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Functional Analysis and Optimization Methods in Hadron Physics pp 1–28Cite as

Theory of Strong Interactions Before the Standard Model

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Abstract

In this chapter we briefly review the attempts to construct a theory of the strong interactions in the 1960s, when it became obvious that a perturbatively renormalizable quantum field theory of these forces was not possible. The derivation of the analytic properties of scattering amplitudes and form factors from the general principles of causality and unitarity in quantum field theory is discussed at a qualitative level, using the Lehmann–Symanzik–Zimmermann (LSZ) formalism. Standard dispersion relations based on Cauchy integral formula are written down for scattering amplitudes and form factors. Finally, some pitfalls in practical applications, related to the so-called instability of analytic continuation, are emphasized.

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Notes

  1. 1.

    We shall use both terms “analytic” and “holomorphic”, which are equivalent for complex-valued functions of complex variables.

  2. 2.

    This relation is known also as Sokhotski–Plemelj, or Plemelj–Privalov relation [16].

  3. 3.

    In the case of unequal masses, the partial waves have also a cut of circular shape in the complex s-plane.

  4. 4.

    Functions satisfying this property are sometimes said to be “analytic functions of real type”, or “real-analytic functions”, not to be confused with the real functions defined only on the real axis.

  5. 5.

    The zeros on the cut can be accounted for by a suitable variation by \(\pm \pi \) of the phase.

  6. 6.

    A compact topological space is the generalization of the sets closed and bounded in the Euclidean spaces. Criteria for compactness for functional spaces have been formulated, for instance by Arzelà–Ascoli theorem [41].

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Authors and Affiliations

  1. Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania

    Irinel Caprini

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  1. Irinel Caprini
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Correspondence to Irinel Caprini .

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Caprini, I. (2019). Theory of Strong Interactions Before the Standard Model. In: Functional Analysis and Optimization Methods in Hadron Physics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-18948-8_1

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