We explore the correspondence between geometric function theory (GFT) and
quantum field theory (QFT). The crossing symmetric dispersion relation provides
the necessary tool to examine the connection between GFT, QFT, and effective
field theories (EFTs), enabling us to connect with the crossing-symmetric
EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of
Typically Real functions are summarized and shown to be of enormous use in
bounding Wilson coefficients in the context of 2-2 scattering. We prove that
two-sided bounds on Wilson coefficients are guaranteed to exist quite generally
for the fully crossing symmetric situation. Numerical implementation of the GFT
constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows
a systematic exploration. A comparison of our findings obtained using GFT
techniques and other results in the literature is made. We study both the
three-channel as well as the two-channel crossing-symmetric cases, the latter
having some crucial differences. We also consider bound state poles as well as
massless poles in EFTs. Finally, we consider nonlinear constraints arising from
the positivity of certain Toeplitz determinants, which occur in the
trigonometric moment problem.
