Summary
A previously proposed meromorphic and mathematically dual representation for an amplitude involving fermions is generalized to the case of an arbitrary number of external particles. The properties of the generalization are discussed and it is shown how it can be employed in the construction of mathematically dual amplitudes for two fermions with many pions, and many-meson amplitudes in which the mesons are composed of positive-parity spin-one-half quarks. All the leading trajectories obtained are then free of ghosts and parity doubling. The explicit forms for the five-point amplitudes are given. The properties of the amplitudes obtained are discussed and possible further applications and generalizations are suggested.
Riassunto
Si generalizza al caso di un numero arbitrario di particelle esterne una rappresentazione, precedentemente proposta, meromorfa e matematicamente duale, per una ampiezza dei fermioni. Si discutono le proprietà di generalizzazione e si dimostra come si possa impiegare nella costruzione di ampiezze matematicamente duali per due fermioni con ampiezze di molti pioni e molti mesoni in cui i mesoni sono composti di quarks di spin 1/2 e parità positiva. Si ottengono così delle traiettorie principali senza fantasmi e di parità raddoppiata. Si forniscono le forme esplicite per le ampiezze a cinque punti, si discutono le proprietà delle ampiezze e si suggeriscono ulteriori applicazioni e generalizzazioni.
Реэюме
Предварительно предложенное мероморфное и математически дуальное представление для амплитуды, включаюшей фермионы, обобшается на случай проиэвольного числа внещних частиц. Обсуждаются свойства зтого обобшения и покаэывается, как его можно испольэовать при конструировании математически дуальных амплитуд для двух фермионов с многими пионами и много-меэонных амплитуд, в которых меэоны обраэованы иэ кварков со спином половина и положительной четностью. Все полученные главные траектории свободны от духов и не содержат удвоения четности. Приводятся точные выражения для пяти-точечных амплитуд. Обсуждаются свойства полученных амплитуд и предлагаются воэможные дальнейщие применения и обобшения.
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References
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An always demandingcomplete Regge behaviour, that is Regge boundedness of all parts of the amplitude in all directions except for small angular regions around the real axes (see ref. (5,8)). An approach violating the above for the same problem as treated in ref. (5) is discussed byR. Carlitz, S. Ellis, P. G. O. Freund andS. Matsuda: Caltech, preprint June 1970.
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Up to whatever parity doubling is intrinsic to the usualn-point dual amplitude itself. SeeS. Fubini andG. Veneziano:Nuovo Cimento,64 A, 811 (1969).
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see ref. (6) and footnote (15)
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The rules involving pions only are the same as those of ref. (20).
It would amusing to use for exampleH 1/(H 1—α(s)) for the ρ propagator (see footnote (16)).
S. Mandelstam: U.C.R.L. preprint 19327 (1969).
A similar approach to elimination of parity doubling has also been pursued byI. Montvay: Budapest preprints (June 1970), whom I wish to thank for sending me copies of his work.
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Venturi, G. Meromorphic mathematically dual amplitudes with fermions. Nuov Cim A 1, 759–776 (1971). https://doi.org/10.1007/BF02734396
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DOI: https://doi.org/10.1007/BF02734396