S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

Abstract

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the $2\to 2$ scattering matrix $S_{2\to 2}$ such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider $3+1$ dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves $k_{\ell }(s)$ that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves $f_{\ell }(s)$, for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

arXiv comment54 pages, 6 figures; v3: clarifications and comments added (footnote 1, end of section 2.3, after eqs.(2.26) and (3.33)), figures 5 and 6 comparing S-wave from primal/dual results added; accepted by JHEP

References in corpus (20)

• The S-matrix Bootstrap I: QFT in AdS
• The S-matrix Bootstrap II: Two Dimensional Amplitudes
• Bootstrapping QCD: the Lake, the Peninsula and the Kink
• Loop Equations and bootstrap methods in the lattice
• S-matrix Bootstrap for Effective Field Theories: Massless Pions
• Landau diagrams in AdS and S-matrices from conformal correlators
• Flux Tube S-matrix Bootstrap
• The S-matrix Bootstrap IV: Multiple Amplitudes
• Carving out the Space of Open-String S-matrix
• Bootstrapping Massive Quantum Field Theories
• Adding flavour to the S-matrix bootstrap
• The O(N) S-matrix Monolith
• Relative entropy in scattering and the S-matrix bootstrap
• Dual S-matrix Bootstrap I: 2D Theory
• A note on the S-matrix bootstrap for the 2d O(N) bosonic model
• SUSY S-matrix Bootstrap and Friends
• S-matrix bootstrap for resonances
• Selection rules for the S-Matrix bootstrap
• The R-matrix bootstrap for the 2d O(N) bosonic model with a boundary
• Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory

Cited by in corpus (16)

• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Bootstrapping Pions at Large $N$
• Positivity and Geometric Function Theory Constraints on Pion Scattering
• Rigorous bounds on the Analytic $S$-matrix
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Nonperturbative Bounds on Scattering of Massive Scalar Particles in $d\ge 2$
• Swampland Conditions for Higher Derivative Couplings from CFT
• Bootstrapping the $a$-anomaly in $4d$ QFTs
• Bounds on photon scattering
• Locality and Analyticity of the Crossing Symmetric Dispersion Relation
• Bootstrapping 2d ${\varphi }^4$ Theory with Hamiltonian Truncation Data
• From conformal correlators to analytic S-matrices: CFT${}_1$/QFT${}_2$
• A Celestial Route to AdS Bulk Locality
• Strings from Almost Nothing
• Probing multi-particle unitarity with the Landau equations
• Primal S-matrix bootstrap with dispersion relations