Quantum field theory and the Bieberbach conjecture

TL;DR

The paper builds a bridge between geometric function theory and quantum field theory by showing that the kernel in a crossing-symmetric dispersion relation can be treated as a univalent function in a domain, enabling de Branges's Bieberbach-type bounds to constrain Wilson coefficients. Using unitarity, it derives two-sided bounds on the scattering amplitude and strong, dimensionless inequalities for the Wilson coefficients ${\mathcal W}_{p,q}$, with explicit checks in 1-loop $\phi^4$, tree-level string amplitudes, and S-matrix bootstrap data. It further connects Szegö's theorem on partial sums to EFT truncations and invokes Grunsky inequalities to obtain nonlinear constraints on EFT coefficients, supported by numerical evidence near $a\sim0$. The work suggests that univalence properties of the crossing kernel (and possibly of the full amplitude) hold in a sizable regime, offering a new analytic toolkit for constraining low-energy EFTs and guiding future explorations in scattering amplitudes and holographic CFT Mellin amplitudes.

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 $φ^4$ 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 $|s|$, fixed $t$, the upper bound reads $|\mathcal{M}(s,t)|\lesssim |s^2|$. We discuss how Szegö'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.

Figures (9)

• Figure 1: Image of the physical cuts. The blue line on $\tilde{z}=1$ indicates the forward limit $s_2=-4/3, s_1\geq 8/3$. The two trajectories start from $\tilde{z}=-1$ and as $s_1$ increases they approach $\tilde{z}=1$.
• Figure 2: Bounds on $\left|\frac{\alpha_n(a)a^{2n}}{\alpha_1(a)a^2}\right|$ as a function of $a$
• Figure 3: Ratio of $\frac{\mathcal{W}_{0,1}}{W_{1,0}}$ obtained from the S-matrix bootstrap. The horizontal axis is the Adler zero $s_0$. The green points are for the pion lake gpv. The blue and red points are for the upper and lower river boundaries bhsstbst while the black points are for the line of minimum averaged total cross section S-matrices bst.
• Figure 4: Constraints on Wilson coefficients using \ref{['eq:n2alphaW']}. Given ${\mathcal{W}}_{0,1},{\mathcal{W}}_{1,0},{\mathcal{W}}_{1,1}, {\mathcal{W}}_{2,0}$ figure shows that bound on the $\mathcal{W}_{0,2}$. Since $\left|\frac{b_2}{b_1}\right|-2$ should be less than zero, $\mathcal{W}_{0,2}$ must lie inside the triangle. Black line is the exact answer. Different values of $a$ are indicated with different colours.
• Figure 5: Bounds on amplitude, as in theorem \ref{['th:ampbound']}, are satisfied by Tree level type II string amplitude and 1-loop $\phi^4$-amplitude.

Theorems & Definitions (18)

• Theorem 2.1
• Theorem 2.2: Sufficient condition
• Theorem 2.3: Necessary condition
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6: Szegö theorem
• Lemma 3.1: Positivity lemma
• proof : Proof.
• Lemma 4.1
• proof : Proof