QFT, EFT and GFT

Abstract 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.

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Quantum field theory and the Bieberbach conjecture

SciPost Physics ◽

10.21468/scipostphys.11.1.002 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Parthiv Haldar ◽

Aninda Sinha ◽

Ahmadullah Zahed

Keyword(s):

Function Theory ◽

Scattering Amplitude ◽

Tree Level ◽

Geometric Function Theory ◽

Quantum Field ◽

Bieberbach Conjecture ◽

Symmetric Representation ◽

Grunsky Inequalities ◽

Geometric Function ◽

S Matrix

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.

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Positivity and geometric function theory constraints on pion scattering

Journal of High Energy Physics ◽

10.1007/jhep12(2021)036 ◽

2021 ◽

Vol 2021 (12) ◽

Author(s):

Ahmadullah Zahed

Keyword(s):

Dispersion Relation ◽

Function Theory ◽

Global Symmetry ◽

Geometric Function Theory ◽

Pion Scattering ◽

Taylor Coefficients ◽

Symmetric Point ◽

Geometric Function ◽

Channel Dispersion ◽

Crucial Ingredient

Abstract This paper presents the fascinating correspondence between the geometric function theory and the scattering amplitudes with O(N) global symmetry. A crucial ingredient to show such correspondence is a fully crossing symmetric dispersion relation in the z-variable, rather than the fixed channel dispersion relation. We have written down fully crossing symmetric dispersion relation for O(N) model in z-variable for three independent combinations of isospin amplitudes. We have presented three independent sum rules or locality constraints for the O(N) model arising from the fully crossing symmetric dispersion relations. We have derived three sets of positivity conditions. We have obtained two-sided bounds on Taylor coefficients of physical Pion amplitudes around the crossing symmetric point (for example, π+π−→ π0π0) applying the positivity conditions and the Bieberbach-Rogosinski inequalities from geometric function theory.

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Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

European Journal of Applied Mathematics ◽

10.1017/s0956792521000127 ◽

2021 ◽

pp. 1-26

Author(s):

ELENA CHERKAEV ◽

MINWOO KIM ◽

MIKYOUNG LIM

Keyword(s):

Composite Materials ◽

Series Expansion ◽

Function Theory ◽

Effective Conductivity ◽

Two Dimensions ◽

Boundary Integral ◽

Geometric Function Theory ◽

Boundary Integral Formulation ◽

Geometric Function ◽

Series Expression

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

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Integral transforms for logharmonic mappings

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02578-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hugo Arbeláez ◽

Víctor Bravo ◽

Rodrigo Hernández ◽

Willy Sierra ◽

Osvaldo Venegas

Keyword(s):

Function Theory ◽

Classical Problem ◽

Integral Transforms ◽

Geometric Function Theory ◽

Integral Transformations ◽

Geometric Function

AbstractBieberbach’s conjecture was very important in the development of geometric function theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof. It is in this context that the integral transformations of the type $f_{\alpha }(z)=\int _{0}^{z}(f(\zeta )/\zeta )^{\alpha }\,d\zeta $ f α ( z ) = ∫ 0 z ( f ( ζ ) / ζ ) α d ζ or $F_{\alpha }(z)=\int _{0}^{z}(f'(\zeta ))^{\alpha }\,d\zeta $ F α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ appear. In this note we extend the classical problem of finding the values of $\alpha \in \mathbb{C}$ α ∈ C for which either $f_{\alpha }$ f α or $F_{\alpha }$ F α are univalent, whenever f belongs to some subclasses of univalent mappings in $\mathbb{D}$ D , to the case of logharmonic mappings by considering the extension of the shear construction introduced by Clunie and Sheil-Small in (Clunie and Sheil-Small in Ann. Acad. Sci. Fenn., Ser. A I 9:3–25, 1984) to this new scenario.

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Some estimates for extremal decomposition of the complex plane

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2018-32-5 ◽

2018 ◽

Vol 32 ◽

pp. 42-47

Author(s):

Iryna Denega

Keyword(s):

Complex Variable ◽

Open Problem ◽

Function Theory ◽

Extremal Problems ◽

Geometric Function Theory ◽

Geometric Function ◽

Inner Radius ◽

Overlapping Domains ◽

Disjoint Domains

In geometric function theory of complex variable extremal problems on non-overlapping domains are well-known classic direction. A lot of such problems are reduced to determination of the maximum of product of inner radii on the system of non-overlapping domains satisfying a certain conditions. In this paper, we consider the well-known problem of maximum of the functional $r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right)$, where $B_{0}$,..., $B_{n}$ are pairwise disjoint domains in $\overline{\mathbb{C}}$, $ a_0=0 $, $|a_{k}|=1$, $k=\overline{1,n}$ are different points of the circle, $\gamma\in (0, n]$, and $r(B,a)$ is the inner radius of the domain $B\subset\overline{\mathbb{C}}$ relative to the point $ a $. This problem was posed as an open problem in the Dubinin paper in 1994. Till now, this problem has not been solved, though some partial solutions are available. In the paper an estimate for the inner radius of the domain that contains the point zero is found. The main result of the paper generalizes the analogous results of [1, 2] to the case of an arbitrary arrangement of systems of points on $\overline{\mathbb{C}}$.

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