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Il Nuovo Cimento A (1965-1970)

, Volume 48, Issue 4, pp 1014–1040 | Cite as

A model for high-energy inelastic processes

  • E. Predazzi
Article

Summary

The problem of expressing the high-energy inelastic amplitudes in terms of the elastic amplitudes that couple through unitarity is discussed. A solution to this problem is given by approximately solving the unitarity equations with assumptions similar to the ones usually made in the absorptive model, together with a high-energy factorization property. The surprising result is obtained that the inelastic amplitudes are completely expressible in terms ofall the elastic ones that have the correct quantum numbers to couple through unitarity. Some general properties of this solution are investigated for πp and Kp charge exchange by using an exact Fourier-Bessel representation which is derived by means of techniques previously developed. Similarly to what happens in the Byers and Yang droplet model, the high-energy inelastic amplitudes in the forward direction show an exponentially decreasing peak (as function of the momentum transfer), if this is present in at least one of the elastic (initial and final) channels. The question of whether the forward peak of the inelastic amplitude is narrower or wider or comparable to the elastic one is seen to require a very detailed parametrization of the elastic data. In principle, in our model, there are no arbitrary parameters left since the inelastic amplitudes are completely determined from the knowledge ofall the elastic ones. In practice, however, the limitation of experimental information at our disposal forces us to introduce an empirical energy-dependent factor. Within the framework of our model, little can be said about this energy-dependent factor. If, however, one makes the extra assumption that all the elastic channels are comparably equal and that the number of channels open at a given (high) energy have the same energy dependence as the multiplicity, then it can be seen that very good agrement is obtained, in this approximation, for π→π0n which is the only case for which a sufficiently large statistics exists. No detailed numerical comparison of the theory with experimental data is attempted in this preliminary paper.

Keywords

Angular Distribution Charge Exchange Spectral Function Inelastic Process Elastic Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Модель для неупругих процессов при высоких энереиях

Резюме

Обсуждается проблема выражения неупругих амплитуд через упругие амплитуды, которые связаны через унитарность. Приводится решение этой проблемы посредством приблизительного решения уравнений унитарности, используя предположения, аналогичные тем, которые обычно делаются в модели с поглощением, совместно со свойством факторизации при высоких энергиях. Получен удивительный результат, что неупругие амплитуды могут быть полностью выражены в терминах от всех упругих амплитуд, которые имеют правильные квантовые числа, для связи через унитарность. Исследуются некоторые общие своиства для перезарядки πp и Kp, используя точное представление Фуряе-Бесселя, которое выводится с помощью предварительно развитой техники. Аналогично тому, как случилось в капельной модели Байерса и Янга, неупругие амплитуды на нулевой угол обнаруживают при высоких энергиях экспоненциально спадаюший пик (как функция переданного импульса), если пик присутствует; по крайней мере, в одном из упругих (начальных и конечных) каналах. Вопрос, является ли пик в неред более узким, или более широким, или сравнимым с упругим пиком, требует очень детальной параметризации упругих данных. В частности, в нашей модели, не остается произвольных параметров, т.к. неупругие амплитуды полностью определяются из знания всех упругих амплитуд. Однако, практически, ограниченность, имеющеися в нашем распоряжении экспериментальной информации, вынуждает нас ввести эмпирический фактор зависящий от энергии. В рамках нашей модели, очень мало можно сказать об этом зависящем от энергии факторе. Если, однако, сделать дополнительное предположение, что все упругие каналы приблизительно равны и что число каналов, открытых при данной (высокой) энергии, имеет ту же энергетическую зависимость, как и множественность, тогда можно видеть, что в этом приближении получается очень хорошее согласие для реакции π→π0n, представляющей единственный случай, для которого существует достаточно больщая статистика. В этой предварительной статье не предпринимается попыток дкя подробного численного сравнения теории с экспериментальными данными.

Riassunto

Si discute il problema di esprimere le ampiezze d’urto inelastiche, nel limite di alte en ergie, per mezzo delle ampiezze elastiche accoppiate attraverso l’unitarietà. Si presenta una soluzione a questo problema risolvendo approssim ativamente le equazioni fornite dalla condizione di unitarietà per mezzo di ipotesi di tipo assorbitivo, insieme con una proprietà di fattorizzazione ad alte energie. Il risultato sorprendente che si ottiene è che le ampiezze inelastiche sono completamente esprimibili in funzione ditutte le ampiezze elastiche aventi i numeri quantici necessari per accoppiarsi attraverso l’unitarietà. Alcune proprietà generali di questa soluzione sono studiate per i processi di scambio carica πp e Kp utilizzando una rappresentazione di Fourier-Bessel esatta che era stata precedentemente derivata. Similmente a quel che si ottiene nel modello a goccia di Byers e Yang, le ampiezze inelastiche ad alte energie nella direzione in avanti mostrano un picco diffrattivo di tipo esponenziale (nel momento trasferito), se questi è presente in almeno uno dei due canali elastici iniziale e finale. Si mostra anche che il problema se la larghezza del picco inelastico sia maggiore o minore o confrontabile con l’elastico, richiede una accurata parametrizzazione dei dati dell’elastico. In linea di principio, nel presente modello non vi sono parametri arbitrari poiché le ampiezze inelastiche sono determinate completamente dall a conoscenza ditutte le ampiezze elastiche. In pratica, tuttavia, la limitatezza dei dati sperimentali disponibili ei costringe ad introdurre un fattore empirico dipendente dall’energia e poco si può dire su questo fattore nell’ambito del modello. Se però si fa l’ipotesi addizionale che i contributi di tutti i canali elastici siano all’incirca uguali e che il numero di canali aperti ad una data (alta) energia abbia la stessa dipendenza dall’energia della molteplicità, si può vedere che un buon accordo si ottiene, in questa approssimazione, per π→π0n che è l’unico processo per il quale si ha a disposizione una statistica sufficiente. In questo lavoro preliminare, non si fa alcun tentativo di fare un particolareggiato confronto della teoria con i dati sperimentali.

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Copyright information

© Società Italiana di Fisica 1967

Authors and Affiliations

  • E. Predazzi
    • 1
  1. 1.The Enrico Fermi Institute for Nuclear StudiesThe University of ChicagoChicago

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