Summary
After reviewing the commonly used dispersion relations, a systematic investigation of more generalized dispersion relations on parametrized curves in the Mandelstam plane fors-u crossing-symmetric amplitudes is made with the aim of obtaining dispersion relations which receive contributions from all three channels, however, in such a way that knowledge of the absorptive parts is only required in regions well inside the various Lehmann ellipses. In addition we require that the dispersion relations receive no contributions from kinematic singularities arising from the parametrization and that they allow partial-wave projections to be made in a relatively simple manner. It is found that dispersion relations on hyperbolic curves in the Mandelstam plane are a natural solution of the problem. The dispersion relations are written in a remarkably simple form similar to the usual fixed-t dispersion relation but with an additionalt-channel contribution. As an interesting application, we derive generalized partial-wave dispersion relations for elastic pion-nucleon scattering, where the left-hand cut contribution is explicitly given by convergent partial-wave series in the crossed channels.
Riassunto
Dopo avere riesaminato le relazioni di dispersione comunemente impiegate, si studiano relazioni di dispersione più generalizzate, definite su curve parametrizzate nel piano di Mandelstam, per le ampiezzes-u a simmetria incrociata, con lo scopo di ottenere relazioni di dispersione che ricevano contributi da tutti e tre i canali; tuttavia, in tal modo, la conoscenza delle parti che determinano l’assorbimento è richiesta soltanto in regioni completamente interne alle varie ellissi di Lehmann. Inoltre si richiede che le relazioni di dispersione non ricevano contributi da singolarità cinematiche derivanti dalla parametrizzazione e che consentano di effettuare in un modo relativamente semplice proiezioni di onde parziali. Si trova che relazioni di dispersione definite su curve iperboliche nel piano di Mandelstam rappresentano una soluzione naturale del problema. Le relazioni di dispersione sono rappresentate in una forma notevolmente semplice, simile a quella delle consuete relazioni di dispersione che si ottengono tenendo fissa la variabilet, ma con un contributo supplementare al canalet. Un’interessante applicazione consiste nell’ottenere relazioni di dispersione generalizzate (per onde parziali) per lo scattering elastico pione-nucleone, nel quale il contributo dei tagli a sinistra viene rappresentato esplicitamente nei canali incrociati da una serie convergente di onde parziali.
Реэюме
Пересмотрев обычно испольэуемые дисперсионные соотноще��ия, про-водится систематическое исследование наиболее обобшенных дисперсионных соотно-щений на параметриэованных кривых в плоскости Манделстама дляs-u кроссинг-симметрич ных амплитуд, с целью получения дисперсионных соотнощений, которые воспринимают вклады от всех трех каналов. Однако в зтом случае требуется энание абсорбционных частей только в областях внутри раэличных зллипсов Лемана. Кроме того, мы требуем, утобы дисперсионные соотнощения не имели вкладов от кинематических сингулярностей, воэникаюших от параметриэации, и чтобы они давали воэможность проиэводить парциальное проектирование относительно простым обраэом. Получается, что дисперсионные соотнощения на гиперболических кривых в плоскости Манделстама преставляют естественное рещение проблемы. Дисперсион-ные соотнощения эаписываются в очень простой форме, аналогично обычному дисперсионному соотнощению при фиксированномt, но с дополнительным вкладомt канала. Как применение зтого подхода, мы выводим обобшенные парциальные дисперсионные соотнощения для упругого пион-нуклонного рассеяния, где вклад левостороннего раэреэа явно выражается череэ сходяшийся парциальный ряд в перекрестных каналах.
Similar content being viewed by others
References
See, for example:P. D. B. Collins andE. J. Squires:Regge Poles in Particle Physics, Springer Tracts in Modern Physics, Vol.45 (Berlin, 1968);J. Hamilton:Pion-Nucleon Interactions, inHigh-Energy Physics, Vol.1, edited byE. H. S. Burhop (New York, 1967), p. 183.
J. Baacke andF. Steiner:Fortschr. Phys.,18, 67 (1970).
R. Oehme:Phys. Rev.,100, 1503 (1955);102, 1174 (1956); see also:R. H. Capps andG. Takeda:Phys. Rev.,103, 1877 (1956);G. F. Chew, M. Goldberger, F. E. Low andY. Nambu:Phys. Rev.,106, 1337 (1957);L. D. Solovyov:Nucl. Phys.,5, 256 (1958);K. Dietz andG. Höhler:Zeits. Phys.,160, 453 (1960);M. Carrassi andG. Passatore:Nuovo Cimento,26, 1254 (1962);27, 1156 (1963).
For πN → πN:F. Steiner: unpublished (1967);J. Baacke, G. Höhler andF. Steiner:Zeits. Phys.,221, 134 (1969);J. Baacke andF. Steiner:Fortschr. Phys.,18, 67 (1970);F. Steiner:Fortschr. Phys.,18, 43 (1970). For photo- and electroproduction:W. Schmidt andG. Schwiderski:Fortschr. Phys.,15, 393 (1967);F. A. Berends, A. Donnachie andD. L. Weaver:Nucl. Phys.,4 B, 1 (1968);G. von Gehlen:Nucl. Phys.,9 B, 17 (1969). For ππ → ππ:S. M. Roy:Phys. Lett.,36 B, 353 (1971);J. L. Basdevant, J. C. Le Guillou andH. Navelet:Nuovo Cimento,7 A, 363 (1972).
