Abstract
We develop a formalism to extract triple crossing symmetric positivity bounds for effective field theories with multiple degrees of freedom, by making use of su symmetric dispersion relations supplemented with positivity of the partial waves, st null constraints and the generalized optical theorem. This generalizes the convex cone approach to constrain the s2 coefficient space to higher orders. Optimal positive bounds can be extracted by semi-definite programs with a continuous decision variable, compared with linear programs for the case of a single field. As an example, we explicitly compute the positivity constraints on bi-scalar theories, and find all the Wilson coefficients can be constrained in a finite region, including the coefficients with odd powers of s, which are absent in the single scalar case.
Highlights
Dispersion relation is included, the structure of positivity bounds becomes much richer
We develop a formalism to extract triple crossing symmetric positivity bounds for effective field theories with multiple degrees of freedom, by making use of su symmetric dispersion relations supplemented with positivity of the partial waves, st null constraints and the generalized optical theorem
An alternative way to use the triple permutation symmetry of a scalar amplitude is to start with a dispersion relation that is triple symmetric [14], and locality enforces an alternative set of null constraints on the triple symmetric dispersion relation
Summary
EFTs contain multiple low energy modes in their spectrum so as to reproduce our phenomenal world. As we shall see shortly, a(ij0k)l(t) and a(ij1k)l(t) can be expressed in terms of dispersive integrals by imposing the st crossing symmetry Bijkl(s, t) = Bikjl(t, s) on the su symmetric dispersion relation Expanding both sides of eq (2.11) on v and t and re-ordering the summations appropriately, we can get. The direct way is to first extract the (potential) positivity region in the parameter space of cmijk,nl from the su symmetric dispersion relation, which can be done with the SDP method that will be described momentarily, and use the null constraints (2.37) to slice out the linear subspace that satisfies the st crossing symmetry, as emphasized by [16] for the case of a single scalar.
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