Abstract
Positivity bounds are powerful tools to constrain effective field theories. Utilizing the partial wave expansion in the dispersion relation and the full crossing symmetry of the scattering amplitude, we derive several sets of generically nonlinear positivity bounds for a generic scalar effective field theory: we refer to these as the P Q, Dsu, Dstu and {overline{D}}^{mathrm{stu}} bounds. While the PQ bounds and Dsu bounds only make use of the s↔u dispersion relation, the Dstu and {overline{D}}^{mathrm{stu}} bounds are obtained by further imposing the s↔t crossing symmetry. In contradistinction to the linear positivity for scalars, these inequalities can be applied to put upper and lower bounds on Wilson coefficients, and are much more constraining as shown in the lowest orders. In particular we are able to exclude theories with soft amplitude behaviour such as weakly broken Galileon theories from admitting a standard UV completion. We also apply these bounds to chiral perturbation theory and we find these bounds are stronger than the previous bounds in constraining its Wilson coefficients.
Highlights
Positivity bounds are constraints on the scattering amplitude that one can derive by assuming the UV theory satisfies some of the most fundamental properties of physics, such as Lorentz invariance, unitarity, crossing symmetry, polynomial boundedness and, crucially, analyticity
Utilizing the partial wave expansion in the dispersion relation and the full crossing symmetry of the scattering amplitude, we derive several sets of generically nonlinear positivity bounds for a generic scalar effective field theory: we refer to these as the P Q, Dsu, Dstu and Dstu bounds
In particular we are able to exclude theories with soft amplitude behaviour such as weakly broken Galileon theories from admitting a standard UV completion. We apply these bounds to chiral perturbation theory and we find these bounds are stronger than the previous bounds in constraining its Wilson coefficients
Summary
We shall introduce the fixed-t dispersion relations that are the basis needed to derive the positivity bounds . Utilizing fixed-t analyticity in the s complex plane and Cauchy’s integral formula, we can derive a twice-subtracted dispersion relation Up to this point, we have only used the UV full amplitude to derive the dispersion relation. Since we can compute the imaginary part of the amplitude to a desired order within the EFT framework from 4m2 to ( Λ), where Λ is the cutoff and 1, so we can subtract out the low energy part of the integral and define λ.
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