Crossing Symmetric Dispersion Relations in Quantum Field Theories

Dec 9, 2020
6 pages
Published in:
  • Phys.Rev.Lett. 126 (2021) 18, 181601
  • Published: May 8, 2021
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Abstract: (APS)
For 2-2 scattering in quantum field theories, the usual fixed t dispersion relation exhibits only two-channel symmetry. This Letter considers a crossing symmetric dispersion relation, reviving certain old ideas from 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 nonperturbative 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.
Note:
  • v3: 6+7 pages, 4 figures, Feynman block discussion added, version to appear in Physical Review Letters
  • Elementary Particles and Fields
  • symmetry: crossing
  • string: massive
  • energy: low
  • dispersion relation
  • field theory
  • effective field theory
  • nonperturbative
  • Froissart bound
  • string model