Rigorous Parametric Dispersion Representation with Three-Channel Symmetry

Auberson, G.; Khuri, N. N.

doi:10.1103/physrevd.6.2953

Cited by 42 publications

(95 citation statements)

References 10 publications

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“…However, a complete classification of circumstances as to when a convex sum of univalent functions leads to a univalent function does not appear to be known in the mathematics literature. Nevertheless, just by using the univalence of the kernel, we will be able to derive analogues of (2) and (3). What we will further show is that as an expansion around a ∼ 0, the Grunsky inequalities hold as the resulting inequality on W is known to hold using either fixed-t or crossing symmetric dispersion relation.…”

Section: Introductionmentioning

confidence: 79%

“…providing a two-sided bound on the absolute value of the function. In the course of 70 years, attempts at proving the Bieberbach conjecture led to the invention of new mathematical results such as (3) and techniques in the area of geometric function theory. Now it is certainly not obvious, but we claim that (2) and ( 3) have analogues in the context of 2-2 scattering in quantum field theory.…”

Section: Introductionmentioning

confidence: 99%

“…Now it is certainly not obvious, but we claim that (2) and ( 3) have analogues in the context of 2-2 scattering in quantum field theory. To see this, we will make use of the crossing symmetric representation of the 2-2 scattering of identical massive scalars, first used in the longforgotten work by Auberson and Khuri [3] and resurrected in [4,6]. If we consider M(s, t) and assume that there are no massless exchanges, then we expect a low energy expansion of the form M(s, t) = p≥0,q≥0…”

Section: Introductionmentioning

confidence: 99%

“…For now, we note that both x, y ∝ z 3 /(z 3 −1) 2 . As such, the appropriate variable isz = z 3 . We write a crossing symmetric dispersion relation in the variablez keeping a fixed.…”

Section: Introductionmentioning

confidence: 99%

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Quantum field theory and the Bieberbach conjecture

2021

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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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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For now, we note that both x, y ∝ z 3 /(z 3 −1) 2 . As such, the appropriate variable isz = z 3 . We write a crossing symmetric dispersion relation in the variablez keeping a fixed.…”

Section: Introductionmentioning

confidence: 99%

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Quantum field theory and the Bieberbach conjecture

2021

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“…Note added: since the original publication of this work, these issues have been clarified in references [83,84]. Their starting point is a dispersion relation [85] which manifests the full s-t-u symmetry. Expanding the dispersion using the OPE leads to an expansion of the correlator in terms of crossing-symmetric objects.…”

Section: Jhep05(2021)243mentioning

confidence: 99%

Dispersive CFT sum rules

Caron-Huot

Mazáč

Rastelli

et al. 2021

J. High Energ. Phys.

167

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We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule “dispersive” if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of “superconvergence” sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably we exhibit a candidate extremal functional for the spin-two gap problem.

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An Approximation Scheme for Constructing π°π° Amplitudes from ACU Requirements

Schwarz

1980

Fortschr. Phys.

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The requirements of analyticity, crossing symmetry and unitarity (ACU) are used as input to construct amplitudes for the scattering of neutral π mesons. They are obtained explicitly as polynomials in certain conformally transformed variables which ensure the correct analyticity structure. Crossing symmetry is expressed as a set of linear relations between the expansion coefficients. The same is true for the threshold behaviour of the partial waves and the behaviour for large energies of either the partial waves and the amplitudes itself. To enforce the positivity of the partial waves (up to a certain l) a new set of variables is introduced. It is obtained by solving the system of linear inequalities expressing the positivity condition. In this way one gets explicit expressions for all amplitudes satisfying the ACU requirements to some approximation. The rigorous constraints obtained from these ACU requirements are used to test the accuracy of this construction. The properties of all amplitudes are investigated in detail. It is indicated how this approximation scheme may be improved to any degree of accuracy and how it may be generalized to the full isospin case.

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Rigorous Parametric Dispersion Representation with Three-Channel Symmetry

Cited by 42 publications

References 10 publications

Quantum field theory and the Bieberbach conjecture

Quantum field theory and the Bieberbach conjecture

Dispersive CFT sum rules

An Approximation Scheme for Constructing π°π° Amplitudes from ACU Requirements

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