Summary
We analyse the possibility of constructing a model for current amplitudes satisfying i) planar duality, ii) complete factorization, iii) the divergence conditions of CVC and current algebra, and iv) the absence of unphysical singularities. The consistency of the points i), ii) and iii) requires the contribution of spurious states in the factorization of the two-current amplitudes. The point iv) is related to the off-shell extrapolation of the amplitudes and also shows the necessity of spurious states. We therefore suggest that spurious states must have a physical interpretation in terms of hadrons. We then consider in detail a specific model for the vector-current operator. The relevant current amplitudes are shown to have a good behaviour in the limit of zero-current momenta; the single-current amplitudes are dominated by the external line insertions and the two-current amplitudes satisfy the Ward-Takahashi divergence conditions. The two-current amplitudes, however, have a bad behaviour for arbitrary momenta, and also show a violation of CVC due to the presence of spurious-pole contributions.
Riassunto
Si analizza la possibilità di costruire un modello per ampiezze di correnti il quale soddisfi alle seguenti condizioni: i) dualità planare, ii) fattorizazione completa, iii) condizioni di divergenza della CVC e dell’algebra delle correnti, e iv) assenza di singolarità non fisiche. La compatibilità dei punti i), ii) e iii) richiede il contributo di stati spuri nella fattorizzazione delle ampiezze per due correnti. Il punto iv) è connesso con l’estrapolazione nelle masse delle ampiezze e mostra anch’esso la necessità degli stati spuri. Si suggerisce pertanto che gli stati spuri debbano avere un’interpretazione fisica in termini di adroni. Si considera poi in dettaglio un modello specifico per l’operatore della corrente vettoriale. Si mostra che le ampiezze hanno un buon comportamento nel limite di impulso zero delle correnti: le ampiezze di corrente singola sono dominate dalle inserzioni sulle linee esterne e le ampiezze per due correnti soddisfano le condizioni di divergenza di Ward-Takahashi. Le ampiezze per due correnti, tuttavia, hanno un brutto comportamento per impulsi arbitrari, mostrando anche una violazione della CVC dovuta alla presenza del contributo di poli spuri.
Реэюме
Мы аналиэируем воэможность конструирования модели для амплитуд токов, удовлетворяюших: 1) плоской дуальности, 2) полной факториэации, 3) условиям расходимости CVC и алгебре токов и 4) отсутствию нефиэических сингуяр-ностей. Согласованность условий 1), 2) и 3) требует вклада ложных состояний в факториэацию двух-токовых амплитуд. Условие 4) свяэано с зкстраполяцией амплитуд вне массовой поверхности, а также укаэывает на необходимость ложных состояний. Мы предполагаем, что ложные состояния должны иметь фиэическую интерпретацию в терминах адронов. Затем мы подробно рассматриваем специальную модель для оператора векторного тока. Покаэывается, что соответствуюшие амплитуды токов имеют хорощее поведение в пределе нулевых токовых импульсов; что одно-токовые амплитуды доминируют благодаря включениям внещних линий и что двух-токовые амплитуды удовлетворяют условиям расходимости Уорда-Така-хащи. Однако двух-токовые амплитуды имеют плохое поведение для проиэвольных моментов, обнаруживая также нарущение СVС вследствие наличия вкладов ложных полюсов.
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Partially supported by G.I.F.T., Departamento de Física Teórica, Barcelona, and by the Consiglio Nazionale delle Ricerche.
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Ademollo, M., Gomis, J. Dual-current amplitudes with spurious states. Nuov Cim A 4, 299–329 (1971). https://doi.org/10.1007/BF02728467
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DOI: https://doi.org/10.1007/BF02728467