consider a few simple sections which are two-or three-dimensional and thus can be nicely plotted.…”

“…These examples are richer than the setup of [1,3,4] where the scattering of the lightest real bosonic particles in a gapped theory was considered but still simpler than the scattering of particles in the fundamental representation of a OðNÞ flavor symmetry [5,6], with UðNÞ symmetry [7] or when we scatter the two lightest particles in Z 2 symmetric 2D theories [2]. The great merit of the simpler examples considered herein is that they are simple enough to be able to be analytically described while rich enough to capture many of the intricate features of these other more elaborate examples.…”

“…We then have a nice three-dimensional section of the allowed S-matrix space parametrized by the three independent couplings g ϕϕb , g ψψb and g ϕψf . This space can be plotted following [2]; the result is the nice hourglass looking coupling space shown in Fig. 2.…”

The O(N) S-matrix monolith

“…which is true if each K a is a function with a pole at θ * and residue given by n a . 5 The contour of integration can be taken to be a big rectangle inside the physical rapidity strip (that is the boundary of the Mandelstam physical sheet). If we impose appropriate crossing transformations on K we can relate the integration over the top part of the rectangle to the bottom part so that we end up with the very same integral (times 2) integrated over the real line alone.…”

“…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.…”