Dual S-matrix bootstrap. Part I. 2D theory
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Springer
Received: August 14, 2020 Accepted: October 4, 2020 Published: November 17, 2020
Andrea L. Guerrieri,a Alexandre Homricha,b,c and Pedro Vieiraa,b a
ICTP South American Institute for Fundamental Research, IFT-UNESP, São Paulo, SP 01440-070, Brazil b Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada c Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: 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. 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. Keywords: Field Theories in Lower Dimensions, Scattering Amplitudes ArXiv ePrint: 2008.02770
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)084
JHEP11(2020)084
Dual S-matrix bootstrap. Part I. 2D theory
Contents 1 Introduction
1
2 Dual optimization and the S-matrix bootstrap
3
application The setup Single component horn Multiple component kinematics Multiple component dual problem Numerical results
8 8 9 10 15 18
4 Discussion
20
A Strong duality
23
B More on dispersion relations B.1 Subtracted dispersions B.2 The 11 → 12 functional
24 24 26
C Dual Lagrangian for multiple components
26
D Dual Z2 bootstrap D.1 Setup the primal problem D.2 Dual construction I: residue constraints D.3 Dual construction II: analyticity and crossing D.4 Dual construction III: unitarity D.5 Dual problem numerics
28 28 30 30 31 33
1
Introduction
Figure 1 is extracted from [1] and [2]. These works explore the allowed space of physical 4D S-matrices. One parametrizes a vast family of S-matrices compatible with given physical and mathematical assumptions and maximize or minimize quantities within this ansatz to find the boundaries of what is possible. The more parameters the ansatz has, the better is the exploration. As the number of parameters become very large, one hopes that these boundaries converge towards the true boundaries of the S-matrix space. Sometimes this works beautifully as illustrated in the figure; sometimes convergence is painful, to say the least, as also illustrated in the figure. In those cases where convergence
–1–
JHEP11(2020)084
3 An 3.1 3.2 3.3 3.4 3.5
|g
max
Tough convergence
200 Nmax 10
Beautiful convergence
150
12 14 16
100
18 20
50
Allowed 1
2
Allowed 3
4
m2b
(b)
Figure 1. a) Maximal cubic coupling showing up in the scattering of
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