Summary
It is shown for the two-dimensional scalar Yukawa interaction, that the renormalized perturbation series has at least a finite radius of convergencewhen regularized (cut-off in space-time and momentum space). This is accomplished by writing down the explicit renormalized series, studying some of its associated combinatorics, and applying Caianiello’s standard arguments on the unrenormalized series (16). The result extends to any super-renormalizable theory of the\(\varphi \bar \psi \psi \) form.
Riassunto
Si dimostra che per l’interazione scalare bidimensionale di Yukawa la serie di perturbazione rinormalizzata ha almeno un raggio di convergenza finito se regolarizzata (taglio nello spazio-tempo e nello spazio dei momenti). Si giunge al risultato scrivendo l’esplicita serie rinormalizzata, studiando alcune delle combinatorie associate, ed applicando gli argomenti base di Caianiello alle serie non rinormalizzata (16). Il risulato è estendibile ad una qualunque teoria superinormalizzabile della forma\(\varphi \bar \psi \psi \)
Реэюме
Для двуме рного скалярн ого вэаимод ействия Юкавы по каэывается, что перено рмирова нный пертурб ационный ряд имеет, по крайней мере, кон ечный радиус сходимос типосле регуляри эации (обреэан ие в координ атно-вре менном и в импул ьсном простр анстве). Ёто проиэв одится путем выписыв ания точ ного перено рмирова нного ряда, иэучая некотор ые иэ его свяэанных комбина торик, и применяя станда ртные аргуме нты Кайа нелло к непере нормирован ному ряду. Ёт от реэул ьтат обобшае тся на любую сверх-пе ренормируе мую теорию, вида\(\varphi \bar \psi \psi \).
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This paper was written with partial support of the Air Force Research and Development Command under Contract No. AF 49(638)-1545.
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Simon, B. Convergence of regularized, renormalized perturbation series for super-renormalizable field theories. Nuovo Cimento A (1965-1970) 59, 199–214 (1969). https://doi.org/10.1007/BF02756356
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DOI: https://doi.org/10.1007/BF02756356