Summary
The leading singularity is studied in scalar triangle graphs. We observe that in some examples of practical interest like the « form factors » the anomalous singularity does not arise when we perform the momentum integrations in the Feynman-Stückelberg amplitudes, indicating that quantum field theory will not necessarily yield the so-called analytic scattering amplitudes. However, the analytic triangle form factor which shows the anomalous singularity and the corresponding Feynman-Stückelberg amplitude coincide everywhere above the normal thresholds, especially in the « physical region » of the variables. But we should be warned against the uncritical use of conventional methods and concepts, even in simple perturbation structures.
Riassunto
Si studia la singolarità principale nei grafici triangolari scalari. Si osserva che in alcuni esempi di interesse pratico come i « fattori di forma » la singolarità anomala non sorge quando si effettuano le integrazioni dell’impulso nelle ampiezze di Feynman-Stückelberg, indicando che la teoria quantistica dei campi non da necessariamente le cosiddette ampiezze di scattering analitiche. Tuttavia il fattore di forma triangolare analitico che presenta la singolarità anomala e la corrispondente ampiezza di Feynman-Stückelberg coincidono dovunque sopra le soglie normali, specialmente nella « regione fisica » delle variabili. Ma bisogna state in guardia contro l’uso indiscriminato dei metodi e concetti convenzionali, anche in semplici strutture perturbative.
Реэюме
Исследуется главная сингулярность в скалярных треугольных графиках. Мы обнаружили, что в некоторых случаях, имеюших практический интерес, подобно « форм-факторам », аномальная сингулярность не воэникает, когда мы проводим импульсные интергирования в амплитудах Фейнмана-Стюкел ьберга, что укаэывает на тот факт, что квантовая теория поля не обяэательно дает так наэываемые аналитические амплитуды рассеяния. Однако, аналитический треугольный форм-фактор, который обнаруживает аномальную сингулярность, и соответствуюшая амплитуда Фейнмана-Стюке льберга совпадают всюду выще нормальных порогов, особенно в « фиэической области » переменных. Но мы хотели бы предостеречь против некритического испольэования обычных методов и представлений, даже в простых структурах воэмушений.
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Rechenberg, H., Sudarshan, E.C.G. Analyticity in quantum field theory. Nuov Cim A 12, 541–568 (1972). https://doi.org/10.1007/BF02736610
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DOI: https://doi.org/10.1007/BF02736610