Crossing Symmetric Dispersion Relations in QFTs

Abstract

For 2-2 scattering in quantum field theories, the usual fixed $t$ dispersion relation exhibits only two-channel symmetry. This paper considers a crossing symmetric dispersion relation, reviving certain old ideas in the 1970s. Rather than the fixed $t$ dispersion relation, this needs a dispersion relation in a different variable $z$, which is related to the Mandelstam invariants $s,t,u$ via a parametric cubic relation making the crossing symmetry in the complex $z$ plane a geometric rotation. The resulting dispersion is manifestly three-channel crossing symmetric. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two sided bounds and derive a general set of new non-perturbative inequalities. We show how these inequalities enable us to locate the first massive string state from a low energy expansion of the four dilaton amplitude in type II string theory. We also show how a generalized (numerical) Froissart bound, valid for all energies, is obtained from this approach.

arXiv commentv3: 6+7 pages, 4 figures, Feynman block discussion added, version to appear in Physical Review Letters

References in corpus (21)

• Causality, Analyticity and an IR Obstruction to UV Completion
• The EFT-Hedron
• Positive Moments for Scattering Amplitudes
• New positivity bounds from full crossing symmetry
• A Mellin space approach to the conformal bootstrap
• Positivity Bounds for Scalar Theories
• Sharp Boundaries for the Swampland
• Conformal Bootstrap in Mellin Space
• Where is String Theory?
• Dispersive CFT Sum Rules
• Bootstrapping QCD: the Lake, the Peninsula and the Kink
• On the Polyakov-Mellin bootstrap
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• Crossing Symmetric Dispersion Relations for Mellin Amplitudes
• On critical exponents without Feynman diagrams
• Relative entropy in scattering and the S-matrix bootstrap
• Dual S-matrix Bootstrap I: 2D Theory
• Froissart bound for/from CFT Mellin amplitudes
• The Low Energy Expansion for Pion-Pion Scattering and Crossing Symmetry in Dispersion Relations
• Froissart Bound on Total Cross-section without Unknown Constants
• Superstring Amplitudes, Unitarity, and Hankel Determinants of Multiple Zeta Values

Cited by in corpus (70)

• Sharp Boundaries for the Swampland
• Analyticity and Unitarity for Cosmological Correlators
• Dispersive CFT Sum Rules
• AdS Bulk Locality from Sharp CFT Bounds
• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• The Inflationary Wavefunction from Analyticity and Factorization
• Into the EFThedron and UV constraints from IR consistency
• QED positivity bounds
• Rigorous Bounds on Light-by-Light Scattering
• Reverse Bootstrapping: IR lessons for UV physics
• Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds
• Natural Selection Rules: New Positivity Bounds for Massive Spinning Particles
• Crossing Symmetric Dispersion Relations for Mellin Amplitudes
• Positivity Bounds on Dark Energy: When Matter Matters
• AdS Virasoro-Shapiro from single-valued periods
• Bootstrapping Pions at Large $N$
• Bounding Violations of the Weak Gravity Conjecture
• Causal Effective Field Theories
• Gravitational Regge bounds
• Quantum field theory and the Bieberbach conjecture
• Cornering Large-$N_c$ QCD with Positivity Bounds
• Triple crossing positivity bounds for multi-field theories
• Positivity and Geometric Function Theory Constraints on Pion Scattering
• Distributions in CFT II. Minkowski Space
• SMEFTs living on the edge: determining the UV theories from positivity and extremality
• What is the $i\epsilon$ for the S-matrix?
• Moments for positivity: using Drell-Yan data to test positivity bounds and reverse-engineer new physics
• Rigorous bounds on the Analytic $S$-matrix
• QFT, EFT and GFT
• Dispersion Formulas in QFTs, CFTs, and Holography
• Testing gravitational wave propagation with multiband detections
• On the unitarity and low energy expansion of the Coon amplitude
• A cautionary case of casual causality
• Scattering from production in 2d
• Positivity bounds from multiple vacua and their cosmological consequences
• Nonperturbative Bounds on Scattering of Massive Scalar Particles in $d\ge 2$
• Swampland Conditions for Higher Derivative Couplings from CFT
• Scalar Fields Near Compact Objects: Resummation versus UV Completion
• Bootstrapping the $a$-anomaly in $4d$ QFTs
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• Bounds on photon scattering
• Finite energy sum rules for gravitational Regge amplitudes
• The EFT Bootstrap at Finite $M_{PL}$
• Celestial insights into the S-matrix bootstrap
• On (Scalar QED) Gravitational Positivity Bounds
• NNLO Positivity Bounds on Chiral Perturbation Theory for a General Number of Flavours
• Positivity Bounds on parity-violating scalar-tensor EFTs
• Effective Field Theory Islands from Perturbative and Nonperturbative Four-Graviton Amplitudes
• Positivity Bounds on Massive Vectors
• Positivity bounds in vector theories
• Non-Forward UV/IR Relations
• Locality and Analyticity of the Crossing Symmetric Dispersion Relation
• Field theory expansions of string theory amplitudes
• Positivity Bounds in Scalar Effective Field Theories at One-loop Level
• Positivity, low twist dominance and CSDR for CFTs
• Dispersion relations, knot polynomials and the $q$-deformed harmonic oscillator
• Positivity bounds at one-loop level: the Higgs sector
• 1d Conformal Field Theory and Dispersion Relations
• A Celestial Route to AdS Bulk Locality
• Charging Up the Functional Bootstrap
• Unitarity and unstable-particle scattering amplitudes
• Positivity in Amplitudes and Quantum Entanglement
• Positivity bounds in scalar-QED EFT at one-loop level
• IR side of bounds on Theories with Spontaneously Broken Lorentz Symmetry
• Quantifying EFT Uncertainties in LHC Searches
• Primal S-matrix bootstrap with dispersion relations
• Bootstrapping leading hadronic muon anomaly
• Five-point partial waves, splitting constraints and hidden zeros
• The IR-Side of Positivity Bounds
• The Rise of Linear Trajectories