Relative entropy in scattering and the S-matrix bootstrap

Abstract

We consider entanglement measures in 2-2 scattering in quantum field theories, focusing on relative entropy which distinguishes two different density matrices. Relative entropy is investigated in several cases which include φ 4 theory, chiral perturbation theory (χP T ) describing pion scattering and dilaton scattering in type II superstring theory. We derive a high energy bound on the relative entropy using known bounds on the elastic differential cross-sections in massive QFTs. In χP T , relative entropy close to threshold has simple expressions in terms of ratios of scattering lengths. Definite sign properties are found for the relative entropy which are over and above the usual positivity of relative entropy in certain cases. We then turn to the recent numerical investigations of the S-matrix bootstrap in the context of pion scattering. By imposing these sign constraints and the ρ resonance, we find restrictions on the allowed S-matrices. By performing hypothesis testing using relative entropy, we isolate two sets of S-matrices living on the boundary which give scattering lengths comparable to experiments but one of which is far from the 1-loop χP T Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and elastic unitarity.

Quantum field theory and the Bieberbach conjecture

“…The green points are for the pion lake [19]. The blue and red points are for the upper and lower river boundaries [20,21] while the black points are for the line of minimum averaged total cross section S-matrices [21].…”

Rigorous bounds on the Analytic $S$-matrix

“…In this Letter, we proposed a dual approach to the S-matrix Bootstrap based solely on the proven analyticity properties of scattering amplitudes. 14 Our strategy consists of decomposing both crossing and unitarity into simpler constraints that can be systematically added to improve the bounds.…”

“…where W(s, t) and E (s) are unconstrained dual variables imposing the analyticity (8) and elastic unitarity (14) in their validity domains, D and s ∈ [4,16]. Here, crossing has already been solved at the level of the primal variables by restricting to even spins only, f 2n+1 (s) = 0.…”

“…First, we simply set the dual variables associated with the elastic unitarity constraint (14) to zero, E (s) = 0. This is because we do not know how to put it in SDP form.…”

“…In the revived non-perturbative S-matrix Bootstrap program of [1,2], the space of scattering amplitudes is carved out numerically by solving an optimization problem in its Primal form: the target to optimize is a physical observable; the constraints are the physical principles of analyticity, crossing, and unitarity. This approach has already beed used to bound several classes of amplitudes, for instance, in relation to integrable systems in two dimensions [3][4][5][6][7][8][9][10][11][12], and in higher dimensions for the case of standard model physics [13][14][15][16][17] and quantum gravity theories [18]. At the same time, any bootstrap scheme in terms of the primal variables involves some type of truncation: as a result the bounds are not strictly rigorous.…”