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Crossing Symmetric Dispersion Relations in QFTs

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arxiv 2012.04877 v3 pith:GQDJGZT3 submitted 2020-12-09 hep-th hep-ph

Crossing Symmetric Dispersion Relations in QFTs

classification hep-th hep-ph
keywords dispersionrelationcrossingsymmetriccertainfieldfixedinequalities
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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