Quantum field theory and the Bieberbach conjecture

Abstract

An intriguing correspondence between ingredients in geometric function theory related to the famous Bieberbach conjecture (de Branges’ theorem) and the non-perturbative crossing symmetric representation of 2-2 scattering amplitudes of identical scalars is pointed out. Using the dispersion relation and unitarity, we are able to derive several inequalities, analogous to those which arise in the discussions of the Bieberbach conjecture. We derive new and strong bounds on the ratio of certain Wilson coefficients and demonstrate that these are obeyed in one-loop \phi^4ϕ4 theory, tree level string theory as well as in the S-matrix bootstrap. Further, we find two sided bounds on the magnitude of the scattering amplitude, which are shown to be respected in all the contexts mentioned above. Translated to the usual Mandelstam variables, for large |s||s|, fixed tt, the upper bound reads |\mathcal{M}(s,t)|\lesssim |s^2||ℳ(s,t)|≲|s2|. We discuss how Szeg"{o}’s theorem corresponds to a check of univalence in an EFT expansion, while how the Grunsky inequalities translate into nontrivial, nonlinear inequalities on the Wilson coefficients.

Dual EFT bootstrap: QCD flux tubes

“…This will clear the way for section 3 where we will be able to use the dual formulation to bootstrap max/min values of the Wilson coefficients in situations where no analytical solution is known. 6 This analytic result fits in the general geometric function theory recently reviewed in [67] and generalised to other interesting physical examples. 7 While further details are provided in [32], we recall that the Schwarz-Pick bounds are saturated by products of Castillejo-Dalitz-Dyson (CDD) factors (known as Blaschke products in complex analysis literature).…”

Positivity and geometric function theory constraints on pion scattering

“…In [4] the correspondence between the famous Bieberbach conjecture (de Branges' theorem) and the non-perturbative crossing symmetric scattering amplitudes was pointed out, which established a close relationship between the Bieberbach-bounds and the bounds on the Wilson coefficients. In [3] it was pointed out that crossing symmetric scattering amplitudes are typically real functions.…”

“…The constraints in EFT Wilson coefficients were worked out in [19][20][21][22][23][24][25][26][27][28][29][30][31], using positivity of the partial wave and null conditions. 4 In our case, we don't use the null constraints; instead, we use positivity (of the partial wave expansion) and bounds in the Taylor series coefficients of the amplitudes in z-variable that appear from the geometric function theory.…”

“…where the measure f (η) is a non-decreasing function. 4 Locality constraints are the same as the null constraints. See [6] for more discussion.…”

“…is discussed in [5,6], and relations to geometric function theory is given in detail in [3,4]. We will write z-variable dispersion relations for all three of the G k (s…”