Abstract
Using duality in optimization theory we formulate a dual approach to the S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix. We then explain how to optimize such bounds numerically, and prove that they provide the same bounds obtained from the usual primal formulation of the S-matrix Bootstrap, at least once convergence is attained from both perspectives. These techniques are then applied to the study of a gapped system with two stable particles of different masses, which serves as a toy model for bootstrapping popular physical systems.
Highlights
Using duality in optimization theory we formulate a dual approach to the Smatrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix
For our first result we consider a scattering amplitude with a single pole corresponding to the exchaAngesooflautsciaolanr pwaroticuleldof bmeasstmob,deexvacetllyoaps ian oduruaanlsantzu(m15)e, raincdamlapximroizce edure — called the dual problem the—Ivnalfiuwgeuhorefet4rhweeerepisnliodstuttehegea2mdaasxaoimffuunmcctoaiobnnssooltufrtmeubvc.1a2tluienogf thveicaobuplleingS|g-|mdeafinterdicasetshewreesidwueould instead rule out unphysical of Sth-empoalet,rwixithstpheacdie↵.e2renTt churevnes cworereswpoonduinldg tao pdip↵erroenat cvhaluteshoef Nbmoaxu. nWdeahraivees of the S-matrix space from two sides, dual and primal, and in this way rigorously bracket the 12For mb 6= m this in particular implies that there is by assumption no three-point coupling where all particles have mass m
This could be due to a symmetry but we do not have to commit to an underlying true boundaries of the sought meachfatneismr hSer-e.matrix space
Summary
For our first result we consider a scattering amplitude with a single pole corresponding to the exchaAngesooflautsciaolanr pwaroticuleldof bmeasstmob,deexvacetllyoaps ian oduruaanlsantzu(m15)e, raincdamlapximroizce edure — called the dual problem the—Ivnalfiuwgeuhorefet4rhweeerepisnliodstuttehegea2mdaasxaoimffuunmcctoaiobnnssooltufrtmeubvc.1a2tluienogf thveicaobuplleingS|g-|mdeafinterdicasetshewreesidwueould instead rule out unphysical of Sth-empoalet,rwixithstpheacdie↵.e2renTt churevnes cworereswpoonduinldg tao pdip↵erroenat cvhaluteshoef Nbmoaxu. nWdeahraivees of the S-matrix space from two sides, dual and primal, and in this way rigorously bracket the 12For mb 6= m this in particular implies that there is by assumption no three-point coupling where all particles have mass m. NWdeahraivees of the S-matrix space from two sides, dual and primal, and in this way rigorously bracket the 12For mb 6= m this in particular implies that there is by assumption no three-point coupling where all particles have mass m. This could be due to a symmetry but we do not have to commit to an underlying true boundaries of the sought meachfatneismr hSer-e.matrix space. This was recently achieved in two dimensions for simple models with a single type of particle tra1n2sforming in some non-trivial global symmetry group [11].3. This paper concerns two dimensional multi-particle systems with arbitrary mass spectra from this dual perspective, clearly one step further in the complexity ladder, closer to the full higher dimensional problem. We will consider a different technical approach, complementary to [11], with some aspects which we hope can be more directly transposable to higher dimensions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
