Functional Analytic Continuation Techniques with Applications in Field Theory

Ciulli, S. and Spearman, T. D. (1982) Functional Analytic Continuation Techniques with Applications in Field Theory. (Preprint)

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Often one has data at points inside the holomorphy domain of a Green’s function, or of an Amplitude or Form—Factor, and wants to obtain information about the spectral function i.e. the discontinuity along the cuts. Data may be experimental or theoretical. In QCD for example the perturbation expansion is valid only for unphysicaL values of the energy: one would like to continue this information to the cuts to find the resonance parameters. However, analytic continuation off open contours is extremely unstable. Also, the straightforward continuation of the truncated perturbation expansion will not do, since this is itself analytic and continuation will thus yield exactly the same result. This problem is solved by functional techniques, first by allowing small imprecisions in the data to remove the uniqueness of the continuation, and then by introducing a stabilizing condition suited to the particular physical problem, which will suppress the functions with incorrect behaviour. The stabilizing condition is expressed in terms of a norm giving a measure of the smoothness of the Discrepancy Function -which is the Amplitude with the resonances removed. The minimal norm computed from the data depends on the trial values of the resonance parameters and enables one to select the best values for these. The corresponding optimal amplitude is also constructed. An explicit solution is obtained for the case of a discrete data set; in the continuous case the problem is expressed in terms of a Fredholm integral equation.

Item Type: Article
Divisions: School of Theoretical Physics > Preprints
Date Deposited: 11 Jul 2018 09:43
Last Modified: 15 Dec 2022 08:43

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