F. Steiner:Fortschr. Phys.,18, 43 (1970).
S. M. Roy:Phys. Lett.,36 B, 353 (1971); see also:H. P. Jakob andF. Steiner:Zeits. Phys.,228, 353 (1969).
For πN → πN:D. V. Shirkov, V. V. Serebryakov andV. A. Meshcheryakov:Dispersion Theories of Strong Interactions at Low Energy (Amsterdam, 1969), and earlier references therein. For γN → πN:G. Białkowski andA. Jurewicz:Ann. of Phys.,31, 436 (1965);A. Jurewicz:Zeits. Phys.,223, 425 (1969);224, 432 (1969). ForNN →NN:M. Cini, S. Fubini andA. Stanghellini:Phys. Rev.,114, 1633 (1959).
D. Atkinson:Phys. Rev.,128, 1908 (1962);C. Lovelace, R. H. Heinz andA. Donnachie:Phys. Lett.,22, 332 (1966);H. Goldberg:Phys. Rev.,171, 1485 (1968);C. Lovelace: inPion-Nucleon Scattering, edited byG. L. Shaw andD. Y. Wong (New York, 1969);J. Engels, G. Höhler andB. Petersson:Nucl. Phys.,15 B, 365 (1970);H. Nielsen, J. Lyng-Petersen andE. Pietarinen:Nucl. Phys.,22 B, 525 (1970);J. Engels:Nucl. Phys.,25 B, 141 (1970).
G. E. Hite andR. Jacob:Phys. Rev. D,5, 422 (1972);G. E. Hite, R. Jacob andF. Steiner:Phys. Rev. D,6, 3333 (1973).
F. Steiner:Phys. Lett.,32 B, 294 (1970).
D. H. Lyth:Rev. Mod. Phys.,37, 709 (1965);G. C. Oades:Suppl. Nuovo Cimento,4, 217 (1966);D. Beder andJ. Finkelstein:Phys. Rev.,160, 1363 (1967);C. B. Chiu andM. DerSarkissian:Nuovo Cimento,55 A, 396 (1968);V. Barger, C. Michael andR. J. N. Phillips:Phys. Rev.,185, 1852 (1969);B. Kayser:Phys. Rev. Lett.,21, 1292 (1968);Phys. Rev. D,1, 306 (1970);H. G. Schlaile: Thesis, Karlsruhe (1970);J. Baacke: Berlin preprint (1971).
See Appendix D of ref. (12).
G. E. Hite andF. Steiner: CERN TH 1590 (1972).
P. W. Greenberg andJ. C. Sandusky:Nuovo Cimento,6 A, 617 (1971);J. C. Sandusky:Nuovo Cimento,6 A, 627 (1971).
For details of the πN kinematics, see ref. (1) or (2).
G. Ebel, M. Gourdin, B. R. Martin, C. Michael, A. Müllensiefen, G. C. Oades, J. L. Petersen, H. Pilkuhn, M. Roos, F. Steiner, J. J. De Swart andD. Wegener:Nucl. Phys.,33 B, 317 (1971).
S. W. McDowell:Phys. Rev.,116, 774 (1959).
H. Lehmann:Nuovo Cimento,10, 579 (1958).
Yu andMoravcsik (17) published recently a paper on dispersion relations for fixed transverse momentum squared,q 2⊥ ≡q 2 sin2 ϑ s; these dispersion relations violate conditionsb)–d) of Sect.2. (See also Appendix D of ref. (12).)
W.-Y. Yu andM. J. Moravcsik:Lett. Nuovo Cimento,4, 228 (1972).
G. Veneziano:Nuovo Cimento,57 A, 190 (1968).
F. Steiner:Fortschr. Phys.,19, 115 (1971).
J. Hamilton:Springer Tracts in Modern Physics,57, 41 (1971);H. Nielsen andG. C. Oades: Aarhus University preprints (1972).
S. Ciulli:Nuovo Cimento,61 A, 787 (1968);62 A, 301 (1969);R. E. Cutkosky andB. B. Deo:Phys. Rev.,174, 1859 (1968).
A. Donnachie, R. Kirsopp andC. Lovelace:Phys. Lett.,26 B, 161 (1968);C. Lovelace: inProceedings of the Heidelberg International Conference on Elementary Particles, edited byH. Filthuth (Amsterdam, 1968), p. 79;C. Lovelace: inPion-Nucleon Scattering, edited byG. L. Shaw andD. Y. Wong (New York, 1969);S. Almehed andC. Lovelace:Nucl. Phys.,40 B, 157 (1972).
C. Lovelace: private communication.
Notice that the second kernel of eq. (B.24) is just the crossing kernel entering the partial-wave crossing relations derived in ref. (19).
Bateman Manuscript Project, Higher Transcendental Functions, Vol.1, p. 124.
Bateman Manuscript Project, Higher Transcendental Functions, Vol.1, p. 666.
For a derivation of eq. (B.44) see Appendix III of ref. (5).
W. R. Frazer andJ. R. Fulco:Phys. Rev.,117, 1603 (1960).
Author information
Authors and Affiliations
Additional information
To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.
Rights and permissions
About this article
Cite this article
Hite, G.E., Steiner, F. New dispersion relations and their application to partial-wave amplitudes. Nuov Cim A 18, 237–270 (1973). https://doi.org/10.1007/BF02722827
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02722827