Summary
This paper deals with some problems in the representation of highly unstable particles (resonances): the usual way of relating a pole of theS-matrix (or field propagator) to each particle is shown not to have quite obvious physical foundations in the case of stable particles; in particular, examples of resonant scattering amplitude and propagator are given explicitly. Another more straightforward way of representing an unstable particle is to describe its continuous mass distribution by means of the spectral function of its field, as was first proposed by Matthews and Salam. Here, these authors’ interpretation of the spectral function is tested and confirmed on the example of the multiple Lce model. But a new definition of the mass and width of the particle has to be proposed in order to rule out some difficulties raised by that of Matthews and Salam. On the other hand, for unstable particles the problem of the correspondence between particles and fields, solved in the case of stable particles (asymptotic field theory), is no longer trivial, and has to be reconsidered. This involves a physical interpretation of renormalization related to the splitting of the spectral function of a field into a peak connected to an unstable particle and a nonresonant background.
Riassunto
In questo articolo si trattano alcuni problemi della rappresentazione di particelle fortemente instabili (risonanze): si dimostra che il metodo usuale di mettere in relazione il polo della matriceS (o propagatore di campo) con ciascuna particelle non ha fondamenti fisici abbastanza ovvi nel caso di particelle stabili; in particolare, si rendono espliciti esempi di ampiezza di scattering risonante e di propagatore. Un altro metodo più diretto per rappresentare una particella instabile è di descrivere la sua distribuzione continua della massa per mezzo della funzione spettrale del suo campo, come proposero per primi Matthews e Salam. Qui si controlla l’interpretazione della funzione spettrale data da questi autori e la si conferma sull’esempio del modello di Lee multiplo. Ma, allo scopo di eliminare alcune difficoltà che insorgono in base alla definizione di Matthews e Salam, si deve proporre una nuova definizione della massa e dell’ampiezza delle particelle. D’altra parte, per particelle instabili il problema della corrispondenza tra particelle e campi, risolto nel caso delle particelle stabili (teoria dei campi asintotica), non è più banale e deve essere studiato di nuovo. Ciò coinvolge un’interpretazione fisica della rinormalizzazione in rapporto con la separazione della funzione spettrale di un campo in un picco connesso ad una particella instabile ed un fondo non risonante.
Реэюме
Эта работа посвяшена некоторым проблемам представления сильно нестабильных частиц (реэонансов). Покаэывается, что обычный способ, который свяэывает полюс S матрицы (или полевого пропагатора) с каждой не имеет вполне очевидных фиэических оснований в случае стабильных частиц. В частности, приводятся примеры амплитуды и пропагатора для реэонансного рассеяния. Другой более непосредственный способ представления нестабильной частицы состоит в том, чтобы описать непрерывный массовый спектр с помошью спектральной функции поля, как было впервые предложено Метьюсом и Саламом. Таким обраэом, интерпретация зтих авторов спектральной функции проверяется и подтверждается на примере многократной модели Ли. Но должно быть предложено новое определение массы и щирины частиц для того, чтобы исключить некоторые трудности, воэникщие у Метьюса и Салама. С другой стороны, для нестабильных частиц проблема соответствия между частицами и полями, рещенная в случае стабильных частиц (асимптотическая теория поля) не является более тривиальной и должна быть эаново рассмотрена. Это включает фиэическую интерпретацию перенормировки, свяэанной с расшеплением спектральной функции поля на пик, соответствуюший нестабильной частице, и нереэонансный фон.
Similar content being viewed by others
References
T. D. Newton andE. P. Wigner:Rev. Mod. Phys.,21, 400 (1949).
G. Calucci, L. Fonda andG. C. Ghirardi:Phys. Rev.,166, 1719 (1968).
G. Calucci andG. C. Ghirardi:Phys. Rev.,169, 1339 (1968).
L. Fonda:Forts. Phys.,20, 135 (1972).
A. Martin:Scattering Theory: Unitarity, Analyticity and Crossing (Berlin, 1969).
R. J. Eden, P. V. Landshoff, D. I. Olive andJ. C. Polkinghorne:The Analytic S-Matrix (Cambridge, 1966).
J. Schwinger:Ann. of Phys.,9, 169 (1960).
P. T. Matthews andA. Salam:Phys. Rev.,112, 283 (1958).
Recently it has been noticed that in this procedure, even if the resonant and asymptotic behaviours are right, there can be some problems in the intermediate physical regions (9): if the true amplitude has a pole, any holomorphic function approaching it in the resonance region should have a very bad behaviour outside this region: the maximum of its modulus would be very high. This would give a very good criterion to decide whether the amplitude has a pole or not. But it can be shown that these violent oscillations on the physical cut outside the resonance interval can easily be removed for example by assigning to the amplitude a multiple pole far from the physical energy range (10). So the ambiguity concerning the amplitudes built from the experimental data will survive: the various possible amplitudes can be very different only in some regions of the complex plane inaccessible to the physical measurements.
I. Caprini, S. Ciulli, A. Pomponiu andI. Sabba-Stefanescu:Phys. Rev. D,5, 1658 (1972).
G. Calucci andG. C. Ghirardi: to be published inPhys. Rev. D.
W. Zimmermann:Nuovo Cimento,13, 503 (1959).
K. Hepp:Journ. Math. Phys.,6, 1762 (1965).
M. Lévy:Nuovo Cimento,13, 115 (1959).
P. T. Matthews andA. Salam:Phys. Rev.,115, 1079 (1959).
R. J. Oakes andC. N. Yang:Phys. Rev. Lett.,11, 174 (1963).
M. Froissart andM. Jacob:Journ. de Phys.,25, 317 (1964).
K. Huang: inPreludes in Theoretical Physics, edited byA. de Shalit, H. Feshbach andL. Van Hove (Amsterdam, 1966).
P. V. Landshoff:Nuovo Cimento,28, 123 (1963).
R. J. Eden andJ. R. Taylor:Phys. Rev. Lett.,11, 516 (1963).
R. J. Eden andJ. R. Taylor:Phys. Rev.,133, B 1575 (1964).
G. F. Chew andF. E. Low:Phys. Rev.,113, 1640 (1959).
J. P. Baton: Thèse de Doctorat d’état (Orsay, 1968).
J. P. Baton andG. Laurens:Phys. Rev.,176, 1574 (1968).
H. Lehmann:Nuovo Cimento,2, 347 (1954).
T. D. Lee:Phys. Rev.,95, 1329 (1954).
G. Källén andW. Pauli:Dan. Mat. Fys. Medd.,30, No. 7 (1955).
J. Lukierski:Bull. Acad. Pol. Sci. Ser. Sci. Math. Astr. Phys., XVI,4, 343 (1968).
R. F. Streater andA. S. Wightman:PCT, Spin and Statistics and All That (New York and Amsterdam, 1964).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hittner, D. On unstable-particle theory. Nuov Cim A 15, 401–429 (1973). https://doi.org/10.1007/BF02734680
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02734680