Abstract
We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 3+1 dimensions due to crossing symmetry, analyticity and unitarity. We extremize cubic couplings, quartic couplings and scattering lengths relevant for the elastic scattering amplitude of two identical scalar particles. In the cases where our results can be compared with the older S-matrix literature they are in excellent agreement. We also extremize a cubic coupling in 2+1 dimensions which we can directly compare to a universal bound for a QFT in AdS. This paper generalizes our previous 1+1 dimensional results of [1] and [2].
Highlights
In [1] and [2] we initiated a bootstrap analysis of massive quantum field theories
For this reason we focus on the two body scattering of the lightest particle in the theory since all the usual cuts of the amplitude correspond to physical kinematics
As we have an analytic function on a domain with a boundary along which it is bounded, so we are able to constrain its values inside this region and in particular the various physical couplings which we define as residues of factorization poles
Summary
In [1] and [2] we initiated a bootstrap analysis of massive quantum field theories. In particular, we obtained bounds on couplings of a quantum field theory compatible with a given spectrum of stable particles. We expect to have maximum values for couplings beyond which the masses of bound states must decrease, or new bound-states should emerge from the continuum, or both This problem is very natural once we make the non-trivial assumption that scattering amplitudes are described by functions that are analytic away from the usual physical poles and cuts. We shall introduce a kind of uniformization coordinates where the full space of physical kinematics is mapped to (a few) unit circles This will allow us to Taylor expand the amplitudes in a convergent and manifestly crossing symmetric way in the full physical plane and to numerically impose unitarity along the physical boundaries. A number of appendices are included to complement the main text presentation
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